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Discrete Dynamics in Nature and Society
Volume 2016, Article ID 8013574, 27 pages
http://dx.doi.org/10.1155/2016/8013574
Research Article

Transmission Dynamics and Optimal Control of Malaria in Kenya

Department of Statistics and Computer Science, Moi University, P.O. Box 3900, Eldoret 30100, Kenya

Received 16 November 2015; Revised 5 January 2016; Accepted 6 April 2016

Academic Editor: Xiaohua Ding

Copyright © 2016 Gabriel Otieno et al. 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

This paper proposes and analyses a mathematical model for the transmission dynamics of malaria with four-time dependent control measures in Kenya: insecticide treated bed nets (ITNs), treatment, indoor residual spray (IRS), and intermittent preventive treatment of malaria in pregnancy (IPTp). We first considered constant control parameters and calculate the basic reproduction number and investigate existence and stability of equilibria as well as stability analysis. We proved that if , the disease-free equilibrium is globally asymptotically stable in . If , the unique endemic equilibrium exists and is globally asymptotically stable. The model also exhibits backward bifurcation at . If , the model admits a unique endemic equilibrium which is globally asymptotically stable in the interior of feasible region . The sensitivity results showed that the most sensitive parameters are mosquito death rate and mosquito biting rates. We then consider the time-dependent control case and use Pontryagin’s Maximum Principle to derive the necessary conditions for the optimal control of the disease using the proposed model. The existence of optimal control problem is proved. Numerical simulations of the optimal control problem using a set of reasonable parameter values suggest that the optimal control strategy for malaria control in endemic areas is the combined use of treatment and IRS; for epidemic prone areas is the use of treatment and IRS; for seasonal areas is the use of treatment; and for low risk areas is the use of ITNs and treatment. Control programs that follow these strategies can effectively reduce the spread of malaria disease in different malaria transmission settings in Kenya.

1. Introduction

Malaria is a leading cause of mortality and morbidity among the under-five group and the pregnant women in Sub-Saharan Africa [1]. These groups are at high risk due to weakened and immature immunity, respectively. With the recent conversion of the Millennium Development Goals (MDGs) to Sustainable Development Goals (SDGs) as part of Global Malaria Action Plan for a malaria-free world by 2030, reducing malaria is critical to post-2015 malaria strategies and for achieving the SDGs such as ensuring healthy lives and promoting well-being for all at all ages. Most Kenyans are vulnerable to malaria because of poverty, inadequate health care infrastructures, and low income of the country. Prompt access to effective treatment for malaria is unacceptably low in Kenya due to the socioeconomic barriers to accessing health care. These challenges call for urgent development of effective and optimal strategies for preventing and controlling spread of malaria.

Malaria transmission is highly variable across Kenya because of the different transmission intensities driven by climate and temperature. Kenya has four malaria epidemiological zones: the endemic areas, the seasonal malaria transmission, the malaria epidemic prone areas, and the low risk malaria areas [2]. Malaria is caused by Plasmodium parasites and it is transmitted from one individual to another by the bite of infected female anopheline mosquitoes [1]. Malaria is an entirely preventable and treatable disease, provided the currently recommended interventions are properly implemented. Controlling malaria transmission involves interrupting the malaria transmission for specific transmission settings and for the most at-risk groups of malaria. World Health organization (WHO) recommended malaria intervention strategies include the use of long-lasting insecticide-treated bed nets (LLINs), indoor residual spray (IRS), chemoprevention for the most vulnerable such as intermittent preventive treatment for pregnant women (IPTp), confirmation of malaria diagnostics through rapid diagnostics tests (RDTs) and microscopy for every suspected case, and timely treatment with artemisinin-based combination therapies (ACTs) [1, 3].

Mathematical modelling has become an important tool in understanding the dynamics of disease transmission and in decision making processes regarding intervention programs for disease control. Mathematical models provide a framework for understanding the transmission dynamics for malaria and can be used for the optimal allocation of different interventions against malaria [4, 5]. Optimal control is a branch of mathematics developed to find optimal ways of controlling a dynamic system [6]. Application of optimal control theory can be an important tool for estimating the efficacy of various policies and control measures and the cost of implementing them. Optimal control approaches have been successfully previously used in decision making for the infectious diseases such as tuberculosis [7, 8], HIV [9], and influenza [10]. Optimal control theory has also been applied in malaria control to assess the impact of antimalarial control measures by formulating the model as an optimal control problem. Most malaria models for analyzing effect of interventions in optimal control used the standard Susceptible-Exposed-Infectious-Recovered (SEIR) model for humans and Susceptible-Exposed-Infectious (SEI) model for mosquitoes. Okosun et al. [11] considered three control variables in assessing the optimal control and cost effectiveness of the interventions but not for different settings. Mwamtobe et al. [12] used three control variables and for only one region in Malawi. Kim et al. [13] used two control efforts in the optimal control model for malaria transmission in South Korea. Agusto et al. [14] used three system control variables. Silva and Torres [15] used one control variable.

IPTp is one of the WHO recommended prevention therapies for the pregnant women. IPTp has been shown to be effective in reducing maternal and infant mortality that are related to malaria for the most at-risk group for malaria [1619]. Very few studies have been carried out in applying optimal control theory to malaria transmission models for different transmission settings. The combined effect of ITNs, IRS, and natural death on reducing the mosquito population has not been demonstrated in optimal control theory in malaria control. The effect of IPTp which is WHO recommended preventive therapy for the most at-risk group for malaria (pregnant women) has not been studied in optimal control theory. No optimal control model for malaria interventions for different transmission settings exits for Kenya. No optimal control model for four control variables incorporating the IPTp malaria intervention studies exits for Kenya. No optimal control model has been stratified by the age group (under five) and specific categories (pregnant women).

In this paper, a model for malaria transmission dynamics with four control strategies is formulated and analyzed. We then formulate an optimal control problem and derive expressions for the optimal control for the malaria transmission model with four control variables and then use the optimal control theory to study the effectiveness of all possible combinations of four malaria preventive measures among the pregnant women and children under five years of age.

2. Model Formulation

The model is formulated by considering the human and mosquito subgroups. The considered model consists of population of susceptible , Exposed humans , infected humans , recovered humans , susceptible mosquitoes , exposed mosquitoes, and infected mosquitoes . The total population sizes at time for humans and mosquitoes are denoted by and , respectively. We employ the SEIRS type model for humans to describe a disease with temporary immunity on recovery from infection. Mosquitoes are assumed not to recover from the parasites so the mosquito population can be described by the SEI model. In the model we incorporate four time-dependent control measures simultaneously: (i) the use of treated bed nets , (ii) treatment of infective humans , (iii) spray of insecticides , and (iv) treatment to protect pregnant women and their newborn children: intermittent preventive treatment for pregnant women (IPTp) . represents the number of individuals not yet infected with the malaria parasite at time , represents individuals who are infected but not yet infectious, is the class representing those who are infected with malaria and are capable of transmitting the disease to susceptible mosquitoes, and represents the class of individuals who have temporarily recovered from the disease.

Figure 1 describes the dynamics of malaria in human and mosquito populations together with interventions.

Figure 1: Malaria model with interventions.

The susceptible humans (pregnant women and children under the age of five) () are recruited at the rate . They either die from natural causes (at a rate ) or move to the exposed class () by acquiring malaria through contact with infectious mosquitoes at a rate or , where is the transmission probability per bite, is the per capita biting rate of mosquitoes, is the contact rate of vector per human per unit time, is the preventive measure using ITNs, is the preventive measure using IPTp, is the infectious mosquitoes at time , is the total number of individuals (pregnant women and children under the age of five), and is the total number of pregnant women. Susceptible class is divided into whole population (children under the age of five and pregnant women) being exposed and the population for the pregnant women being exposed. Exposed individuals move to the infectious class after the development of clinical symptoms at the rate . Infectious individuals are assumed to recover at a rate , where is the rate of spontaneous recovery, is the control on treatment of infected individuals, and is the efficacy of treatment. Infectious individuals who do not recover die at a rate . Individuals infected with malaria suffer a disease induced death at rate of and natural death . Infected individuals then progress to partially immune group where upon recovery the partially immune individual loses immunity at the rate and becomes susceptible again.

Susceptible mosquitoes () are recruited at the rate and acquire malaria infection (following contact with humans infected with malaria) at the rate . They either die from natural causes (at a rate ) or move to the exposed class by acquiring malaria through contacts with infected humans at a rate , where is the probability for a vector to get infected after biting an infectious human and are individuals infected with malaria at time . The mosquito population is reduced, due to the use of insecticides spray, at a rate , where represents the control due to IRS and represents the efficacy of IRS. Mosquito population is also reduced as a result of natural death () and at the rate , where represents the control due to ITNs and is the efficacy due to ITNs. Newly infected mosquitoes are moved into the exposed class () at a rate and progresses to the class of symptomatic mosquitoes ().

The state variables of the model are represented and described in Table 1. Table 2 describes and shows parameters of the model. Table 3 describes and represents malaria prevention and control strategies practiced in Kenya.

Table 1: State variables of the malaria model.
Table 2: Description of parameter variables of the malaria model.
Table 3: Prevention and control variables in the model.

The following assumptions have been used in the formulation of the model:(i)Population for human and mosquito being constant (no immigrants).(ii)No recovery for infected mosquitoes.(iii)Mosquitoes not dying due to disease infection.(iv)All parameters in the model being nonnegative.Putting the above formulations and assumptions together gives the following system of nonlinear differential equations describing the dynamics of malaria in human and mosquito populations together with interventions:With initial conditions is the per capita incidence rate among mosquitoes (force of infection for susceptible mosquitoes), is the force of infection for susceptible humans (pregnant and under 5), and is the force of infection for susceptible pregnant humans.

The total population size for the human is and for mosquito is and their differential equations are given by and , respectively.

2.1. Mathematical Analysis of the Malaria Model with Intervention Strategies

We will assume that the control parameters are constant so as to determine the basic reproduction number, steady states, and their stability as well as the bifurcation analysis.

We describe the basic properties and analysis of the formulated malaria model with control strategies through mathematical analysis of the formulated model.

2.1.1. Basic Properties of the Model: Positivity and Invariant Regions

All the state variables and parameters for model (1) are nonnegative for all .

The feasible solutions set for model (1) given by is positively invariant and hence model (1) is biologically, epidemiologically meaningful and mathematically well posed in the domain .

System (1) has always a disease-free equilibrium given by

2.1.2. Basic Reproduction Number

The matrices and for the new infection terms and the remaining transfer terms at disease-free equilibrium [26], respectively, are given byIt follows that the basic reproduction of model (1), denoted by , is given by

2.1.3. Stability Analysis of Disease-Free Equilibrium Point

(1) Local Stability of Disease-Free Equilibrium Point

Theorem 1. The disease-free equilibrium point for system (1) is locally asymptotically stable if .

Proof. The Jacobian matrix of the malaria model (1) at the disease-free equilibrium point is given byThe eigenvalues of the Jacobian matrix are the solutions of the characteristic equationExpanding the determinant into a characteristic equation we haveHence we havewhereThus, applying the Routh-Hurwitz criteria [27] to polynomial (10), we have that all determinants of the Hurwitz matrices are positive. Hence all the eigenvalues of the Jacobian have negative real part, implying that the DFE point is (at least) locally asymptotically stable .

Next, we study the global behavior of the disease-free equilibrium for system (1).

(2) Global Stability of Disease-Free Equilibrium Point

Theorem 2. The DFE, , of system of (1) is globally asymptotically stable if .

Proof. We consider the following Lyapunov function:whereComputing the derivative of along the solution of the system of differential equation (1),Thus we have established that if and the equality holds if and only if and . If then when and are sufficiently close to and , respectively, except when
On the boundary when , and and , as .
Therefore the largest compact invariant when is the singleton in . By LaSalle’s invariant principle [28], is globally asymptotically stable for in .

Next, we investigate the endemic equilibrium and its stability of system (1).

2.1.4. Stability Analysis of Endemic Equilibrium Point

First we determine the existence of the endemic equilibrium points.

The endemic equilibrium () of the model is given byTo derive , we solve model (1) by equating it to zero.

Substituting and solving for (as an expression of parameters only) through some algebraic manipulation giveorwhereWe use the quadratic formula to find the roots of (17); that is,HenceFrom the quadratic equation (17) we analyze the possibility of multiple endemic equilibria. It is important to note that the coefficient is always positive with and having different signs. All the negative terms in are in the form of where is the control that is bounded by 1.

It follows that(i)there is a unique endemic equilibrium if and or ,(ii)there is a unique endemic equilibrium if (i.e., if ),(iii)there are two endemic equilibria if , , and ,(iv)there are no endemic equilibria otherwise.Note that the hypothesis is equivalent to .

Thus the results of this section can be summarized in the following theorem.

Theorem 3. If , is an equilibrium of system (1) and it is locally asymptotically stable. Furthermore, there exists an endemic equilibrium if conditions in (i) are satisfied or two endemic equilibria if conditions in (iii) are satisfied. If , then is unstable and there exists a unique endemic equilibrium.

Item (iii) indicates the possibility of backward bifurcation in model (1) when . In the next section we will prove the occurrence of multiple equilibria for comes from the backward bifurcation and this will give information on the local stability of the endemic equilibrium.

(1) Local Stability Analysis of Endemic Equilibrium Point. We use centre manifold theory [29] to investigate the stability of the endemic equilibria for model (1) where we carry out bifurcation analysis of system (1) at . We consider a transmission rate as bifurcation parameter so that .

To apply the theory, we introduce dimensionless state variables into system (1).

The system of (1) can be written as We let , , , , , , and

Therefore system (1) in vector form can be written aswhere and are transposed matrices and with , Let be the bifurcation parameter; the system is linearized at disease-free equilibrium point when with . That is, The Jacobian matrix of (1) calculated at is given by Centre manifold approach [29] is then applied.

A right eigenvector associated with the eigenvalue zero is . Solving the system we have the following right eigenvector:The left eigenvectors satisfying are . Solving the system, the left eigenvector will be given byComputing for the sign of and as indicated in the theorem givesso that is always positive.

Therefore the following result is established.

Theorem 4. Model (1) exhibits backward bifurcation at whenever and and . Whenever and , then model (1) exhibits a forward bifurcation at .

Finally, we will investigate the global stability of the endemic equilibrium in the feasible region.

(2) Global Stability Analysis of Endemic Equilibrium Point. Global stability results for the endemic equilibrium can be obtained when it is unique and whenever it exists. We have established in Theorem 4 that if this implies the existence and uniqueness of the endemic equilibrium.

The global behavior of the endemic equilibrium of model (1) when it exists is explored by proving that such an equilibrium is globally asymptotic stable in the interior of the feasible region . We will use the geometric approach to global stability as described by Li and Muldowney [30]. The following conditions are required for the global stability of the endemic equilibrium, : (i) the uniqueness of in the interior of (condition ); (ii) the existence of an absorbing compact set in the interior of (condition ); and (iii) the fulfillment of a Bendixson criterion (i.e., inequality (A.6)).

Theorem 5. If and and , then the endemic equilibrium of the malaria model (1) is globally asymptotically stable in the interior of .

Proof. Following Li and Muldowney [30], for system (1), under the assumption of , satisfies conditions , the existence of the endemic equilibrium has also been shown, and the instability of DFE implies the uniform persistence [31]; that is, there exists a constant such that any solutions with in the interior of satisfyThe uniform persistence together with boundedness of is equivalent to the existence of a compact set in the interior of which is absorbing for (4) as described by Hutson and Schmitt [32]. Thus, is verified. Moreover, is the only equilibrium in the interior of , so that is also verified.
What remains is to find conditions for which the Bendixson criterion given by (A.6) is verified. To this aim, let us begin by observing that, from the Jacobian matrix associated with a general solution of reduced system (1), we get the second additive compound matrix :where where
Choose now matrix . Then , and the matrix can be written in block form aswhereThe vector norm in is here chosen to be Let denote the Lozinskii measure with respect to this norm. Using the method of estimating in [29], we havewhere and are matrix norms with respect to the vector norm and denotes the Lozinskii measure with respect to norm. Since is a scalar, its Lozinskii measure with respect to any norm in is equal to .
ThereforeThereforeWe rewrite the last two equations of system (1) for and as Substituting (39) into (37) and (40) into (38) we haveFor the uniform persistence constant , there exists a time independent of , the compact absorbing set, such that