# An Optimal Control Approach to Malaria Prevention via Insecticide-Treated Nets

Authors: Cristiana J. Silva; Delfim F. M. Torres
Journal: Conference Papers in Mathematics
 (2013)

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## Abstract

Malaria is a life threatening disease, entirely preventable and treatable, provided that the currently recommended interventions are properly implemented. These interventions include vector control through the use of insecticide-treated nets (ITNs). However, ITN possession is not necessarily translated into use. Human behavior change interventions, including information, education, communication (IEC) campaigns and postdistribution hang-up campaigns, are strongly recommended. In this paper, we consider a recent mathematical model for the effects of ITNs on the transmission dynamics of malaria infection, which takes into account the human behavior. We introduce in this model asupervision control, representing IEC campaigns for improving the ITN usage. We propose and solve an optimal control problem where the aim is to minimize the number of infected humans while keeping the cost as low as possible. Numerical results are provided, which show the effectiveness of the optimal control interventions.


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## Body

## 1. Introduction

Malaria is a life threatening disease caused byPlasmodium parasites and transmitted from one individual to another by the bite of infected female anopheline mosquitoes [1, 2]. In the human body, the parasites multiply in the liver and then infect red blood cells. Following World Health Organization (WHO) 2012 report, an estimated 3.3 billion people were at risk of malaria in 2011, with populations living in Sub-Saharan Africa having the highest risk of acquiring malaria [3]. Malaria is an entirely preventable and treatable disease, provided the currently recommended interventions are properly implemented. Following WHO, these interventions include (i) vector control through the use of insecticide-treated nets (ITNs), indoor residual spraying and, in some specific settings, larval control, (ii) chemoprevention for the most vulnerable populations, particularly pregnant women and infants, (iii) confirmation of malaria diagnosis through microscopy or rapid diagnostic tests for every suspected case, and (iv) timely treatment with appropriate antimalarial medicines [3]. An ITN is a mosquito net that repels, disables, and/or kills mosquitoes coming into contact with insecticide on the netting material. ITNs are considered one of the most effective interventions against malaria [4]. In 2007, WHO recommended full ITN coverage of all people at risk of malaria, even in high-transmission settings [5]. By 2011, 32 countries in the African region and 78 other countries worldwide, had adopted the WHO recommendation. A total of 89 countries, including 39 in Africa, distribute ITNs free of charge. Between 2004 and 2010, the numbers of ITNs delivered annually by manufacturers to malaria-endemic countries in Sub-Saharan Africa increased from 6 million to 145 million. However, the numbers delivered in 2011 and 2012 are below the number of ITNs required to protect all population at risk. There is an urgent need to identify new funding sources to maintain and expand coverage levels of interventions so that outbreaks of disease can be avoided and international targets for reducing malaria cases and deaths can be attained [3].A number of studies reported that ITN possession is not necessarily translated into use. Human behavior change interventions, including information, education, and communication (IEC) campaigns and postdistribution hang-up campaigns, are strongly recommended, especially where there is evidence of their effectiveness in improving ITN usage [3, 6, 7]. In this paper, we consider the model from [8] for the effects of ITNs on the transmission dynamics of malaria infection. Other articles considered the impact of intervention strategies using ITN (see, e.g., [9, 10]). However, only in [8], the human behavior is incorporated into the model. We introduce in the model of [8] a supervision control, u, which represents IEC campaigns for improving the ITN usage. The reader interested in the use of optimal control to infectious diseases is referred to [11, 12] and the references cited therein. For the state of the art in malaria research see [2].The text is organized as follows. In Section2, we present the mathematical model for malaria transmission with one control function u. In Section 3, we propose an optimal control problem for the minimization of the number of infected humans while controlling the cost of control interventions. Finally, in Section 4 some numerical results are analyzed and interpreted from the epidemiological point of view.


## 2. Controlled Model

We consider a mathematical model presented in [8] for the effects of ITN on the transmission of malaria infection and introduce a time-dependent supervision control u. The model considers transmission of malaria infection of mosquito (also referred as vector) and human (also referred as host) population. The host population is divided into two compartments, susceptible (Sh) and infected (Ih), with a total population (Nh) given by Nh=Sh+Ih. Analogously, the vector population is divided into two compartments, susceptible (Sv) and infected (Iv), with a total population (Nv) given by Nv=Sv+Iv. The model is constructed under the following assumptions: all newborns individuals are assumed to be susceptible and no infected individuals are assumed to come from outside the community. The human and mosquito recruitment rates are denoted by Λh and Λv, respectively. The disease is fast progressing, and thus the exposed stage is minimal and is not considered. Infected individuals can die from the disease or become susceptible after recovery while the mosquito population does not recover from infection. ITNs contribute for the mortality of mosquitoes. The average number of bites per mosquito, per unit of time (mosquito-human contact rate), is given by
(1)β=βmax(1-b),
where βmax denotes the maximum transmission rate and b the proportion of ITN usage. It is assumed that the minimum transmission rate is zero. The value of β is the same for human and mosquito population, so the average number of bites per human per unit of time is βNv/Nh (see [8] and the references cited therein). Thus, the force of infection for susceptible humans (λh) and susceptible vectors (λv) are given by
(2)λh=p1βIvNh,λv=p2βIhNh,
where p1 and p2 are the transmission probability per bite from infectious mosquitoes to humans and from infected humans to mosquitoes, respectively. The death rate of the mosquitoes is modeled by μvb=μv1+μmaxb, where μv1 is the natural death rate and μmaxb is the death rate due to pesticide on ITNs. The coefficient 1-u represents the effort of susceptible humans that become infected by infectious mosquitoes bites, such as educational programs/campaigns for the correct use of ITNs, supervision teams that visit every house in a certain region and assure that every person has access to an ITN, know how to use it correctly, and recognize its importance on the reduction of malaria disease transmission. The values of the parameters Λh, Λv, μh, δh,γh, μv1,μmaxb, βmax, p1, and p2 are taken from [8] (see Table 1).Table 1
Parameter values.
Symbol
Description
Value
Λ
h
Recruitment rate in humans
1
0
3
/
(
70
×
365
)
Λ
v
Recruitment rate in mosquitoes
1
0
4
/
21
μ
h
Natural mortality rate in humans
1
/
(
70
×
365
)
δ
h
Disease induced mortality rate in humans
1
0
-
3
b
Proportion of treated net usage
0.25; 0.3; 0.4; 0.5; 0.6; 0.7; 0.75
γ
h
Recovery rate of infected humans to be susceptible
1/4
μ
v
1
Natural mortality rate of mosquitoes
1/21
μ
max
b
Mortality rate of mosquitoes due to treated net
1/21
β
max
Maximum mosquito-human contact rate
0.1
p
1
Probability of disease transmission from mosquito
1
p
2
Probability of disease transmission from human to mosquito
1
A
1
Weight constant on infected humans
25
C
Weight constant on control
50
S
h
(
0
)
Susceptible individuals initial value
800
I
h
(
0
)
Infected individuals initial value
200
S
v
(
0
)
Susceptible vectors initial value
4000
I
v
(
0
)
Infectious vectors initial value
900The state system of the controlled malaria model is given by(3)S˙h(t)=Λh-(1-u(t))λhSh(t)+γhIh(t)-μhSh(t),I˙h(t)=(1-u(t))λhSh(t)-(μh+γh+δh)Ih(t),S˙v(t)=Λv-λvSv(t)-μvbSv(t),I˙v(t)=p2λvSv(t)-μvbIv(t).The rate of change of the total human and mosquito populations is given by(4)N˙h(t)=Λh-μhNh(t)-δhIh(t),N˙v(t)=Λv-μvbNv(t).


## 3. Optimal Control Problem

We formulate an optimal control problem that describes the goal and restrictions of the epidemic. In [8] it is found that the ITN usage must attain 75% (b=0.75) of the host population in order to extinct malaria. Therefore, educational campaigns must continue encouraging the population to use ITNs. Moreover, it is very important to assure that ITNs are in good conditions and each individual knows how to use them properly. Having this in mind, we introduce a supervision control function, u, where the coefficient 1-u represents the effort to reduce the number of susceptible humans that become infected by infectious mosquitoes bites, assuring that ITNs are correctly used by the fraction b of the host population.We consider the state system (3) of ordinary differential equations in ℝ4 with the set of admissible control functions given by
(5)Ω={u(·)∈L∞(0,tf)∣0⩽u(t)⩽1,∀t∈[0,tf]}.
The objective functional is given by
(6)J1(u)=∫0tf‍A1Ih(t)+C2u2(t)dt,
where the weight coefficient, C, is a measure of the relative cost of the interventions associated to the control u and A1 is the weight coefficient for the class Ih. The aim is to minimize the infected humans while keeping the cost low. More precisely, we propose the optimal control problem of determining (Sh*,Ih*,Sv*,Iv*) associated to an admissible control u*(·)∈Ω on the time interval [0,tf], satisfying (3), the initial conditions Sh(0), Ih(0), Sv(0), and Iv(0) (see Table 1) and minimizing the cost functional (6), that is,
(7)J1(u*(·))=minΩJ1(u(·)).
The existence of an optimal control u*(·) comes from the convexity of the Lagrangian of (6) with respect to the control and the regularity of the system (3) (see, e.g., [13, 14] for existence results of optimal solutions). Applying the Pontryagin maximum principle [15], we derive the optimal solution (u*,Sh*,Ih*,Sv*,Iv*) of the proposed optimal control problem (see the Appendix).More generally, one could take the following cost functional:(8)J2(u)=∫0tf‍A1Ih(t)+A2Iv(t)+C2u2(t)dt,
where A2 is the weight constant on infectious mosquitoes (for numerical simulations we considered A2=25). It turns out that when we include in the objective functional the number of infectious mosquitoes, the distribution of the total host population Nh, and vector population Nv by the categories Sh, Ih, and Sv, Iv, respectively, is the same for both cost functionals J1 and J2 (see Figures 1 and 2). On the other hand, the effort on the control is higher for the cost functional J2 (see Figure 3). Therefore, we choose to use the cost functional J1 in our numerical simulations (Section 4).Susceptible and infected individuals for different cost functionalsJ1 and J2 (parameter/constant values from Table 1 and b=0.75).
(a)
Susceptible humans forJ1 and J2
(b)
Infected humans forJ1 and J2Susceptible and infectious mosquitoes for different cost functionsJ1 and J2 (parameter/constant values from Table 1 and b=0.75).
(a)
Susceptible mosquitoes forJ1 and J2
(b)
Infectious mosquitoes forJ1 and J2Figure 3
Optimal controlu for different cost functionals J1 and J2 (parameter/constant values from Table 1 and b=0.75).


## 4. Numerical Results and Discussion

Our numerical results were obtained and confirmed following different approaches. The first approach consisted in using IPOPT [16] and the algebraic modeling language AMPL [17]. In a second approach, we used the PROPT MATLAB Optimal Control Software [18]. The results were coincident and are easily confirmed by the ones obtained using an iterative method that consists in solving the system of eight ODEs given by (3) and (A.6) in the Appendix. For that, one first solves system (3) with a guess for the control over the time interval [0,tf] using a forward fourth-order Runge-Kutta scheme and the transversality conditions (A.5) in Appendix. Then, system (A.6) is solved by a backward fourth-order Runge-Kutta scheme using the current iteration solution of (3). The control is updated by using a convex combination of the previous control and the value from (A.8) (see the Appendix). The iterative method ends when the values of the approximations at the previous iteration are close to the ones at the present iteration. For details, see [19, 20].First of all, we considerb=0.75 and show that when we apply the supervision control u, better results are obtained, that is, the number of infected humans vanishes faster when compared to the case where no control is used. If the control intervention u is applied, then the number of infected individuals vanishes after approximately 30 days. If no control is considered, then it takes approximately 70 days to assure that there are no infected humans (see Figure 4 for the fraction of susceptible and infected humans and Figure 5 for the optimal control).Susceptible and infected humans forb=0.75 with and without control.
(a)
Susceptible humans
(b)
Infected humansFigure 5
Optimal controlu for b=0.75.For smaller proportions of ITN usage thanb=0.75, similar results on the reduction of infected humans are attained when we consider the optimal supervision control u (see Figures 6 and 7). We note that the control u does not contribute significantly for the decrease of Iv (see Figure 8).Susceptible and infected humans forb=0.25;0.3;0.4;0.5;0.6;0.7;0.75.
(a)
Susceptible humans
(b)
Infected humansFigure 7
Optimal controlu for b=0.25;0.3;0.4;0.5;0.6;0.7;0.75.Susceptible and infectious mosquitoes forb=0.75 with and without control.
(a)
Susceptible mosquitoes
(b)
Infectious mosquitoes



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*Source: 658468-2013-07-30.xml*