Optimal Control of a Vector-Host Epidemic Model … D. E. Mahmudah, A. Suryanto and Trisilowati...

*Corresponding author. Email: [email protected]

Applied Mathematical Sciences, Vol. 7, 2013, no. 99, 4919 - 4927

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2013.37415

Optimal Control of a Vector-Host Epidemic Model

with Direct Transmission

D. E. Mahmudah, A. Suryanto* and Trisilowati

Department of Mathematics

Faculty of Sciences - Brawijaya University

Jl. Veteran Malang 65145, Indonesia

Copyright © 2013 D. E. Mahmudah, A. Suryanto and Trisilowati. This is an open access

article distributed under the Creative Commons Attribution License, which permits unrestricted

use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we extend an existing vector-host epidemic model with direct

transmission to assess the impact of two control measures, i.e. the preventive

control to minimize vector-host contacts and the insecticide control to the vector.

The aim is to derive optimal prevention strategies with minimal implementation

cost. The characterization of optimal control is performed analytically by applying

Pontryagin Minimum Principle. The obtained optimality system is then solved

numerically to investigate that there are cost effective control efforts in reducing

the incidence of infectious hosts and vectors.

Keywords: A vector-host epidemic model, Direct transmission, Optimal control,

Pontryagin minimum principle, Forward-backward sweep method

1 Introduction

Many mathematical models have been used to understand the spread of

vector-host epidemic diseases such as malaria, dengue, chikungunya,

elephantiasis, and dysentery; see e.g. [5]. Various kinds of strategies for disease

4920 D. E. Mahmudah, A. Suryanto and Trisilowati

control have been carried out with the aim of minimizing infective host or as

prevention efforts. For example, optimal strategies for controlling the spread of

malaria disease using prevention and treatment on malaria have been studied in

[3]. Similarly, Agusto et al. [1] have investigated an optimal prevention strategies

of malaria disease using treatment, insecticide treated bed nets and spray of

mosquito insecticide as the system control variables. Okosun and Makinde [8]

have studied the impact of drug resistance in malaria transmission to investigate

the role of drug resistance individuals in malaria transmission and its optimal

control problem.

Generally, insects or vectors become the main intermediaries of infectious

disease to the host population; see e.g. [5] and [7]. In such vector-host epidemic

model, transmission disease is usually caused by contacts between infective

vectors and susceptible hosts only, see e.g. [1], [5], [7] and [9]. However, the

direct transmission such as a contact between infective host and susceptible host

population should be incorporated in the model [4]. By considering the direct

transmission, in the normalized variables, Cai and Li [4] proposed the following

vector-host epidemic model

,

)(12

21

VHVV

VHVV

HHHVHH

HHVHHHH

IISdt

dI

SISdt

dS

IISISmdt

dI

SIISmISdt

dS

(1)

where and represents the proportion of susceptible and infective host

population, respectively while and are respectively the proportion of

susceptible and infective vector population. In model (1), , and are

respectively the rate of direct transmission from infectious host to susceptible

host, the rate of transmission from vector to host and the of transmission from

host to vector; is the host birth rate which is assumed to be equal to the host

mortality rate; is the vector birth rate which is assumed to be equal to the vector

mortality rate; is the recovery rate of infectious hosts; and is the ratio of

vector population and host population. Cai and Li [4] have shown that

1,1,0,,,:,,, 4 VVHHVVHHVVHH ISISISISRISIS

is invariant region. It is also shown that system has a disease free equilibrium

)0,1,0,1(0 E which is asymptotically stable if .10 R Beside the disease free

Optimal control of a vector-host epidemic model 4921

equilibrium, system has also an endemic equilibrium )ˆ,ˆ,ˆ,ˆ(ˆVVHH ISISE

which is asymptotically stable whenever ,10 R where

.

ˆ

ˆˆ,

ˆˆ,

ˆ)(ˆ

)ˆ)(ˆ(ˆ

212

1 H

HV

HV

HH

HHH

I

II

IS

ImI

IIS

Here 0R is the basic reproduction number which is defined by

,120

mR

and HI is the root of the following quadratic equation

,0ˆ)ˆ( 012

2 aIaIa HH

with ,)(,0 121112 maa .0)1)(( 00 Ra

In this paper, we extend model (1) by including the preventive contact

control in the form of medical treatment to minimize vector host contacts as well

as the level of larvacide and adulticide used for vectors control at vector’s habitat

to minimize the infected host at minimal cost.

2 Optimal Control Problem

To extend the basic vector-host epidemic model with direct transmission (1),

we introduce two control measures, i.e. the preventive contact control in the form

of medical treatment to minimize vector host contacts ( 1u ) and the level of

larvacide and adulticide used for vector’s control at vector’s habitat ( 2u ) as

vector’s control with 10 1 u and 10 2 u . By including these two control

measures, we obtained the following model


.

)()1(

)1(

2

2

121

211

VVHVV

VVHVV

HHHVHH

HHVHHHH

IcuIISdt

dI

ScuSISdt

dS

IISISmudt

dI

SIISmuISdt

dS

(2)

Now, we consider an optimal control problem to minimize the objective

functional

T

H dttDutCutBIuuJ

0

22

2121 ,)()()(),( (3)

subject to system (2) where T is the final time and all parameter B, C, and D are

positive balancing cost factors. Our aim is to minimize the number of infected

host ),(tIH while minimizing the cost for applying control )(1 tu and ).(2 tu

Thus, we seek an optimal control pair *1u and *

2u such that

UuuuuJuuJ 2121*2

*1 ,,min, , (4)

where 2,1,1,0,0:|, 21 iTuuuU i is control set. For this purposes

we shall use Pontryagin’s Maximum Principle. This principle converts (2), (3) and

(4) into a problem of minimizing a Hamiltonian ( ), defined by

VVHVIVVHVS

HHHVHI

HHVHHHS

H

IcuIISScuSIS

IISISmu

SIISmuIS

tDutCutBIH

VV

H

H

22

121

211

22

21

1

1

)()()(

where ),(),(),( tttVVHHHH SSIISS and )(t

VV II are costate

variables. Applying Pontryagin’s Minimum Principle and the existence result of

the optimal control, we obtain the following theorem.


Theorem 1. Given an optimal control *

2

*

1 ,uu and solutions **** ,,, VVHH ISIS of

the corresponding state system (2) that minimizes ),( 21 uuJ over U. Then there

exists costate variables ,,,,VVHH ISIS , such that

HH

HIHVSVH

SIImuImuI

dt

d

121211 11

VVHH

HIVSVIHSH

ISSSSB

dt

d

11

VV

VIHSH

SIcuI

dt

d

2

,11 22121 VHH

VIIHSH

IcuSmuSmu

dt

d

with transversality conditions ,0)()()()( TTTTVVHH ISIS and the

controls *1u and *

2u satisfy the optimality condition,

,1,

2,0maxmin

2*1

C

ISmu

VHSI HH

.1,2

,0maxmin*2

D

cIcSu

VIVS VV

Proof. The existence of optimal control can be proved using the same argument as

in Theorem 1 of [2] which is based on the boundedness of solution of system (1)

without control variables. The differential equations governing the costate

variables are obtained by taking the derivative of the Hamiltonian function,

evaluated at the optimal control. Then the costate system can be written as

HH

HIHVSVH

H

SIImuImuI

dS

dH

dt

d

121211 11

V

VHH

H

IV

SVIHSHH

I

S

SSSBdI

dH

dt

d

11


VV

VIHSH

V

SIcuI

dS

dH

dt

d

2

,11 22121 VHH

VIIHSH

V

IcuSmuSmu

dI

dH

dt

d

with transversality conditions .0)()()()( TTTTVVHH ISIS

On the interior of the control set, where ,10 iu for ,2,1i we have

,20 **

2

**

2

*

1

1HH IVHSVH ISmISmCu

u

H

.20 ***

2

2

VIVS cIcSDuu

HVV

Solving for the optimal control, we obtain

,

2

**2*

1C

ISmu

VHSI HH

.2

***2

D

cIcSu

VIVS VV

Hence, the optimal controls are

,1,

2,0maxmin

**2*

1

C

ISmu

VHSI HH

.1,2

,0maxmin

***2

D

cIcSu

VIVS VV

3 Numerical Simulations

In this section, we investigate numerically the optimal solution to the

vector-host epidemic model with direct transmission by Sweep Backward-

Forward method [6]. The optimality system is solved using the fourth order Runge-


Kutta. The state system is solved forward in time with initial conditions

7.00,1.00,9.00 VHH SIS and 3.00 VI while the costate system

is solved backward in time. The controls values are update at the end of iteration

using the formula for optimal controls. For the following numerical simulations,

we use parameters given in Table 1.

Table 1. Values of parameters in the vector-host epidemic model.

Paramater 1 2 m

Value 0.4427 0.9619 0.0046 0.7749 0.8173 0.8687 0.05

0 100 200 300 4000.88

0.9

0.92

0.94

0.96

0.98

Time (days)

Susceptib

les H

ost (S

H)

(a)

0 100 200 300 4000.02

0.04

0.06

0.08

0.1

Time (days)

Infe

ctio

us H

ost (I

H)

(b)

0 100 200 300 4000

0.2

0.4

0.6

Time (days)

Susceptib

les V

ecto

r (S

V)

(c)

0 100 200 300 4000.2

0.4

0.6

0.8

1

Time (days)

Infe

ctio

us V

ecto

r (I

V)

(d)

no control

w ith control

no control

w ith control

no control

w ith control

no control

w ith control

Figure 1. Numerical solution of system (1) with and without control.

Using parameter in Table 1, we find that the reproduction number is

16984.110 R which means that without control, any solution will be

convergent to the endemic equilibrium. However, when control variables are

introduced in system (1) then different behavior of solutions are obtained. For


example if we take the weight constant values in the objective functional are

chosen to be 10,10,1 DCB , then we notice that the presence of controls

may reduce the number of infective host as well as the number of infective vector

significantly. The obtained optimal control )(*1 tu and )(*

2 tu can be seen in

Figure 2.

0 100 200 300 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (days)

Optim

al C

ontr

ol (

u 1*)

(a)

0 100 200 300 4000

0.1

0.2

0.3

0.4

Time (days)

Optim

al C

ontr

ol (

u 2*)

(b)

Figure 2. The optimal controls profile: (a) function )(*1 tu and (b) function

)(*2 tu .

4 Conclusion

In this work we have studied the effects of preventive control to reduce

vector-host contacts and the insecticide control to the vector in a vector-host

epidemic model with direct transmission. Using Pontryagin Minimum Principle

we have established the existence of optimal control pair which minimize the

proportion of both infective host and infective vector as well as minimize the cost

of applying controls. The numerical simulations with and without control show

that the control strategy helps to reduce the number of both infective host and

infective vector significantly.

Acknowledgment

Part of this research is financially supported by the Directorate General of

Higher Education, Ministry of Education and Culture of Republic Indonesia via

DIPA of Brawijaya University No.: DIPA-023.04.2.414989/2013, and based on

letter of decision of Brawijaya University Rector No.: 295/SK/2013.


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Received: July 25, 2013

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