Research Article Disease Control of Delay SEIR Model with...

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Hindawi Publishing Corporation Computational and Mathematical Methods in Medicine Volume 2013, Article ID 830237, 11 pages http://dx.doi.org/10.1155/2013/830237 Research Article Disease Control of Delay SEIR Model with Nonlinear Incidence Rate and Vertical Transmission Yan Cheng, 1,2 Qiuhui Pan, 3 and Mingfeng He 1 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China 2 Department of Mathematics, Tonghua Normal University, Tonghua 136000, China 3 School of Innovation Experiment, Dalian University of Technology, Dalian 116024, China Correspondence should be addressed to Mingfeng He; [email protected] Received 19 July 2013; Revised 30 September 2013; Accepted 3 October 2013 Academic Editor: Seiya Imoto Copyright © 2013 Yan Cheng et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e aim of this paper is to develop two delayed SEIR epidemic models with nonlinear incidence rate, continuous treatment, and impulsive vaccination for a class of epidemic with latent period and vertical transition. For continuous treatment, we obtain a basic reproductive number R 0 and prove the global stability by using the Lyapunov functional method. We obtain two thresholds R ∗ and R ∗ for impulsive vaccination and prove that if R ∗ < 1, then the disease-free periodic solution is globally attractive and if R ∗ > 1, then the disease is permanent by using the comparison theorem of impulsive differential equation. Numerical simulations indicate that pulse vaccination strategy or a longer latent period will make the population size infected by a disease decrease. 1. Introduction Mathematical models describing the population dynamics of infectious diseases have been playing an important role in understanding epidemiological patterns and disease control. Researchers have studied the epidemic models by ordinary differential equations [1–3] and the references cited therein. A customarily epidemic model is susceptible, infectious, and recovered model (SIR for short) [4–7]. But in real life, many diseases have a period of incubation time inside the hosts before the hosts become infectious; if we include incubation period of the hosts, the model is described as SEIR model. As tuberculosis (TB), measles and so on, a susceptible individual becomes exposed (infected but not infective) by adequate contact with an infectious individual. SEIR infections disease model has been studied by many authors for its important biological meaning [8–13]. In [13], the authors considered the following delayed SEIR epidemic model: () = Λ − () − () () 1 + () + () , () = () () 1 + () − − ( − ) ( − ) 1 + ( − ) − , () = − ( − ) ( − ) 1 + ( − ) − ( + ) , () = − ( + ) () , (1) where (), (), (), and () represent the number of indi- viduals who are susceptible, exposed, infected, and removed, respectively. e parameters Λ, , , and are positive constants, and here Λ is the constant recruitment rate into the population, is the contact rate, is the birth and death rate, is the removal rate. >0 represents a time delay describing the latent period of the disease and the term ( − (−)(− ))/(1 + ( − )) represents the individuals surviving in the latent period and becoming infective at time . e sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. In the study of epidemic model, the spread of an infec- tious disease is a crucial issue, which depends on both the population behavior and the infectivity of the disease. ese two aspects are captured in the incidence rate of a disease. In many epidemiological models, the incidence rate is described as mass action incidence with bilinear interactions given by , where is the probability of transmission per

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Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2013 Article ID 830237 11 pageshttpdxdoiorg1011552013830237

Research ArticleDisease Control of Delay SEIR Model with Nonlinear IncidenceRate and Vertical Transmission

Yan Cheng12 Qiuhui Pan3 and Mingfeng He1

1 School of Mathematical Sciences Dalian University of Technology Dalian 116024 China2Department of Mathematics Tonghua Normal University Tonghua 136000 China3 School of Innovation Experiment Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Mingfeng He mfhedluteducn

Received 19 July 2013 Revised 30 September 2013 Accepted 3 October 2013

Academic Editor Seiya Imoto

Copyright copy 2013 Yan Cheng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The aim of this paper is to develop two delayed SEIR epidemic models with nonlinear incidence rate continuous treatment andimpulsive vaccination for a class of epidemic with latent period and vertical transition For continuous treatment we obtain a basicreproductive number R

0and prove the global stability by using the Lyapunov functional method We obtain two thresholds Rlowast

and Rlowastfor impulsive vaccination and prove that if Rlowast lt 1 then the disease-free periodic solution is globally attractive and if

Rlowastgt 1 then the disease is permanent by using the comparison theorem of impulsive differential equation Numerical simulations

indicate that pulse vaccination strategy or a longer latent period will make the population size infected by a disease decrease

1 Introduction

Mathematical models describing the population dynamics ofinfectious diseases have been playing an important role inunderstanding epidemiological patterns and disease controlResearchers have studied the epidemic models by ordinarydifferential equations [1ndash3] and the references cited thereinA customarily epidemic model is susceptible infectious andrecovered model (SIR for short) [4ndash7] But in real life manydiseases have a period of incubation time inside the hostsbefore the hosts become infectious if we include incubationperiod of the hosts the model is described as SEIRmodel Astuberculosis (TB) measles and so on a susceptible individualbecomes exposed (infected but not infective) by adequatecontact with an infectious individual SEIR infections diseasemodel has been studied by many authors for its importantbiological meaning [8ndash13] In [13] the authors considered thefollowing delayed SEIR epidemic model

1198781015840(119905) = Λ minus 120583119878 (119905) minus

120573119878 (119905) 119868 (119905)

1 + 120572119868 (119905)

+ 120575119877 (119905)

1198641015840(119905) =

120573119878 (119905) 119868 (119905)

1 + 120572119868 (119905)

minus

120573119890minus120583120591119878 (119905 minus 120591) 119868 (119905 minus 120591)

1 + 120572119868 (119905 minus 120591)

minus 120583119864

1198681015840(119905) =

120573119890minus120583120591119878 (119905 minus 120591) 119868 (119905 minus 120591)

1 + 120572119868 (119905 minus 120591)

minus (120583 + 120574) 119868

1198771015840(119905) = 120574119868 minus (120583 + 120575) 119877 (119905)

(1)where 119878(119905) 119864(119905) 119868(119905) and 119877(119905) represent the number of indi-viduals who are susceptible exposed infected and removedrespectively The parameters Λ 120573 120574 and 120583 are positiveconstants and hereΛ is the constant recruitment rate into thepopulation 120573 is the contact rate 120583 is the birth and death rate120574 is the removal rate 120591 gt 0 represents a time delay describingthe latent period of the disease and the term (120573119890

minus120583120591119878(119905minus120591)119868(119905minus

120591))(1 + 120572119868(119905 minus 120591)) represents the individuals surviving in thelatent period 120591 and becoming infective at time 119905The sufficientconditions are obtained for the global asymptotic stability ofthe endemic equilibrium

In the study of epidemic model the spread of an infec-tious disease is a crucial issue which depends on both thepopulation behavior and the infectivity of the disease Thesetwo aspects are captured in the incidence rate of a disease Inmany epidemiological models the incidence rate is describedas mass action incidence with bilinear interactions givenby 120573119878119868 where 120573 is the probability of transmission per

2 Computational and Mathematical Methods in Medicine

contact and 119878 and 119868 represent the susceptible and infectedpopulations respectively This contact law is more appro-priate for a few of infected individuals when the size ofinfected individuals is increasing the underlying assumptionof homogeneous mixing may not be valid In fact with theincrease of infected populations the susceptible individualwill takemeasure to prevent unbounded contact rates In [14]Anderson andMay proposed a saturated incidence rate of theform 120573119878119868(1 + 120572119878) in which 120573119878119868measures the infection forceof the disease and 1(1 + 120572119878) measures the inhibition effectfrom the behavioral change of the susceptible individualsThesame as the nonlinear incidence rates of the form 119896119868

119901119878119902 were

investigated by Liu et al [15 16]In real life some diseases may be transferred through

horizontal transmission and vertical transmission (disease isthe passing of an infection to offspring of infected parents)The offspring of infected parentsmay already be infectedwiththe disease at birth so many infections in nature transmitthrough both horizontal and vertical modes such as tuber-culosis (TB) rubella hepatitis B and AIDS [17ndash21]

Vaccination and treatment are important strategy for theelimination of infectious diseases Recently pulse vaccinationhas been confirmed as an effective method to prevent thespread of the disease [22ndash24] Theoretical results show thatthe pulse vaccination strategy can be distinguished fromthe conventional strategies in leading to disease eradicationat relatively low values of vaccination [25] The study ofvaccination treatment and associated behavioral changesrelated to disease transmission has been the subject of intensetheoretical analysis

The literature on SEIR model with nonlinear incidenceconstant infectious period impulsive vaccination dealingwith the analysis of disease that is vertically and horizontallytransmitted is not extensive [17 19] But in fact under thesituation of disease with vertical transmission the contin-uous treatment should be considered for the infected andimpulsive vaccination to the susceptible newborns of thesusceptible exposed and the removed and newborns ofinfected which not be vertical infected

Motivated by the literature above we introduce delayepidemic models with nonlinear incidence rates of the form120573119878119901119868 and we also considered the constant latency period

and vertically and horizontally in (2) The purpose of thispaper is to study the nonlinear dynamics of system andwe consider two different strategies to the model which areconstant treatment and pulse vaccination to the newbornsand susceptible

1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898

1015840(119878 + 119877 + 119864) +

119889

119879

119868

(2)

The basic assumptions are as follows

(i) The total population size at time 119905 (day) is denotedby 119873 = 119878 + 119864 + 119868 + 119877 For = 0 this shows thatthe total population has a constant size Without lossof generality we assume in this paper 119873 = 1 Thenewborns of 119878 119864 and 119877 are susceptible individualsand the newborns of 119868 who are not vertically infectedare also susceptible individuals

(ii) The positive constant 119887 (per day) denotes the deathrate and birth rate of susceptible exposed and recov-ered individuals The positive constant 120575 (per day)denotes the death rate and birth rate of infectiveindividuals The positive constant 120574 (per day) is thenatural recovery rate of infective individuals Thepositive constant 119902 (119902 le 1) (per day) is the verticaltransmission rate and note 1199021015840 = 1 minus 119902 1199021015840 lt 119902

(per day) and then 0 lt 1199021015840lt 1 Fraction 1198981015840 of all

newborns with mothers in the susceptible exposedand recovered classes are vaccinated and appearedin the recovered class while the remaining fraction119898 = 1 minus 119898

1015840 appears in the susceptible class suppose119887119898 gt 119902

1015840120575 (119889119879) is the proportion of those cured

successfully(iii) The incidence rate is described by a nonlinear func-

tion 120573119878119901119868 where 120573 (per day) is a positive constant

describing the infection rate 120591 gt 0 is the length ofthe latent period and the term 120573119890

minus119887120591119878119901(119905 minus 120591)119868(119905 minus 120591)

reflects the fact that an individual is surviving in thelatent period 120591 and becoming infective at time 119905

The remaining part of this paper is organized as followsIn Section 2 we investigate the global stability of the endemicequilibrium of (2) by using Rouches theorem and Lyapunov-LaSalle type theorem The global asymptotic stability ofdisease-free periodic solution and the conditions for thepermanence of the disease by comparison techniques aredescribed in Section 3 Numerical simulations are presentedin Section 4 In Section 5 we conclude this paper with someremarks

2 Continuous Treatment Strategy ofthe SEIR Model

In this section we consider a continuous treatment of SEIRmodel with constant latent period and nonlinear incidencerate By using 119878 + 119877 + 119864 = 1 minus 119868 notice that first andthird equations of system (2) do not contain the variables 119864and 119877 therefore system (2) is equivalent to the following 2-dimensional system

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

(3)

21 Disease-Free Equilibrium and Its Stability First we define

R0=

120573119898119901119890minus119887120591

120575 minus 119902120575 + 120574 + (119889119879)

(4)

Computational and Mathematical Methods in Medicine 3

Theorem 1 If R0lt 1 the disease-free equilibrium 119864

0(119898 0)

of system (3) is locally asymptotically stable for all 120591 ge 0 ifR0gt 1 the disease-free equilibrium 119864

0(119898 0) is unstable

Proof Steady states of system satisfy the following system ofequations

119887119898 (1 minus 119868) minus 120573119878119901119868 minus 119887119878 + 119902

1015840120575119868 = 0

120573119890minus119887120591119878119901119868 + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868 = 0

(5)

Obviously 1198640(119898 0) is the disease-free equilibrium of (3)

In order to analyze the behavior of the system (3) near 1198640 we

linearize the system about the equilibrium point let 119878(119905) =119883(119905) + 119898 119868(119905) = 119884(119905)

1198831015840(119905) = minus119887119883 (119905) minus (120573119898

119901+ 119887119898 minus 119902

1015840120575)119884 (119905)

1198841015840(119905) = minus(120575 minus 119902120575 + 120574 +

119889

119879

)119884 (119905) + 120573119890minus119887120591119898119901119884 (119905 minus 120591)

(6)

1205821= minus119887 lt 0 is one of the eigenvalues of the linearization of

system (6) near the steady state 1198640 and the other eigenvalue

1205822is determined by equation

120582 minus 120573119898119901119890minus(119887+120582)120591

+ 120575 minus 119902120575 + 120574 +

119889

119879

= 0 (7)

Let

119891 (120582) = 120582 minus 120573119898119901119890minus(119887+120582)120591

+ 120575 minus 119902120575 + 120574 +

119889

119879

(8)

ifR0gt 1 it is easy to show that for 120582 real

119891 (0) = (120575 minus 119902120575 + 120574 +

119889

119879

) (1 minusR0) lt 0

lim120582rarr+infin

119891 (120582) = +infin

(9)

hence 119891(120582) = 0 has a positive real root Therefore ifR0gt 1

the disease-free equilibrium 1198640(119898 0) is unstable

If R0lt 1 we prove that the disease-free equilibrium

1198640(119898 0) is locally stable Otherwise Re 120582 ge 0 We note that

Re 120582 = (120575 minus 119902120575 + 120574 + 119889

119879

) (R0119890minusRe120582120591 cos (119897119898120582120591) minus 1)

le (120575 minus 119902120575 + 120574 +

119889

119879

) (R0minus 1)

(10)

a contradiction Hence the disease-free equilibrium 1198640is

locally asymptotically stable ifR0lt 1

Theorem 2 If R0lt 1 the disease-free equilibrium 119864

0(119898 0)

of system (3) is globally asymptotically stable for all 120591 ge 0

To proof the global stability of the disease-free equilib-rium 119864

0(119898 0) we choose Lyapunov function

119881 (119905) = 119883 (119905) + 119884 (119905) + 120573119890minus119887120591119898119901int

119905

119905minus120591

119884 (120585) 119889120585 (11)

and it is easy to prove 1198811015840(119905) lt 0 lim119905rarrinfin

119881(119905) = 0 it followsthat lim

119905rarrinfin119883(119905) = 0 lim

119905rarrinfin119884(119905) = 0

22 Endemic Equilibrium and Its Stability If R0gt 1 then

system (3) has a unique positive equilibrium 1198641(119878lowast 119868lowast)

where

119878lowast= (

(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591

120573

)

1119901

119868lowast=

119887

(119887119898 minus 1199021015840120575) + (120575 minus 119902120575 + 120574 + (119889119879)) 119890

119887120591

times (119898 minus (

(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591

120573

)

1119901

)

(12)

Theorem 3 If R0gt 1 conditions (17) and (22) are satisfied

then for 120591 ge 0 the endemic equilibrium 1198641(119878lowast 119868lowast) of system (3)

is locally asymptotically stable

Proof Let 119878(119905) = 119883(119905) + 119878lowast 119868(119905) = 119884(119905) + 119868

lowast the linearizedsystem is obtained

1198831015840(119905) = minus (119887 + 120573119901119868

lowast119878lowast(119901minus1)

)119883 (119905)

minus (120573119878lowast119901+ 119887119898 minus 119902

1015840120575)119884 (119905)

1198841015840(119905) = 120573119901119890

minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119884 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

)119884 (119905)

(13)

From the linearized system we obtain the characteristic equ-ation

1205822+ 119901120582 + 119903 + (119892120582 + 119902) 119890

minus120582120591= 0 (14)

where

119901 = 120575 minus 119902120575 + 120574 +

119889

119879

+ 120573119901119878lowast(119901minus1)

119868lowast+ 119887

119903 = (119887 + 120573119901119868lowast119878lowast(119901minus1)

) (120575 minus 119902120575 + 120574 +

119889

119879

)

119892 = minus120573119878lowast119901119890minus119887120591

119902 = minus (119887 + 120573119901119868lowast119878lowast(119901minus1)

) 120573119890minus119887120591119878lowast119901

+ (120573119878lowast119901+ 119887119898 minus 119902

1015840120575) 120573119901119890

minus119887120591119868lowast119878lowast(119901minus1)

(15)

For 120591 = 0 the characteristic equation becomes

1205822+ (119901 + 119892) 120582 + (119903 + 119902) = 0 (16)

and we can see that both roots are negative and real if andonly if

119901 + 119892 gt 0 119903 + 119902 gt 0 (17)

Now for 120591 = 0 if 120582 = 120596119894 is a root of (16) we have

minus1205962+ 119902119890minus120596120591119894

+ 119901120596119894 + 119903 + 119892120596119890minus120596120591119894

= 0 (18)

4 Computational and Mathematical Methods in Medicine

Separating the real and imaginary parts we have

119903 minus 1205962+ 119892120596 sin (120596120591) + 119902 cos120596120591 = 0

119901120596 + 119892120596 cos120596120591 minus 119902 sin120596120591 = 0(19)

Adding both equations and regrouping by powers of 120596 weobtain the following fourth degree polynomial

1205964+ (1199012minus 1198922minus 2119903) 120596

2+ 1199032minus 1199022= 0 (20)

from which we have

1205962=

1198922minus 1199012+ 2119903 plusmn radic(119892

2minus 1199012+ 2119903)2minus 4 (119903

2minus 1199022)

2

(21)

It follows that if

1199012minus 1198922minus 2119903 gt 0 119903

2minus 1199022gt 0 (22)

are satisfied (20) does not have positive solutions and thecharacteristic equation (14) does not have purely imaginaryroots Inequalities in (17) and (22) guarantee that all roots of(14) have no positive roots According to Rouchersquos theoremTheorem 3 is proved

Subsequently we discuss the sufficient conditions underwhich the endemic equilibrium is globally asymptoticallystable for the system (3) For 119878 + 119864 + 119868 + 119877 = 1 hence thedynamics of system (3) in the first octant of 119877

4is equivalent

to that of the following system

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

(23)

The initial conditions for system (23) take the form

119878 (120585) = 1205931(120585) 119864 (120585) = 120593

2(120585) 119868 (120585) = 120593

3(120585)

1205931 (0) gt 0 120593

2 (0) gt 0 1205933 (0) gt 0

(24)

where (1205931(120585) 1205932(120585) 1205933(120585)) isin 119862([minus120591 0] 119877

3

+0) the space of

continuous functions mapping the interval [minus120591 0] into 1198773+0

where 1198773+0= (1199091 1199092 1199093) 119909119894ge 0 119894 = 1 2 3

For continuity of the initial conditions we require

119864 (0) = int

0

minus120591

120573119890119887120585119878(120585)119901119868 (120585) 119889120585 (25)

It is well known by the fundamental theory of functionaldifferential equations [26] the system (23) has a uniquesolution (119878(119905) 119864(119905) 119868(119905)) satisfying the initial conditions Itis easy to show that all solutions of system (23) with initialconditions are defined on [0 +infin) and remain positive for all119905 ge 0

Lemma 4 (see [25]) Let the initial condition be 119878(120585) = 119878(0) gt0 119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) Then119878(119905) le max1 119878(0) + 119864(0) + 119868(0) = 119872

Theorem 5 Let the initial condition be 119878(120585) = 119878(0) gt 0119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) FurthersupposeR

0gt 1 then for any infectious period 120591 satisfying

120591 gt max

1

119887

ln120573119872119901minus1

(120588119872 + 2119901119868lowast+ 3120588119901119868

lowast)

120573119901119868lowast119872119901minus1

minus 4119887120588 minus 120588 (119887119898 minus 1199021015840120575)

1

119887

ln120573119872119901minus1

(120588 + 1) (119868lowast119901 +119872)

4120588119887 + 2119887 minus 120573119901119868lowast119872119901minus1

1

119887

ln120573119872119901minus1

(119868lowast119901 + 2119872119901 + 3119872)

2(120575 minus 119902120575 + 120574 +

119889

119879

) + 120588 (119887119898 minus 1199021015840120575)

(26)

where119872 = max1 119878(0)+119864(0)+119868(0) the endemic equilibriumis globally asymptotically stable

Proof Let 119878(119905) = 119883(119905)+119878lowast 119864(119905) = 119884(119905)+119864lowast 119868(119905) = 119885(119905)+ 119868lowastthe linearized system is

1198831015840(119905) = minus (119887 + 120573119901119868

lowast119878lowast(119901minus1)

)119883 (119905)

+ (1199021015840120575 minus 120573119878

lowast119901minus 119887119898)119885 (119905)

1198841015840(119905) = 120573119901119868

lowast119878lowast(119901minus1)

119883(119905) + 120573119878lowast119901119885 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)

1198851015840(119905) = 120573119901119890

minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

)119885 (119905)

(27)

The trivial solution of system (27) is globally asymptot-ically stable and is equivalent to the fact that the endemicequilibrium (119878

lowast 119864lowast 119868lowast) of system (23) is globally asymptot-

ically stable We will employ Lyapunov functional techniqueto prove it

Now let us introduce the following functions

1198811 (119905) =

1

2

120588(119883 (119905) + 119884 (119905))2+

1

2

(1198842(119905) + 119885

2(119905))

1198812(119905) = (120588 + 1) 120573119901119890

minus119887120591119868lowast119872(119901minus1)

int

119905

119905minus120591

1198832(120585) 119889120585

+ (120588 + 1) 120573119890minus119887120591119872119901int

119905

119905minus120591

1198852(120585) 119889120585

(28)

Computational and Mathematical Methods in Medicine 5

where 120588 gt 0 is an arbitrary real constant Choosing 120588 =

120573119901119878lowast119901(119887119898 minus 119902

1015840120575) the derivative of 119881

1(119905) is

1198811015840

1(119905) = 120588 (119883 (119905) + 119884 (119905)) (119883

1015840(119905) + 119884

1015840(119905))

+ 119884 (119905) 1198841015840(119905) + 119885 (119905) 119885

1015840(119905)

= 120588 (119883 (119905) + 119884 (119905))

times [minus119887119883 (119905) + (1199021015840120575 minus 119887119898)119885 (119905) minus 119887119884 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591) minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) ]

+ 119884 (119905) [120573119901119868lowast119878lowast(119901minus1)

119883 (119905) + 120573119878lowast119901119885 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591)

minus120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)]

+ 119885 (119905) [120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)

minus (120575 minus 119902120575 + 120574 +

119889

119879

)119885 (119905)]

= minus 1198871205881198832(119905) minus (119887120588 + 119887) 119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

]1198852(119905)

+ [120573119901119868lowast119878lowast(119901minus1)

minus 2120588119887]119883 (119905) 119884 (119905)

+ 120588 (1199021015840120575 minus 119887119898)119885 (119905)119883 (119905)

minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905)119883 (119905 minus 120591)

minus 120573120588119890minus119887120591119878lowast119901119883 (119905) 119885 (119905 minus 120591)

minus 120573120588119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)

minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119884 (119905)119883 (119905 minus 120591)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119884 (119905)119883 (119905 minus 120591)

minus 120573119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)

+ 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)119885 (119905)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)119885 (119905)

(29)

and applying Cauchy-Chwartz inequality to all productterms we obtain the following expression

1198811015840

1(119905) le minus [2119887120588 minus

1

2

120573119901119868lowast119878lowast(119901minus1)

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573120588119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119883

2(119905)

minus [2120588119887 + 119887 minus

1

2

119901120573119868lowast119872119901minus1

minus

1

2

(120588 + 1) 120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119885

2(119905)

+ (120588 + 1) 120573119901119890minus119887120591119868lowast119872(119901minus1)

1198832(119905 minus 120591)

+ (120588 + 1) 120573119890minus1198871205911198721199011198852(119905 minus 120591)

(30)

We choose Lyapunov function to be the form 119881(119905) = 1198811(119905) +

1198812(119905) and we get

1198811015840(119905) = 119881

1015840

1(119905) + (120588 + 1) 120573119901119890

minus119887120591119868lowast119872(119901minus1)

times (1198832(119905) minus 119883

2(119905 minus 120591)) + (120588 + 1) 120573119890

minus119887120591119872119901

times (1198852(119905) minus 119885

2(119905 minus 120591))

(31)

Substituting this in the inequality for 1198811(119905) we get

1198811015840(119905) le minus [2119887120588 +

1

2

120588 (119887119898 minus 1199021015840120575) minus

1

2

120573119901119868lowast119872lowast(119901minus1)

minus

1

2

120573119890minus119887120591119872119901minus1

(120588119872 + 2119901119868lowast+ 3120588119901119868

lowast)]1198832(119905)

minus [2120588119887 + 119887 minus

1

2

119901120573119868lowast119872119901minus1

minus

1

2

(120588 + 1) 120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573119890minus119887120591119872119901minus1

(119868lowast119901 + 2119872120588 + 3119872)]119885

2(119905)

(32)

The right-hand expression of the above inequality is alwaysnegative provided that (26) holds A direct application of theLyapunov-LaSalle type theorem shows that lim

119905rarrinfin119883(119905) rarr

0 lim119905rarrinfin

119884(119905) rarr 0 lim119905rarrinfin

119885(119905) rarr 0 The proof iscomplete

3 Continuous Treatment andPulse Vaccination Strategies

When continuous treatment and pulse vaccination strategiesare included in the SEIR epidemic model with the nonlinear

6 Computational and Mathematical Methods in Medicine

infectious force and vertical transmission it can be written asfollows

1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898

1015840(119878 + 119877 + 119864) +

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905

+) = 119864 (119905)

119868 (119905+) = 119868 (119905) 119877 (119905

+) = 119877 (119905) + 120579119878 (119905)

119905 = 119896119879 119896 isin 119885+

(33)

and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate

We also consider the following reduced systems

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905

+) = 119868 (119905)

119905 = 119896119879 119896 isin 119885+

(34)

Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877

2

+|119878 ge

0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)

We first state two important lemmas which are useful inour following discussions

Lemma 6 (see [12]) Consider the following impulsive differ-ential equation

1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879

119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879

(35)

where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by

119906lowast(119905) =

119886

119887

+ (119906 minus

119886

119887

) 119890minus119887(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879 (36)

which is globally asymptotically stable where

119906 =

119886

119887

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

(37)

Lemma 7 (see [27 28]) Consider the following equation

1199061015840(119905) = 119886

1119906 (119905 minus 120591) minus 119886

2119906 (119905) (38)

where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has

(1) if 1198861lt 1198862 then lim

119905rarrinfin119906(119905) = 0

(2) if 1198861gt 1198862 then lim

119905rarrinfin119906(119905) = +infin

31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy

1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885

+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885

+

(39)

By Lemma 6 we obtain the periodic solution of system (39)

119878lowast(119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

119896119879 lt 119905 le (119896 + 1) 119879 (40)

and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)

Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that

Rlowast=

119898119901119890minus119887120591120573

120575 minus 119902120575 + 120574 + (119889119879)

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

lt 1 (41)

Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small

such that

119890minus119887120591120573(

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760)

119901

lt 120575 minus 119902120575 + 120574 +

119889

119879

(42)

From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus

119887119878 and then we consider the following comparison systemwith pulse

1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879

119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879

119908 (0+) = 119878 (0

+)

(43)

In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and

lim119905rarrinfin

119908 (119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

= 119878lowast(119905)

119896119879 lt 119905 le (119896 + 1) 119879

(44)

There exists an integer 1198961 such that

119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)

Computational and Mathematical Methods in Medicine 7

That is

119878 (119905) lt 119878lowast(119905) + 120576

0le

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760= 119878

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961

(46)

Furthermore from the second equation we have

1198681015840(119905) le 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(47)

and then we consider the following comparison equation

1199081015840

1(119905) = 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(48)

then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840

1(119905) lt 0 lim

119896rarrinfin119868(119905) = 0 So

there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576

1for all

119905 gt 1198962119879

When 119905 gt 1198962119879 from the first equation of system (34) we

have

1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761) minus (120573119878

119901minus1

1205761+ 119887) 119878 (119905) (49)

Consider the following comparison impulsive differentialequation for all 119905 gt 119896

2119879

1199081015840

2(119905) = (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761)

minus (120573119878

119901minus1

1205761+ 119887)119908

2 (119905) 119905 = 119896119879

1199082(119905+) = (1 minus 120579)119908

2(119905) 119905 = 119896119879

1199082(0+) = 119878 (0

+)

(50)

By Lemma 6 we have the unique periodic solution of system(50) given by

119908lowast

2=

119887119898 minus (119887119898 minus 1199021015840120575) 1205761

120573119878

119901minus1

1205761+ 119887

(1 minus

120579119890minus(1205761120573119878119901minus1

+119887)(119905minus119899119879)

1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1

+119887)119879

)

119896119879 lt 119905 le (119896 + 1) 119879

(51)

By the comparison theorem there exists an integer 1198963gt 1198962

such that

119878 (119905) gt 1199082(119905) gt 119908

lowast

2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)

Because 1205760and 1205761are sufficiently small it follows from (46)

and (52) that lim119905rarrinfin

119878(119905) = 119878lowast(119905) Therefore the disease-free

solution (119878lowast(119905) 0) of system (34) is globally attractive The

proof is completed

Denote

120579lowast= (119890119887119879minus 1)((

120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

)

1119901

minus 1)

120591lowast=

1

119887

ln120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(53)

Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the

infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free

periodic solution is globally attractive provided that 120579 gt 120579lowast

Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879

)(1 minus (1 minus 120579)119890minus119887119879

))119901le

119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution

is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890

119887120591(120575 minus 119902120575 +

120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast

Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough

32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence

Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872

119894ge 119898119894 119894 = 1 2 (both are

independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies

1198981le 119878 (119905) le 119872

1 119898

2le 119868 (119905) le 119872

2 (54)

Denote

Rlowast=

120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(55)

Theorem 12 If Rlowastgt 1 then there is a positive constant

119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)

satisfies 119868(119905) ge 119898119868for all t sufficiently large

Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1

for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898

119878

119868(119905) ge 119898119868for all 119905 gt 119905

0

Firstly from the first equation of system (34) we have

1198781015840(119905) gt 119902

1015840120575 minus (120573 + 119887) 119878 (56)

8 Computational and Mathematical Methods in Medicine

Consider the following comparison equations

1198831015840(119905) = 119902

1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879

119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879

(57)

By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905

0(1199050is

sufficiently large) such that

119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576

ge

1199021015840120575

120573 + 119887

(

(1 minus 120579) (1 minus 119890minus(120573+119887)119879

)

1 minus (1 minus 120579) 119890minus(120573+119887)119879

) minus 120576 = 119898119878gt 0

(58)

Now we will prove that there exist 119898119868gt 0 and a suff-

iciently large 1199050such that 119868(119905) ge 119898

119868for all 119905 gt 119905

0 Since the

proof is rather long it will be convenient to divide it into twosteps

Step 1 Since Rlowastgt 1 there exist 119898lowast

119868gt 0 120576 gt 0 sufficiently

small such that

120573120578119901119890minus119887120591

minus (120575 + 120574 +

119889

119879

minus 119902120575) gt 0 (59)

where 120578 = (1199021015840120575(120573119898

lowast

119868+ 119887))((1 minus 120579)(1 minus 119890

minus119887119879)(1 minus (1 minus

120579)119890minus119887119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt

119898lowast

119868for all 119905 ge 119905

0 Suppose that the claim is not valid There

exists a 1199050gt 0 such that 119868(119905) lt 119898

lowast

119868for all 119905 ge 119905

0 and then

follows from the first equation of system (34) that for 119905 ge 1199050

1198781015840(119905) gt 119902

1015840120575 minus (120573119898

lowast

119868+ 119887) 119878 (119905) (60)

Consider the comparison impulsive system for 119905 ge 1199050

1198831015840

1(119905) = 119902

1015840120575 minus (120573119898

lowast

119868+ 119887)119883 (119905) 119905 = 119896119879

1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879

(61)

According to Lemma 6 there exists 1198791gt 1199050such that

119878 (119905) gt 119883lowast

1(119905) minus 120576

ge

1199021015840120575

120573119898lowast

119868+ 119887

(

(1 minus 120579) (1 minus 119890minus120573(119898

lowast

119868+119887)119879

)

1 minus (1 minus 120579) 119890minus120573(119898

lowast

119868+119887)119879

) minus 120576 = 120578

(62)

for all 1198791gt 1199050 The second equation of system (34) can be

translated into the following form

1198681015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

minus120573119890minus119887120591 119889

119889119905

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585

(63)

Define a function 119881(119905) such that

119881 (119905) = 119868 (119905) + 120573119890minus119887120591

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585 (64)

then the derivative of 119881(119905) along the solution of system (34)is

1198811015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

= (120575 minus 119902120575 + 120574 +

119889

119879

)(

120573119878119901(119905) 119890minus119887120591

120575 minus 119902120575 + 120574 + (119889119879)

minus 1) 119868 (119905)

119905 gt 1198791

(65)

From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that

119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591

Hence there exists a 1199051gt 0 such that 119868(119905

1) ge 119898

lowast

119868

Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898

lowast

119868for all 119905 gt 119905

1 then our aim is obtained

Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case

setting 119905lowast = inf119905gt1199051

119868(119905) le 119898lowast

119868 there are two possible cases for

119905lowast

Define

119898119868= min

119898lowast

119868

2

1199021 119902

1= 119898lowast

119868119890minus(120575minus119902120575+120574+(119889119879))120591

(66)

We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion

is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast

119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578

for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that

119868 (119905) ge

119898lowast

119868

2

for 119905lowast lt 119905 lt 119905lowast + 119892 (67)

In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591

Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588

Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast

119868 and it is obvious that 119868(119905) ge 119902

1

for 119905lowast lt 119905 lt 119905lowast + 119892

Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902

1

Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021

Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898

119868for 119905lowast + 120591 lt 119905 lt 119905

lowast+ 120588 Since this kind of

interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898

119868for all large 119905 in

the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have

proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868

for all large 119905 The proof is completed

Computational and Mathematical Methods in Medicine 9

Set

120579lowast=

(119890119887119879minus 1) (1 minus ((119890

119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

)

((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

+ (119890119887119879minus 1)

120591lowast=

1

119887

ln120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(68)

From Theorems 12 we also easily obtain the followingresults

Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast

(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +

120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591

lowast

4 Numerical Simulations

In this section we present some numerical simulations todemonstrate our theoretical results established in this paper

Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible

Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease

Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591

5 Conclusion

In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model

0 20 40 60 80 100t

120579 = 01

120579 = 05

02

03

04

05

06

07

08

09

S(t)

(a)

0 20 40 60 80 100t

120579 = 01

120579 = 05

0

005

01

015

02

025

03

035

04

045

05

I(t)

(b)

Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =

0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05

10 Computational and Mathematical Methods in Medicine

0 20 40 60 80 100t

03

035

04

045

05

055

06

065

07

075S(t)

120591 = 01

120591 = 08

(a)

0 20 40 60 80 100t

005

01

015

02

025

03

035

04

045

05

I(t)

120591 = 01

120591 = 08

(b)

Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =

0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08

006008

04

08

12

16

20

002004

01

0

005

01

015

02

025

03

035

120579

120591

I

Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]

are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R

0and prove the global

stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R

lowastgt 1 the disease is permanent by using

the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (no 1124319)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991

[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007

[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006

[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

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Page 2: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

2 Computational and Mathematical Methods in Medicine

contact and 119878 and 119868 represent the susceptible and infectedpopulations respectively This contact law is more appro-priate for a few of infected individuals when the size ofinfected individuals is increasing the underlying assumptionof homogeneous mixing may not be valid In fact with theincrease of infected populations the susceptible individualwill takemeasure to prevent unbounded contact rates In [14]Anderson andMay proposed a saturated incidence rate of theform 120573119878119868(1 + 120572119878) in which 120573119878119868measures the infection forceof the disease and 1(1 + 120572119878) measures the inhibition effectfrom the behavioral change of the susceptible individualsThesame as the nonlinear incidence rates of the form 119896119868

119901119878119902 were

investigated by Liu et al [15 16]In real life some diseases may be transferred through

horizontal transmission and vertical transmission (disease isthe passing of an infection to offspring of infected parents)The offspring of infected parentsmay already be infectedwiththe disease at birth so many infections in nature transmitthrough both horizontal and vertical modes such as tuber-culosis (TB) rubella hepatitis B and AIDS [17ndash21]

Vaccination and treatment are important strategy for theelimination of infectious diseases Recently pulse vaccinationhas been confirmed as an effective method to prevent thespread of the disease [22ndash24] Theoretical results show thatthe pulse vaccination strategy can be distinguished fromthe conventional strategies in leading to disease eradicationat relatively low values of vaccination [25] The study ofvaccination treatment and associated behavioral changesrelated to disease transmission has been the subject of intensetheoretical analysis

The literature on SEIR model with nonlinear incidenceconstant infectious period impulsive vaccination dealingwith the analysis of disease that is vertically and horizontallytransmitted is not extensive [17 19] But in fact under thesituation of disease with vertical transmission the contin-uous treatment should be considered for the infected andimpulsive vaccination to the susceptible newborns of thesusceptible exposed and the removed and newborns ofinfected which not be vertical infected

Motivated by the literature above we introduce delayepidemic models with nonlinear incidence rates of the form120573119878119901119868 and we also considered the constant latency period

and vertically and horizontally in (2) The purpose of thispaper is to study the nonlinear dynamics of system andwe consider two different strategies to the model which areconstant treatment and pulse vaccination to the newbornsand susceptible

1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898

1015840(119878 + 119877 + 119864) +

119889

119879

119868

(2)

The basic assumptions are as follows

(i) The total population size at time 119905 (day) is denotedby 119873 = 119878 + 119864 + 119868 + 119877 For = 0 this shows thatthe total population has a constant size Without lossof generality we assume in this paper 119873 = 1 Thenewborns of 119878 119864 and 119877 are susceptible individualsand the newborns of 119868 who are not vertically infectedare also susceptible individuals

(ii) The positive constant 119887 (per day) denotes the deathrate and birth rate of susceptible exposed and recov-ered individuals The positive constant 120575 (per day)denotes the death rate and birth rate of infectiveindividuals The positive constant 120574 (per day) is thenatural recovery rate of infective individuals Thepositive constant 119902 (119902 le 1) (per day) is the verticaltransmission rate and note 1199021015840 = 1 minus 119902 1199021015840 lt 119902

(per day) and then 0 lt 1199021015840lt 1 Fraction 1198981015840 of all

newborns with mothers in the susceptible exposedand recovered classes are vaccinated and appearedin the recovered class while the remaining fraction119898 = 1 minus 119898

1015840 appears in the susceptible class suppose119887119898 gt 119902

1015840120575 (119889119879) is the proportion of those cured

successfully(iii) The incidence rate is described by a nonlinear func-

tion 120573119878119901119868 where 120573 (per day) is a positive constant

describing the infection rate 120591 gt 0 is the length ofthe latent period and the term 120573119890

minus119887120591119878119901(119905 minus 120591)119868(119905 minus 120591)

reflects the fact that an individual is surviving in thelatent period 120591 and becoming infective at time 119905

The remaining part of this paper is organized as followsIn Section 2 we investigate the global stability of the endemicequilibrium of (2) by using Rouches theorem and Lyapunov-LaSalle type theorem The global asymptotic stability ofdisease-free periodic solution and the conditions for thepermanence of the disease by comparison techniques aredescribed in Section 3 Numerical simulations are presentedin Section 4 In Section 5 we conclude this paper with someremarks

2 Continuous Treatment Strategy ofthe SEIR Model

In this section we consider a continuous treatment of SEIRmodel with constant latent period and nonlinear incidencerate By using 119878 + 119877 + 119864 = 1 minus 119868 notice that first andthird equations of system (2) do not contain the variables 119864and 119877 therefore system (2) is equivalent to the following 2-dimensional system

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

(3)

21 Disease-Free Equilibrium and Its Stability First we define

R0=

120573119898119901119890minus119887120591

120575 minus 119902120575 + 120574 + (119889119879)

(4)

Computational and Mathematical Methods in Medicine 3

Theorem 1 If R0lt 1 the disease-free equilibrium 119864

0(119898 0)

of system (3) is locally asymptotically stable for all 120591 ge 0 ifR0gt 1 the disease-free equilibrium 119864

0(119898 0) is unstable

Proof Steady states of system satisfy the following system ofequations

119887119898 (1 minus 119868) minus 120573119878119901119868 minus 119887119878 + 119902

1015840120575119868 = 0

120573119890minus119887120591119878119901119868 + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868 = 0

(5)

Obviously 1198640(119898 0) is the disease-free equilibrium of (3)

In order to analyze the behavior of the system (3) near 1198640 we

linearize the system about the equilibrium point let 119878(119905) =119883(119905) + 119898 119868(119905) = 119884(119905)

1198831015840(119905) = minus119887119883 (119905) minus (120573119898

119901+ 119887119898 minus 119902

1015840120575)119884 (119905)

1198841015840(119905) = minus(120575 minus 119902120575 + 120574 +

119889

119879

)119884 (119905) + 120573119890minus119887120591119898119901119884 (119905 minus 120591)

(6)

1205821= minus119887 lt 0 is one of the eigenvalues of the linearization of

system (6) near the steady state 1198640 and the other eigenvalue

1205822is determined by equation

120582 minus 120573119898119901119890minus(119887+120582)120591

+ 120575 minus 119902120575 + 120574 +

119889

119879

= 0 (7)

Let

119891 (120582) = 120582 minus 120573119898119901119890minus(119887+120582)120591

+ 120575 minus 119902120575 + 120574 +

119889

119879

(8)

ifR0gt 1 it is easy to show that for 120582 real

119891 (0) = (120575 minus 119902120575 + 120574 +

119889

119879

) (1 minusR0) lt 0

lim120582rarr+infin

119891 (120582) = +infin

(9)

hence 119891(120582) = 0 has a positive real root Therefore ifR0gt 1

the disease-free equilibrium 1198640(119898 0) is unstable

If R0lt 1 we prove that the disease-free equilibrium

1198640(119898 0) is locally stable Otherwise Re 120582 ge 0 We note that

Re 120582 = (120575 minus 119902120575 + 120574 + 119889

119879

) (R0119890minusRe120582120591 cos (119897119898120582120591) minus 1)

le (120575 minus 119902120575 + 120574 +

119889

119879

) (R0minus 1)

(10)

a contradiction Hence the disease-free equilibrium 1198640is

locally asymptotically stable ifR0lt 1

Theorem 2 If R0lt 1 the disease-free equilibrium 119864

0(119898 0)

of system (3) is globally asymptotically stable for all 120591 ge 0

To proof the global stability of the disease-free equilib-rium 119864

0(119898 0) we choose Lyapunov function

119881 (119905) = 119883 (119905) + 119884 (119905) + 120573119890minus119887120591119898119901int

119905

119905minus120591

119884 (120585) 119889120585 (11)

and it is easy to prove 1198811015840(119905) lt 0 lim119905rarrinfin

119881(119905) = 0 it followsthat lim

119905rarrinfin119883(119905) = 0 lim

119905rarrinfin119884(119905) = 0

22 Endemic Equilibrium and Its Stability If R0gt 1 then

system (3) has a unique positive equilibrium 1198641(119878lowast 119868lowast)

where

119878lowast= (

(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591

120573

)

1119901

119868lowast=

119887

(119887119898 minus 1199021015840120575) + (120575 minus 119902120575 + 120574 + (119889119879)) 119890

119887120591

times (119898 minus (

(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591

120573

)

1119901

)

(12)

Theorem 3 If R0gt 1 conditions (17) and (22) are satisfied

then for 120591 ge 0 the endemic equilibrium 1198641(119878lowast 119868lowast) of system (3)

is locally asymptotically stable

Proof Let 119878(119905) = 119883(119905) + 119878lowast 119868(119905) = 119884(119905) + 119868

lowast the linearizedsystem is obtained

1198831015840(119905) = minus (119887 + 120573119901119868

lowast119878lowast(119901minus1)

)119883 (119905)

minus (120573119878lowast119901+ 119887119898 minus 119902

1015840120575)119884 (119905)

1198841015840(119905) = 120573119901119890

minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119884 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

)119884 (119905)

(13)

From the linearized system we obtain the characteristic equ-ation

1205822+ 119901120582 + 119903 + (119892120582 + 119902) 119890

minus120582120591= 0 (14)

where

119901 = 120575 minus 119902120575 + 120574 +

119889

119879

+ 120573119901119878lowast(119901minus1)

119868lowast+ 119887

119903 = (119887 + 120573119901119868lowast119878lowast(119901minus1)

) (120575 minus 119902120575 + 120574 +

119889

119879

)

119892 = minus120573119878lowast119901119890minus119887120591

119902 = minus (119887 + 120573119901119868lowast119878lowast(119901minus1)

) 120573119890minus119887120591119878lowast119901

+ (120573119878lowast119901+ 119887119898 minus 119902

1015840120575) 120573119901119890

minus119887120591119868lowast119878lowast(119901minus1)

(15)

For 120591 = 0 the characteristic equation becomes

1205822+ (119901 + 119892) 120582 + (119903 + 119902) = 0 (16)

and we can see that both roots are negative and real if andonly if

119901 + 119892 gt 0 119903 + 119902 gt 0 (17)

Now for 120591 = 0 if 120582 = 120596119894 is a root of (16) we have

minus1205962+ 119902119890minus120596120591119894

+ 119901120596119894 + 119903 + 119892120596119890minus120596120591119894

= 0 (18)

4 Computational and Mathematical Methods in Medicine

Separating the real and imaginary parts we have

119903 minus 1205962+ 119892120596 sin (120596120591) + 119902 cos120596120591 = 0

119901120596 + 119892120596 cos120596120591 minus 119902 sin120596120591 = 0(19)

Adding both equations and regrouping by powers of 120596 weobtain the following fourth degree polynomial

1205964+ (1199012minus 1198922minus 2119903) 120596

2+ 1199032minus 1199022= 0 (20)

from which we have

1205962=

1198922minus 1199012+ 2119903 plusmn radic(119892

2minus 1199012+ 2119903)2minus 4 (119903

2minus 1199022)

2

(21)

It follows that if

1199012minus 1198922minus 2119903 gt 0 119903

2minus 1199022gt 0 (22)

are satisfied (20) does not have positive solutions and thecharacteristic equation (14) does not have purely imaginaryroots Inequalities in (17) and (22) guarantee that all roots of(14) have no positive roots According to Rouchersquos theoremTheorem 3 is proved

Subsequently we discuss the sufficient conditions underwhich the endemic equilibrium is globally asymptoticallystable for the system (3) For 119878 + 119864 + 119868 + 119877 = 1 hence thedynamics of system (3) in the first octant of 119877

4is equivalent

to that of the following system

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

(23)

The initial conditions for system (23) take the form

119878 (120585) = 1205931(120585) 119864 (120585) = 120593

2(120585) 119868 (120585) = 120593

3(120585)

1205931 (0) gt 0 120593

2 (0) gt 0 1205933 (0) gt 0

(24)

where (1205931(120585) 1205932(120585) 1205933(120585)) isin 119862([minus120591 0] 119877

3

+0) the space of

continuous functions mapping the interval [minus120591 0] into 1198773+0

where 1198773+0= (1199091 1199092 1199093) 119909119894ge 0 119894 = 1 2 3

For continuity of the initial conditions we require

119864 (0) = int

0

minus120591

120573119890119887120585119878(120585)119901119868 (120585) 119889120585 (25)

It is well known by the fundamental theory of functionaldifferential equations [26] the system (23) has a uniquesolution (119878(119905) 119864(119905) 119868(119905)) satisfying the initial conditions Itis easy to show that all solutions of system (23) with initialconditions are defined on [0 +infin) and remain positive for all119905 ge 0

Lemma 4 (see [25]) Let the initial condition be 119878(120585) = 119878(0) gt0 119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) Then119878(119905) le max1 119878(0) + 119864(0) + 119868(0) = 119872

Theorem 5 Let the initial condition be 119878(120585) = 119878(0) gt 0119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) FurthersupposeR

0gt 1 then for any infectious period 120591 satisfying

120591 gt max

1

119887

ln120573119872119901minus1

(120588119872 + 2119901119868lowast+ 3120588119901119868

lowast)

120573119901119868lowast119872119901minus1

minus 4119887120588 minus 120588 (119887119898 minus 1199021015840120575)

1

119887

ln120573119872119901minus1

(120588 + 1) (119868lowast119901 +119872)

4120588119887 + 2119887 minus 120573119901119868lowast119872119901minus1

1

119887

ln120573119872119901minus1

(119868lowast119901 + 2119872119901 + 3119872)

2(120575 minus 119902120575 + 120574 +

119889

119879

) + 120588 (119887119898 minus 1199021015840120575)

(26)

where119872 = max1 119878(0)+119864(0)+119868(0) the endemic equilibriumis globally asymptotically stable

Proof Let 119878(119905) = 119883(119905)+119878lowast 119864(119905) = 119884(119905)+119864lowast 119868(119905) = 119885(119905)+ 119868lowastthe linearized system is

1198831015840(119905) = minus (119887 + 120573119901119868

lowast119878lowast(119901minus1)

)119883 (119905)

+ (1199021015840120575 minus 120573119878

lowast119901minus 119887119898)119885 (119905)

1198841015840(119905) = 120573119901119868

lowast119878lowast(119901minus1)

119883(119905) + 120573119878lowast119901119885 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)

1198851015840(119905) = 120573119901119890

minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

)119885 (119905)

(27)

The trivial solution of system (27) is globally asymptot-ically stable and is equivalent to the fact that the endemicequilibrium (119878

lowast 119864lowast 119868lowast) of system (23) is globally asymptot-

ically stable We will employ Lyapunov functional techniqueto prove it

Now let us introduce the following functions

1198811 (119905) =

1

2

120588(119883 (119905) + 119884 (119905))2+

1

2

(1198842(119905) + 119885

2(119905))

1198812(119905) = (120588 + 1) 120573119901119890

minus119887120591119868lowast119872(119901minus1)

int

119905

119905minus120591

1198832(120585) 119889120585

+ (120588 + 1) 120573119890minus119887120591119872119901int

119905

119905minus120591

1198852(120585) 119889120585

(28)

Computational and Mathematical Methods in Medicine 5

where 120588 gt 0 is an arbitrary real constant Choosing 120588 =

120573119901119878lowast119901(119887119898 minus 119902

1015840120575) the derivative of 119881

1(119905) is

1198811015840

1(119905) = 120588 (119883 (119905) + 119884 (119905)) (119883

1015840(119905) + 119884

1015840(119905))

+ 119884 (119905) 1198841015840(119905) + 119885 (119905) 119885

1015840(119905)

= 120588 (119883 (119905) + 119884 (119905))

times [minus119887119883 (119905) + (1199021015840120575 minus 119887119898)119885 (119905) minus 119887119884 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591) minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) ]

+ 119884 (119905) [120573119901119868lowast119878lowast(119901minus1)

119883 (119905) + 120573119878lowast119901119885 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591)

minus120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)]

+ 119885 (119905) [120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)

minus (120575 minus 119902120575 + 120574 +

119889

119879

)119885 (119905)]

= minus 1198871205881198832(119905) minus (119887120588 + 119887) 119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

]1198852(119905)

+ [120573119901119868lowast119878lowast(119901minus1)

minus 2120588119887]119883 (119905) 119884 (119905)

+ 120588 (1199021015840120575 minus 119887119898)119885 (119905)119883 (119905)

minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905)119883 (119905 minus 120591)

minus 120573120588119890minus119887120591119878lowast119901119883 (119905) 119885 (119905 minus 120591)

minus 120573120588119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)

minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119884 (119905)119883 (119905 minus 120591)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119884 (119905)119883 (119905 minus 120591)

minus 120573119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)

+ 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)119885 (119905)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)119885 (119905)

(29)

and applying Cauchy-Chwartz inequality to all productterms we obtain the following expression

1198811015840

1(119905) le minus [2119887120588 minus

1

2

120573119901119868lowast119878lowast(119901minus1)

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573120588119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119883

2(119905)

minus [2120588119887 + 119887 minus

1

2

119901120573119868lowast119872119901minus1

minus

1

2

(120588 + 1) 120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119885

2(119905)

+ (120588 + 1) 120573119901119890minus119887120591119868lowast119872(119901minus1)

1198832(119905 minus 120591)

+ (120588 + 1) 120573119890minus1198871205911198721199011198852(119905 minus 120591)

(30)

We choose Lyapunov function to be the form 119881(119905) = 1198811(119905) +

1198812(119905) and we get

1198811015840(119905) = 119881

1015840

1(119905) + (120588 + 1) 120573119901119890

minus119887120591119868lowast119872(119901minus1)

times (1198832(119905) minus 119883

2(119905 minus 120591)) + (120588 + 1) 120573119890

minus119887120591119872119901

times (1198852(119905) minus 119885

2(119905 minus 120591))

(31)

Substituting this in the inequality for 1198811(119905) we get

1198811015840(119905) le minus [2119887120588 +

1

2

120588 (119887119898 minus 1199021015840120575) minus

1

2

120573119901119868lowast119872lowast(119901minus1)

minus

1

2

120573119890minus119887120591119872119901minus1

(120588119872 + 2119901119868lowast+ 3120588119901119868

lowast)]1198832(119905)

minus [2120588119887 + 119887 minus

1

2

119901120573119868lowast119872119901minus1

minus

1

2

(120588 + 1) 120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573119890minus119887120591119872119901minus1

(119868lowast119901 + 2119872120588 + 3119872)]119885

2(119905)

(32)

The right-hand expression of the above inequality is alwaysnegative provided that (26) holds A direct application of theLyapunov-LaSalle type theorem shows that lim

119905rarrinfin119883(119905) rarr

0 lim119905rarrinfin

119884(119905) rarr 0 lim119905rarrinfin

119885(119905) rarr 0 The proof iscomplete

3 Continuous Treatment andPulse Vaccination Strategies

When continuous treatment and pulse vaccination strategiesare included in the SEIR epidemic model with the nonlinear

6 Computational and Mathematical Methods in Medicine

infectious force and vertical transmission it can be written asfollows

1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898

1015840(119878 + 119877 + 119864) +

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905

+) = 119864 (119905)

119868 (119905+) = 119868 (119905) 119877 (119905

+) = 119877 (119905) + 120579119878 (119905)

119905 = 119896119879 119896 isin 119885+

(33)

and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate

We also consider the following reduced systems

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905

+) = 119868 (119905)

119905 = 119896119879 119896 isin 119885+

(34)

Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877

2

+|119878 ge

0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)

We first state two important lemmas which are useful inour following discussions

Lemma 6 (see [12]) Consider the following impulsive differ-ential equation

1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879

119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879

(35)

where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by

119906lowast(119905) =

119886

119887

+ (119906 minus

119886

119887

) 119890minus119887(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879 (36)

which is globally asymptotically stable where

119906 =

119886

119887

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

(37)

Lemma 7 (see [27 28]) Consider the following equation

1199061015840(119905) = 119886

1119906 (119905 minus 120591) minus 119886

2119906 (119905) (38)

where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has

(1) if 1198861lt 1198862 then lim

119905rarrinfin119906(119905) = 0

(2) if 1198861gt 1198862 then lim

119905rarrinfin119906(119905) = +infin

31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy

1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885

+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885

+

(39)

By Lemma 6 we obtain the periodic solution of system (39)

119878lowast(119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

119896119879 lt 119905 le (119896 + 1) 119879 (40)

and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)

Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that

Rlowast=

119898119901119890minus119887120591120573

120575 minus 119902120575 + 120574 + (119889119879)

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

lt 1 (41)

Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small

such that

119890minus119887120591120573(

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760)

119901

lt 120575 minus 119902120575 + 120574 +

119889

119879

(42)

From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus

119887119878 and then we consider the following comparison systemwith pulse

1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879

119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879

119908 (0+) = 119878 (0

+)

(43)

In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and

lim119905rarrinfin

119908 (119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

= 119878lowast(119905)

119896119879 lt 119905 le (119896 + 1) 119879

(44)

There exists an integer 1198961 such that

119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)

Computational and Mathematical Methods in Medicine 7

That is

119878 (119905) lt 119878lowast(119905) + 120576

0le

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760= 119878

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961

(46)

Furthermore from the second equation we have

1198681015840(119905) le 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(47)

and then we consider the following comparison equation

1199081015840

1(119905) = 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(48)

then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840

1(119905) lt 0 lim

119896rarrinfin119868(119905) = 0 So

there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576

1for all

119905 gt 1198962119879

When 119905 gt 1198962119879 from the first equation of system (34) we

have

1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761) minus (120573119878

119901minus1

1205761+ 119887) 119878 (119905) (49)

Consider the following comparison impulsive differentialequation for all 119905 gt 119896

2119879

1199081015840

2(119905) = (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761)

minus (120573119878

119901minus1

1205761+ 119887)119908

2 (119905) 119905 = 119896119879

1199082(119905+) = (1 minus 120579)119908

2(119905) 119905 = 119896119879

1199082(0+) = 119878 (0

+)

(50)

By Lemma 6 we have the unique periodic solution of system(50) given by

119908lowast

2=

119887119898 minus (119887119898 minus 1199021015840120575) 1205761

120573119878

119901minus1

1205761+ 119887

(1 minus

120579119890minus(1205761120573119878119901minus1

+119887)(119905minus119899119879)

1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1

+119887)119879

)

119896119879 lt 119905 le (119896 + 1) 119879

(51)

By the comparison theorem there exists an integer 1198963gt 1198962

such that

119878 (119905) gt 1199082(119905) gt 119908

lowast

2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)

Because 1205760and 1205761are sufficiently small it follows from (46)

and (52) that lim119905rarrinfin

119878(119905) = 119878lowast(119905) Therefore the disease-free

solution (119878lowast(119905) 0) of system (34) is globally attractive The

proof is completed

Denote

120579lowast= (119890119887119879minus 1)((

120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

)

1119901

minus 1)

120591lowast=

1

119887

ln120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(53)

Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the

infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free

periodic solution is globally attractive provided that 120579 gt 120579lowast

Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879

)(1 minus (1 minus 120579)119890minus119887119879

))119901le

119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution

is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890

119887120591(120575 minus 119902120575 +

120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast

Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough

32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence

Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872

119894ge 119898119894 119894 = 1 2 (both are

independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies

1198981le 119878 (119905) le 119872

1 119898

2le 119868 (119905) le 119872

2 (54)

Denote

Rlowast=

120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(55)

Theorem 12 If Rlowastgt 1 then there is a positive constant

119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)

satisfies 119868(119905) ge 119898119868for all t sufficiently large

Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1

for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898

119878

119868(119905) ge 119898119868for all 119905 gt 119905

0

Firstly from the first equation of system (34) we have

1198781015840(119905) gt 119902

1015840120575 minus (120573 + 119887) 119878 (56)

8 Computational and Mathematical Methods in Medicine

Consider the following comparison equations

1198831015840(119905) = 119902

1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879

119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879

(57)

By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905

0(1199050is

sufficiently large) such that

119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576

ge

1199021015840120575

120573 + 119887

(

(1 minus 120579) (1 minus 119890minus(120573+119887)119879

)

1 minus (1 minus 120579) 119890minus(120573+119887)119879

) minus 120576 = 119898119878gt 0

(58)

Now we will prove that there exist 119898119868gt 0 and a suff-

iciently large 1199050such that 119868(119905) ge 119898

119868for all 119905 gt 119905

0 Since the

proof is rather long it will be convenient to divide it into twosteps

Step 1 Since Rlowastgt 1 there exist 119898lowast

119868gt 0 120576 gt 0 sufficiently

small such that

120573120578119901119890minus119887120591

minus (120575 + 120574 +

119889

119879

minus 119902120575) gt 0 (59)

where 120578 = (1199021015840120575(120573119898

lowast

119868+ 119887))((1 minus 120579)(1 minus 119890

minus119887119879)(1 minus (1 minus

120579)119890minus119887119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt

119898lowast

119868for all 119905 ge 119905

0 Suppose that the claim is not valid There

exists a 1199050gt 0 such that 119868(119905) lt 119898

lowast

119868for all 119905 ge 119905

0 and then

follows from the first equation of system (34) that for 119905 ge 1199050

1198781015840(119905) gt 119902

1015840120575 minus (120573119898

lowast

119868+ 119887) 119878 (119905) (60)

Consider the comparison impulsive system for 119905 ge 1199050

1198831015840

1(119905) = 119902

1015840120575 minus (120573119898

lowast

119868+ 119887)119883 (119905) 119905 = 119896119879

1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879

(61)

According to Lemma 6 there exists 1198791gt 1199050such that

119878 (119905) gt 119883lowast

1(119905) minus 120576

ge

1199021015840120575

120573119898lowast

119868+ 119887

(

(1 minus 120579) (1 minus 119890minus120573(119898

lowast

119868+119887)119879

)

1 minus (1 minus 120579) 119890minus120573(119898

lowast

119868+119887)119879

) minus 120576 = 120578

(62)

for all 1198791gt 1199050 The second equation of system (34) can be

translated into the following form

1198681015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

minus120573119890minus119887120591 119889

119889119905

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585

(63)

Define a function 119881(119905) such that

119881 (119905) = 119868 (119905) + 120573119890minus119887120591

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585 (64)

then the derivative of 119881(119905) along the solution of system (34)is

1198811015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

= (120575 minus 119902120575 + 120574 +

119889

119879

)(

120573119878119901(119905) 119890minus119887120591

120575 minus 119902120575 + 120574 + (119889119879)

minus 1) 119868 (119905)

119905 gt 1198791

(65)

From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that

119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591

Hence there exists a 1199051gt 0 such that 119868(119905

1) ge 119898

lowast

119868

Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898

lowast

119868for all 119905 gt 119905

1 then our aim is obtained

Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case

setting 119905lowast = inf119905gt1199051

119868(119905) le 119898lowast

119868 there are two possible cases for

119905lowast

Define

119898119868= min

119898lowast

119868

2

1199021 119902

1= 119898lowast

119868119890minus(120575minus119902120575+120574+(119889119879))120591

(66)

We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion

is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast

119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578

for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that

119868 (119905) ge

119898lowast

119868

2

for 119905lowast lt 119905 lt 119905lowast + 119892 (67)

In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591

Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588

Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast

119868 and it is obvious that 119868(119905) ge 119902

1

for 119905lowast lt 119905 lt 119905lowast + 119892

Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902

1

Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021

Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898

119868for 119905lowast + 120591 lt 119905 lt 119905

lowast+ 120588 Since this kind of

interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898

119868for all large 119905 in

the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have

proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868

for all large 119905 The proof is completed

Computational and Mathematical Methods in Medicine 9

Set

120579lowast=

(119890119887119879minus 1) (1 minus ((119890

119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

)

((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

+ (119890119887119879minus 1)

120591lowast=

1

119887

ln120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(68)

From Theorems 12 we also easily obtain the followingresults

Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast

(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +

120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591

lowast

4 Numerical Simulations

In this section we present some numerical simulations todemonstrate our theoretical results established in this paper

Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible

Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease

Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591

5 Conclusion

In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model

0 20 40 60 80 100t

120579 = 01

120579 = 05

02

03

04

05

06

07

08

09

S(t)

(a)

0 20 40 60 80 100t

120579 = 01

120579 = 05

0

005

01

015

02

025

03

035

04

045

05

I(t)

(b)

Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =

0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05

10 Computational and Mathematical Methods in Medicine

0 20 40 60 80 100t

03

035

04

045

05

055

06

065

07

075S(t)

120591 = 01

120591 = 08

(a)

0 20 40 60 80 100t

005

01

015

02

025

03

035

04

045

05

I(t)

120591 = 01

120591 = 08

(b)

Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =

0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08

006008

04

08

12

16

20

002004

01

0

005

01

015

02

025

03

035

120579

120591

I

Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]

are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R

0and prove the global

stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R

lowastgt 1 the disease is permanent by using

the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (no 1124319)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991

[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007

[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006

[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

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Page 3: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

Computational and Mathematical Methods in Medicine 3

Theorem 1 If R0lt 1 the disease-free equilibrium 119864

0(119898 0)

of system (3) is locally asymptotically stable for all 120591 ge 0 ifR0gt 1 the disease-free equilibrium 119864

0(119898 0) is unstable

Proof Steady states of system satisfy the following system ofequations

119887119898 (1 minus 119868) minus 120573119878119901119868 minus 119887119878 + 119902

1015840120575119868 = 0

120573119890minus119887120591119878119901119868 + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868 = 0

(5)

Obviously 1198640(119898 0) is the disease-free equilibrium of (3)

In order to analyze the behavior of the system (3) near 1198640 we

linearize the system about the equilibrium point let 119878(119905) =119883(119905) + 119898 119868(119905) = 119884(119905)

1198831015840(119905) = minus119887119883 (119905) minus (120573119898

119901+ 119887119898 minus 119902

1015840120575)119884 (119905)

1198841015840(119905) = minus(120575 minus 119902120575 + 120574 +

119889

119879

)119884 (119905) + 120573119890minus119887120591119898119901119884 (119905 minus 120591)

(6)

1205821= minus119887 lt 0 is one of the eigenvalues of the linearization of

system (6) near the steady state 1198640 and the other eigenvalue

1205822is determined by equation

120582 minus 120573119898119901119890minus(119887+120582)120591

+ 120575 minus 119902120575 + 120574 +

119889

119879

= 0 (7)

Let

119891 (120582) = 120582 minus 120573119898119901119890minus(119887+120582)120591

+ 120575 minus 119902120575 + 120574 +

119889

119879

(8)

ifR0gt 1 it is easy to show that for 120582 real

119891 (0) = (120575 minus 119902120575 + 120574 +

119889

119879

) (1 minusR0) lt 0

lim120582rarr+infin

119891 (120582) = +infin

(9)

hence 119891(120582) = 0 has a positive real root Therefore ifR0gt 1

the disease-free equilibrium 1198640(119898 0) is unstable

If R0lt 1 we prove that the disease-free equilibrium

1198640(119898 0) is locally stable Otherwise Re 120582 ge 0 We note that

Re 120582 = (120575 minus 119902120575 + 120574 + 119889

119879

) (R0119890minusRe120582120591 cos (119897119898120582120591) minus 1)

le (120575 minus 119902120575 + 120574 +

119889

119879

) (R0minus 1)

(10)

a contradiction Hence the disease-free equilibrium 1198640is

locally asymptotically stable ifR0lt 1

Theorem 2 If R0lt 1 the disease-free equilibrium 119864

0(119898 0)

of system (3) is globally asymptotically stable for all 120591 ge 0

To proof the global stability of the disease-free equilib-rium 119864

0(119898 0) we choose Lyapunov function

119881 (119905) = 119883 (119905) + 119884 (119905) + 120573119890minus119887120591119898119901int

119905

119905minus120591

119884 (120585) 119889120585 (11)

and it is easy to prove 1198811015840(119905) lt 0 lim119905rarrinfin

119881(119905) = 0 it followsthat lim

119905rarrinfin119883(119905) = 0 lim

119905rarrinfin119884(119905) = 0

22 Endemic Equilibrium and Its Stability If R0gt 1 then

system (3) has a unique positive equilibrium 1198641(119878lowast 119868lowast)

where

119878lowast= (

(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591

120573

)

1119901

119868lowast=

119887

(119887119898 minus 1199021015840120575) + (120575 minus 119902120575 + 120574 + (119889119879)) 119890

119887120591

times (119898 minus (

(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591

120573

)

1119901

)

(12)

Theorem 3 If R0gt 1 conditions (17) and (22) are satisfied

then for 120591 ge 0 the endemic equilibrium 1198641(119878lowast 119868lowast) of system (3)

is locally asymptotically stable

Proof Let 119878(119905) = 119883(119905) + 119878lowast 119868(119905) = 119884(119905) + 119868

lowast the linearizedsystem is obtained

1198831015840(119905) = minus (119887 + 120573119901119868

lowast119878lowast(119901minus1)

)119883 (119905)

minus (120573119878lowast119901+ 119887119898 minus 119902

1015840120575)119884 (119905)

1198841015840(119905) = 120573119901119890

minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119884 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

)119884 (119905)

(13)

From the linearized system we obtain the characteristic equ-ation

1205822+ 119901120582 + 119903 + (119892120582 + 119902) 119890

minus120582120591= 0 (14)

where

119901 = 120575 minus 119902120575 + 120574 +

119889

119879

+ 120573119901119878lowast(119901minus1)

119868lowast+ 119887

119903 = (119887 + 120573119901119868lowast119878lowast(119901minus1)

) (120575 minus 119902120575 + 120574 +

119889

119879

)

119892 = minus120573119878lowast119901119890minus119887120591

119902 = minus (119887 + 120573119901119868lowast119878lowast(119901minus1)

) 120573119890minus119887120591119878lowast119901

+ (120573119878lowast119901+ 119887119898 minus 119902

1015840120575) 120573119901119890

minus119887120591119868lowast119878lowast(119901minus1)

(15)

For 120591 = 0 the characteristic equation becomes

1205822+ (119901 + 119892) 120582 + (119903 + 119902) = 0 (16)

and we can see that both roots are negative and real if andonly if

119901 + 119892 gt 0 119903 + 119902 gt 0 (17)

Now for 120591 = 0 if 120582 = 120596119894 is a root of (16) we have

minus1205962+ 119902119890minus120596120591119894

+ 119901120596119894 + 119903 + 119892120596119890minus120596120591119894

= 0 (18)

4 Computational and Mathematical Methods in Medicine

Separating the real and imaginary parts we have

119903 minus 1205962+ 119892120596 sin (120596120591) + 119902 cos120596120591 = 0

119901120596 + 119892120596 cos120596120591 minus 119902 sin120596120591 = 0(19)

Adding both equations and regrouping by powers of 120596 weobtain the following fourth degree polynomial

1205964+ (1199012minus 1198922minus 2119903) 120596

2+ 1199032minus 1199022= 0 (20)

from which we have

1205962=

1198922minus 1199012+ 2119903 plusmn radic(119892

2minus 1199012+ 2119903)2minus 4 (119903

2minus 1199022)

2

(21)

It follows that if

1199012minus 1198922minus 2119903 gt 0 119903

2minus 1199022gt 0 (22)

are satisfied (20) does not have positive solutions and thecharacteristic equation (14) does not have purely imaginaryroots Inequalities in (17) and (22) guarantee that all roots of(14) have no positive roots According to Rouchersquos theoremTheorem 3 is proved

Subsequently we discuss the sufficient conditions underwhich the endemic equilibrium is globally asymptoticallystable for the system (3) For 119878 + 119864 + 119868 + 119877 = 1 hence thedynamics of system (3) in the first octant of 119877

4is equivalent

to that of the following system

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

(23)

The initial conditions for system (23) take the form

119878 (120585) = 1205931(120585) 119864 (120585) = 120593

2(120585) 119868 (120585) = 120593

3(120585)

1205931 (0) gt 0 120593

2 (0) gt 0 1205933 (0) gt 0

(24)

where (1205931(120585) 1205932(120585) 1205933(120585)) isin 119862([minus120591 0] 119877

3

+0) the space of

continuous functions mapping the interval [minus120591 0] into 1198773+0

where 1198773+0= (1199091 1199092 1199093) 119909119894ge 0 119894 = 1 2 3

For continuity of the initial conditions we require

119864 (0) = int

0

minus120591

120573119890119887120585119878(120585)119901119868 (120585) 119889120585 (25)

It is well known by the fundamental theory of functionaldifferential equations [26] the system (23) has a uniquesolution (119878(119905) 119864(119905) 119868(119905)) satisfying the initial conditions Itis easy to show that all solutions of system (23) with initialconditions are defined on [0 +infin) and remain positive for all119905 ge 0

Lemma 4 (see [25]) Let the initial condition be 119878(120585) = 119878(0) gt0 119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) Then119878(119905) le max1 119878(0) + 119864(0) + 119868(0) = 119872

Theorem 5 Let the initial condition be 119878(120585) = 119878(0) gt 0119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) FurthersupposeR

0gt 1 then for any infectious period 120591 satisfying

120591 gt max

1

119887

ln120573119872119901minus1

(120588119872 + 2119901119868lowast+ 3120588119901119868

lowast)

120573119901119868lowast119872119901minus1

minus 4119887120588 minus 120588 (119887119898 minus 1199021015840120575)

1

119887

ln120573119872119901minus1

(120588 + 1) (119868lowast119901 +119872)

4120588119887 + 2119887 minus 120573119901119868lowast119872119901minus1

1

119887

ln120573119872119901minus1

(119868lowast119901 + 2119872119901 + 3119872)

2(120575 minus 119902120575 + 120574 +

119889

119879

) + 120588 (119887119898 minus 1199021015840120575)

(26)

where119872 = max1 119878(0)+119864(0)+119868(0) the endemic equilibriumis globally asymptotically stable

Proof Let 119878(119905) = 119883(119905)+119878lowast 119864(119905) = 119884(119905)+119864lowast 119868(119905) = 119885(119905)+ 119868lowastthe linearized system is

1198831015840(119905) = minus (119887 + 120573119901119868

lowast119878lowast(119901minus1)

)119883 (119905)

+ (1199021015840120575 minus 120573119878

lowast119901minus 119887119898)119885 (119905)

1198841015840(119905) = 120573119901119868

lowast119878lowast(119901minus1)

119883(119905) + 120573119878lowast119901119885 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)

1198851015840(119905) = 120573119901119890

minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

)119885 (119905)

(27)

The trivial solution of system (27) is globally asymptot-ically stable and is equivalent to the fact that the endemicequilibrium (119878

lowast 119864lowast 119868lowast) of system (23) is globally asymptot-

ically stable We will employ Lyapunov functional techniqueto prove it

Now let us introduce the following functions

1198811 (119905) =

1

2

120588(119883 (119905) + 119884 (119905))2+

1

2

(1198842(119905) + 119885

2(119905))

1198812(119905) = (120588 + 1) 120573119901119890

minus119887120591119868lowast119872(119901minus1)

int

119905

119905minus120591

1198832(120585) 119889120585

+ (120588 + 1) 120573119890minus119887120591119872119901int

119905

119905minus120591

1198852(120585) 119889120585

(28)

Computational and Mathematical Methods in Medicine 5

where 120588 gt 0 is an arbitrary real constant Choosing 120588 =

120573119901119878lowast119901(119887119898 minus 119902

1015840120575) the derivative of 119881

1(119905) is

1198811015840

1(119905) = 120588 (119883 (119905) + 119884 (119905)) (119883

1015840(119905) + 119884

1015840(119905))

+ 119884 (119905) 1198841015840(119905) + 119885 (119905) 119885

1015840(119905)

= 120588 (119883 (119905) + 119884 (119905))

times [minus119887119883 (119905) + (1199021015840120575 minus 119887119898)119885 (119905) minus 119887119884 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591) minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) ]

+ 119884 (119905) [120573119901119868lowast119878lowast(119901minus1)

119883 (119905) + 120573119878lowast119901119885 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591)

minus120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)]

+ 119885 (119905) [120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)

minus (120575 minus 119902120575 + 120574 +

119889

119879

)119885 (119905)]

= minus 1198871205881198832(119905) minus (119887120588 + 119887) 119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

]1198852(119905)

+ [120573119901119868lowast119878lowast(119901minus1)

minus 2120588119887]119883 (119905) 119884 (119905)

+ 120588 (1199021015840120575 minus 119887119898)119885 (119905)119883 (119905)

minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905)119883 (119905 minus 120591)

minus 120573120588119890minus119887120591119878lowast119901119883 (119905) 119885 (119905 minus 120591)

minus 120573120588119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)

minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119884 (119905)119883 (119905 minus 120591)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119884 (119905)119883 (119905 minus 120591)

minus 120573119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)

+ 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)119885 (119905)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)119885 (119905)

(29)

and applying Cauchy-Chwartz inequality to all productterms we obtain the following expression

1198811015840

1(119905) le minus [2119887120588 minus

1

2

120573119901119868lowast119878lowast(119901minus1)

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573120588119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119883

2(119905)

minus [2120588119887 + 119887 minus

1

2

119901120573119868lowast119872119901minus1

minus

1

2

(120588 + 1) 120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119885

2(119905)

+ (120588 + 1) 120573119901119890minus119887120591119868lowast119872(119901minus1)

1198832(119905 minus 120591)

+ (120588 + 1) 120573119890minus1198871205911198721199011198852(119905 minus 120591)

(30)

We choose Lyapunov function to be the form 119881(119905) = 1198811(119905) +

1198812(119905) and we get

1198811015840(119905) = 119881

1015840

1(119905) + (120588 + 1) 120573119901119890

minus119887120591119868lowast119872(119901minus1)

times (1198832(119905) minus 119883

2(119905 minus 120591)) + (120588 + 1) 120573119890

minus119887120591119872119901

times (1198852(119905) minus 119885

2(119905 minus 120591))

(31)

Substituting this in the inequality for 1198811(119905) we get

1198811015840(119905) le minus [2119887120588 +

1

2

120588 (119887119898 minus 1199021015840120575) minus

1

2

120573119901119868lowast119872lowast(119901minus1)

minus

1

2

120573119890minus119887120591119872119901minus1

(120588119872 + 2119901119868lowast+ 3120588119901119868

lowast)]1198832(119905)

minus [2120588119887 + 119887 minus

1

2

119901120573119868lowast119872119901minus1

minus

1

2

(120588 + 1) 120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573119890minus119887120591119872119901minus1

(119868lowast119901 + 2119872120588 + 3119872)]119885

2(119905)

(32)

The right-hand expression of the above inequality is alwaysnegative provided that (26) holds A direct application of theLyapunov-LaSalle type theorem shows that lim

119905rarrinfin119883(119905) rarr

0 lim119905rarrinfin

119884(119905) rarr 0 lim119905rarrinfin

119885(119905) rarr 0 The proof iscomplete

3 Continuous Treatment andPulse Vaccination Strategies

When continuous treatment and pulse vaccination strategiesare included in the SEIR epidemic model with the nonlinear

6 Computational and Mathematical Methods in Medicine

infectious force and vertical transmission it can be written asfollows

1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898

1015840(119878 + 119877 + 119864) +

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905

+) = 119864 (119905)

119868 (119905+) = 119868 (119905) 119877 (119905

+) = 119877 (119905) + 120579119878 (119905)

119905 = 119896119879 119896 isin 119885+

(33)

and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate

We also consider the following reduced systems

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905

+) = 119868 (119905)

119905 = 119896119879 119896 isin 119885+

(34)

Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877

2

+|119878 ge

0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)

We first state two important lemmas which are useful inour following discussions

Lemma 6 (see [12]) Consider the following impulsive differ-ential equation

1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879

119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879

(35)

where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by

119906lowast(119905) =

119886

119887

+ (119906 minus

119886

119887

) 119890minus119887(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879 (36)

which is globally asymptotically stable where

119906 =

119886

119887

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

(37)

Lemma 7 (see [27 28]) Consider the following equation

1199061015840(119905) = 119886

1119906 (119905 minus 120591) minus 119886

2119906 (119905) (38)

where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has

(1) if 1198861lt 1198862 then lim

119905rarrinfin119906(119905) = 0

(2) if 1198861gt 1198862 then lim

119905rarrinfin119906(119905) = +infin

31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy

1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885

+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885

+

(39)

By Lemma 6 we obtain the periodic solution of system (39)

119878lowast(119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

119896119879 lt 119905 le (119896 + 1) 119879 (40)

and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)

Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that

Rlowast=

119898119901119890minus119887120591120573

120575 minus 119902120575 + 120574 + (119889119879)

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

lt 1 (41)

Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small

such that

119890minus119887120591120573(

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760)

119901

lt 120575 minus 119902120575 + 120574 +

119889

119879

(42)

From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus

119887119878 and then we consider the following comparison systemwith pulse

1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879

119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879

119908 (0+) = 119878 (0

+)

(43)

In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and

lim119905rarrinfin

119908 (119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

= 119878lowast(119905)

119896119879 lt 119905 le (119896 + 1) 119879

(44)

There exists an integer 1198961 such that

119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)

Computational and Mathematical Methods in Medicine 7

That is

119878 (119905) lt 119878lowast(119905) + 120576

0le

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760= 119878

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961

(46)

Furthermore from the second equation we have

1198681015840(119905) le 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(47)

and then we consider the following comparison equation

1199081015840

1(119905) = 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(48)

then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840

1(119905) lt 0 lim

119896rarrinfin119868(119905) = 0 So

there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576

1for all

119905 gt 1198962119879

When 119905 gt 1198962119879 from the first equation of system (34) we

have

1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761) minus (120573119878

119901minus1

1205761+ 119887) 119878 (119905) (49)

Consider the following comparison impulsive differentialequation for all 119905 gt 119896

2119879

1199081015840

2(119905) = (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761)

minus (120573119878

119901minus1

1205761+ 119887)119908

2 (119905) 119905 = 119896119879

1199082(119905+) = (1 minus 120579)119908

2(119905) 119905 = 119896119879

1199082(0+) = 119878 (0

+)

(50)

By Lemma 6 we have the unique periodic solution of system(50) given by

119908lowast

2=

119887119898 minus (119887119898 minus 1199021015840120575) 1205761

120573119878

119901minus1

1205761+ 119887

(1 minus

120579119890minus(1205761120573119878119901minus1

+119887)(119905minus119899119879)

1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1

+119887)119879

)

119896119879 lt 119905 le (119896 + 1) 119879

(51)

By the comparison theorem there exists an integer 1198963gt 1198962

such that

119878 (119905) gt 1199082(119905) gt 119908

lowast

2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)

Because 1205760and 1205761are sufficiently small it follows from (46)

and (52) that lim119905rarrinfin

119878(119905) = 119878lowast(119905) Therefore the disease-free

solution (119878lowast(119905) 0) of system (34) is globally attractive The

proof is completed

Denote

120579lowast= (119890119887119879minus 1)((

120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

)

1119901

minus 1)

120591lowast=

1

119887

ln120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(53)

Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the

infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free

periodic solution is globally attractive provided that 120579 gt 120579lowast

Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879

)(1 minus (1 minus 120579)119890minus119887119879

))119901le

119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution

is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890

119887120591(120575 minus 119902120575 +

120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast

Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough

32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence

Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872

119894ge 119898119894 119894 = 1 2 (both are

independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies

1198981le 119878 (119905) le 119872

1 119898

2le 119868 (119905) le 119872

2 (54)

Denote

Rlowast=

120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(55)

Theorem 12 If Rlowastgt 1 then there is a positive constant

119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)

satisfies 119868(119905) ge 119898119868for all t sufficiently large

Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1

for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898

119878

119868(119905) ge 119898119868for all 119905 gt 119905

0

Firstly from the first equation of system (34) we have

1198781015840(119905) gt 119902

1015840120575 minus (120573 + 119887) 119878 (56)

8 Computational and Mathematical Methods in Medicine

Consider the following comparison equations

1198831015840(119905) = 119902

1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879

119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879

(57)

By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905

0(1199050is

sufficiently large) such that

119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576

ge

1199021015840120575

120573 + 119887

(

(1 minus 120579) (1 minus 119890minus(120573+119887)119879

)

1 minus (1 minus 120579) 119890minus(120573+119887)119879

) minus 120576 = 119898119878gt 0

(58)

Now we will prove that there exist 119898119868gt 0 and a suff-

iciently large 1199050such that 119868(119905) ge 119898

119868for all 119905 gt 119905

0 Since the

proof is rather long it will be convenient to divide it into twosteps

Step 1 Since Rlowastgt 1 there exist 119898lowast

119868gt 0 120576 gt 0 sufficiently

small such that

120573120578119901119890minus119887120591

minus (120575 + 120574 +

119889

119879

minus 119902120575) gt 0 (59)

where 120578 = (1199021015840120575(120573119898

lowast

119868+ 119887))((1 minus 120579)(1 minus 119890

minus119887119879)(1 minus (1 minus

120579)119890minus119887119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt

119898lowast

119868for all 119905 ge 119905

0 Suppose that the claim is not valid There

exists a 1199050gt 0 such that 119868(119905) lt 119898

lowast

119868for all 119905 ge 119905

0 and then

follows from the first equation of system (34) that for 119905 ge 1199050

1198781015840(119905) gt 119902

1015840120575 minus (120573119898

lowast

119868+ 119887) 119878 (119905) (60)

Consider the comparison impulsive system for 119905 ge 1199050

1198831015840

1(119905) = 119902

1015840120575 minus (120573119898

lowast

119868+ 119887)119883 (119905) 119905 = 119896119879

1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879

(61)

According to Lemma 6 there exists 1198791gt 1199050such that

119878 (119905) gt 119883lowast

1(119905) minus 120576

ge

1199021015840120575

120573119898lowast

119868+ 119887

(

(1 minus 120579) (1 minus 119890minus120573(119898

lowast

119868+119887)119879

)

1 minus (1 minus 120579) 119890minus120573(119898

lowast

119868+119887)119879

) minus 120576 = 120578

(62)

for all 1198791gt 1199050 The second equation of system (34) can be

translated into the following form

1198681015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

minus120573119890minus119887120591 119889

119889119905

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585

(63)

Define a function 119881(119905) such that

119881 (119905) = 119868 (119905) + 120573119890minus119887120591

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585 (64)

then the derivative of 119881(119905) along the solution of system (34)is

1198811015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

= (120575 minus 119902120575 + 120574 +

119889

119879

)(

120573119878119901(119905) 119890minus119887120591

120575 minus 119902120575 + 120574 + (119889119879)

minus 1) 119868 (119905)

119905 gt 1198791

(65)

From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that

119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591

Hence there exists a 1199051gt 0 such that 119868(119905

1) ge 119898

lowast

119868

Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898

lowast

119868for all 119905 gt 119905

1 then our aim is obtained

Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case

setting 119905lowast = inf119905gt1199051

119868(119905) le 119898lowast

119868 there are two possible cases for

119905lowast

Define

119898119868= min

119898lowast

119868

2

1199021 119902

1= 119898lowast

119868119890minus(120575minus119902120575+120574+(119889119879))120591

(66)

We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion

is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast

119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578

for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that

119868 (119905) ge

119898lowast

119868

2

for 119905lowast lt 119905 lt 119905lowast + 119892 (67)

In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591

Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588

Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast

119868 and it is obvious that 119868(119905) ge 119902

1

for 119905lowast lt 119905 lt 119905lowast + 119892

Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902

1

Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021

Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898

119868for 119905lowast + 120591 lt 119905 lt 119905

lowast+ 120588 Since this kind of

interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898

119868for all large 119905 in

the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have

proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868

for all large 119905 The proof is completed

Computational and Mathematical Methods in Medicine 9

Set

120579lowast=

(119890119887119879minus 1) (1 minus ((119890

119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

)

((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

+ (119890119887119879minus 1)

120591lowast=

1

119887

ln120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(68)

From Theorems 12 we also easily obtain the followingresults

Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast

(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +

120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591

lowast

4 Numerical Simulations

In this section we present some numerical simulations todemonstrate our theoretical results established in this paper

Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible

Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease

Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591

5 Conclusion

In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model

0 20 40 60 80 100t

120579 = 01

120579 = 05

02

03

04

05

06

07

08

09

S(t)

(a)

0 20 40 60 80 100t

120579 = 01

120579 = 05

0

005

01

015

02

025

03

035

04

045

05

I(t)

(b)

Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =

0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05

10 Computational and Mathematical Methods in Medicine

0 20 40 60 80 100t

03

035

04

045

05

055

06

065

07

075S(t)

120591 = 01

120591 = 08

(a)

0 20 40 60 80 100t

005

01

015

02

025

03

035

04

045

05

I(t)

120591 = 01

120591 = 08

(b)

Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =

0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08

006008

04

08

12

16

20

002004

01

0

005

01

015

02

025

03

035

120579

120591

I

Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]

are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R

0and prove the global

stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R

lowastgt 1 the disease is permanent by using

the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (no 1124319)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991

[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007

[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006

[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

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Page 4: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

4 Computational and Mathematical Methods in Medicine

Separating the real and imaginary parts we have

119903 minus 1205962+ 119892120596 sin (120596120591) + 119902 cos120596120591 = 0

119901120596 + 119892120596 cos120596120591 minus 119902 sin120596120591 = 0(19)

Adding both equations and regrouping by powers of 120596 weobtain the following fourth degree polynomial

1205964+ (1199012minus 1198922minus 2119903) 120596

2+ 1199032minus 1199022= 0 (20)

from which we have

1205962=

1198922minus 1199012+ 2119903 plusmn radic(119892

2minus 1199012+ 2119903)2minus 4 (119903

2minus 1199022)

2

(21)

It follows that if

1199012minus 1198922minus 2119903 gt 0 119903

2minus 1199022gt 0 (22)

are satisfied (20) does not have positive solutions and thecharacteristic equation (14) does not have purely imaginaryroots Inequalities in (17) and (22) guarantee that all roots of(14) have no positive roots According to Rouchersquos theoremTheorem 3 is proved

Subsequently we discuss the sufficient conditions underwhich the endemic equilibrium is globally asymptoticallystable for the system (3) For 119878 + 119864 + 119868 + 119877 = 1 hence thedynamics of system (3) in the first octant of 119877

4is equivalent

to that of the following system

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

(23)

The initial conditions for system (23) take the form

119878 (120585) = 1205931(120585) 119864 (120585) = 120593

2(120585) 119868 (120585) = 120593

3(120585)

1205931 (0) gt 0 120593

2 (0) gt 0 1205933 (0) gt 0

(24)

where (1205931(120585) 1205932(120585) 1205933(120585)) isin 119862([minus120591 0] 119877

3

+0) the space of

continuous functions mapping the interval [minus120591 0] into 1198773+0

where 1198773+0= (1199091 1199092 1199093) 119909119894ge 0 119894 = 1 2 3

For continuity of the initial conditions we require

119864 (0) = int

0

minus120591

120573119890119887120585119878(120585)119901119868 (120585) 119889120585 (25)

It is well known by the fundamental theory of functionaldifferential equations [26] the system (23) has a uniquesolution (119878(119905) 119864(119905) 119868(119905)) satisfying the initial conditions Itis easy to show that all solutions of system (23) with initialconditions are defined on [0 +infin) and remain positive for all119905 ge 0

Lemma 4 (see [25]) Let the initial condition be 119878(120585) = 119878(0) gt0 119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) Then119878(119905) le max1 119878(0) + 119864(0) + 119868(0) = 119872

Theorem 5 Let the initial condition be 119878(120585) = 119878(0) gt 0119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) FurthersupposeR

0gt 1 then for any infectious period 120591 satisfying

120591 gt max

1

119887

ln120573119872119901minus1

(120588119872 + 2119901119868lowast+ 3120588119901119868

lowast)

120573119901119868lowast119872119901minus1

minus 4119887120588 minus 120588 (119887119898 minus 1199021015840120575)

1

119887

ln120573119872119901minus1

(120588 + 1) (119868lowast119901 +119872)

4120588119887 + 2119887 minus 120573119901119868lowast119872119901minus1

1

119887

ln120573119872119901minus1

(119868lowast119901 + 2119872119901 + 3119872)

2(120575 minus 119902120575 + 120574 +

119889

119879

) + 120588 (119887119898 minus 1199021015840120575)

(26)

where119872 = max1 119878(0)+119864(0)+119868(0) the endemic equilibriumis globally asymptotically stable

Proof Let 119878(119905) = 119883(119905)+119878lowast 119864(119905) = 119884(119905)+119864lowast 119868(119905) = 119885(119905)+ 119868lowastthe linearized system is

1198831015840(119905) = minus (119887 + 120573119901119868

lowast119878lowast(119901minus1)

)119883 (119905)

+ (1199021015840120575 minus 120573119878

lowast119901minus 119887119898)119885 (119905)

1198841015840(119905) = 120573119901119868

lowast119878lowast(119901minus1)

119883(119905) + 120573119878lowast119901119885 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)

1198851015840(119905) = 120573119901119890

minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

)119885 (119905)

(27)

The trivial solution of system (27) is globally asymptot-ically stable and is equivalent to the fact that the endemicequilibrium (119878

lowast 119864lowast 119868lowast) of system (23) is globally asymptot-

ically stable We will employ Lyapunov functional techniqueto prove it

Now let us introduce the following functions

1198811 (119905) =

1

2

120588(119883 (119905) + 119884 (119905))2+

1

2

(1198842(119905) + 119885

2(119905))

1198812(119905) = (120588 + 1) 120573119901119890

minus119887120591119868lowast119872(119901minus1)

int

119905

119905minus120591

1198832(120585) 119889120585

+ (120588 + 1) 120573119890minus119887120591119872119901int

119905

119905minus120591

1198852(120585) 119889120585

(28)

Computational and Mathematical Methods in Medicine 5

where 120588 gt 0 is an arbitrary real constant Choosing 120588 =

120573119901119878lowast119901(119887119898 minus 119902

1015840120575) the derivative of 119881

1(119905) is

1198811015840

1(119905) = 120588 (119883 (119905) + 119884 (119905)) (119883

1015840(119905) + 119884

1015840(119905))

+ 119884 (119905) 1198841015840(119905) + 119885 (119905) 119885

1015840(119905)

= 120588 (119883 (119905) + 119884 (119905))

times [minus119887119883 (119905) + (1199021015840120575 minus 119887119898)119885 (119905) minus 119887119884 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591) minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) ]

+ 119884 (119905) [120573119901119868lowast119878lowast(119901minus1)

119883 (119905) + 120573119878lowast119901119885 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591)

minus120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)]

+ 119885 (119905) [120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)

minus (120575 minus 119902120575 + 120574 +

119889

119879

)119885 (119905)]

= minus 1198871205881198832(119905) minus (119887120588 + 119887) 119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

]1198852(119905)

+ [120573119901119868lowast119878lowast(119901minus1)

minus 2120588119887]119883 (119905) 119884 (119905)

+ 120588 (1199021015840120575 minus 119887119898)119885 (119905)119883 (119905)

minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905)119883 (119905 minus 120591)

minus 120573120588119890minus119887120591119878lowast119901119883 (119905) 119885 (119905 minus 120591)

minus 120573120588119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)

minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119884 (119905)119883 (119905 minus 120591)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119884 (119905)119883 (119905 minus 120591)

minus 120573119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)

+ 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)119885 (119905)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)119885 (119905)

(29)

and applying Cauchy-Chwartz inequality to all productterms we obtain the following expression

1198811015840

1(119905) le minus [2119887120588 minus

1

2

120573119901119868lowast119878lowast(119901minus1)

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573120588119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119883

2(119905)

minus [2120588119887 + 119887 minus

1

2

119901120573119868lowast119872119901minus1

minus

1

2

(120588 + 1) 120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119885

2(119905)

+ (120588 + 1) 120573119901119890minus119887120591119868lowast119872(119901minus1)

1198832(119905 minus 120591)

+ (120588 + 1) 120573119890minus1198871205911198721199011198852(119905 minus 120591)

(30)

We choose Lyapunov function to be the form 119881(119905) = 1198811(119905) +

1198812(119905) and we get

1198811015840(119905) = 119881

1015840

1(119905) + (120588 + 1) 120573119901119890

minus119887120591119868lowast119872(119901minus1)

times (1198832(119905) minus 119883

2(119905 minus 120591)) + (120588 + 1) 120573119890

minus119887120591119872119901

times (1198852(119905) minus 119885

2(119905 minus 120591))

(31)

Substituting this in the inequality for 1198811(119905) we get

1198811015840(119905) le minus [2119887120588 +

1

2

120588 (119887119898 minus 1199021015840120575) minus

1

2

120573119901119868lowast119872lowast(119901minus1)

minus

1

2

120573119890minus119887120591119872119901minus1

(120588119872 + 2119901119868lowast+ 3120588119901119868

lowast)]1198832(119905)

minus [2120588119887 + 119887 minus

1

2

119901120573119868lowast119872119901minus1

minus

1

2

(120588 + 1) 120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573119890minus119887120591119872119901minus1

(119868lowast119901 + 2119872120588 + 3119872)]119885

2(119905)

(32)

The right-hand expression of the above inequality is alwaysnegative provided that (26) holds A direct application of theLyapunov-LaSalle type theorem shows that lim

119905rarrinfin119883(119905) rarr

0 lim119905rarrinfin

119884(119905) rarr 0 lim119905rarrinfin

119885(119905) rarr 0 The proof iscomplete

3 Continuous Treatment andPulse Vaccination Strategies

When continuous treatment and pulse vaccination strategiesare included in the SEIR epidemic model with the nonlinear

6 Computational and Mathematical Methods in Medicine

infectious force and vertical transmission it can be written asfollows

1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898

1015840(119878 + 119877 + 119864) +

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905

+) = 119864 (119905)

119868 (119905+) = 119868 (119905) 119877 (119905

+) = 119877 (119905) + 120579119878 (119905)

119905 = 119896119879 119896 isin 119885+

(33)

and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate

We also consider the following reduced systems

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905

+) = 119868 (119905)

119905 = 119896119879 119896 isin 119885+

(34)

Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877

2

+|119878 ge

0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)

We first state two important lemmas which are useful inour following discussions

Lemma 6 (see [12]) Consider the following impulsive differ-ential equation

1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879

119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879

(35)

where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by

119906lowast(119905) =

119886

119887

+ (119906 minus

119886

119887

) 119890minus119887(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879 (36)

which is globally asymptotically stable where

119906 =

119886

119887

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

(37)

Lemma 7 (see [27 28]) Consider the following equation

1199061015840(119905) = 119886

1119906 (119905 minus 120591) minus 119886

2119906 (119905) (38)

where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has

(1) if 1198861lt 1198862 then lim

119905rarrinfin119906(119905) = 0

(2) if 1198861gt 1198862 then lim

119905rarrinfin119906(119905) = +infin

31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy

1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885

+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885

+

(39)

By Lemma 6 we obtain the periodic solution of system (39)

119878lowast(119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

119896119879 lt 119905 le (119896 + 1) 119879 (40)

and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)

Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that

Rlowast=

119898119901119890minus119887120591120573

120575 minus 119902120575 + 120574 + (119889119879)

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

lt 1 (41)

Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small

such that

119890minus119887120591120573(

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760)

119901

lt 120575 minus 119902120575 + 120574 +

119889

119879

(42)

From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus

119887119878 and then we consider the following comparison systemwith pulse

1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879

119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879

119908 (0+) = 119878 (0

+)

(43)

In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and

lim119905rarrinfin

119908 (119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

= 119878lowast(119905)

119896119879 lt 119905 le (119896 + 1) 119879

(44)

There exists an integer 1198961 such that

119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)

Computational and Mathematical Methods in Medicine 7

That is

119878 (119905) lt 119878lowast(119905) + 120576

0le

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760= 119878

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961

(46)

Furthermore from the second equation we have

1198681015840(119905) le 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(47)

and then we consider the following comparison equation

1199081015840

1(119905) = 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(48)

then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840

1(119905) lt 0 lim

119896rarrinfin119868(119905) = 0 So

there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576

1for all

119905 gt 1198962119879

When 119905 gt 1198962119879 from the first equation of system (34) we

have

1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761) minus (120573119878

119901minus1

1205761+ 119887) 119878 (119905) (49)

Consider the following comparison impulsive differentialequation for all 119905 gt 119896

2119879

1199081015840

2(119905) = (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761)

minus (120573119878

119901minus1

1205761+ 119887)119908

2 (119905) 119905 = 119896119879

1199082(119905+) = (1 minus 120579)119908

2(119905) 119905 = 119896119879

1199082(0+) = 119878 (0

+)

(50)

By Lemma 6 we have the unique periodic solution of system(50) given by

119908lowast

2=

119887119898 minus (119887119898 minus 1199021015840120575) 1205761

120573119878

119901minus1

1205761+ 119887

(1 minus

120579119890minus(1205761120573119878119901minus1

+119887)(119905minus119899119879)

1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1

+119887)119879

)

119896119879 lt 119905 le (119896 + 1) 119879

(51)

By the comparison theorem there exists an integer 1198963gt 1198962

such that

119878 (119905) gt 1199082(119905) gt 119908

lowast

2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)

Because 1205760and 1205761are sufficiently small it follows from (46)

and (52) that lim119905rarrinfin

119878(119905) = 119878lowast(119905) Therefore the disease-free

solution (119878lowast(119905) 0) of system (34) is globally attractive The

proof is completed

Denote

120579lowast= (119890119887119879minus 1)((

120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

)

1119901

minus 1)

120591lowast=

1

119887

ln120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(53)

Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the

infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free

periodic solution is globally attractive provided that 120579 gt 120579lowast

Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879

)(1 minus (1 minus 120579)119890minus119887119879

))119901le

119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution

is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890

119887120591(120575 minus 119902120575 +

120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast

Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough

32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence

Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872

119894ge 119898119894 119894 = 1 2 (both are

independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies

1198981le 119878 (119905) le 119872

1 119898

2le 119868 (119905) le 119872

2 (54)

Denote

Rlowast=

120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(55)

Theorem 12 If Rlowastgt 1 then there is a positive constant

119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)

satisfies 119868(119905) ge 119898119868for all t sufficiently large

Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1

for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898

119878

119868(119905) ge 119898119868for all 119905 gt 119905

0

Firstly from the first equation of system (34) we have

1198781015840(119905) gt 119902

1015840120575 minus (120573 + 119887) 119878 (56)

8 Computational and Mathematical Methods in Medicine

Consider the following comparison equations

1198831015840(119905) = 119902

1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879

119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879

(57)

By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905

0(1199050is

sufficiently large) such that

119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576

ge

1199021015840120575

120573 + 119887

(

(1 minus 120579) (1 minus 119890minus(120573+119887)119879

)

1 minus (1 minus 120579) 119890minus(120573+119887)119879

) minus 120576 = 119898119878gt 0

(58)

Now we will prove that there exist 119898119868gt 0 and a suff-

iciently large 1199050such that 119868(119905) ge 119898

119868for all 119905 gt 119905

0 Since the

proof is rather long it will be convenient to divide it into twosteps

Step 1 Since Rlowastgt 1 there exist 119898lowast

119868gt 0 120576 gt 0 sufficiently

small such that

120573120578119901119890minus119887120591

minus (120575 + 120574 +

119889

119879

minus 119902120575) gt 0 (59)

where 120578 = (1199021015840120575(120573119898

lowast

119868+ 119887))((1 minus 120579)(1 minus 119890

minus119887119879)(1 minus (1 minus

120579)119890minus119887119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt

119898lowast

119868for all 119905 ge 119905

0 Suppose that the claim is not valid There

exists a 1199050gt 0 such that 119868(119905) lt 119898

lowast

119868for all 119905 ge 119905

0 and then

follows from the first equation of system (34) that for 119905 ge 1199050

1198781015840(119905) gt 119902

1015840120575 minus (120573119898

lowast

119868+ 119887) 119878 (119905) (60)

Consider the comparison impulsive system for 119905 ge 1199050

1198831015840

1(119905) = 119902

1015840120575 minus (120573119898

lowast

119868+ 119887)119883 (119905) 119905 = 119896119879

1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879

(61)

According to Lemma 6 there exists 1198791gt 1199050such that

119878 (119905) gt 119883lowast

1(119905) minus 120576

ge

1199021015840120575

120573119898lowast

119868+ 119887

(

(1 minus 120579) (1 minus 119890minus120573(119898

lowast

119868+119887)119879

)

1 minus (1 minus 120579) 119890minus120573(119898

lowast

119868+119887)119879

) minus 120576 = 120578

(62)

for all 1198791gt 1199050 The second equation of system (34) can be

translated into the following form

1198681015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

minus120573119890minus119887120591 119889

119889119905

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585

(63)

Define a function 119881(119905) such that

119881 (119905) = 119868 (119905) + 120573119890minus119887120591

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585 (64)

then the derivative of 119881(119905) along the solution of system (34)is

1198811015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

= (120575 minus 119902120575 + 120574 +

119889

119879

)(

120573119878119901(119905) 119890minus119887120591

120575 minus 119902120575 + 120574 + (119889119879)

minus 1) 119868 (119905)

119905 gt 1198791

(65)

From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that

119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591

Hence there exists a 1199051gt 0 such that 119868(119905

1) ge 119898

lowast

119868

Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898

lowast

119868for all 119905 gt 119905

1 then our aim is obtained

Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case

setting 119905lowast = inf119905gt1199051

119868(119905) le 119898lowast

119868 there are two possible cases for

119905lowast

Define

119898119868= min

119898lowast

119868

2

1199021 119902

1= 119898lowast

119868119890minus(120575minus119902120575+120574+(119889119879))120591

(66)

We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion

is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast

119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578

for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that

119868 (119905) ge

119898lowast

119868

2

for 119905lowast lt 119905 lt 119905lowast + 119892 (67)

In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591

Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588

Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast

119868 and it is obvious that 119868(119905) ge 119902

1

for 119905lowast lt 119905 lt 119905lowast + 119892

Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902

1

Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021

Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898

119868for 119905lowast + 120591 lt 119905 lt 119905

lowast+ 120588 Since this kind of

interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898

119868for all large 119905 in

the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have

proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868

for all large 119905 The proof is completed

Computational and Mathematical Methods in Medicine 9

Set

120579lowast=

(119890119887119879minus 1) (1 minus ((119890

119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

)

((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

+ (119890119887119879minus 1)

120591lowast=

1

119887

ln120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(68)

From Theorems 12 we also easily obtain the followingresults

Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast

(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +

120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591

lowast

4 Numerical Simulations

In this section we present some numerical simulations todemonstrate our theoretical results established in this paper

Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible

Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease

Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591

5 Conclusion

In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model

0 20 40 60 80 100t

120579 = 01

120579 = 05

02

03

04

05

06

07

08

09

S(t)

(a)

0 20 40 60 80 100t

120579 = 01

120579 = 05

0

005

01

015

02

025

03

035

04

045

05

I(t)

(b)

Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =

0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05

10 Computational and Mathematical Methods in Medicine

0 20 40 60 80 100t

03

035

04

045

05

055

06

065

07

075S(t)

120591 = 01

120591 = 08

(a)

0 20 40 60 80 100t

005

01

015

02

025

03

035

04

045

05

I(t)

120591 = 01

120591 = 08

(b)

Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =

0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08

006008

04

08

12

16

20

002004

01

0

005

01

015

02

025

03

035

120579

120591

I

Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]

are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R

0and prove the global

stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R

lowastgt 1 the disease is permanent by using

the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (no 1124319)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991

[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007

[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006

[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

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Page 5: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

Computational and Mathematical Methods in Medicine 5

where 120588 gt 0 is an arbitrary real constant Choosing 120588 =

120573119901119878lowast119901(119887119898 minus 119902

1015840120575) the derivative of 119881

1(119905) is

1198811015840

1(119905) = 120588 (119883 (119905) + 119884 (119905)) (119883

1015840(119905) + 119884

1015840(119905))

+ 119884 (119905) 1198841015840(119905) + 119885 (119905) 119885

1015840(119905)

= 120588 (119883 (119905) + 119884 (119905))

times [minus119887119883 (119905) + (1199021015840120575 minus 119887119898)119885 (119905) minus 119887119884 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591) minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) ]

+ 119884 (119905) [120573119901119868lowast119878lowast(119901minus1)

119883 (119905) + 120573119878lowast119901119885 (119905)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905 minus 120591)

minus120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)]

+ 119885 (119905) [120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)

minus (120575 minus 119902120575 + 120574 +

119889

119879

)119885 (119905)]

= minus 1198871205881198832(119905) minus (119887120588 + 119887) 119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

]1198852(119905)

+ [120573119901119868lowast119878lowast(119901minus1)

minus 2120588119887]119883 (119905) 119884 (119905)

+ 120588 (1199021015840120575 minus 119887119898)119885 (119905)119883 (119905)

minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883 (119905)119883 (119905 minus 120591)

minus 120573120588119890minus119887120591119878lowast119901119883 (119905) 119885 (119905 minus 120591)

minus 120573120588119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)

minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119884 (119905)119883 (119905 minus 120591)

minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119884 (119905)119883 (119905 minus 120591)

minus 120573119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)

+ 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)

119883(119905 minus 120591)119885 (119905)

+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)119885 (119905)

(29)

and applying Cauchy-Chwartz inequality to all productterms we obtain the following expression

1198811015840

1(119905) le minus [2119887120588 minus

1

2

120573119901119868lowast119878lowast(119901minus1)

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573120588119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119883

2(119905)

minus [2120588119887 + 119887 minus

1

2

119901120573119868lowast119872119901minus1

minus

1

2

(120588 + 1) 120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119885

2(119905)

+ (120588 + 1) 120573119901119890minus119887120591119868lowast119872(119901minus1)

1198832(119905 minus 120591)

+ (120588 + 1) 120573119890minus1198871205911198721199011198852(119905 minus 120591)

(30)

We choose Lyapunov function to be the form 119881(119905) = 1198811(119905) +

1198812(119905) and we get

1198811015840(119905) = 119881

1015840

1(119905) + (120588 + 1) 120573119901119890

minus119887120591119868lowast119872(119901minus1)

times (1198832(119905) minus 119883

2(119905 minus 120591)) + (120588 + 1) 120573119890

minus119887120591119872119901

times (1198852(119905) minus 119885

2(119905 minus 120591))

(31)

Substituting this in the inequality for 1198811(119905) we get

1198811015840(119905) le minus [2119887120588 +

1

2

120588 (119887119898 minus 1199021015840120575) minus

1

2

120573119901119868lowast119872lowast(119901minus1)

minus

1

2

120573119890minus119887120591119872119901minus1

(120588119872 + 2119901119868lowast+ 3120588119901119868

lowast)]1198832(119905)

minus [2120588119887 + 119887 minus

1

2

119901120573119868lowast119872119901minus1

minus

1

2

(120588 + 1) 120573119890minus119887120591119872119901minus1

(119868lowast119901 +119872)]119884

2(119905)

minus [120575 minus 119902120575 + 120574 +

119889

119879

+

1

2

120588 (119887119898 minus 1199021015840120575)

minus

1

2

120573119890minus119887120591119872119901minus1

(119868lowast119901 + 2119872120588 + 3119872)]119885

2(119905)

(32)

The right-hand expression of the above inequality is alwaysnegative provided that (26) holds A direct application of theLyapunov-LaSalle type theorem shows that lim

119905rarrinfin119883(119905) rarr

0 lim119905rarrinfin

119884(119905) rarr 0 lim119905rarrinfin

119885(119905) rarr 0 The proof iscomplete

3 Continuous Treatment andPulse Vaccination Strategies

When continuous treatment and pulse vaccination strategiesare included in the SEIR epidemic model with the nonlinear

6 Computational and Mathematical Methods in Medicine

infectious force and vertical transmission it can be written asfollows

1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898

1015840(119878 + 119877 + 119864) +

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905

+) = 119864 (119905)

119868 (119905+) = 119868 (119905) 119877 (119905

+) = 119877 (119905) + 120579119878 (119905)

119905 = 119896119879 119896 isin 119885+

(33)

and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate

We also consider the following reduced systems

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905

+) = 119868 (119905)

119905 = 119896119879 119896 isin 119885+

(34)

Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877

2

+|119878 ge

0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)

We first state two important lemmas which are useful inour following discussions

Lemma 6 (see [12]) Consider the following impulsive differ-ential equation

1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879

119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879

(35)

where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by

119906lowast(119905) =

119886

119887

+ (119906 minus

119886

119887

) 119890minus119887(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879 (36)

which is globally asymptotically stable where

119906 =

119886

119887

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

(37)

Lemma 7 (see [27 28]) Consider the following equation

1199061015840(119905) = 119886

1119906 (119905 minus 120591) minus 119886

2119906 (119905) (38)

where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has

(1) if 1198861lt 1198862 then lim

119905rarrinfin119906(119905) = 0

(2) if 1198861gt 1198862 then lim

119905rarrinfin119906(119905) = +infin

31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy

1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885

+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885

+

(39)

By Lemma 6 we obtain the periodic solution of system (39)

119878lowast(119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

119896119879 lt 119905 le (119896 + 1) 119879 (40)

and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)

Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that

Rlowast=

119898119901119890minus119887120591120573

120575 minus 119902120575 + 120574 + (119889119879)

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

lt 1 (41)

Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small

such that

119890minus119887120591120573(

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760)

119901

lt 120575 minus 119902120575 + 120574 +

119889

119879

(42)

From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus

119887119878 and then we consider the following comparison systemwith pulse

1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879

119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879

119908 (0+) = 119878 (0

+)

(43)

In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and

lim119905rarrinfin

119908 (119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

= 119878lowast(119905)

119896119879 lt 119905 le (119896 + 1) 119879

(44)

There exists an integer 1198961 such that

119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)

Computational and Mathematical Methods in Medicine 7

That is

119878 (119905) lt 119878lowast(119905) + 120576

0le

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760= 119878

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961

(46)

Furthermore from the second equation we have

1198681015840(119905) le 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(47)

and then we consider the following comparison equation

1199081015840

1(119905) = 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(48)

then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840

1(119905) lt 0 lim

119896rarrinfin119868(119905) = 0 So

there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576

1for all

119905 gt 1198962119879

When 119905 gt 1198962119879 from the first equation of system (34) we

have

1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761) minus (120573119878

119901minus1

1205761+ 119887) 119878 (119905) (49)

Consider the following comparison impulsive differentialequation for all 119905 gt 119896

2119879

1199081015840

2(119905) = (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761)

minus (120573119878

119901minus1

1205761+ 119887)119908

2 (119905) 119905 = 119896119879

1199082(119905+) = (1 minus 120579)119908

2(119905) 119905 = 119896119879

1199082(0+) = 119878 (0

+)

(50)

By Lemma 6 we have the unique periodic solution of system(50) given by

119908lowast

2=

119887119898 minus (119887119898 minus 1199021015840120575) 1205761

120573119878

119901minus1

1205761+ 119887

(1 minus

120579119890minus(1205761120573119878119901minus1

+119887)(119905minus119899119879)

1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1

+119887)119879

)

119896119879 lt 119905 le (119896 + 1) 119879

(51)

By the comparison theorem there exists an integer 1198963gt 1198962

such that

119878 (119905) gt 1199082(119905) gt 119908

lowast

2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)

Because 1205760and 1205761are sufficiently small it follows from (46)

and (52) that lim119905rarrinfin

119878(119905) = 119878lowast(119905) Therefore the disease-free

solution (119878lowast(119905) 0) of system (34) is globally attractive The

proof is completed

Denote

120579lowast= (119890119887119879minus 1)((

120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

)

1119901

minus 1)

120591lowast=

1

119887

ln120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(53)

Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the

infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free

periodic solution is globally attractive provided that 120579 gt 120579lowast

Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879

)(1 minus (1 minus 120579)119890minus119887119879

))119901le

119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution

is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890

119887120591(120575 minus 119902120575 +

120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast

Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough

32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence

Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872

119894ge 119898119894 119894 = 1 2 (both are

independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies

1198981le 119878 (119905) le 119872

1 119898

2le 119868 (119905) le 119872

2 (54)

Denote

Rlowast=

120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(55)

Theorem 12 If Rlowastgt 1 then there is a positive constant

119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)

satisfies 119868(119905) ge 119898119868for all t sufficiently large

Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1

for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898

119878

119868(119905) ge 119898119868for all 119905 gt 119905

0

Firstly from the first equation of system (34) we have

1198781015840(119905) gt 119902

1015840120575 minus (120573 + 119887) 119878 (56)

8 Computational and Mathematical Methods in Medicine

Consider the following comparison equations

1198831015840(119905) = 119902

1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879

119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879

(57)

By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905

0(1199050is

sufficiently large) such that

119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576

ge

1199021015840120575

120573 + 119887

(

(1 minus 120579) (1 minus 119890minus(120573+119887)119879

)

1 minus (1 minus 120579) 119890minus(120573+119887)119879

) minus 120576 = 119898119878gt 0

(58)

Now we will prove that there exist 119898119868gt 0 and a suff-

iciently large 1199050such that 119868(119905) ge 119898

119868for all 119905 gt 119905

0 Since the

proof is rather long it will be convenient to divide it into twosteps

Step 1 Since Rlowastgt 1 there exist 119898lowast

119868gt 0 120576 gt 0 sufficiently

small such that

120573120578119901119890minus119887120591

minus (120575 + 120574 +

119889

119879

minus 119902120575) gt 0 (59)

where 120578 = (1199021015840120575(120573119898

lowast

119868+ 119887))((1 minus 120579)(1 minus 119890

minus119887119879)(1 minus (1 minus

120579)119890minus119887119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt

119898lowast

119868for all 119905 ge 119905

0 Suppose that the claim is not valid There

exists a 1199050gt 0 such that 119868(119905) lt 119898

lowast

119868for all 119905 ge 119905

0 and then

follows from the first equation of system (34) that for 119905 ge 1199050

1198781015840(119905) gt 119902

1015840120575 minus (120573119898

lowast

119868+ 119887) 119878 (119905) (60)

Consider the comparison impulsive system for 119905 ge 1199050

1198831015840

1(119905) = 119902

1015840120575 minus (120573119898

lowast

119868+ 119887)119883 (119905) 119905 = 119896119879

1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879

(61)

According to Lemma 6 there exists 1198791gt 1199050such that

119878 (119905) gt 119883lowast

1(119905) minus 120576

ge

1199021015840120575

120573119898lowast

119868+ 119887

(

(1 minus 120579) (1 minus 119890minus120573(119898

lowast

119868+119887)119879

)

1 minus (1 minus 120579) 119890minus120573(119898

lowast

119868+119887)119879

) minus 120576 = 120578

(62)

for all 1198791gt 1199050 The second equation of system (34) can be

translated into the following form

1198681015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

minus120573119890minus119887120591 119889

119889119905

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585

(63)

Define a function 119881(119905) such that

119881 (119905) = 119868 (119905) + 120573119890minus119887120591

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585 (64)

then the derivative of 119881(119905) along the solution of system (34)is

1198811015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

= (120575 minus 119902120575 + 120574 +

119889

119879

)(

120573119878119901(119905) 119890minus119887120591

120575 minus 119902120575 + 120574 + (119889119879)

minus 1) 119868 (119905)

119905 gt 1198791

(65)

From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that

119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591

Hence there exists a 1199051gt 0 such that 119868(119905

1) ge 119898

lowast

119868

Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898

lowast

119868for all 119905 gt 119905

1 then our aim is obtained

Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case

setting 119905lowast = inf119905gt1199051

119868(119905) le 119898lowast

119868 there are two possible cases for

119905lowast

Define

119898119868= min

119898lowast

119868

2

1199021 119902

1= 119898lowast

119868119890minus(120575minus119902120575+120574+(119889119879))120591

(66)

We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion

is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast

119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578

for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that

119868 (119905) ge

119898lowast

119868

2

for 119905lowast lt 119905 lt 119905lowast + 119892 (67)

In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591

Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588

Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast

119868 and it is obvious that 119868(119905) ge 119902

1

for 119905lowast lt 119905 lt 119905lowast + 119892

Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902

1

Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021

Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898

119868for 119905lowast + 120591 lt 119905 lt 119905

lowast+ 120588 Since this kind of

interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898

119868for all large 119905 in

the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have

proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868

for all large 119905 The proof is completed

Computational and Mathematical Methods in Medicine 9

Set

120579lowast=

(119890119887119879minus 1) (1 minus ((119890

119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

)

((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

+ (119890119887119879minus 1)

120591lowast=

1

119887

ln120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(68)

From Theorems 12 we also easily obtain the followingresults

Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast

(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +

120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591

lowast

4 Numerical Simulations

In this section we present some numerical simulations todemonstrate our theoretical results established in this paper

Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible

Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease

Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591

5 Conclusion

In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model

0 20 40 60 80 100t

120579 = 01

120579 = 05

02

03

04

05

06

07

08

09

S(t)

(a)

0 20 40 60 80 100t

120579 = 01

120579 = 05

0

005

01

015

02

025

03

035

04

045

05

I(t)

(b)

Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =

0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05

10 Computational and Mathematical Methods in Medicine

0 20 40 60 80 100t

03

035

04

045

05

055

06

065

07

075S(t)

120591 = 01

120591 = 08

(a)

0 20 40 60 80 100t

005

01

015

02

025

03

035

04

045

05

I(t)

120591 = 01

120591 = 08

(b)

Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =

0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08

006008

04

08

12

16

20

002004

01

0

005

01

015

02

025

03

035

120579

120591

I

Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]

are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R

0and prove the global

stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R

lowastgt 1 the disease is permanent by using

the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (no 1124319)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991

[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007

[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006

[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

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Page 6: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

6 Computational and Mathematical Methods in Medicine

infectious force and vertical transmission it can be written asfollows

1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198641015840(119905) = 120573119878

119901119868 minus 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898

1015840(119878 + 119877 + 119864) +

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905

+) = 119864 (119905)

119868 (119905+) = 119868 (119905) 119877 (119905

+) = 119877 (119905) + 120579119878 (119905)

119905 = 119896119879 119896 isin 119885+

(33)

and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate

We also consider the following reduced systems

1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878

119901119868 minus 119887119878 + 119902

1015840120575119868

1198681015840(119905) = 120573119890

minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus

119889

119879

119868

119905 = 119896119879 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905

+) = 119868 (119905)

119905 = 119896119879 119896 isin 119885+

(34)

Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877

2

+|119878 ge

0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)

We first state two important lemmas which are useful inour following discussions

Lemma 6 (see [12]) Consider the following impulsive differ-ential equation

1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879

119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879

(35)

where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by

119906lowast(119905) =

119886

119887

+ (119906 minus

119886

119887

) 119890minus119887(119905minus119896119879)

119896119879 lt 119905 le (119896 + 1) 119879 (36)

which is globally asymptotically stable where

119906 =

119886

119887

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

(37)

Lemma 7 (see [27 28]) Consider the following equation

1199061015840(119905) = 119886

1119906 (119905 minus 120591) minus 119886

2119906 (119905) (38)

where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has

(1) if 1198861lt 1198862 then lim

119905rarrinfin119906(119905) = 0

(2) if 1198861gt 1198862 then lim

119905rarrinfin119906(119905) = +infin

31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy

1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885

+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885

+

(39)

By Lemma 6 we obtain the periodic solution of system (39)

119878lowast(119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

119896119879 lt 119905 le (119896 + 1) 119879 (40)

and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)

Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that

Rlowast=

119898119901119890minus119887120591120573

120575 minus 119902120575 + 120574 + (119889119879)

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

lt 1 (41)

Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small

such that

119890minus119887120591120573(

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760)

119901

lt 120575 minus 119902120575 + 120574 +

119889

119879

(42)

From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus

119887119878 and then we consider the following comparison systemwith pulse

1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879

119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879

119908 (0+) = 119878 (0

+)

(43)

In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and

lim119905rarrinfin

119908 (119905) = 119898 minus

119898120579119890minus119887(119905minus119899119879)

1 minus (1 minus 120579) 119890minus119887119879

= 119878lowast(119905)

119896119879 lt 119905 le (119896 + 1) 119879

(44)

There exists an integer 1198961 such that

119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)

Computational and Mathematical Methods in Medicine 7

That is

119878 (119905) lt 119878lowast(119905) + 120576

0le

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760= 119878

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961

(46)

Furthermore from the second equation we have

1198681015840(119905) le 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(47)

and then we consider the following comparison equation

1199081015840

1(119905) = 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(48)

then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840

1(119905) lt 0 lim

119896rarrinfin119868(119905) = 0 So

there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576

1for all

119905 gt 1198962119879

When 119905 gt 1198962119879 from the first equation of system (34) we

have

1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761) minus (120573119878

119901minus1

1205761+ 119887) 119878 (119905) (49)

Consider the following comparison impulsive differentialequation for all 119905 gt 119896

2119879

1199081015840

2(119905) = (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761)

minus (120573119878

119901minus1

1205761+ 119887)119908

2 (119905) 119905 = 119896119879

1199082(119905+) = (1 minus 120579)119908

2(119905) 119905 = 119896119879

1199082(0+) = 119878 (0

+)

(50)

By Lemma 6 we have the unique periodic solution of system(50) given by

119908lowast

2=

119887119898 minus (119887119898 minus 1199021015840120575) 1205761

120573119878

119901minus1

1205761+ 119887

(1 minus

120579119890minus(1205761120573119878119901minus1

+119887)(119905minus119899119879)

1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1

+119887)119879

)

119896119879 lt 119905 le (119896 + 1) 119879

(51)

By the comparison theorem there exists an integer 1198963gt 1198962

such that

119878 (119905) gt 1199082(119905) gt 119908

lowast

2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)

Because 1205760and 1205761are sufficiently small it follows from (46)

and (52) that lim119905rarrinfin

119878(119905) = 119878lowast(119905) Therefore the disease-free

solution (119878lowast(119905) 0) of system (34) is globally attractive The

proof is completed

Denote

120579lowast= (119890119887119879minus 1)((

120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

)

1119901

minus 1)

120591lowast=

1

119887

ln120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(53)

Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the

infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free

periodic solution is globally attractive provided that 120579 gt 120579lowast

Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879

)(1 minus (1 minus 120579)119890minus119887119879

))119901le

119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution

is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890

119887120591(120575 minus 119902120575 +

120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast

Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough

32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence

Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872

119894ge 119898119894 119894 = 1 2 (both are

independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies

1198981le 119878 (119905) le 119872

1 119898

2le 119868 (119905) le 119872

2 (54)

Denote

Rlowast=

120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(55)

Theorem 12 If Rlowastgt 1 then there is a positive constant

119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)

satisfies 119868(119905) ge 119898119868for all t sufficiently large

Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1

for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898

119878

119868(119905) ge 119898119868for all 119905 gt 119905

0

Firstly from the first equation of system (34) we have

1198781015840(119905) gt 119902

1015840120575 minus (120573 + 119887) 119878 (56)

8 Computational and Mathematical Methods in Medicine

Consider the following comparison equations

1198831015840(119905) = 119902

1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879

119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879

(57)

By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905

0(1199050is

sufficiently large) such that

119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576

ge

1199021015840120575

120573 + 119887

(

(1 minus 120579) (1 minus 119890minus(120573+119887)119879

)

1 minus (1 minus 120579) 119890minus(120573+119887)119879

) minus 120576 = 119898119878gt 0

(58)

Now we will prove that there exist 119898119868gt 0 and a suff-

iciently large 1199050such that 119868(119905) ge 119898

119868for all 119905 gt 119905

0 Since the

proof is rather long it will be convenient to divide it into twosteps

Step 1 Since Rlowastgt 1 there exist 119898lowast

119868gt 0 120576 gt 0 sufficiently

small such that

120573120578119901119890minus119887120591

minus (120575 + 120574 +

119889

119879

minus 119902120575) gt 0 (59)

where 120578 = (1199021015840120575(120573119898

lowast

119868+ 119887))((1 minus 120579)(1 minus 119890

minus119887119879)(1 minus (1 minus

120579)119890minus119887119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt

119898lowast

119868for all 119905 ge 119905

0 Suppose that the claim is not valid There

exists a 1199050gt 0 such that 119868(119905) lt 119898

lowast

119868for all 119905 ge 119905

0 and then

follows from the first equation of system (34) that for 119905 ge 1199050

1198781015840(119905) gt 119902

1015840120575 minus (120573119898

lowast

119868+ 119887) 119878 (119905) (60)

Consider the comparison impulsive system for 119905 ge 1199050

1198831015840

1(119905) = 119902

1015840120575 minus (120573119898

lowast

119868+ 119887)119883 (119905) 119905 = 119896119879

1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879

(61)

According to Lemma 6 there exists 1198791gt 1199050such that

119878 (119905) gt 119883lowast

1(119905) minus 120576

ge

1199021015840120575

120573119898lowast

119868+ 119887

(

(1 minus 120579) (1 minus 119890minus120573(119898

lowast

119868+119887)119879

)

1 minus (1 minus 120579) 119890minus120573(119898

lowast

119868+119887)119879

) minus 120576 = 120578

(62)

for all 1198791gt 1199050 The second equation of system (34) can be

translated into the following form

1198681015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

minus120573119890minus119887120591 119889

119889119905

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585

(63)

Define a function 119881(119905) such that

119881 (119905) = 119868 (119905) + 120573119890minus119887120591

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585 (64)

then the derivative of 119881(119905) along the solution of system (34)is

1198811015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

= (120575 minus 119902120575 + 120574 +

119889

119879

)(

120573119878119901(119905) 119890minus119887120591

120575 minus 119902120575 + 120574 + (119889119879)

minus 1) 119868 (119905)

119905 gt 1198791

(65)

From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that

119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591

Hence there exists a 1199051gt 0 such that 119868(119905

1) ge 119898

lowast

119868

Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898

lowast

119868for all 119905 gt 119905

1 then our aim is obtained

Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case

setting 119905lowast = inf119905gt1199051

119868(119905) le 119898lowast

119868 there are two possible cases for

119905lowast

Define

119898119868= min

119898lowast

119868

2

1199021 119902

1= 119898lowast

119868119890minus(120575minus119902120575+120574+(119889119879))120591

(66)

We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion

is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast

119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578

for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that

119868 (119905) ge

119898lowast

119868

2

for 119905lowast lt 119905 lt 119905lowast + 119892 (67)

In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591

Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588

Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast

119868 and it is obvious that 119868(119905) ge 119902

1

for 119905lowast lt 119905 lt 119905lowast + 119892

Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902

1

Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021

Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898

119868for 119905lowast + 120591 lt 119905 lt 119905

lowast+ 120588 Since this kind of

interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898

119868for all large 119905 in

the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have

proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868

for all large 119905 The proof is completed

Computational and Mathematical Methods in Medicine 9

Set

120579lowast=

(119890119887119879minus 1) (1 minus ((119890

119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

)

((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

+ (119890119887119879minus 1)

120591lowast=

1

119887

ln120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(68)

From Theorems 12 we also easily obtain the followingresults

Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast

(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +

120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591

lowast

4 Numerical Simulations

In this section we present some numerical simulations todemonstrate our theoretical results established in this paper

Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible

Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease

Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591

5 Conclusion

In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model

0 20 40 60 80 100t

120579 = 01

120579 = 05

02

03

04

05

06

07

08

09

S(t)

(a)

0 20 40 60 80 100t

120579 = 01

120579 = 05

0

005

01

015

02

025

03

035

04

045

05

I(t)

(b)

Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =

0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05

10 Computational and Mathematical Methods in Medicine

0 20 40 60 80 100t

03

035

04

045

05

055

06

065

07

075S(t)

120591 = 01

120591 = 08

(a)

0 20 40 60 80 100t

005

01

015

02

025

03

035

04

045

05

I(t)

120591 = 01

120591 = 08

(b)

Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =

0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08

006008

04

08

12

16

20

002004

01

0

005

01

015

02

025

03

035

120579

120591

I

Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]

are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R

0and prove the global

stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R

lowastgt 1 the disease is permanent by using

the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (no 1124319)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991

[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007

[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006

[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

Submit your manuscripts athttpwwwhindawicom

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Page 7: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

Computational and Mathematical Methods in Medicine 7

That is

119878 (119905) lt 119878lowast(119905) + 120576

0le

119898(1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

+ 1205760= 119878

119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961

(46)

Furthermore from the second equation we have

1198681015840(119905) le 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(47)

and then we consider the following comparison equation

1199081015840

1(119905) = 120573119890

minus119887120591119878

119901

119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +

119889

119879

) 119868 (119905)

119905 = 119896119879 119896 gt 1198961

(48)

then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840

1(119905) lt 0 lim

119896rarrinfin119868(119905) = 0 So

there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576

1for all

119905 gt 1198962119879

When 119905 gt 1198962119879 from the first equation of system (34) we

have

1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761) minus (120573119878

119901minus1

1205761+ 119887) 119878 (119905) (49)

Consider the following comparison impulsive differentialequation for all 119905 gt 119896

2119879

1199081015840

2(119905) = (119887119898 minus (119887119898 minus 119902

1015840120575) 1205761)

minus (120573119878

119901minus1

1205761+ 119887)119908

2 (119905) 119905 = 119896119879

1199082(119905+) = (1 minus 120579)119908

2(119905) 119905 = 119896119879

1199082(0+) = 119878 (0

+)

(50)

By Lemma 6 we have the unique periodic solution of system(50) given by

119908lowast

2=

119887119898 minus (119887119898 minus 1199021015840120575) 1205761

120573119878

119901minus1

1205761+ 119887

(1 minus

120579119890minus(1205761120573119878119901minus1

+119887)(119905minus119899119879)

1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1

+119887)119879

)

119896119879 lt 119905 le (119896 + 1) 119879

(51)

By the comparison theorem there exists an integer 1198963gt 1198962

such that

119878 (119905) gt 1199082(119905) gt 119908

lowast

2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)

Because 1205760and 1205761are sufficiently small it follows from (46)

and (52) that lim119905rarrinfin

119878(119905) = 119878lowast(119905) Therefore the disease-free

solution (119878lowast(119905) 0) of system (34) is globally attractive The

proof is completed

Denote

120579lowast= (119890119887119879minus 1)((

120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

)

1119901

minus 1)

120591lowast=

1

119887

ln120573119898119901

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

1 minus 119890minus119887119879

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(53)

Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the

infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free

periodic solution is globally attractive provided that 120579 gt 120579lowast

Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879

)(1 minus (1 minus 120579)119890minus119887119879

))119901le

119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution

is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890

119887120591(120575 minus 119902120575 +

120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast

Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough

32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence

Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872

119894ge 119898119894 119894 = 1 2 (both are

independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies

1198981le 119878 (119905) le 119872

1 119898

2le 119868 (119905) le 119872

2 (54)

Denote

Rlowast=

120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(55)

Theorem 12 If Rlowastgt 1 then there is a positive constant

119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)

satisfies 119868(119905) ge 119898119868for all t sufficiently large

Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1

for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898

119878

119868(119905) ge 119898119868for all 119905 gt 119905

0

Firstly from the first equation of system (34) we have

1198781015840(119905) gt 119902

1015840120575 minus (120573 + 119887) 119878 (56)

8 Computational and Mathematical Methods in Medicine

Consider the following comparison equations

1198831015840(119905) = 119902

1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879

119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879

(57)

By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905

0(1199050is

sufficiently large) such that

119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576

ge

1199021015840120575

120573 + 119887

(

(1 minus 120579) (1 minus 119890minus(120573+119887)119879

)

1 minus (1 minus 120579) 119890minus(120573+119887)119879

) minus 120576 = 119898119878gt 0

(58)

Now we will prove that there exist 119898119868gt 0 and a suff-

iciently large 1199050such that 119868(119905) ge 119898

119868for all 119905 gt 119905

0 Since the

proof is rather long it will be convenient to divide it into twosteps

Step 1 Since Rlowastgt 1 there exist 119898lowast

119868gt 0 120576 gt 0 sufficiently

small such that

120573120578119901119890minus119887120591

minus (120575 + 120574 +

119889

119879

minus 119902120575) gt 0 (59)

where 120578 = (1199021015840120575(120573119898

lowast

119868+ 119887))((1 minus 120579)(1 minus 119890

minus119887119879)(1 minus (1 minus

120579)119890minus119887119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt

119898lowast

119868for all 119905 ge 119905

0 Suppose that the claim is not valid There

exists a 1199050gt 0 such that 119868(119905) lt 119898

lowast

119868for all 119905 ge 119905

0 and then

follows from the first equation of system (34) that for 119905 ge 1199050

1198781015840(119905) gt 119902

1015840120575 minus (120573119898

lowast

119868+ 119887) 119878 (119905) (60)

Consider the comparison impulsive system for 119905 ge 1199050

1198831015840

1(119905) = 119902

1015840120575 minus (120573119898

lowast

119868+ 119887)119883 (119905) 119905 = 119896119879

1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879

(61)

According to Lemma 6 there exists 1198791gt 1199050such that

119878 (119905) gt 119883lowast

1(119905) minus 120576

ge

1199021015840120575

120573119898lowast

119868+ 119887

(

(1 minus 120579) (1 minus 119890minus120573(119898

lowast

119868+119887)119879

)

1 minus (1 minus 120579) 119890minus120573(119898

lowast

119868+119887)119879

) minus 120576 = 120578

(62)

for all 1198791gt 1199050 The second equation of system (34) can be

translated into the following form

1198681015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

minus120573119890minus119887120591 119889

119889119905

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585

(63)

Define a function 119881(119905) such that

119881 (119905) = 119868 (119905) + 120573119890minus119887120591

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585 (64)

then the derivative of 119881(119905) along the solution of system (34)is

1198811015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

= (120575 minus 119902120575 + 120574 +

119889

119879

)(

120573119878119901(119905) 119890minus119887120591

120575 minus 119902120575 + 120574 + (119889119879)

minus 1) 119868 (119905)

119905 gt 1198791

(65)

From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that

119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591

Hence there exists a 1199051gt 0 such that 119868(119905

1) ge 119898

lowast

119868

Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898

lowast

119868for all 119905 gt 119905

1 then our aim is obtained

Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case

setting 119905lowast = inf119905gt1199051

119868(119905) le 119898lowast

119868 there are two possible cases for

119905lowast

Define

119898119868= min

119898lowast

119868

2

1199021 119902

1= 119898lowast

119868119890minus(120575minus119902120575+120574+(119889119879))120591

(66)

We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion

is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast

119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578

for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that

119868 (119905) ge

119898lowast

119868

2

for 119905lowast lt 119905 lt 119905lowast + 119892 (67)

In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591

Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588

Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast

119868 and it is obvious that 119868(119905) ge 119902

1

for 119905lowast lt 119905 lt 119905lowast + 119892

Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902

1

Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021

Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898

119868for 119905lowast + 120591 lt 119905 lt 119905

lowast+ 120588 Since this kind of

interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898

119868for all large 119905 in

the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have

proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868

for all large 119905 The proof is completed

Computational and Mathematical Methods in Medicine 9

Set

120579lowast=

(119890119887119879minus 1) (1 minus ((119890

119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

)

((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

+ (119890119887119879minus 1)

120591lowast=

1

119887

ln120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(68)

From Theorems 12 we also easily obtain the followingresults

Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast

(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +

120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591

lowast

4 Numerical Simulations

In this section we present some numerical simulations todemonstrate our theoretical results established in this paper

Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible

Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease

Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591

5 Conclusion

In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model

0 20 40 60 80 100t

120579 = 01

120579 = 05

02

03

04

05

06

07

08

09

S(t)

(a)

0 20 40 60 80 100t

120579 = 01

120579 = 05

0

005

01

015

02

025

03

035

04

045

05

I(t)

(b)

Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =

0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05

10 Computational and Mathematical Methods in Medicine

0 20 40 60 80 100t

03

035

04

045

05

055

06

065

07

075S(t)

120591 = 01

120591 = 08

(a)

0 20 40 60 80 100t

005

01

015

02

025

03

035

04

045

05

I(t)

120591 = 01

120591 = 08

(b)

Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =

0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08

006008

04

08

12

16

20

002004

01

0

005

01

015

02

025

03

035

120579

120591

I

Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]

are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R

0and prove the global

stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R

lowastgt 1 the disease is permanent by using

the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (no 1124319)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991

[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007

[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006

[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

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Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 8: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

8 Computational and Mathematical Methods in Medicine

Consider the following comparison equations

1198831015840(119905) = 119902

1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879

119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879

(57)

By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905

0(1199050is

sufficiently large) such that

119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576

ge

1199021015840120575

120573 + 119887

(

(1 minus 120579) (1 minus 119890minus(120573+119887)119879

)

1 minus (1 minus 120579) 119890minus(120573+119887)119879

) minus 120576 = 119898119878gt 0

(58)

Now we will prove that there exist 119898119868gt 0 and a suff-

iciently large 1199050such that 119868(119905) ge 119898

119868for all 119905 gt 119905

0 Since the

proof is rather long it will be convenient to divide it into twosteps

Step 1 Since Rlowastgt 1 there exist 119898lowast

119868gt 0 120576 gt 0 sufficiently

small such that

120573120578119901119890minus119887120591

minus (120575 + 120574 +

119889

119879

minus 119902120575) gt 0 (59)

where 120578 = (1199021015840120575(120573119898

lowast

119868+ 119887))((1 minus 120579)(1 minus 119890

minus119887119879)(1 minus (1 minus

120579)119890minus119887119879

)) minus 120576We claim that for any 119905

0gt 0 it is impossible that 119868(119905) lt

119898lowast

119868for all 119905 ge 119905

0 Suppose that the claim is not valid There

exists a 1199050gt 0 such that 119868(119905) lt 119898

lowast

119868for all 119905 ge 119905

0 and then

follows from the first equation of system (34) that for 119905 ge 1199050

1198781015840(119905) gt 119902

1015840120575 minus (120573119898

lowast

119868+ 119887) 119878 (119905) (60)

Consider the comparison impulsive system for 119905 ge 1199050

1198831015840

1(119905) = 119902

1015840120575 minus (120573119898

lowast

119868+ 119887)119883 (119905) 119905 = 119896119879

1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879

(61)

According to Lemma 6 there exists 1198791gt 1199050such that

119878 (119905) gt 119883lowast

1(119905) minus 120576

ge

1199021015840120575

120573119898lowast

119868+ 119887

(

(1 minus 120579) (1 minus 119890minus120573(119898

lowast

119868+119887)119879

)

1 minus (1 minus 120579) 119890minus120573(119898

lowast

119868+119887)119879

) minus 120576 = 120578

(62)

for all 1198791gt 1199050 The second equation of system (34) can be

translated into the following form

1198681015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

minus120573119890minus119887120591 119889

119889119905

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585

(63)

Define a function 119881(119905) such that

119881 (119905) = 119868 (119905) + 120573119890minus119887120591

int

119905

119905minus120591

119878119901(120585) 119868 (120585) 119889120585 (64)

then the derivative of 119881(119905) along the solution of system (34)is

1198811015840(119905) = (120573119878

119901(119905) 119890minus119887120591

minus 120575 + 119902120575 minus 120574 minus

119889

119879

) 119868 (119905)

= (120575 minus 119902120575 + 120574 +

119889

119879

)(

120573119878119901(119905) 119890minus119887120591

120575 minus 119902120575 + 120574 + (119889119879)

minus 1) 119868 (119905)

119905 gt 1198791

(65)

From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that

119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591

Hence there exists a 1199051gt 0 such that 119868(119905

1) ge 119898

lowast

119868

Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898

lowast

119868for all 119905 gt 119905

1 then our aim is obtained

Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case

setting 119905lowast = inf119905gt1199051

119868(119905) le 119898lowast

119868 there are two possible cases for

119905lowast

Define

119898119868= min

119898lowast

119868

2

1199021 119902

1= 119898lowast

119868119890minus(120575minus119902120575+120574+(119889119879))120591

(66)

We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion

is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast

119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578

for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that

119868 (119905) ge

119898lowast

119868

2

for 119905lowast lt 119905 lt 119905lowast + 119892 (67)

In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591

Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588

Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast

119868 and it is obvious that 119868(119905) ge 119902

1

for 119905lowast lt 119905 lt 119905lowast + 119892

Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902

1

Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021

Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898

119868for 119905lowast + 120591 lt 119905 lt 119905

lowast+ 120588 Since this kind of

interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898

119868for all large 119905 in

the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have

proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868

for all large 119905 The proof is completed

Computational and Mathematical Methods in Medicine 9

Set

120579lowast=

(119890119887119879minus 1) (1 minus ((119890

119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

)

((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

+ (119890119887119879minus 1)

120591lowast=

1

119887

ln120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(68)

From Theorems 12 we also easily obtain the followingresults

Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast

(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +

120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591

lowast

4 Numerical Simulations

In this section we present some numerical simulations todemonstrate our theoretical results established in this paper

Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible

Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease

Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591

5 Conclusion

In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model

0 20 40 60 80 100t

120579 = 01

120579 = 05

02

03

04

05

06

07

08

09

S(t)

(a)

0 20 40 60 80 100t

120579 = 01

120579 = 05

0

005

01

015

02

025

03

035

04

045

05

I(t)

(b)

Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =

0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05

10 Computational and Mathematical Methods in Medicine

0 20 40 60 80 100t

03

035

04

045

05

055

06

065

07

075S(t)

120591 = 01

120591 = 08

(a)

0 20 40 60 80 100t

005

01

015

02

025

03

035

04

045

05

I(t)

120591 = 01

120591 = 08

(b)

Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =

0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08

006008

04

08

12

16

20

002004

01

0

005

01

015

02

025

03

035

120579

120591

I

Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]

are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R

0and prove the global

stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R

lowastgt 1 the disease is permanent by using

the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (no 1124319)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991

[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007

[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006

[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 9: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

Computational and Mathematical Methods in Medicine 9

Set

120579lowast=

(119890119887119879minus 1) (1 minus ((119890

119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

)

((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)

1119901

+ (119890119887119879minus 1)

120591lowast=

1

119887

ln120573

119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))

(

(1 minus 120579) (1 minus 119890minus119887119879

)

1 minus (1 minus 120579) 119890minus119887119879

)

119901

(68)

From Theorems 12 we also easily obtain the followingresults

Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast

(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +

120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591

lowast

4 Numerical Simulations

In this section we present some numerical simulations todemonstrate our theoretical results established in this paper

Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible

Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease

Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591

5 Conclusion

In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model

0 20 40 60 80 100t

120579 = 01

120579 = 05

02

03

04

05

06

07

08

09

S(t)

(a)

0 20 40 60 80 100t

120579 = 01

120579 = 05

0

005

01

015

02

025

03

035

04

045

05

I(t)

(b)

Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =

0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05

10 Computational and Mathematical Methods in Medicine

0 20 40 60 80 100t

03

035

04

045

05

055

06

065

07

075S(t)

120591 = 01

120591 = 08

(a)

0 20 40 60 80 100t

005

01

015

02

025

03

035

04

045

05

I(t)

120591 = 01

120591 = 08

(b)

Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =

0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08

006008

04

08

12

16

20

002004

01

0

005

01

015

02

025

03

035

120579

120591

I

Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]

are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R

0and prove the global

stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R

lowastgt 1 the disease is permanent by using

the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (no 1124319)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991

[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007

[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006

[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 10: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

10 Computational and Mathematical Methods in Medicine

0 20 40 60 80 100t

03

035

04

045

05

055

06

065

07

075S(t)

120591 = 01

120591 = 08

(a)

0 20 40 60 80 100t

005

01

015

02

025

03

035

04

045

05

I(t)

120591 = 01

120591 = 08

(b)

Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =

0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08

006008

04

08

12

16

20

002004

01

0

005

01

015

02

025

03

035

120579

120591

I

Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]

are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R

0and prove the global

stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R

lowastgt 1 the disease is permanent by using

the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (no 1124319)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991

[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007

[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006

[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 11: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

Computational and Mathematical Methods in Medicine 11

Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012

[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008

[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007

[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002

[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012

[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008

[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005

[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007

[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010

[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985

[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013

[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002

[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004

[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991

[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991

[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991

[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994

[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011

[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977

[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993

[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001

[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 12: Research Article Disease Control of Delay SEIR Model with ...downloads.hindawi.com/journals/cmmm/2013/830237.pdf · Research Article Disease Control of Delay SEIR Model with Nonlinear

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


The BAC ConneCTion€¦ · Delay the Disease Delay the Disease is a fitness program designed to empower people with Parkinson’s disease by optimizing their physical function. Call

The BAC ConneCTion€¦ · Delay the Disease Delay the Disease is a fitness program designed to empower people with Parkinson’s disease by optimizing their physical function. Call

Epidemiology SEIR/SIR Models

Epidemiology SEIR/SIR Models

Final SEIR Nursery Products Hawes Composting Facility

Final SEIR Nursery Products Hawes Composting Facility

WORKING PAPER An SEIR Infectious Disease Model with ... › wp-content › uploads › BFI_WP_202025.pdfAn SEIR Infectious Disease Model with Testing and Conditional Quarantine David

WORKING PAPER An SEIR Infectious Disease Model with ... › wp-content › uploads › BFI_WP_202025.pdfAn SEIR Infectious Disease Model with Testing and Conditional Quarantine David

Mathematical Modeling and Simulation Study of SEIR disease and ...

Mathematical Modeling and Simulation Study of SEIR disease and ...

Modelling and Analysis of SEIR with Delay Differential Equation · person to another through contact with blood and body fluids, breathing in an airborne virus or being bitten by

Modelling and Analysis of SEIR with Delay Differential Equation · person to another through contact with blood and body fluids, breathing in an airborne virus or being bitten by

SEIR epidemic model with delay - Cambridge University Press · death rate due to causes other than the disease, y is the expected number of contacts per unit of time multiplied by

SEIR epidemic model with delay - Cambridge University Press · death rate due to causes other than the disease, y is the expected number of contacts per unit of time multiplied by

Effectiveness of control strategies for Coronavirus Disease 2018: a SEIR dynamic ... · 1 Effectiveness of control strategies for Coronavirus Disease 2018: a SEIR dynamic modeling

Effectiveness of control strategies for Coronavirus Disease 2018: a SEIR dynamic ... · 1 Effectiveness of control strategies for Coronavirus Disease 2018: a SEIR dynamic modeling

AN SEIR INFECTIOUS DISEASE MODEL WITH …An SEIR Infectious Disease Model with Testing and Conditional Quarantine David W. Berger, Kyle F. Herkenhoff, and Simon Mongey NBER Working

AN SEIR INFECTIOUS DISEASE MODEL WITH …An SEIR Infectious Disease Model with Testing and Conditional Quarantine David W. Berger, Kyle F. Herkenhoff, and Simon Mongey NBER Working

Tracking Epidemics with State-space SEIR and Google …faculty.chicagobooth.edu/nicholas.polson/research/papers/Track.pdf · Tracking Epidemics with State-space SEIR and Google Flu

Tracking Epidemics with State-space SEIR and Google …faculty.chicagobooth.edu/nicholas.polson/research/papers/Track.pdf · Tracking Epidemics with State-space SEIR and Google Flu

Section 3 SEIR/EIS Notices and Distribution … Feeder Project FEIR SEIR/EIS Notices and Distribution Documentation ... Air Resources Board, ... New Deal Resources in Hawk’s Nest

Section 3 SEIR/EIS Notices and Distribution … Feeder Project FEIR SEIR/EIS Notices and Distribution Documentation ... Air Resources Board, ... New Deal Resources in Hawk’s Nest

Optimal Control of Epidemiological SEIR models with L1 ...

Optimal Control of Epidemiological SEIR models with L1 ...

Draft SEIS/SEIR: Sacramento River Deep Water Ship Channel: 1 … · 2013-11-19 · Proposed Project Footprint SEIS/SEIR Sacramento River Deep Water Ship Channel 0 0.75 1.5 2.25 3

Draft SEIS/SEIR: Sacramento River Deep Water Ship Channel: 1 … · 2013-11-19 · Proposed Project Footprint SEIS/SEIR Sacramento River Deep Water Ship Channel 0 0.75 1.5 2.25 3

CHAPTER 4.0: REVISIONS TO THE DRAFT SEIR

CHAPTER 4.0: REVISIONS TO THE DRAFT SEIR

Single Pipeline Performance · Single Pipeline Performance 0-delay 1-delay 2-delay 3-delay 4-delay 5-delay 6-delay 7-delay 8-delay Figure 7. Single Pipeline Performance Figure 8.

Single Pipeline Performance · Single Pipeline Performance 0-delay 1-delay 2-delay 3-delay 4-delay 5-delay 6-delay 7-delay 8-delay Figure 7. Single Pipeline Performance Figure 8.

Analysis of SDEs Applied to SEIR Epidemic Models by ... · estimates. Stochastic modeling enables estimation of parameters from disease outbreak data to be equipped with standard

Analysis of SDEs Applied to SEIR Epidemic Models by ... · estimates. Stochastic modeling enables estimation of parameters from disease outbreak data to be equipped with standard

A DELAY DIFFERENTIAL EQUATIONS MODEL FOR DISEASE ...

A DELAY DIFFERENTIAL EQUATIONS MODEL FOR DISEASE ...

Analysis and Control of an SEIR Epidemic System With Nonlinear

Analysis and Control of an SEIR Epidemic System With Nonlinear

Extinction and Recurrence of Multi-group SEIR …...Extinction and Recurrence of Multi-group SEIR Epidemic Models with Stochastic Perturbations ∗ Qingshan Yang 1, Xuerong Mao 2†

Extinction and Recurrence of Multi-group SEIR …...Extinction and Recurrence of Multi-group SEIR Epidemic Models with Stochastic Perturbations ∗ Qingshan Yang 1, Xuerong Mao 2†

AMERICAN CASE MANAGEMENT ASSOCIATION 17th Annual …Ohio Health Delay the Disease Columbus, Ohio David Zid, BA ACE, APG Co-Founder/Director Ohio Health Delay the Disease Hillard, Ohio

AMERICAN CASE MANAGEMENT ASSOCIATION 17th Annual …Ohio Health Delay the Disease Columbus, Ohio David Zid, BA ACE, APG Co-Founder/Director Ohio Health Delay the Disease Hillard, Ohio