AvianInfluenzaA(H7N9)ModelBasedonPoultryTransport … · 2019. 7. 30. · uenza A (H7N9) virus was...

Research ArticleAvian Influenza A (H7N9) Model Based on Poultry TransportNetwork in China

Juping Zhang12 Wenjun Jing12 Wenyi Zhang3 and Zhen Jin 12

1Complex Systems Research Center Shanxi University Taiyuan Shanxi 030006 China2Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and PreventionShanxi University Taiyuan Shanxi 030006 China3Institute of Disease Control and Prevention of PLA Beijing 100071 China

Correspondence should be addressed to Zhen Jin jinzhn263net

Received 1 June 2018 Accepted 27 September 2018 Published 4 November 2018

Academic Editor Konstantin Blyuss

Copyright copy 2018 Juping Zhang et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to analyze the spread of avian influenza A (H7N9) we construct an avian influenza transmission model from poultry(including poultry farm backyard poultry farm live-poultry wholesale market and wet market) to human according to poultrytransport network We obtain the threshold value for the prevalence of avian influenza A (H7N9) and also give the existence andnumber of the boundary equilibria and endemic equilibria in different conditionsWe can see that poultry transport network plays animportant role in controlling avian influenza A (H7N9) Finally numerical simulations are presented to illustrate the effects ofpoultry in different places on avian influenza In order to reduce human infections in China our results suggest that closing the retaillive-poultry market or preventing the poultry of backyard poultry farm into the live-poultry market is feasible in a suitable condition

1 Introduction

Avian influenza A (H7N9) is a subtype of influenza virusesthat have been detected in birds in the past Until 2013outbreak in China no human infections with H7N9 viruseshad ever been reported But from March 31 to August 312013 134 cases had been reported in mainlandChina resulting in 45 deaths [1] However the virus cameback in November 2013 again Afterwards the disease cameback in November every year In fact the second outbreakoccurred from November 2013 to May 2014 +e thirdoutbreak occurred from November 2014 to June 2015 +efourth outbreak occurred fromNovember 2015 to June 2016And the fifth outbreak occurred from September 2016 toMay 2017 (NHFPC [1])+e disease causes a high death rateIn China from March 2013 to May 2017 H7N9 has resultedin 1263 human cases including 459 deaths with a death rateof nearly 37 In China from September 2016 to May 2017provinces with human cases are shown as Figure 1 H7N9virus does not induce clinical signs in poultry and is clas-sified as a low pathogenicity avian influenza virus (LPAIV)[2] However the virus can infect humans and most of the

reported cases of human H7N9 infection have resulted insevere respiratory illness [3]

Jones et al [4] demonstrated that interspecies trans-mission of H7N9 virus occurs readily between societyfinches and bobwhite quail but only sporadically betweenfinches and chickens and transmission occurs throughshared water Pantin-Jackwood et al [3] showed that quailand chickens are susceptible to infection shed large amountsof virus and are likely important in the spread of the virus tohumans and it is therefore conceivable that passerine birdsmay serve as vectors for transmission of H7N9 virus todomestic poultry [4] Zhang et al [5] concluded that mi-grant birds are the original infection source Many authorsinvestigated the epidemic model which describes thetransmission of avian influenza among birds and humans[8ndash15] Liu et al [16] constructed two avian influenza bird-to-human transmission models with different growth lawsof the avian population one with logistic growth and theother with Allee effect and analyzed their dynamical be-havior Lin et al [17] developed three different SIRSmodels to fit the observed human cases between March2013 and July 2015 in China and found that environmental

HindawiComputational and Mathematical Methods in MedicineVolume 2018 Article ID 7383170 14 pageshttpsdoiorg10115520187383170

transmission via viral shedding of infected chickens hadcontributed to the spread of H7N9 human cases in ChinaChen and Wen [18] took into account gene mutation inpoultry Guo et al [19] proposed and analyzed an SE-SEISavian-human inshyuenza model Mu and Yang [20] analyzedan SI-SEIR avian-human inshyuenza model with latent pe-riod and nonlinear recovery rate Gourley et al [21] ana-lyzed the patchy model for the spatiotemporal distributionof a migratory bird species Bourouiba et al [22] in-vestigated the role of migratory birds in the spread of H5N1avian inshyuenza among birds by considering a system ofdelay dierential equations for the numbers of birds onpatches where the delays represent the shyight times betweenpatches In China in 2013 to control the outbreak localauthorities of the provinces and municipalities such asJiangsu Shanghai and Zhejiang temporarily closed theretail live-poultry markets which proved to be an eectivecontrol measure Data indicate that the novel avian in-shyuenza A (H7N9) virus was most likely transmittedfrom the secondary wholesale market to the retail live-

poultry market and then to humans [6 7] How is avianinshyuenza A (H7N9) transmitted from live-poultry to hu-man in China In order to reveal the fact the globalnetwork model of avian inshyuenza A (H7N9) is constructedbased on poultry transport network e relationship be-tween the global system and subsystem is analyzed ecorresponding risk indices are obtained We study theimpact of subsystems on the risk index of the globalsystem When the disease occurs it can provide theoreticalguidance for the global and local transport of poultry

In this paper we construct an avian inshyuenza A (H7N9)transmission model from live poultry (including poultryfarm backyard poultry farm live-poultry wholesale marketand wet market) to human for the heterogenous environ-ments which aect the spread of H7N9 e remainingpart of this paper is organized as follows in Section 2 werst establish the model based on poultry transport networkWe derive the threshold value of the model In Sections 3and 4 we discuss the dierent boundary and endemicequilibrium in the dierent thresholds Section 5 gives the

Tibet

Inner Mongoria

Gansu

Sichuan

Jilin

Hebei

Hunan

Hubei

Guangxi

Shaanxi Henan

Jiangxi

Shanxi

Anhui

FujianGuizhou

Liaoning

Shandong

Guangdong

Jiangsu

ZhejiangChongqing

BeijingTianjing

Shanghai

01ndash56ndash20

21ndash40

Infectious cases

41ndash80gt80

Figure 1 Provinces with avion inshyuenza A (H7N9) from September 2016 to May 2017

2 Computational and Mathematical Methods in Medicine

effect of different transmission rate on H7N9 by numericalsimulation Finally concluding remarks are made in Section6

2 Model Based on Poultry Transport Network

+e avian population is classified into poultry farmbackyard poultry farm live-poultry wholesale market andwet market (the retail live-poultry market) According tothe present situation in China the backyard poultryfeeding is regarded as a large node which is considered tobe connected with all other nodes (except poultry farm) innetwork +e relationship diagram of poultry transportand contacts between human and poultry are described inFigure 2 Let Ni

fa(t) Njpa(t) and Nk

ma(t) be the totalnumber of poultry in ith poultry farm jth live-poultrywholesale market and kth wet market at time t re-spectively where Ni

fa(t) Nipa(t) and Ni

ma(t) are classifiedinto two subclasses susceptible and infective denotedby Si

fa(t) and Iifa(t) Sj

pa(t) and Ijpa(t) and Sk

ma(t) andIkma(t) respectively Suppose there are L poultry farms

M live-poultry wholesales and K wet markets namelyi 1 L j 1 M k 1 K And they are in-dependent of each other Let Nh(t) be the total number ofhuman at time t +e human population is classifiedinto three subclasses susceptible infective and re-covered denoted by Sh(t) Ih(t) and Rh(t) respectivelyAll new recruitments of human population and avianpopulation are susceptible+e avian influenza virus is notcontagious from an infective human to a susceptiblehuman It is only contagious from an infective avian toa susceptible avian and a susceptible human An infectedavian keeps in the state of disease and cannot recover butan infected human can recover and the recovered humanhas permanent immunity We neglect death rates of thepoultry individuals during the transport process +edetailed description of dynamical transmission of H7N9avian influenza is described in the following flowchart(Figure 3)

+e corresponding dynamical model can be seen in thefollowing equation

dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa i 1 L

dSba(t)

dt Ab minus βbSbaIba minusdbSba minus 1113944

j

ljSba minus 1113944k

ckSba

dIba(t)

dt βbSbaIba minusdbIba minus αbIba minus 1113944

j

ljIba minus 1113944k

ckIba

dSjpa(t)

dt 1113944

i

aijSifa + ljSba minus βpS

jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa j 1 M

dIjpa(t)

dt 1113944

i

aijIifa + ljIba + βpS

jpaI

jpa minus dpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j

bjkSjpa + ckSba minus βmS

kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j

bjkIjpa + ckIba + βmS

kmaI

kma minus dmI

kma minus αmI

kma k 1 K

dSh(t)

dt Ah minus 1113944

k

βkhShIkma minus βhShIba minusdhSh

dIh(t)

dt 1113944

k

βkhShIkma + βhShIba minusdhIh minus αhIh minus chIh

dRh(t)

dt chIh minus dhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

Computational and Mathematical Methods in Medicine 3

e interpretations of parameters of system (1) aredescribed in Table 1 e parameters in system (1) are allnonnegative constants

e variation of the number of poultry in ith poultryfarm Ni

fa(t) is

dNifa(t)dt

Aif minus dfNifa minus αfI

ifa minus sum

j

aijNifa (2)

and thus

Nifa(t) le

Aifdf +sumjaij

Wifa (3)

Similarly the variation of the number of poultry inbackyard poultry farm Nba(t) isdNba(t)dt

Ab minusdbNba minus αbIba minus sumj

ljNba minus sumk

ckNba

(4)

and thus

Nba(t) leAb

db +sumjlj +sumkckWba (5)

e variation of the number of poultry in jth live-poultrywholesale market Nj

pa(t) is

Poultryfarm

Poultryfarm

Poultryfarm

Poultryfarm

Poultryfarm

Wholesalemarket

Wholesalemarket

Wet market

Backyardpoultry farm

Wet market Wet market Wet market

HumanHumanHumanHumanHuman

Figure 2 A possible network of H7N9 avian inshyuenza

Sba

Iba

Rh

ljSba

Ab

ckSba

Aif βfSifaIifa

aijSifa dpSjpa

dfIifa

βpSjpaIjpa

βmSkmaIkma

βbSbaIba

βkhShIkma

βhShIba

dfSifa

dbSba

dbIba

bjkSjpa bjkIjpa

dpIjpa

aijIifa

dmSkma

dhSh dhIh αhIh

γhIh

αbIba

ljIbaαpIjpa

αfIifa

ckIba

αmIkma

dmIkma

Ah dhRh

Sifa Iifa

Sh Ih

Sjpa

Skma Ikma

Ijpa

Figure 3 Detailed transfer diagram on the dynamical transmission of H7N9 avian inshyuenza


dNjpa(t)

dt 1113944

i

aijNifa + ljNba minusdpN

jpa minus αpI

jpa minus 1113944

k

bjkNjpa

(6)

and thus

Njpa(t) le

1113936iaijNifa + ljNba

dp + 1113936kbjk

le1113936i aijA

if1113872 1113873 df + 1113936jaij1113872 11138731113872 1113873 + ljAb1113872 1113873 db + 1113936jlj + 1113936kck1113872 11138731113872 11138731113872 1113873

dp + 1113936kbjk

Wjpa (7)

+e variation of the number of poultry in kth wet marketNk

ma(t) is

dNkma(t)

dt 1113944

j

bjkNjpa + ckNba minus dmN

kma minus αmI

kma (8)

and thus

Nkma(t) le

1113936jbjkNjpa + ckNba

dm

le1113936j bjk 1113936i aijA

if1113872 1113873 df + 1113936jaij1113872 11138731113872 1113873 + ljAb1113872 1113873 db + 1113936jlj + 1113936kck1113872 11138731113872 11138731113872 11138731113872 1113873 dp + 1113936kbjk1113872 11138731113872 1113873 + ckAb( 1113857 db + 1113936jlj + 1113936kck1113872 11138731113872 1113873

dm W

kma

(9)

+e variation of the number of human Nh(t) is

dNh(t)

dt Ah minusdhNh minus αhIh (10)

and thus

Nh(t) leAh

dh (11)

For convenience we denote the positive solution

(S1fa SLfa I1fa IL

fa Sba Iba S1pa SMpa I1pa IM

pa

S1ma Skma I1ma Ik

ma Sh Ih) of system (1) by (S I)Let G≔ (S I) isin R2(L+M+K)+4

+ Sifa + Ii

fa leWifa Sba + Iba le1113966

Wba Sjpa + I

jpa le W

jpa Sk

ma + Ikma leWk

ma Sh + Ih le (Ahdh)

then G is a positively invariant for system (1)

In order to find the disease-free equilibrium of system(1) we consider

dSifa(t)

dt A

if minusdfS

ifa minus 1113944

j

aijSifa

dSba(t)

dt Ab minus dbSba minus 1113944

j


ckSba

dSjpa(t)

dt 1113944

i

aijSifa + ljSba minus dpS

jpa minus 1113944

k

bjkSjpa

dSkma(t)

dt 1113944

j

bjkSjpa + ckSba minus dmS

kma

dSh(t)

dt Ah minusdhSh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

Table 1 Parameters of system (1)

Parameter InterpretationAif All new recruitments of the avian in ith poultry farm

Ab All new recruitments of the avian in backyard poultry farmdf bpm +e natural death rate (including slaughter) of the avian in different placesαf bpm +e disease-related death rate of the infected avian in different placesβf bpm +e transmission rate from infective avian to susceptible avian in different placesaij +e transport rate of individuals from ith poultry farm to jth live-poultry wholesale marketbjk +e transport rate of individuals from jth live-poultry wholesale market to kth wet marketlj +e transport rate of individuals from backyard poultry farm to jth live-poultry wholesale marketck +e transport rate of individuals from backyard poultry farm to kth wet marketAh All new recruitments of the humandh +e natural death rate of the humanβkh +e transmission rate from the infective avian in kth wet market to the susceptible humanβh +e transmission rate from the infective avian in backyard farm to the susceptible humanαh +e disease-related death rate of the infected humanch +e recovery rate of the infective human


System (12) has the unique positive equilibriumS0 (S

i0fa1113980111397911139781113981L

S0ba Sj0pa1113980radic11139791113978radic1113981M

Sk0ma1113980radic11139791113978radic1113981K

S0h) where Si0fa Ai

f (df + 1113936jaij)

S0ba Ab(db + 1113936jlj + 1113936kck) Sj0pa (1113936iaijS

i0fa + ljS

0ba)(dp+

1113936kbjk) Sk0ma (1113936jbjkSj0

pa + ckS0ba)dm and S0h Ahdh +usE0 (S

i0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 Sj0pa1113980radic11139791113978radic1113981M

01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) is the dis-

ease-free equilibrium of system (1)According to the concepts of the next generation matrix

and reproduction number presented in [23 24] we define

F

F11 0 0 00 F22 0 00 0 F33 00 0 0 F44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V

V11 0 0 00 V22 0 00 0 V33 00 0 0 V44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(13)

where

F11

βfS10fa 0 middot middot middot 0

0 βfS20fa middot middot middot 0

⋮ ⋮ middot middot middot ⋮

0 0 middot middot middotβfSL0fa

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F22 βbS0ba1113872 1113873

F33

βpS10pa 0 middot middot middot 0

0 βpS20pa middot middot middot 0⋮ ⋮ middot middot middot ⋮0 0 middot middot middotβpSM0

pa

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F44

βmS10ma 0 middot middot middot 0

0 βmS20ma middot middot middot 0



⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V11

df +αf + 1113944j

aij 0 middot middot middot 0

0 df +αf + 1113944j

aij middot middot middot 0

⋮ ⋮ middot middot middot ⋮0 0 middot middot middot df +αf + 1113944

j

aij

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V33

dp +αp + 1113944k

bjk 0 middot middot middot 0

0 dp +αp + 1113944k

bjk middot middot middot 0

⋮ ⋮ middot middot middot ⋮0 0 middot middot middot dp +αp + 1113944

k

bjk

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V44

dm +αm 0 middot middot middot 00 dm +αm middot middot middot 0⋮ ⋮ middot middot middot ⋮0 0 middot middot middot dm +αm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V22 db +αb + 1113944j

lj + 1113944k

ck⎛⎝ ⎞⎠

(14)

Set R0 ρ(FVminus1) where ρ represents the spectral radiusof the matrix+en R0 is called the reproduction number forsystem (1) where

R0 max1le ileL1lejleM1lekleK

Rif0 R

jp0 R

km0 Rb01113966 1113967

Rif0

βfSi0fa

df + αf + 1113936jaij

Rjp0

βpSj0pa

dp + αp + 1113936kbjk

Rkm0

βmSk0ma

dm + αm

Rb0 βbS0ba

db + αb + 1113936jlj + 1113936kck

(15)

If R0 lt 1 then system (1) has the disease-free equilibriumE0 and E0 is locally asymptotically stable

Remark 1 If we do not consider backyard poultry farmthen system (1) becomes

dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa i 1 L

dSjpa(t)

dt 1113944

i

aijSifa minus βpS

jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i

aijIifa + βpS

jpaI

jpa minus dpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

j 1 M

dSkma(t)

dt 1113944

j

bjkSjpa minus βmS

kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j

bjkIjpa + βmS

kmaI

kma minus dmI

kma minus αmI

kma k 1 K

dSh(t)

dt Ah minus 1113944

k

βkhShIkma minus dhSh

dIh(t)

dt 1113944

k

βkhShIkma minusdhIh minus αhIh minus chIh

dRh(t)

dt chIh minus dhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

A similar analysis is available for the above system


3 Analysis of Subsystems of System (1)

Consider the poultry of the poultry farm subsystem given bythe first two equations of system (1) as follows

dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Let the right-hand side of system (17) equals to zerowhen Ii

fa ne 0 we obtain

Silowastfa

df + αf + 1113936jaij

βf

Iilowastfa

Aifβf minus df + 1113936jaij1113872 1113873 df + αf + 1113936jaij1113872 1113873

βf df + αf + 1113936jaij1113872 1113873

(18)

If Rif0 gt 1 system (17) has the positive equilibrium

(Silowastfa Iilowast

fa) If Rif0 lt 1 system (17) has only the disease-free

equilibrium (Si0fa 0)

Remark 2

(1) If min1le ileL

Rif01113864 1113865gt 1 then each farm has the positive

equilibrium(2) If max

1le ileLRif01113864 1113865gt 1 then some of the poultry farms have

the positive equilibrium and the others have only thedisease-free equilibrium

Consider the poultry of the backyard poultry farmsubsystem given by the third and fourth equations of system(1) as follows

dSba(t)

dt Ab minus βbSbaIba minus dbSba minus 1113944

j


ckSba

dIba(t)


j


ckIba

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)

Let the right-hand side of system (19) equals to zerowhen Iba ne 0 we obtain

Slowastba

db + αb + 1113936jlj + 1113936kck

βb

Ilowastba

Ab minus db + 1113936jlj + 1113936kck1113872 1113873Slowastba

βbSlowastba

(20)

If Rb0 gt 1 system (19) has the positive equilibrium(Silowast

ba Iilowastba) If Rb0 lt 1 system (19) has only the disease-free

equilibrium (S0ba 0)Consider the poultry of the live-poultry wholesale

market subsystem given by the fifth and sixth equations ofsystem (1) as follows

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)

Let the right-hand side of system (21) equals to zerowhen I

jpa ne 0 we can divide it into two cases

If Iifa 0 and Iba 0 then we have

Sjlowastpa

dp + αp + 1113936kbjk

βp

Ijlowastpa

1113936iaijSi0fa + ljS

0ba minus dp + 1113936kbjk1113872 1113873Sjlowast

pa

dp + αp + 1113936kbjk

(22)

If Rjp0 gt 1 then system (21) has the positive equilibrium

(Sjlowastpa I

jlowastpa)

If Iifa ne 0 or Iba ne 0 then we obtain

Ijlowastpa

1113936iaijIilowastfa + ljI

lowastba

dp + αp + 1113936kbjk minus βpSjlowastpa

b1Sjlowast2pa + b2S

jlowastpa + b3 0

(23)

where

b1 minusβp dp + 1113944k

bjk⎛⎝ ⎞⎠lt 0

b2 βp 1113944i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠ + βp 1113944i

aijIilowastfa + ljI

lowastba

⎛⎝ ⎞⎠

+ dp + 1113944k

bjk⎛⎝ ⎞⎠ dp + αp + 1113944

k

bjk⎛⎝ ⎞⎠gt 0

b3 minus dp + αp + 1113944k

bjk⎛⎝ ⎞⎠ 1113944

i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠lt 0

(24)

Because b22 minus 4b1b3 gt 0 the solutions of the aboveequation are

Sjlowastpa1 minusb2 +

b22 minus 4b1b3

1113969

2b1gt 0

Sjlowastpa2 minusb2 minus

b22 minus 4b1b3

1113969

2b1gt 0

(25)

If Rj2p0 ((dp + αp + 1113936kbjk)(βpS

jlowastpa2))gt 1 system (21)

has two positive equilibria (Sjlowastpa1 I

jlowastpa1) and (Sjlowast

pa2 Ijlowastpa2) If

Rj2p0 lt 1 and R

j1p0 ((dp + αp + 1113936kbjk)(βpS

jlowastpa1))gt 1 system

(21) has one positive equilibrium (Sjlowastpa1 I

jlowastpa1) If R

j1p0 lt 1

system (21) has no positive equilibriumConsider the poultry of the wet market (the retail live-

poultry market) subsystem given by the seventh and eighthequations of system (1) as follows


dSkma(t)

dt 1113944

j


kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

Let the right-hand side of system (26) equals to zerowhen Ik

ma ne 0 we can divide it into two casesIf I

jpa 0 and Iba 0 then we have

Sklowastma

dm + αmβm

Iklowastma

1113936jbjkSj0pa + ckS0ba minus dmSklowast

ma

dm + αm

(27)

If Rkm0 gt 1 then system (26) has the positive equilib-

rium (Sklowastma Iklowast

ma)If I

jpa ne 0 or Iba ne 0 then we have

Iklowastma

1113936jbjkIjlowastpa + ckIlowastba

dm + αm minus βmSklowastma

g1Sklowast2ma + g2S

klowastma + g3 0

(28)

where

g1 minusdmβm lt 0

g2 βm 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠ + βm 1113944j

bjkIjlowastpa + ckI

lowastba

⎛⎝ ⎞⎠

+ dm dm + αm( 1113857gt 0

g3 minus dm + αm( 1113857 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠lt 0

(29)

Because g22 minus 4g1g3 gt 0 the solutions of the above

equation are

Sklowastma1 minusg2 +

g22 minus 4g1g3

1113969

2g1gt 0

Sklowastma2 minusg2 minus

g22 minus 4g1g3

1113969

2g1gt 0

(30)

If Rk2m0 ((dm + αm)(βmSklowast

ma2))gt 1 system (26) has two

positive equilibria (Sklowastma1 Iklowast

ma1) and (Sklowastma2 Iklowast

ma2) If Rk2m0 lt 1

and Rk1m0 ((dm + αm)(βmSklowast

ma1))gt 1 system (26) has one

positive equilibrium (Sklowastma1 Iklowast

ma1) If Rk1m0 lt 1 system (26) has

no positive equilibriumConsider the human subsystem given by the last three

equations of system (1) as follows

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k

βkhShIkma + βhShIba minus dhIh minus αhIh minus chIh

dRh(t)

dt chIh minusdhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(31)

Since the first two equations of system (31) are in-dependent of the variable Rh we only need to analyze thefirst two equations of system (31) Let the right-hand side ofsystem (31) equals to zero when Ih ne 0 if Ik

ma ne 0 or Iba ne 0then we have

Slowasth

Ah

1113936kβkhIklowastma + βhIlowastba + dh

Ilowasth

1113936kβkhIklowastma + βhIlowastba1113872 1113873Sh

dh + αh + ch

(32)

4 Analysis of the Full System (1)

We analyze the following equivalent system

dSifa(t)

dt A

if minus βfS

ifaI

ifa minusdfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

dSba(t)


j


ckSba

dIba(t)


j


ckIba

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j


kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)


For the sake of discussion without loss of generality weassume that a node has at least one link with the nodes in thenext layer So we have the following cases

Case 1 If R0 max1le ileL1lejleM1lekleK

Rif0 R

jp0 Rk

m0 Rb01113966 1113967lt 1

system (33) has only the disease-free equilibrium E0

(Si0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) Namely when

all poultry has no avian influenza human will not be infectedwith avian influenza

Case 2 If max1le ileL1lejleM

Rif0 R

jp01113966 1113967lt 1 Rb0 lt 1 and

min1lekleK

Rkm01113966 1113967gt 1 system (33) has the boundary equilibrium

Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S

j0pa1113980radic11139791113978radic1113981M

01113980111397911139781113981M

Sklowastma1113980radic11139791113978radic1113981K

Iklowastma1113980radic11139791113978radic1113981K

Smlowasth I

mlowasth

⎛⎜⎜⎝ ⎞⎟⎟⎠

(34)

+is shows that avian influenza A (H7N9) virus is mostlikely transmitted from the retail live-poultry market tohumans when poultry has no disease in other types of farms

Case 3 If max1le ileL

Rif01113864 1113865lt 1 Rb0 lt 1 and min

1lejleMR

jp01113966 1113967gt 1 sys-

tem (33) has the boundary equilibrium as described nextIf min

1lekleKRk1m01113966 1113967gt 1 and max

1lekleKRk2m01113966 1113967lt 1 system (33) has

one boundary equilibrium

Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S

jlowastpa1113980radic11139791113978radic1113981M

Ijlowastpa1113980111397911139781113981M

Skplowastma11113980radic11139791113978radic1113981K

Ikplowastma11113980radic11139791113978radic1113981K

Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

(35)

If min1lekleK

Rk2m01113966 1113967gt 1 system (33) has two boundary

equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth2 I

pmlowasth2

⎛⎜⎜⎝ ⎞⎟⎟⎠

(36)

+is shows that avian influenza A (H7N9) virus is mostlikely transmitted from the secondary wholesale market tothe retail live-poultry market and then to humans [6 7] Andthere may be two boundary equilibria


Rif01113864 1113865lt 1 and Rb0 gt 1 system (33) has the

boundary equilibrium as described nextIf min

1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min

1lekleKRk1m01113966 1113967gt 1 and

max1lekleK

Rk2m01113966 1113967lt 1 system (33) has one boundary equilibrium

Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

Slowastba Ilowastba S

jblowastpa11113980radic11139791113978radic1113981M

Ijblowastpa11113980radic11139791113978radic1113981M

Skbplowastma111113980radicradic11139791113978radicradic1113981

K

Ikbplowastma111113980radic11139791113978radic1113981

K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(37)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1

system (33) has two boundary equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(38)

If min1lejleM

Rj2p01113966 1113967gt 1 max

1lekleKRk2m01113966 1113967lt 1 and min

1lekleKRk1m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(39)

If min1lejleM

Rj2p01113966 1113967gt 1 and min

1lekleKRk2m01113966 1113967gt 1 system (33) has

four boundary equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth22 I

bpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(40)

When the poultry of poultry farms has no avian in-fluenza and the poultry of backyard poultry farm has avianinfluenza we can obtain four cases In four cases human ismost likely transmitted from the backyard poultry farm tothe secondary wholesale market then to the retail live-poultry market and finally to humans or direct trans-mission from backyard poultry to humans

Case 5 If min1le ileL

Rif01113864 1113865gt 1 and Rb0 lt 1 system (33) has the


1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK



Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S

jflowastpa11113980radic11139791113978radic1113981M

Ijflowastpa11113980radic11139791113978radic1113981M

Skfplowastma111113980radicradic11139791113978radicradic1113981

K

Ikfplowastma111113980radic11139791113978radic1113981

K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(41)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(42)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(43)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth22 I

fpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(44)

When the poultry of poultry farms has avian influenzaand the poultry of backyard poultry farm has no avianinfluenza we can obtain four cases In four cases human ismost likely transmitted from the poultry farm to the sec-ondary wholesale market then to the retail live-poultrymarket and finally to humans


Rif01113864 1113865gt 1 and Rb0 gt 1 system (33) has the

positive equilibrium as described nextIf min

1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK

Rk2m01113966 1113967lt 1 system (33) has one positive equilibrium

Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L


jfblowastpa11113980radic11139791113978radic1113981M

Ijfblowastpa11113980radic11139791113978radic1113981M

Skfbplowastma111113980radicradic11139791113978radicradic1113981

K

Ikfbplowastma111113980radicradic11139791113978radicradic1113981

K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(45)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1

system (33) has two positive equilibria

E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(46)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(47)

If min1lejleM

Rj2p01113966 1113967gt 1 and min


four positive equilibria

E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth22 I

fbpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(48)

When the poultry of poultry farms has avian influenzaand the poultry of backyard poultry farm has avian in-fluenza we can obtain four cases In four cases human ismost likely transmitted from the poultry farm and back-yard poultry farm to the secondary wholesale market thento the retail live-poultry market and finally to humans ordirect transmission from backyard poultry to humans

Remark 3 If we assume that there is an edge between eachnode of the upper layer and each node of the next layer thatis each node of the upper layer transport poultry to eachnode of the next layer in the network when max

1le ileLRif01113864 1113865gt 1

max1lejleM

Rjp01113966 1113967gt 1 or max

1lekleKRkm01113966 1113967gt 1 according to the actual


situation it can be calculated and analyzed by a similarmethod Hence we omit them here

5 Numerical Simulations

In this section we first use L 3 M 2 and K 3 submodelto simulate +e course of the infected human is typically 1ndash4weeks and we assume that it is 25 weeks on average+us therecovery rate of the infective human is ch 16month +edisease-related death rate of the infected human is αh 037+e disease-induced death rates of poultry are assumed to beαf pm 4 times 10minus5 and αb 5 times 10minus4 We assume that humancan survive 70 years and the poultry can survive 2 months inthe farm 1 week in wholesale market 1 month in wet marketand 8 months in backyard farm respectively +ese rates alsoreferred to removal due to slaughtering Hence these ratesreferred to removal due to slaughtering and the natural deathWe take the parameter values as dh 119 times 10minus3monthdf 08month dp dm 1month and db 0125monthrespectively

We estimate that the number of susceptible poultrypopulation is between 107 and 108 the number of infectivepoultry population is between 0 and 1000 in farm thenumber of susceptible poultry population is between 104 and105 the number of infective poultry population is between0 and 500 in live-poultry wholesale market the number ofsusceptible poultry population is between 102 and 103 thenumber of infective poultry population is between 0 and 100in wet market and the number of susceptible humanpopulation is between 107 and 108 in the region So wechoose the initial values as (S1fa(0) I1fa(0) S2fa(0) I2fa(0)

S3fa(0) I3fa(0)) (5times 107100049 times 10790045times 107800)(S1pa(0) I1pa(0) S2pa(0) I2pa(0)) (7 times 1042005times 104100)(S1ma(0) I1ma(0) S2ma(0) I2ma(0) S3ma(0) I3ma(0)) (103 50

103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)

Ih(0) Rh(0)) (10700)+e difficulty in parameter estimations is that there is no

scientifically or officially reported data of live-poultry trans-portation in China +e values of aij bjk lj and ck used insimulations may be estimated based on living habits of peopleof regions the density of human population and so on Nowwe assume that the transport rates of the backyard poultry arethe same to each node namely lj 01 and ck 01 wherej 1 2 k 1 2 3 Let a11 003 a12 004 a21 003

a22 004 a31 005 and a32 002 and b11 003 b12

003 b13 004 b21 005 b22 002 and b23 003 Weassumed the replenishment rate to be 2 months which is themean lifetime of farm poultry Let A1

f 25 times 107A2f 245 times 107 A3

f 225 times 107 Ab 833 and Ah 1000respectively

+e transmission rates from the infective poultry in kthwet market to the susceptible human are βkh 118 times 10minus9k 1 2 3 +e transmission rate from the infective poultryin backyard farm to the susceptible human isβh 166 times 10minus8

+e transmission rates from infective poultry to sus-ceptible poultry in different places are βf 278 times 10minus8βb 469 times 10minus4 βp 288 times 10minus8 and βm 188 times 10minus8

respectively +en R0 09784lt 1 Solution Ih(t) is as-ymptotically stable and converges to the disease-free equi-librium in Figure 4

+e transmission rates from infective poultry to sus-ceptible poultry in different places are βb 479 times 10minus4βp 288 times 10minus8 and βm 188 times 10minus8 respectively +eseparameters are fixed βf is varied Let βf 318 times 10minus8βf 418 times 10minus8 and βf 518 times 10minus8 From Figure 5(a) wecan see that the beginning is almost the same but the later isdifferent +erefore the transmission rate βf has a smallimpact in the earliest stages but has an important impact onthe late disease

+e transmission rates from infective poultry to sus-ceptible poultry in different places are βf 318 times 10minus8βp 288 times 10minus8 and βm 188 times 10minus8 respectively +eseparameters are fixed βb is varied Let βb 479 times 10minus4βb 579 times 10minus4 and βb 779 times 10minus4 +is only affects thenumber of infected humans whereas it has no effect on thearrival time of the peak (Figure 5(b)) +erefore preventingpoultry of backyard poultry farm into the live-poultrymarket is feasible in a suitable condition

+e transmission rates from infective poultry to sus-ceptible poultry in different places are βf 318 times 10minus8βb 479 times 10minus4 and βm 188 times 10minus8 respectively +eseparameters are fixed βp is varied Let βp 288 times 10minus8βp 688 times 10minus8 and βp 988 times 10minus8+e bigger the βp themore the human infected (Figure 5(c)) +e secondarywholesale market plays an amplifier role

+e transmission rates from infective poultry to sus-ceptible poultry in different places are βf 318 times 10minus8βb 479 times 10minus4 and βp 288 times 10minus8 respectively +eseparameters are fixed βm is varied Let βm 188 times 10minus8βm 688 times 10minus8 and βm 988 times 10minus8 +e impact is rel-atively small (Figure 5(d))

6 Conclusion

In this paper we construct the avian influenza transmissionmodel from poultry (including poultry farm backyardpoultry farm live-poultry wholesale market and wet mar-ket) to human We obtain the threshold value for theprevalence of avian influenza and the number of theboundary equilibria and endemic equilibria in differentconditions Numerical simulations show the effects of dif-ferent transmission rates of different layer on the infectedhuman And we can obtain the following cases

(1) +e poultry of poultry farm backyard poultry farmand poultry wholesale market have no avian in-fluenza but there is a possible outbreak of avianinfluenza in wet market (the retail live-poultrymarket) and avian influenza A (H7N9) virus is mostlikely transmitted from the retail live-poultry marketto humans

(2) +e poultry of poultry farm and backyard poultryfarm has no avian influenza but there is a possibleoutbreak of avian influenza in poultry wholesalemarket and then avian influenza A (H7N9) virus is


0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)

Figure 4 Solution Ih(t) is asymptotically stable and converges to the disease-free state value

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)

βf = 318 lowast 10minus8



(a)

Infe

ctiv

e hum

ans (I h

)

βb = 479 lowast 10minus4



0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)

βp = 288 lowast 10minus8



0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)

βm = 188 lowast 10minus8



0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)

Figure 5 e plots display the changes of Ih(t) with βf bpm varying


most likely transmitted from the poultry wholesalemarket to the retail live-poultry market and finally tohumans

(3) +e poultry of poultry farm has avian influenza andthe poultry of backyard poultry farm has no avianinfluenza but there is a possible outbreak of avianinfluenza in poultry farm and then avian influenza A(H7N9) virus is most likely transmitted from poultrymarket to poultry wholesale market then to the retaillive-poultry market and finally to humans

(4) +e poultry of poultry farm has no avian influenzaand the poultry of backyard poultry farm has avianinfluenza but there is a possible outbreak of avianinfluenza in backyard poultry farm and then avianinfluenza A (H7N9) virus is most likely transmittedfrom backyard poultry farm to poultry wholesalemarket then to the retail live-poultry market andfinally to humans or direct transmission frombackyard poultry to humans

(5) +e poultry of poultry farm and backyard poultryfarm has avian influenza but there is a possibleoutbreak of avian influenza in poultry farm andbackyard poultry farm and then avian influenza A(H7N9) virus is most likely transmitted from poultryfarm and backyard poultry farm to poultry wholesalemarket then to the retail live-poultry market andfinally to humans or direct transmission frombackyard poultry to humans

Hence the poultry of some nodes on network has avianinfluenza and then all edges connected to the node should becut off It has a great inhibitory on preventing the spread ofdisease So the network of poultry transportation plays animportant role in controlling avian influenza A (H7N9)Moreover we find that there may have been avian influenza A(H7N9) among humans when there is avian influenza A(H7N9) in the retail live-poultry market so closing the live-poultry market can reduce the spread of disease to humans ata certain time In addition we find that there may have beenavian influenza A (H7N9) among humans when there is avianinfluenza A (H7N9) in the backyard poultry farm But thespread of backyard poultry to human is quit complex It canbe either direct infection or indirect infection In China thereare many backyard poultry so there are still some difficultiesin the prevention and control of avian influenza A (H7N9)

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was partly supported by the National SciencesFoundation of China (11331009 11314091 and 11501339)

References

[1] National Health and Family Planning Commission of China(NHFPC) ldquoNational notifiable disease situationrdquo httpennhfpcgovcndiseaseshtml in English httpwwwnhfpcgovcnzhuzhanyqxxlistsshtml in Chinese

[2] Q Li L Zhou M Zhou et al ldquoEpidemiology of humaninfections with avian influenza a (H7N9) virus in ChinardquoNew England Journal of Medicine vol 370 pp 520ndash5322014

[3] M J Pantin-Jackwood P J Miller E Spackman et al ldquoRoleof poultry in the spread of novel H7N9 influenza virus inChinardquo Journal of Virology vol 88 no 10 pp 5381ndash53902014

[4] J C Jones S Sonnberg R J Webby and R G WebsterldquoInfluenza a (H7N9) virus transmission between finches andpoultryrdquo Emerging Infectious Diseases vol 21 no 4pp 619ndash628 2015

[5] J Zhang Z Jin G Q Sun X D Sun Y M Wang andB Huang ldquoDetermination of original infection source ofH7N9 avian influenza by dynamical modelrdquo Scientific Reportsvol 4 p 4846 2014

[6] Y Chen W Liang S Yang et al ldquoHuman infections with theemerging avian influenza a H7N9 virus from wet marketpoultry clinical analysis and characterisation of viral ge-nomerdquo e Lancet vol 381 no 9881 pp 1916ndash1925 2013

[7] C Bao L Cui M Zhou L Hong G F Gao and H WangldquoLive-animal markets and influenza A (H7N9) virus in-fectionrdquo New England Journal of Medicine vol 368 no 24pp 2337ndash2339 2013

[8] S Iwami Y Takeuchi and X N Liu ldquoAvian-human influenzaepidemic modelrdquo Mathematical Biosciences vol 207 no 1pp 1ndash25 2007

[9] K I Kim Z G Lin and L Zhang ldquoAvian-human influenzaepidemic model with diffusionrdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 313ndash322 2010

[10] Y-H Hsieh J H Wu J Fang Y Yang and J LouldquoQuantification of bird-to-bird and bird-to-human infectionsduring 2013 novel H7N9 avian influenza outbreak in ChinardquoPLoS One vol 9 no 12 Article ID e111834 2014

[11] J C Jones S Sonnberg Z A Kocer et al ldquoPossible role ofsongbirds and parakeets in transmission of influenzaa (H7N9) virus to humansrdquo Emerging Infectious Diseasesvol 20 no 3 pp 380ndash385 2014

[12] X Ma and W Wang ldquoA discrete model of avian influenzawith seasonal reproduction and transmissionrdquo Journal ofBiological Dynamics vol 4 no 3 pp 296ndash314 2010

[13] X S Wang and J Wu ldquoPeriodic systems of delay differentialequations and avian influenza dynamicsrdquo Journal of Math-ematical Sciences vol 201 no 5 pp 693ndash704 2014

[14] N K Vaidya and L M Wahl ldquoAvian influenza dynamicsunder periodic environmental conditionsrdquo SIAM Journal onApplied Mathematics vol 75 no 2 pp 443ndash467 2015

[15] S Liu L Pang S Ruan and X Zhang ldquoGlobal dynamics ofavian influenza epidemic models with psychological effectrdquoComputational and Mathematical Methods in Medicinevol 2015 Article ID 913726 12 pages 2015

[16] S Liu S Ruan and X Zhang ldquoNonlinear dynamics of avianinfluenza epidemic modelsrdquo Mathematical Biosciencesvol 283 pp 118ndash135 2017

[17] Q Y Lin Z G Lin A P Y Chiu and D He ldquoSeasonality ofinfluenza a (H7N9) virus in China fitting simple epidemicmodels to human casesrdquo PLoS One vol 11 no 3 Article IDe0151333 2016


[18] Y Chen and Y Wen ldquoGlobal dynamic analysis of a H7N9avian-human influenza model in an outbreak regionrdquo Journalof eoretical Biology vol 367 pp 180ndash188 2015

[19] S M Guo J Wang M Ghosh and X Z Li ldquoAnalysis of avianinfluenza a (H7N9) model based on the low pathogenicity inpoultryrdquo Journal of Biological Systems vol 25 no 2 pp 1ndash162017

[20] R Mu and Y Yang ldquoGlobal dynamics of an avian influenzaA (H7N9) epidemic model with latent period and non-linear recovery raterdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 732169411 pages 2018

[21] S A Gourley R Liu and J Wu ldquoSpatiotemporal distribu-tions of migratory birds patchy models with delayrdquo SIAMJournal on Applied Dynamical Systems vol 9 no 2pp 589ndash610 2010

[22] L Bourouiba S A Gourley R Liu and J Wu ldquo+e in-teraction of migratory birds and domestic poultry and its rolein sustaining avian influenzardquo SIAM Journal on AppliedMathematics vol 71 no 2 pp 487ndash516 2011

[23] O Diekmann J A P Heesterbeek and J A J Metz ldquoOn thedefinition and the computation of the basic reproduction ratioin the models for infectious disease in heterogeneous pop-ulationsrdquo Journal of Mathematical Biology vol 28 no 4pp 365ndash382 1990

[24] P van den Driessche and J Watmough ldquoReproductionnumbers and sub-threshold endemic equilibria for com-partmental models of disease transmissionrdquo MathematicalBiosciences vol 180 no 1-2 pp 29ndash48 2002


Stem Cells International

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Disease Markers


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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

transmission via viral shedding of infected chickens hadcontributed to the spread of H7N9 human cases in ChinaChen and Wen [18] took into account gene mutation inpoultry Guo et al [19] proposed and analyzed an SE-SEISavian-human inshyuenza model Mu and Yang [20] analyzedan SI-SEIR avian-human inshyuenza model with latent pe-riod and nonlinear recovery rate Gourley et al [21] ana-lyzed the patchy model for the spatiotemporal distributionof a migratory bird species Bourouiba et al [22] in-vestigated the role of migratory birds in the spread of H5N1avian inshyuenza among birds by considering a system ofdelay dierential equations for the numbers of birds onpatches where the delays represent the shyight times betweenpatches In China in 2013 to control the outbreak localauthorities of the provinces and municipalities such asJiangsu Shanghai and Zhejiang temporarily closed theretail live-poultry markets which proved to be an eectivecontrol measure Data indicate that the novel avian in-shyuenza A (H7N9) virus was most likely transmittedfrom the secondary wholesale market to the retail live-

poultry market and then to humans [6 7] How is avianinshyuenza A (H7N9) transmitted from live-poultry to hu-man in China In order to reveal the fact the globalnetwork model of avian inshyuenza A (H7N9) is constructedbased on poultry transport network e relationship be-tween the global system and subsystem is analyzed ecorresponding risk indices are obtained We study theimpact of subsystems on the risk index of the globalsystem When the disease occurs it can provide theoreticalguidance for the global and local transport of poultry

In this paper we construct an avian inshyuenza A (H7N9)transmission model from live poultry (including poultryfarm backyard poultry farm live-poultry wholesale marketand wet market) to human for the heterogenous environ-ments which aect the spread of H7N9 e remainingpart of this paper is organized as follows in Section 2 werst establish the model based on poultry transport networkWe derive the threshold value of the model In Sections 3and 4 we discuss the dierent boundary and endemicequilibrium in the dierent thresholds Section 5 gives the

Tibet

Inner Mongoria

Gansu

Sichuan

Jilin

Hebei

Hunan

Hubei

Guangxi

Shaanxi Henan

Jiangxi

Shanxi

Anhui

FujianGuizhou

Liaoning

Shandong

Guangdong

Jiangsu

ZhejiangChongqing

BeijingTianjing

Shanghai

01ndash56ndash20

21ndash40

Infectious cases

41ndash80gt80

Figure 1 Provinces with avion inshyuenza A (H7N9) from September 2016 to May 2017














dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa i 1 L

dSba(t)


j


ckSba

dIba(t)


j


ckIba

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa j 1 M

dIjpa(t)

dt 1113944

i


jpaI

jpa minus dpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j


kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minus dmI

kma minus αmI

kma k 1 K

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minus dhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)




fa(t) is

dNifa(t)dt


ifa minus sum

j

aijNifa (2)

and thus

Nifa(t) le

Aifdf +sumjaij

Wifa (3)



ljNba minus sumk

ckNba

(4)

and thus

Nba(t) leAb



pa(t) is

Poultryfarm

Poultryfarm

Poultryfarm

Poultryfarm

Poultryfarm

Wholesalemarket

Wholesalemarket

Wet market





Sba

Iba

Rh

ljSba

Ab

ckSba

Aif βfSifaIifa

aijSifa dpSjpa

dfIifa

βpSjpaIjpa

βmSkmaIkma

βbSbaIba

βkhShIkma

βhShIba

dfSifa

dbSba

dbIba

bjkSjpa bjkIjpa

dpIjpa

aijIifa

dmSkma

dhSh dhIh αhIh

γhIh

αbIba

ljIbaαpIjpa

αfIifa

ckIba

αmIkma

dmIkma

Ah dhRh

Sifa Iifa

Sh Ih

Sjpa

Skma Ikma

Ijpa



dNjpa(t)

dt 1113944

i


jpa minus αpI

jpa minus 1113944

k

bjkNjpa

(6)

and thus

Njpa(t) le


dp + 1113936kbjk

le1113936i aijA


dp + 1113936kbjk

Wjpa (7)


ma(t) is

dNkma(t)

dt 1113944

j


kma minus αmI

kma (8)

and thus

Nkma(t) le


dm

le1113936j bjk 1113936i aijA


dm W

kma

(9)


dNh(t)


and thus

Nh(t) leAh

dh (11)


(S1fa SLfa I1fa IL


pa

S1ma Skma I1ma Ik


+ Sifa + Ii


Wba Sjpa + I

jpa le W

jpa Sk

ma + Ikma leWk




dSifa(t)

dt A

if minusdfS

ifa minus 1113944

j

aijSifa

dSba(t)


j


ckSba

dSjpa(t)

dt 1113944

i


jpa minus 1113944

k

bjkSjpa

dSkma(t)

dt 1113944

j


kma

dSh(t)

dt Ah minusdhSh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)






i0fa1113980111397911139781113981L



S0h) where Si0fa Ai

f (df + 1113936jaij)


i0fa + ljS

0ba)(dp+



i0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) is the dis-



F

F11 0 0 00 F22 0 00 0 F33 00 0 0 F44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V

V11 0 0 00 V22 0 00 0 V33 00 0 0 V44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(13)

where

F11





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F22 βbS0ba1113872 1113873

F33



pa

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F44





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V11

df +αf + 1113944j


0 df +αf + 1113944j



j

aij

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V33

dp +αp + 1113944k


0 dp +αp + 1113944k



k

bjk

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V44


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V22 db +αb + 1113944j

lj + 1113944k

ck⎛⎝ ⎞⎠

(14)



Rif0 R

jp0 R

km0 Rb01113966 1113967

Rif0

βfSi0fa

df + αf + 1113936jaij

Rjp0

βpSj0pa

dp + αp + 1113936kbjk

Rkm0

βmSk0ma

dm + αm

Rb0 βbS0ba

db + αb + 1113936jlj + 1113936kck

(15)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa i 1 L

dSjpa(t)

dt 1113944

i

aijSifa minus βpS

jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i

aijIifa + βpS

jpaI

jpa minus dpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

j 1 M

dSkma(t)

dt 1113944

j

bjkSjpa minus βmS

kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j

bjkIjpa + βmS

kmaI

kma minus dmI

kma minus αmI

kma k 1 K

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minus dhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)





dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


fa ne 0 we obtain

Silowastfa

df + αf + 1113936jaij

βf

Iilowastfa


βf df + αf + 1113936jaij1113872 1113873

(18)





Remark 2

(1) If min1le ileL






dSba(t)


j


ckSba

dIba(t)


j


ckIba

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


Slowastba

db + αb + 1113936jlj + 1113936kck

βb

Ilowastba


βbSlowastba

(20)





dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)




Sjlowastpa

dp + αp + 1113936kbjk

βp

Ijlowastpa



pa

dp + αp + 1113936kbjk

(22)


(Sjlowastpa I

jlowastpa)


Ijlowastpa


lowastba


b1Sjlowast2pa + b2S

jlowastpa + b3 0

(23)

where



b2 βp 1113944i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠ + βp 1113944i

aijIilowastfa + ljI

lowastba

⎛⎝ ⎞⎠

+ dp + 1113944k

bjk⎛⎝ ⎞⎠ dp + αp + 1113944

k



bjk⎛⎝ ⎞⎠ 1113944

i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠lt 0

(24)



b22 minus 4b1b3

1113969

2b1gt 0


b22 minus 4b1b3

1113969

2b1gt 0

(25)





pa2 Ijlowastpa2) If

Rj2p0 lt 1 and R

j1p0 ((dp + αp + 1113936kbjk)(βpS



jlowastpa1) If R

j1p0 lt 1




dSkma(t)

dt 1113944

j


kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Sklowastma

dm + αmβm

Iklowastma


ma

dm + αm

(27)



ma)If I


Iklowastma



g1Sklowast2ma + g2S

klowastma + g3 0

(28)

where

g1 minusdmβm lt 0

g2 βm 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠ + βm 1113944j

bjkIjlowastpa + ckI

lowastba

⎛⎝ ⎞⎠

+ dm dm + αm( 1113857gt 0

g3 minus dm + αm( 1113857 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠lt 0

(29)


equation are


g22 minus 4g1g3

1113969

2g1gt 0


g22 minus 4g1g3

1113969

2g1gt 0

(30)





ma2) If Rk2m0 lt 1







dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minusdhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(31)



Slowasth

Ah


Ilowasth


dh + αh + ch

(32)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minusdfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

dSba(t)


j


ckSba

dIba(t)


j


ckIba

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j


kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)




Rif0 R

jp0 Rk

m0 Rb01113966 1113967lt 1


(Si0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) Namely when



Rif0 R

jp01113966 1113967lt 1 Rb0 lt 1 and

min1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S

j0pa1113980radic11139791113978radic1113981M

01113980111397911139781113981M



Smlowasth I

mlowasth

⎛⎜⎜⎝ ⎞⎟⎟⎠

(34)




1lejleMR

jp01113966 1113967gt 1 sys-





Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

(35)

If min1lekleK


equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth2 I

pmlowasth2

⎛⎜⎜⎝ ⎞⎟⎟⎠

(36)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(37)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(38)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(39)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth22 I

bpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(40)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK



Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(41)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(42)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(43)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth22 I

fpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(44)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(45)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(46)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(47)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth22 I

fbpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(48)



1le ileLRif01113864 1113865gt 1

max1lejleM

Rjp01113966 1113967gt 1 or max








103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)



a22 004 a31 005 and a32 002 and b11 003 b12











6 Conclusion





0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of































dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa i 1 L

dSba(t)


j


ckSba

dIba(t)


j


ckIba

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa j 1 M

dIjpa(t)

dt 1113944

i


jpaI

jpa minus dpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j


kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minus dmI

kma minus αmI

kma k 1 K

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minus dhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)




fa(t) is

dNifa(t)dt


ifa minus sum

j

aijNifa (2)

and thus

Nifa(t) le

Aifdf +sumjaij

Wifa (3)



ljNba minus sumk

ckNba

(4)

and thus

Nba(t) leAb



pa(t) is

Poultryfarm

Poultryfarm

Poultryfarm

Poultryfarm

Poultryfarm

Wholesalemarket

Wholesalemarket

Wet market





Sba

Iba

Rh

ljSba

Ab

ckSba

Aif βfSifaIifa

aijSifa dpSjpa

dfIifa

βpSjpaIjpa

βmSkmaIkma

βbSbaIba

βkhShIkma

βhShIba

dfSifa

dbSba

dbIba

bjkSjpa bjkIjpa

dpIjpa

aijIifa

dmSkma

dhSh dhIh αhIh

γhIh

αbIba

ljIbaαpIjpa

αfIifa

ckIba

αmIkma

dmIkma

Ah dhRh

Sifa Iifa

Sh Ih

Sjpa

Skma Ikma

Ijpa



dNjpa(t)

dt 1113944

i


jpa minus αpI

jpa minus 1113944

k

bjkNjpa

(6)

and thus

Njpa(t) le


dp + 1113936kbjk

le1113936i aijA


dp + 1113936kbjk

Wjpa (7)


ma(t) is

dNkma(t)

dt 1113944

j


kma minus αmI

kma (8)

and thus

Nkma(t) le


dm

le1113936j bjk 1113936i aijA


dm W

kma

(9)


dNh(t)


and thus

Nh(t) leAh

dh (11)


(S1fa SLfa I1fa IL


pa

S1ma Skma I1ma Ik


+ Sifa + Ii


Wba Sjpa + I

jpa le W

jpa Sk

ma + Ikma leWk




dSifa(t)

dt A

if minusdfS

ifa minus 1113944

j

aijSifa

dSba(t)


j


ckSba

dSjpa(t)

dt 1113944

i


jpa minus 1113944

k

bjkSjpa

dSkma(t)

dt 1113944

j


kma

dSh(t)

dt Ah minusdhSh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)






i0fa1113980111397911139781113981L



S0h) where Si0fa Ai

f (df + 1113936jaij)


i0fa + ljS

0ba)(dp+



i0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) is the dis-



F

F11 0 0 00 F22 0 00 0 F33 00 0 0 F44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V

V11 0 0 00 V22 0 00 0 V33 00 0 0 V44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(13)

where

F11





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F22 βbS0ba1113872 1113873

F33



pa

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F44





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V11

df +αf + 1113944j


0 df +αf + 1113944j



j

aij

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V33

dp +αp + 1113944k


0 dp +αp + 1113944k



k

bjk

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V44


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V22 db +αb + 1113944j

lj + 1113944k

ck⎛⎝ ⎞⎠

(14)



Rif0 R

jp0 R

km0 Rb01113966 1113967

Rif0

βfSi0fa

df + αf + 1113936jaij

Rjp0

βpSj0pa

dp + αp + 1113936kbjk

Rkm0

βmSk0ma

dm + αm

Rb0 βbS0ba

db + αb + 1113936jlj + 1113936kck

(15)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa i 1 L

dSjpa(t)

dt 1113944

i

aijSifa minus βpS

jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i

aijIifa + βpS

jpaI

jpa minus dpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

j 1 M

dSkma(t)

dt 1113944

j

bjkSjpa minus βmS

kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j

bjkIjpa + βmS

kmaI

kma minus dmI

kma minus αmI

kma k 1 K

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minus dhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)





dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


fa ne 0 we obtain

Silowastfa

df + αf + 1113936jaij

βf

Iilowastfa


βf df + αf + 1113936jaij1113872 1113873

(18)





Remark 2

(1) If min1le ileL






dSba(t)


j


ckSba

dIba(t)


j


ckIba

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


Slowastba

db + αb + 1113936jlj + 1113936kck

βb

Ilowastba


βbSlowastba

(20)





dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)




Sjlowastpa

dp + αp + 1113936kbjk

βp

Ijlowastpa



pa

dp + αp + 1113936kbjk

(22)


(Sjlowastpa I

jlowastpa)


Ijlowastpa


lowastba


b1Sjlowast2pa + b2S

jlowastpa + b3 0

(23)

where



b2 βp 1113944i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠ + βp 1113944i

aijIilowastfa + ljI

lowastba

⎛⎝ ⎞⎠

+ dp + 1113944k

bjk⎛⎝ ⎞⎠ dp + αp + 1113944

k



bjk⎛⎝ ⎞⎠ 1113944

i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠lt 0

(24)



b22 minus 4b1b3

1113969

2b1gt 0


b22 minus 4b1b3

1113969

2b1gt 0

(25)





pa2 Ijlowastpa2) If

Rj2p0 lt 1 and R

j1p0 ((dp + αp + 1113936kbjk)(βpS



jlowastpa1) If R

j1p0 lt 1




dSkma(t)

dt 1113944

j


kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Sklowastma

dm + αmβm

Iklowastma


ma

dm + αm

(27)



ma)If I


Iklowastma



g1Sklowast2ma + g2S

klowastma + g3 0

(28)

where

g1 minusdmβm lt 0

g2 βm 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠ + βm 1113944j

bjkIjlowastpa + ckI

lowastba

⎛⎝ ⎞⎠

+ dm dm + αm( 1113857gt 0

g3 minus dm + αm( 1113857 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠lt 0

(29)


equation are


g22 minus 4g1g3

1113969

2g1gt 0


g22 minus 4g1g3

1113969

2g1gt 0

(30)





ma2) If Rk2m0 lt 1







dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minusdhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(31)



Slowasth

Ah


Ilowasth


dh + αh + ch

(32)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minusdfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

dSba(t)


j


ckSba

dIba(t)


j


ckIba

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j


kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)




Rif0 R

jp0 Rk

m0 Rb01113966 1113967lt 1


(Si0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) Namely when



Rif0 R

jp01113966 1113967lt 1 Rb0 lt 1 and

min1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S

j0pa1113980radic11139791113978radic1113981M

01113980111397911139781113981M



Smlowasth I

mlowasth

⎛⎜⎜⎝ ⎞⎟⎟⎠

(34)




1lejleMR

jp01113966 1113967gt 1 sys-





Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

(35)

If min1lekleK


equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth2 I

pmlowasth2

⎛⎜⎜⎝ ⎞⎟⎟⎠

(36)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(37)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(38)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(39)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth22 I

bpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(40)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK



Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(41)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(42)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(43)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth22 I

fpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(44)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(45)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(46)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(47)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth22 I

fbpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(48)



1le ileLRif01113864 1113865gt 1

max1lejleM

Rjp01113966 1113967gt 1 or max








103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)



a22 004 a31 005 and a32 002 and b11 003 b12











6 Conclusion





0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















fa(t) is

dNifa(t)dt


ifa minus sum

j

aijNifa (2)

and thus

Nifa(t) le

Aifdf +sumjaij

Wifa (3)



ljNba minus sumk

ckNba

(4)

and thus

Nba(t) leAb



pa(t) is

Poultryfarm

Poultryfarm

Poultryfarm

Poultryfarm

Poultryfarm

Wholesalemarket

Wholesalemarket

Wet market





Sba

Iba

Rh

ljSba

Ab

ckSba

Aif βfSifaIifa

aijSifa dpSjpa

dfIifa

βpSjpaIjpa

βmSkmaIkma

βbSbaIba

βkhShIkma

βhShIba

dfSifa

dbSba

dbIba

bjkSjpa bjkIjpa

dpIjpa

aijIifa

dmSkma

dhSh dhIh αhIh

γhIh

αbIba

ljIbaαpIjpa

αfIifa

ckIba

αmIkma

dmIkma

Ah dhRh

Sifa Iifa

Sh Ih

Sjpa

Skma Ikma

Ijpa



dNjpa(t)

dt 1113944

i


jpa minus αpI

jpa minus 1113944

k

bjkNjpa

(6)

and thus

Njpa(t) le


dp + 1113936kbjk

le1113936i aijA


dp + 1113936kbjk

Wjpa (7)


ma(t) is

dNkma(t)

dt 1113944

j


kma minus αmI

kma (8)

and thus

Nkma(t) le


dm

le1113936j bjk 1113936i aijA


dm W

kma

(9)


dNh(t)


and thus

Nh(t) leAh

dh (11)


(S1fa SLfa I1fa IL


pa

S1ma Skma I1ma Ik


+ Sifa + Ii


Wba Sjpa + I

jpa le W

jpa Sk

ma + Ikma leWk




dSifa(t)

dt A

if minusdfS

ifa minus 1113944

j

aijSifa

dSba(t)


j


ckSba

dSjpa(t)

dt 1113944

i


jpa minus 1113944

k

bjkSjpa

dSkma(t)

dt 1113944

j


kma

dSh(t)

dt Ah minusdhSh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)






i0fa1113980111397911139781113981L



S0h) where Si0fa Ai

f (df + 1113936jaij)


i0fa + ljS

0ba)(dp+



i0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) is the dis-



F

F11 0 0 00 F22 0 00 0 F33 00 0 0 F44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V

V11 0 0 00 V22 0 00 0 V33 00 0 0 V44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(13)

where

F11





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F22 βbS0ba1113872 1113873

F33



pa

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F44





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V11

df +αf + 1113944j


0 df +αf + 1113944j



j

aij

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V33

dp +αp + 1113944k


0 dp +αp + 1113944k



k

bjk

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V44


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V22 db +αb + 1113944j

lj + 1113944k

ck⎛⎝ ⎞⎠

(14)



Rif0 R

jp0 R

km0 Rb01113966 1113967

Rif0

βfSi0fa

df + αf + 1113936jaij

Rjp0

βpSj0pa

dp + αp + 1113936kbjk

Rkm0

βmSk0ma

dm + αm

Rb0 βbS0ba

db + αb + 1113936jlj + 1113936kck

(15)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa i 1 L

dSjpa(t)

dt 1113944

i

aijSifa minus βpS

jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i

aijIifa + βpS

jpaI

jpa minus dpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

j 1 M

dSkma(t)

dt 1113944

j

bjkSjpa minus βmS

kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j

bjkIjpa + βmS

kmaI

kma minus dmI

kma minus αmI

kma k 1 K

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minus dhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)





dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


fa ne 0 we obtain

Silowastfa

df + αf + 1113936jaij

βf

Iilowastfa


βf df + αf + 1113936jaij1113872 1113873

(18)





Remark 2

(1) If min1le ileL






dSba(t)


j


ckSba

dIba(t)


j


ckIba

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


Slowastba

db + αb + 1113936jlj + 1113936kck

βb

Ilowastba


βbSlowastba

(20)





dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)




Sjlowastpa

dp + αp + 1113936kbjk

βp

Ijlowastpa



pa

dp + αp + 1113936kbjk

(22)


(Sjlowastpa I

jlowastpa)


Ijlowastpa


lowastba


b1Sjlowast2pa + b2S

jlowastpa + b3 0

(23)

where



b2 βp 1113944i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠ + βp 1113944i

aijIilowastfa + ljI

lowastba

⎛⎝ ⎞⎠

+ dp + 1113944k

bjk⎛⎝ ⎞⎠ dp + αp + 1113944

k



bjk⎛⎝ ⎞⎠ 1113944

i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠lt 0

(24)



b22 minus 4b1b3

1113969

2b1gt 0


b22 minus 4b1b3

1113969

2b1gt 0

(25)





pa2 Ijlowastpa2) If

Rj2p0 lt 1 and R

j1p0 ((dp + αp + 1113936kbjk)(βpS



jlowastpa1) If R

j1p0 lt 1




dSkma(t)

dt 1113944

j


kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Sklowastma

dm + αmβm

Iklowastma


ma

dm + αm

(27)



ma)If I


Iklowastma



g1Sklowast2ma + g2S

klowastma + g3 0

(28)

where

g1 minusdmβm lt 0

g2 βm 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠ + βm 1113944j

bjkIjlowastpa + ckI

lowastba

⎛⎝ ⎞⎠

+ dm dm + αm( 1113857gt 0

g3 minus dm + αm( 1113857 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠lt 0

(29)


equation are


g22 minus 4g1g3

1113969

2g1gt 0


g22 minus 4g1g3

1113969

2g1gt 0

(30)





ma2) If Rk2m0 lt 1







dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minusdhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(31)



Slowasth

Ah


Ilowasth


dh + αh + ch

(32)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minusdfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

dSba(t)


j


ckSba

dIba(t)


j


ckIba

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j


kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)




Rif0 R

jp0 Rk

m0 Rb01113966 1113967lt 1


(Si0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) Namely when



Rif0 R

jp01113966 1113967lt 1 Rb0 lt 1 and

min1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S

j0pa1113980radic11139791113978radic1113981M

01113980111397911139781113981M



Smlowasth I

mlowasth

⎛⎜⎜⎝ ⎞⎟⎟⎠

(34)




1lejleMR

jp01113966 1113967gt 1 sys-





Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

(35)

If min1lekleK


equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth2 I

pmlowasth2

⎛⎜⎜⎝ ⎞⎟⎟⎠

(36)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(37)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(38)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(39)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth22 I

bpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(40)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK



Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(41)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(42)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(43)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth22 I

fpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(44)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(45)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(46)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(47)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth22 I

fbpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(48)



1le ileLRif01113864 1113865gt 1

max1lejleM

Rjp01113966 1113967gt 1 or max








103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)



a22 004 a31 005 and a32 002 and b11 003 b12











6 Conclusion





0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















dNjpa(t)

dt 1113944

i


jpa minus αpI

jpa minus 1113944

k

bjkNjpa

(6)

and thus

Njpa(t) le


dp + 1113936kbjk

le1113936i aijA


dp + 1113936kbjk

Wjpa (7)


ma(t) is

dNkma(t)

dt 1113944

j


kma minus αmI

kma (8)

and thus

Nkma(t) le


dm

le1113936j bjk 1113936i aijA


dm W

kma

(9)


dNh(t)


and thus

Nh(t) leAh

dh (11)


(S1fa SLfa I1fa IL


pa

S1ma Skma I1ma Ik


+ Sifa + Ii


Wba Sjpa + I

jpa le W

jpa Sk

ma + Ikma leWk




dSifa(t)

dt A

if minusdfS

ifa minus 1113944

j

aijSifa

dSba(t)


j


ckSba

dSjpa(t)

dt 1113944

i


jpa minus 1113944

k

bjkSjpa

dSkma(t)

dt 1113944

j


kma

dSh(t)

dt Ah minusdhSh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)






i0fa1113980111397911139781113981L



S0h) where Si0fa Ai

f (df + 1113936jaij)


i0fa + ljS

0ba)(dp+



i0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) is the dis-



F

F11 0 0 00 F22 0 00 0 F33 00 0 0 F44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V

V11 0 0 00 V22 0 00 0 V33 00 0 0 V44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(13)

where

F11





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F22 βbS0ba1113872 1113873

F33



pa

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F44





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V11

df +αf + 1113944j


0 df +αf + 1113944j



j

aij

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V33

dp +αp + 1113944k


0 dp +αp + 1113944k



k

bjk

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V44


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V22 db +αb + 1113944j

lj + 1113944k

ck⎛⎝ ⎞⎠

(14)



Rif0 R

jp0 R

km0 Rb01113966 1113967

Rif0

βfSi0fa

df + αf + 1113936jaij

Rjp0

βpSj0pa

dp + αp + 1113936kbjk

Rkm0

βmSk0ma

dm + αm

Rb0 βbS0ba

db + αb + 1113936jlj + 1113936kck

(15)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa i 1 L

dSjpa(t)

dt 1113944

i

aijSifa minus βpS

jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i

aijIifa + βpS

jpaI

jpa minus dpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

j 1 M

dSkma(t)

dt 1113944

j

bjkSjpa minus βmS

kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j

bjkIjpa + βmS

kmaI

kma minus dmI

kma minus αmI

kma k 1 K

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minus dhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)





dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


fa ne 0 we obtain

Silowastfa

df + αf + 1113936jaij

βf

Iilowastfa


βf df + αf + 1113936jaij1113872 1113873

(18)





Remark 2

(1) If min1le ileL






dSba(t)


j


ckSba

dIba(t)


j


ckIba

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


Slowastba

db + αb + 1113936jlj + 1113936kck

βb

Ilowastba


βbSlowastba

(20)





dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)




Sjlowastpa

dp + αp + 1113936kbjk

βp

Ijlowastpa



pa

dp + αp + 1113936kbjk

(22)


(Sjlowastpa I

jlowastpa)


Ijlowastpa


lowastba


b1Sjlowast2pa + b2S

jlowastpa + b3 0

(23)

where



b2 βp 1113944i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠ + βp 1113944i

aijIilowastfa + ljI

lowastba

⎛⎝ ⎞⎠

+ dp + 1113944k

bjk⎛⎝ ⎞⎠ dp + αp + 1113944

k



bjk⎛⎝ ⎞⎠ 1113944

i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠lt 0

(24)



b22 minus 4b1b3

1113969

2b1gt 0


b22 minus 4b1b3

1113969

2b1gt 0

(25)





pa2 Ijlowastpa2) If

Rj2p0 lt 1 and R

j1p0 ((dp + αp + 1113936kbjk)(βpS



jlowastpa1) If R

j1p0 lt 1




dSkma(t)

dt 1113944

j


kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Sklowastma

dm + αmβm

Iklowastma


ma

dm + αm

(27)



ma)If I


Iklowastma



g1Sklowast2ma + g2S

klowastma + g3 0

(28)

where

g1 minusdmβm lt 0

g2 βm 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠ + βm 1113944j

bjkIjlowastpa + ckI

lowastba

⎛⎝ ⎞⎠

+ dm dm + αm( 1113857gt 0

g3 minus dm + αm( 1113857 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠lt 0

(29)


equation are


g22 minus 4g1g3

1113969

2g1gt 0


g22 minus 4g1g3

1113969

2g1gt 0

(30)





ma2) If Rk2m0 lt 1







dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minusdhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(31)



Slowasth

Ah


Ilowasth


dh + αh + ch

(32)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minusdfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

dSba(t)


j


ckSba

dIba(t)


j


ckIba

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j


kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)




Rif0 R

jp0 Rk

m0 Rb01113966 1113967lt 1


(Si0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) Namely when



Rif0 R

jp01113966 1113967lt 1 Rb0 lt 1 and

min1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S

j0pa1113980radic11139791113978radic1113981M

01113980111397911139781113981M



Smlowasth I

mlowasth

⎛⎜⎜⎝ ⎞⎟⎟⎠

(34)




1lejleMR

jp01113966 1113967gt 1 sys-





Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

(35)

If min1lekleK


equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth2 I

pmlowasth2

⎛⎜⎜⎝ ⎞⎟⎟⎠

(36)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(37)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(38)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(39)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth22 I

bpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(40)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK



Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(41)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(42)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(43)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth22 I

fpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(44)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(45)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(46)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(47)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth22 I

fbpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(48)



1le ileLRif01113864 1113865gt 1

max1lejleM

Rjp01113966 1113967gt 1 or max








103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)



a22 004 a31 005 and a32 002 and b11 003 b12











6 Conclusion





0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















i0fa1113980111397911139781113981L



S0h) where Si0fa Ai

f (df + 1113936jaij)


i0fa + ljS

0ba)(dp+



i0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) is the dis-



F

F11 0 0 00 F22 0 00 0 F33 00 0 0 F44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V

V11 0 0 00 V22 0 00 0 V33 00 0 0 V44

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(13)

where

F11





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F22 βbS0ba1113872 1113873

F33



pa

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F44





⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V11

df +αf + 1113944j


0 df +αf + 1113944j



j

aij

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V33

dp +αp + 1113944k


0 dp +αp + 1113944k



k

bjk

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V44


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

V22 db +αb + 1113944j

lj + 1113944k

ck⎛⎝ ⎞⎠

(14)



Rif0 R

jp0 R

km0 Rb01113966 1113967

Rif0

βfSi0fa

df + αf + 1113936jaij

Rjp0

βpSj0pa

dp + αp + 1113936kbjk

Rkm0

βmSk0ma

dm + αm

Rb0 βbS0ba

db + αb + 1113936jlj + 1113936kck

(15)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa i 1 L

dSjpa(t)

dt 1113944

i

aijSifa minus βpS

jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i

aijIifa + βpS

jpaI

jpa minus dpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

j 1 M

dSkma(t)

dt 1113944

j

bjkSjpa minus βmS

kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j

bjkIjpa + βmS

kmaI

kma minus dmI

kma minus αmI

kma k 1 K

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minus dhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)





dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


fa ne 0 we obtain

Silowastfa

df + αf + 1113936jaij

βf

Iilowastfa


βf df + αf + 1113936jaij1113872 1113873

(18)





Remark 2

(1) If min1le ileL






dSba(t)


j


ckSba

dIba(t)


j


ckIba

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


Slowastba

db + αb + 1113936jlj + 1113936kck

βb

Ilowastba


βbSlowastba

(20)





dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)




Sjlowastpa

dp + αp + 1113936kbjk

βp

Ijlowastpa



pa

dp + αp + 1113936kbjk

(22)


(Sjlowastpa I

jlowastpa)


Ijlowastpa


lowastba


b1Sjlowast2pa + b2S

jlowastpa + b3 0

(23)

where



b2 βp 1113944i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠ + βp 1113944i

aijIilowastfa + ljI

lowastba

⎛⎝ ⎞⎠

+ dp + 1113944k

bjk⎛⎝ ⎞⎠ dp + αp + 1113944

k



bjk⎛⎝ ⎞⎠ 1113944

i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠lt 0

(24)



b22 minus 4b1b3

1113969

2b1gt 0


b22 minus 4b1b3

1113969

2b1gt 0

(25)





pa2 Ijlowastpa2) If

Rj2p0 lt 1 and R

j1p0 ((dp + αp + 1113936kbjk)(βpS



jlowastpa1) If R

j1p0 lt 1




dSkma(t)

dt 1113944

j


kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Sklowastma

dm + αmβm

Iklowastma


ma

dm + αm

(27)



ma)If I


Iklowastma



g1Sklowast2ma + g2S

klowastma + g3 0

(28)

where

g1 minusdmβm lt 0

g2 βm 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠ + βm 1113944j

bjkIjlowastpa + ckI

lowastba

⎛⎝ ⎞⎠

+ dm dm + αm( 1113857gt 0

g3 minus dm + αm( 1113857 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠lt 0

(29)


equation are


g22 minus 4g1g3

1113969

2g1gt 0


g22 minus 4g1g3

1113969

2g1gt 0

(30)





ma2) If Rk2m0 lt 1







dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minusdhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(31)



Slowasth

Ah


Ilowasth


dh + αh + ch

(32)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minusdfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

dSba(t)


j


ckSba

dIba(t)


j


ckIba

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j


kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)




Rif0 R

jp0 Rk

m0 Rb01113966 1113967lt 1


(Si0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) Namely when



Rif0 R

jp01113966 1113967lt 1 Rb0 lt 1 and

min1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S

j0pa1113980radic11139791113978radic1113981M

01113980111397911139781113981M



Smlowasth I

mlowasth

⎛⎜⎜⎝ ⎞⎟⎟⎠

(34)




1lejleMR

jp01113966 1113967gt 1 sys-





Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

(35)

If min1lekleK


equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth2 I

pmlowasth2

⎛⎜⎜⎝ ⎞⎟⎟⎠

(36)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(37)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(38)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(39)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth22 I

bpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(40)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK



Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(41)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(42)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(43)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth22 I

fpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(44)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(45)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(46)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(47)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth22 I

fbpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(48)



1le ileLRif01113864 1113865gt 1

max1lejleM

Rjp01113966 1113967gt 1 or max








103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)



a22 004 a31 005 and a32 002 and b11 003 b12











6 Conclusion





0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















dSifa(t)

dt A

if minus βfS

ifaI

ifa minus dfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minusdfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


fa ne 0 we obtain

Silowastfa

df + αf + 1113936jaij

βf

Iilowastfa


βf df + αf + 1113936jaij1113872 1113873

(18)





Remark 2

(1) If min1le ileL






dSba(t)


j


ckSba

dIba(t)


j


ckIba

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


Slowastba

db + αb + 1113936jlj + 1113936kck

βb

Ilowastba


βbSlowastba

(20)





dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)




Sjlowastpa

dp + αp + 1113936kbjk

βp

Ijlowastpa



pa

dp + αp + 1113936kbjk

(22)


(Sjlowastpa I

jlowastpa)


Ijlowastpa


lowastba


b1Sjlowast2pa + b2S

jlowastpa + b3 0

(23)

where



b2 βp 1113944i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠ + βp 1113944i

aijIilowastfa + ljI

lowastba

⎛⎝ ⎞⎠

+ dp + 1113944k

bjk⎛⎝ ⎞⎠ dp + αp + 1113944

k



bjk⎛⎝ ⎞⎠ 1113944

i

aijSilowastfa + ljS

lowastba

⎛⎝ ⎞⎠lt 0

(24)



b22 minus 4b1b3

1113969

2b1gt 0


b22 minus 4b1b3

1113969

2b1gt 0

(25)





pa2 Ijlowastpa2) If

Rj2p0 lt 1 and R

j1p0 ((dp + αp + 1113936kbjk)(βpS



jlowastpa1) If R

j1p0 lt 1




dSkma(t)

dt 1113944

j


kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Sklowastma

dm + αmβm

Iklowastma


ma

dm + αm

(27)



ma)If I


Iklowastma



g1Sklowast2ma + g2S

klowastma + g3 0

(28)

where

g1 minusdmβm lt 0

g2 βm 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠ + βm 1113944j

bjkIjlowastpa + ckI

lowastba

⎛⎝ ⎞⎠

+ dm dm + αm( 1113857gt 0

g3 minus dm + αm( 1113857 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠lt 0

(29)


equation are


g22 minus 4g1g3

1113969

2g1gt 0


g22 minus 4g1g3

1113969

2g1gt 0

(30)





ma2) If Rk2m0 lt 1







dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minusdhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(31)



Slowasth

Ah


Ilowasth


dh + αh + ch

(32)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minusdfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

dSba(t)


j


ckSba

dIba(t)


j


ckIba

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j


kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)




Rif0 R

jp0 Rk

m0 Rb01113966 1113967lt 1


(Si0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) Namely when



Rif0 R

jp01113966 1113967lt 1 Rb0 lt 1 and

min1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S

j0pa1113980radic11139791113978radic1113981M

01113980111397911139781113981M



Smlowasth I

mlowasth

⎛⎜⎜⎝ ⎞⎟⎟⎠

(34)




1lejleMR

jp01113966 1113967gt 1 sys-





Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

(35)

If min1lekleK


equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth2 I

pmlowasth2

⎛⎜⎜⎝ ⎞⎟⎟⎠

(36)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(37)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(38)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(39)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth22 I

bpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(40)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK



Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(41)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(42)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(43)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth22 I

fpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(44)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(45)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(46)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(47)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth22 I

fbpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(48)



1le ileLRif01113864 1113865gt 1

max1lejleM

Rjp01113966 1113967gt 1 or max








103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)



a22 004 a31 005 and a32 002 and b11 003 b12











6 Conclusion





0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















dSkma(t)

dt 1113944

j


kmaI

kma minus dmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Sklowastma

dm + αmβm

Iklowastma


ma

dm + αm

(27)



ma)If I


Iklowastma



g1Sklowast2ma + g2S

klowastma + g3 0

(28)

where

g1 minusdmβm lt 0

g2 βm 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠ + βm 1113944j

bjkIjlowastpa + ckI

lowastba

⎛⎝ ⎞⎠

+ dm dm + αm( 1113857gt 0

g3 minus dm + αm( 1113857 1113944j

bjkSjlowastpa + ckS

lowastba

⎛⎝ ⎞⎠lt 0

(29)


equation are


g22 minus 4g1g3

1113969

2g1gt 0


g22 minus 4g1g3

1113969

2g1gt 0

(30)





ma2) If Rk2m0 lt 1







dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


dRh(t)

dt chIh minusdhRh

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(31)



Slowasth

Ah


Ilowasth


dh + αh + ch

(32)



dSifa(t)

dt A

if minus βfS

ifaI

ifa minusdfS

ifa minus 1113944

j

aijSifa

dIifa(t)

dt βfS

ifaI

ifa minus dfI

ifa minus αfI

ifa minus 1113944

j

aijIifa

dSba(t)


j


ckSba

dIba(t)


j


ckIba

dSjpa(t)

dt 1113944

i


jpaI

jpa minus dpS

jpa minus 1113944

k

bjkSjpa

dIjpa(t)

dt 1113944

i


jpaI

jpa minusdpI

jpa minus αpI

jpa minus 1113944

k

bjkIjpa

dSkma(t)

dt 1113944

j


kmaI

kma minusdmS

kma

dIkma(t)

dt 1113944

j


kmaI

kma minusdmI

kma minus αmI

kma

dSh(t)

dt Ah minus 1113944

k


dIh(t)

dt 1113944

k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)




Rif0 R

jp0 Rk

m0 Rb01113966 1113967lt 1


(Si0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) Namely when



Rif0 R

jp01113966 1113967lt 1 Rb0 lt 1 and

min1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S

j0pa1113980radic11139791113978radic1113981M

01113980111397911139781113981M



Smlowasth I

mlowasth

⎛⎜⎜⎝ ⎞⎟⎟⎠

(34)




1lejleMR

jp01113966 1113967gt 1 sys-





Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

(35)

If min1lekleK


equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth2 I

pmlowasth2

⎛⎜⎜⎝ ⎞⎟⎟⎠

(36)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(37)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(38)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(39)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth22 I

bpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(40)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK



Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(41)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(42)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(43)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth22 I

fpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(44)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(45)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(46)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(47)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth22 I

fbpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(48)



1le ileLRif01113864 1113865gt 1

max1lejleM

Rjp01113966 1113967gt 1 or max








103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)



a22 004 a31 005 and a32 002 and b11 003 b12











6 Conclusion





0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















Rif0 R

jp0 Rk

m0 Rb01113966 1113967lt 1


(Si0fa1113980111397911139781113981L

01113980111397911139781113981L


01113980111397911139781113981M


01113980111397911139781113981K

S0h 0) Namely when



Rif0 R

jp01113966 1113967lt 1 Rb0 lt 1 and

min1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S

j0pa1113980radic11139791113978radic1113981M

01113980111397911139781113981M



Smlowasth I

mlowasth

⎛⎜⎜⎝ ⎞⎟⎟⎠

(34)




1lejleMR

jp01113966 1113967gt 1 sys-





Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

(35)

If min1lekleK


equilibria

E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth1 I

pmlowasth1

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L

S0ba 0 S


Ijlowastpa1113980111397911139781113981M



Spmlowasth2 I

pmlowasth2

⎛⎜⎜⎝ ⎞⎟⎟⎠

(36)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(37)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(38)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(39)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth11 I

bpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth12 I

bpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth21 I

bpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Si0fa1113980111397911139781113981L

01113980111397911139781113981L





K


K

Sbpmlowasth22 I

bpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(40)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK



Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(41)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(42)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(43)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth22 I

fpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(44)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(45)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(46)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(47)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth22 I

fbpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(48)



1le ileLRif01113864 1113865gt 1

max1lejleM

Rjp01113966 1113967gt 1 or max








103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)



a22 004 a31 005 and a32 002 and b11 003 b12











6 Conclusion





0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(41)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(42)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(43)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth11 I

fpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth12 I

fpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth21 I

fpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L

S0ba 0 S




K


K

Sfpmlowasth22 I

fpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(44)





1lejleMR

j1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 min


max1lekleK


Elowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

(45)

If min1lejleM

Rj1p01113966 1113967gt 1 max

1lejleMR

j2p01113966 1113967lt 1 and min

1lekleKRk2m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

(46)

If min1lejleM

Rj2p01113966 1113967gt 1 max


1lekleKRk1m01113966 1113967gt 1


E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

(47)

If min1lejleM

Rj2p01113966 1113967gt 1 and min



E1lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth11 I

fbpmlowasth11

⎛⎜⎜⎝ ⎞⎟⎟⎠

E2lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth12 I

fbpmlowasth12

⎛⎜⎜⎝ ⎞⎟⎟⎠

E3lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth21 I

fbpmlowasth21

⎛⎜⎜⎝ ⎞⎟⎟⎠

E4lowast

Silowastfa1113980111397911139781113981L

Iilowastfa1113980111397911139781113981L





K


K

Sfbpmlowasth22 I

fbpmlowasth22

⎛⎜⎜⎝ ⎞⎟⎟⎠

(48)



1le ileLRif01113864 1113865gt 1

max1lejleM

Rjp01113966 1113967gt 1 or max








103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)



a22 004 a31 005 and a32 002 and b11 003 b12











6 Conclusion





0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of
























103 50 103 50) (Sb(0) Ib(0)) (104100) and (Sh(0)



a22 004 a31 005 and a32 002 and b11 003 b12











6 Conclusion





0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















0 20 40 60 80 1000

50

100

150

200

250

Time (months)

Infe

ctiv

e hum

ans (I h

)


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

Time (months)

Infe

ctiv

e hum

ans (I h

)




(a)

Infe

ctiv

e hum

ans (I h

)




0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (months)

(b)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

1000

Time (months)

(c)

Infe

ctiv

e hum

ans (I h

)




0 20 40 60 80 1000

200

400

600

800

1000

Time (months)

(d)








Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of
























Data Availability




Acknowledgments


References































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















AvianInfluenzaA(H7N9)ModelBasedonPoultryTransport … · 2019. 7. 30. · uenza A (H7N9) virus was...

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