Stability Analysis and Optimal Control of a Vector-Borne Disease with Nonlinear Incidence

Abstract

The paper considers a model for the transmission dynamics of a vector-borne disease with nonlinear incidence rate. It is proved that the global dynamics of the disease are completely determined by the basic reproduction number. In order to assess the effectiveness of disease control measures, the sensitivity analysis of the basic reproductive number 𝑅₀ and the endemic proportions with respect to epidemiological and demographic parameters are provided. From the results of the sensitivity analysis, the model is modified to assess the impact of three control measures; the preventive control to minimize vector human contacts, the treatment control to the infected human, and the insecticide control to the vector. Analytically the existence of the optimal control is established by the use of an optimal control technique and numerically it is solved by an iterative method. Numerical simulations and optimal analysis of the model show that restricted and proper use of control measures might considerably decrease the number of infected humans in a viable way.

1. Introduction

Vector-borne diseases are infectious diseases caused by viruses, bacteria, protozoa, or rickettsia which are primarily transmitted by disease transmitting biological agents, called vectors. Vector-borne diseases, in particular, mosquito-borne diseases such as malaria, dengue fever, and West Nile Virus that are transmitted to humans by blood-sucker mosquito, have been big problem for the public health in the world. The literature dealing with the mathematical theory and dynamics of vector-borne diseases are quite extensive. Many mathematical models concerning the emergence and reemergence of the vector-host infectious disease have been proposed and analyzed in the literature [1, 2].

Mathematical modeling became considerable important tool in the study of epidemiology because it helped us to understand the observed epidemiological patterns, disease control and provide understanding of the underlying mechanisms which influence the spread of disease and may suggest control strategies. The model formulation and its simulation with parameter estimation allow us to test for sensitivity and comparison of conjunctures. The foundations of the modern mathematical epidemiology based on the compartment models were laid in the early 20th century [3].

The incidence of a disease is the number of infection per unit time and plays an important role in the study of mathematical epidemiology. In classical epidemiological bilinear incidence rate 𝛽𝑆𝐼 and standard incidence rate 𝛽(𝑆/𝑁)𝐼 are frequently used, where 𝛽 is the probability of transmission per contact, 𝑆 is susceptible, and 𝐼 is infective individuals. However, actual data and evidence observed for many diseases show that dynamics of disease transmission are not always as simple as shown in these rates. There are a number of biological mechanisms which may result in nonlinearities in the transmission rates. In 1978, Capasso and Serio [4] introduced a saturated incidence rate 𝑔(𝐼)𝑆 in an epidemic models. This is important because the number of effective contacts between infective and susceptible individuals may saturate at high infective levels due to overcrowding of infective individuals or due to protective measures endorsed by susceptible individuals. A variety of nonlinear incidence rates have been used in epidemic models [5–10]. In [10], an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors, (mosquitoes), such as malaria, yellow fever, dengue and so on.

Optimal control theory is a powerful mathematical tool to make decision involving complex dynamical systems [11]. For example, what percentage of the population should be vaccinated as time evolves in a given epidemic model to minimize both the number of infected people and the cost of implementing the vaccination strategy. The desired outcome depends on the particular situation. New drug treatments and combinations of drugs are under constant development. The optimal treatment scheme for patients remains the subject of intense debate. Further, optimal control methods have been used to study the dynamics of some diseases (see [12, 13] and the references therein).

Recently, a number of mathematical models have been proposed to study the transmission dynamics of vector-borne diseases. Cai and Li [1] describes the dynamics of a vector-borne disease considering that the infection moves from person to person directly with no environmental source and intermediate vector or host. There have been applications of optimal control methods to epidemiological models, namely, Blayneh et al. [14], Okosun and Makinde [15], Lashari and Zaman [16, 17], and so forth. Lashari and Zaman [17] used personal protection, blood screening, and vector-reduction strategies as optimal control to reduce the transmission of a vector-borne disease. Kar and Batabyal [18] analyzed a nonlinear epidemic model and used optimal control technique to reduce the disease burden with a vaccination program.

In this work, we consider a vector host epidemic model with nonlinear incidence rate. Our aim is to carry out qualitative behavior and present a rigorous analysis of the resulting model to investigate the parameters to show how they affect the vector-borne disease transmission. We perform sensitivity analysis of the basic reproductive number and the endemic equilibrium with respect to epidemiological and demographic parameters. From the sensitivity analysis, we find that the reproductive number is most sensitive to the biting and mortality rates of mosquito. Further, the treatment rate of infectious humans is also a sensitive parameter for equilibrium proportion of infectious humans. These suggest us to develop strategies that target the mosquito biting rate, mosquito death rate, and treatment of infectious individuals in controlling the disease. Based on sensitivity analysis, we formulate an optimal control problem to minimize the number of infected human using three main efforts as control measures. Unfortunately, there is no vaccine nor specific treatment against vector-borne disease is available; that is why the main measures to limit the impact of such epidemic have to be considered. Therefore, we look at time-dependent prevention, treatment efforts and breeding sites destruction, for which optimal control theory is applied.

This paper is organized as follows. The model is developed in Section 2. The analysis of global stability of the equilibria of the model is investigated in Section 3. Section 4 focuses on the sensitivity analysis. Section 5 describes the extended model with three control measures and numerical simulations are presented in Section 6. Finally, conclusions are summarized in Section 7.

2. Model Formulation

The total human population, denoted by 𝑁_ℎ(𝑡), is split into susceptible individuals (𝑆_ℎ(𝑡)) and infected individuals (𝐼_ℎ(𝑡)) so that 𝑁_ℎ(𝑡)=𝑆_ℎ(𝑡)+𝐼_ℎ(𝑡). Whereas, the total vector population, denoted by 𝑁_𝑣(𝑡), is subdivided into susceptible vectors (𝑆_𝑣(𝑡)) and infectious vectors (𝐼_𝑣(𝑡)). Thus 𝑁_𝑣(𝑡)=𝑆_𝑣(𝑡)+𝐼_𝑣(𝑡).

The dynamics of the disease are described by the following system of differential equations: 𝑑𝑆_ℎ𝑑𝑡=Λ_ℎ−𝑏𝛽₁𝑆_ℎ𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝑆_ℎ𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ𝑆_ℎ+𝛾_ℎ𝐼_ℎ,𝑑𝐼_ℎ=𝑑𝑡𝑏𝛽₁𝑆_ℎ𝐼_𝑣1+𝛼₁𝐼_𝑣+𝛽₃𝑆_ℎ𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ𝐼_ℎ−𝛾_ℎ𝐼_ℎ,𝑑𝑆_𝑣𝑑𝑡=Λ_𝑣−𝑏𝛽₂𝐼_ℎ𝑆_𝑣1+𝛼₃𝐼_ℎ−𝜇_𝑣𝑆_𝑣,𝑑𝐼_𝑣=𝑑𝑡𝑏𝛽₂𝐼_ℎ𝑆_𝑣1+𝛼₃𝐼_ℎ−𝜇_𝑣𝐼_𝑣. (2.1)

Susceptible humans are recruited at a rate Λ_ℎ, whereas susceptible vectors are generated by Λ_𝑣. We assume that the number of bites per vector per host per unit time is 𝜑, the proportion of infected bites that gives rise to the infection is 𝑟, and the ratio of vector numbers to host numbers is 𝜉. Let 𝑏=𝜑𝑟𝜉, let 𝛽₁ be the transmission rate from vector to human, and let 𝛽₂ be the transmission rate from human to vector. 𝛽₃ is the transmission probability from human to human. 𝜇_ℎ is natural death rate of human, 𝜇_𝑣 is death rate of vectors, respectively. We assume that infectious individuals do not acquire permanent immunity and become susceptible again by the rate 𝛾_ℎ. Further we assume that incidence terms for human population and vector population that transmit disease are saturation interactions and are given by 𝑏𝛽₁𝑆_ℎ𝐼_𝑣/(1+𝛼₁𝐼_𝑣), 𝛽₃𝑆_ℎ𝐼_ℎ/(1+𝛼₂𝐼_ℎ), and 𝑏𝛽₂𝐼_ℎ𝑆_𝑣/(1+𝛼₃𝐼_ℎ), where 𝛼₁, 𝛼₂, and 𝛼₃ determine the level at which the force of infection saturates.

Obviously, Δ={(𝑆_ℎ,𝐼_ℎ,𝑆_𝑣,𝐼_𝑣)∈𝑅⁴∶𝑆_ℎ+𝐼_ℎ=Λ_ℎ/𝜇_ℎ,𝑆_𝑣+𝐼_𝑣=Λ_𝑣/𝜇_𝑣} is positively invariant, system (2.1) is dissipative, and the global attractor is contained in Δ.

The total dynamics of vector population are 𝑑𝑁_𝑣/𝑑𝑡=Λ_𝑣−𝜇_𝑣𝑁_𝑣. Thus we can assume without loss of generality that 𝑁_𝑣=Λ_𝑣/𝜇_𝑣 for all 𝑡≥0 provided that 𝑆_𝑣(0)+𝐼_𝑣(0)=Λ_𝑣/𝜇_𝑣. On Δ, 𝑆_𝑣=Λ_𝑣/𝜇_𝑣−𝐼_𝑣. Therefore, we attack system (2.1) by studying the subsystem 𝑑𝑆_ℎ𝑑𝑡=Λ_ℎ−𝑏𝛽₁𝑆_ℎ𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝑆_ℎ𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ𝑆_ℎ+𝛾_ℎ𝐼_ℎ,𝑑𝐼_ℎ=𝑑𝑡𝑏𝛽₁𝑆_ℎ𝐼_𝑣1+𝛼₁𝐼_𝑣+𝛽₃𝑆_ℎ𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ𝐼_ℎ−𝛾_ℎ𝐼_ℎ,𝑑𝐼_𝑣=𝑑𝑡𝑏𝛽₂𝜇_𝑣Λ_𝑣−𝜇_𝑣𝐼_𝑣𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣𝐼_𝑣. (2.2)

From biological considerations, we study system (2.2) in the closed set Ω={(𝑆_ℎ,𝐼_ℎ,𝐼_𝑣)∈𝑅³₊∶𝑆_ℎ+𝐼_ℎ=Λ_ℎ/𝜇_ℎ,𝐼_𝑣≤Λ_𝑣/𝜇_𝑣}, where 𝑅³₊ denotes the nonnegative cone of 𝑅³ including its lower dimensional faces. It can be easily verified that Ω is positively invariant with respect to (2.2).

3. Mathematical Analysis of the Model

The dynamics of the disease are described by the basic reproduction number 𝑅₀. The threshold quantity 𝑅₀ is called the reproduction number, which is defined as the average number of secondary infections produced by an infected individual in a completely susceptible population. The basic reproduction number of model (2.2) is given by the expression 𝑅₀=𝛽₃Λ_ℎ𝜇_ℎ𝜇_ℎ+𝛾_ℎ+𝑏²𝛽₁𝛽₂Λ_ℎΛ_𝑣𝜇²_𝑣𝜇_ℎ𝜇_ℎ+𝛾_ℎ. (3.1) Direct calculation shows that system (2.2) has two equilibrium states. For 𝑅₀≤1, the only equilibrium is disease-free equilibrium 𝐸₀=(Λ_ℎ/𝜇_ℎ,0,0). For 𝑅₀>1, there is an additional equilibrium 𝐸^∗(𝑆^∗_ℎ,𝐼^∗_ℎ,𝐼^∗_𝑣) which is called endemic equilibrium, where 𝑆^∗_ℎ=Λ_ℎ−𝜇_ℎ𝐼^∗_ℎ𝜇_ℎ,𝐼^∗_𝑣=𝑏𝛽₂Λ_𝑣𝐼^∗_ℎ𝜇²_𝑣+𝛼₃𝜇²_𝑣+𝑏𝛽₂𝜇_𝑣𝐼^∗_ℎ, (3.2) and 𝐼^∗_ℎ is the root of the following quadratic equation. 𝑎₁𝐼_ℎ^2∗+𝑎₂𝐼^∗_ℎ+𝑎₃=0, (3.3) with 𝑎₁=𝛼₂𝜇_ℎ𝑏²𝛽₁𝛽₂Λ_𝑣+𝛼₃𝜇²_𝑣+𝑏𝛽₂𝜇_𝑣+𝛼₁𝑏𝛽₂Λ_𝑣𝛽₃𝜇_ℎ+𝛼₂𝜇_ℎ𝜇_ℎ+𝛾_ℎ,𝑎₂=𝜇_ℎ𝑏²𝛽₁𝛽₂Λ_𝑣+𝛽₃𝜇²_𝑣+𝛼₂𝜇²_𝑣𝜇_ℎ𝜇_ℎ+𝛾_ℎ−𝑏²𝛽₁𝛽₂Λ_ℎΛ_𝑣+𝛼₃𝜇²_𝑣+𝑏𝛽₂𝜇_𝑣+𝛼₁𝑏𝛽₂Λ_𝑣𝜇_ℎ𝜇_ℎ+𝛾_ℎ−Λ_ℎ𝛽₃,𝑎₃=𝜇_ℎ𝜇_ℎ+𝛾_ℎ𝜇²_𝑣1−𝑅₀. (3.4) From (3.3), we see that 𝑅₀>1 if and only if 𝑎₃<0. Since 𝑎₁>0, (3.3) has a unique positive root in feasible region. If 𝑅₀<1, then 𝑎₃>0. Also, it can be easily seen that 𝑎₂>0 for 𝑅₀<1. Thus, by considering the shape of the graph of (3.3) (and noting that 𝑎₃>0), we have that there will be zero (positive) endemic equilibrium in this case. Therefore, we can conclude that if 𝑅₀<1, (3.3) has no positive root in the feasible region. If, 𝑅₀>1, (3.3) has a unique positive root in the feasible region. This result is summarized below.

Theorem 3.1. System (2.2) always has the infection-free equilibrium 𝐸₀. If ℛ₀>1, system (2.2) has a unique endemic equilibrium 𝐸^∗=(𝑆^∗_ℎ,𝐼^∗_ℎ,𝐼^∗_𝑣) defined by (3.2) and (3.3).

3.1. Global Stability of Disease-Free Equilibrium

In this subsection, we analyze the global behavior of the equilibria for system (2.2). The following theorem provides the global property of the disease-free equilibrium 𝐸₀ of the system.

Theorem 3.2. If 𝑅₀≤1, then the infection-free equilibrium 𝐸₀ is globally asymptotically stable in the interior of Ω.

Proof. To establish the global stability of the disease-free equilibrium, we construct the following Lyapunov function: 𝐿(𝑡)=𝐼_ℎ(𝑡)+𝑏𝛽₁Λ_ℎ𝜇_ℎ𝜇_𝑣𝐼_𝑣(𝑡). (3.5) Calculating the time derivative of 𝐿 along the solutions of system (2.2), we obtain𝐿^′(𝑡)=𝐼^′_ℎ(𝑡)+𝑏𝛽₁Λ_ℎ𝜇_ℎ𝜇_𝑣𝐼^′_𝑣(𝑡)=𝑏𝛽₁𝑆_ℎ𝐼_𝑣1+𝛼₁𝐼_𝑣+𝛽₃𝑆_ℎ𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ+𝛾_ℎ𝐼_ℎ+𝑏𝛽₁Λ_ℎ𝜇_ℎ𝜇_𝑣𝑏𝛽₂Λ_𝑣𝜇_𝑣1+𝛼₃𝐼_ℎ𝐼_ℎ−𝑏𝛽₂𝐼_𝑣𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣𝐼_𝑣≤𝑏𝛽₁Λ_ℎ𝜇_ℎ𝐼_𝑣+𝛽₃Λ_ℎ𝜇_ℎ𝐼_ℎ−𝜇_ℎ+𝛾_ℎ𝐼_ℎ+𝑏𝛽₁Λ_ℎ𝜇_ℎ𝜇_𝑣𝑏𝛽₂Λ_𝑣𝜇_𝑣𝐼_ℎ−𝑏𝛽₂𝐼_𝑣𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣𝐼_𝑣=−𝜇_ℎ+𝛾_ℎ𝐼_ℎ1−𝑅₀−𝑏²𝛽₁𝛽₂Λ_ℎ𝜇_ℎ𝜇_𝑣𝐼_𝑣𝐼_ℎ1+𝛼₃𝐼_ℎ. (3.6) Thus 𝐿^′(𝑡) is negative if 𝑅₀≤1. When 𝑅₀<1, the derivative 𝐿^′=0 if and only if 𝐼_ℎ=0, while in the case 𝑅₀=1, the derivative 𝐿^′=0 if and only if 𝐼_ℎ=0 or 𝐼_𝑣=0. Consequently, the largest compact invariant set in {(𝑆_ℎ,𝐼_ℎ,𝐼_𝑣)∈Ω,𝐿^′=0}, when 𝑅₀≤1, is the singelton 𝐸₀. Hence, LaSalle’s invariance principle [19] implies that 𝐸₀ is globally asymptotically stable in Ω. This completes the proof.

3.2. Global Stability of the Endemic Equilibrium

Here, we use the geometrical approach of Li and Muldowney to investigate the global stability of the endemic equilibrium 𝐸^∗ in the feasible region Ω. We have omitted the detailed introduction of this approach and we refer the interested readers to see [20]. We summarize this approach below.

Consider a 𝐶¹ map 𝑓∶𝑥↦𝑓(𝑥) from an open set 𝐷⊂𝑅^𝑛 to 𝑅^𝑛 such that each solution 𝑥(𝑡,𝑥₀) to the differential equation 𝑥^′=𝑓(𝑥) (3.7) is uniquely determined by the initial value 𝑥(0,𝑥₀). We have following assumptions:(𝐻₁)𝐷 is simply connected;(𝐻₂) there exists a compact absorbing set 𝐾⊂𝐷;(𝐻₃) Equation (3.7) has unique equilibrium 𝑥 in 𝐷.

Let 𝑃∶𝑥↦𝑃(𝑥) be a nonsingular (^𝑛₂)×(^𝑛₂) matrix-valued function which is 𝐶¹ in 𝐷 and a vector norm |⋅| on 𝑅^𝑁, where 𝑁=(^𝑛₂).

Let 𝜇 be the Lozinskiĭ measure with respect to the |⋅|. Define a quantity 𝑞₂ as 𝑞₂=limsup_𝑡→∞sup_𝑥0∈𝐾1𝑡^𝑡₀𝜇𝐵𝑥𝑠,𝑥₀𝑑𝑠, (3.8) where 𝐵=𝑃_𝑓𝑃⁻¹+𝑃𝐽^[2]𝑃⁻¹, the matrix 𝑃_𝑓 is obtained by replacing each entry 𝑝 of 𝑃 by its derivative in the direction of 𝑓, (𝑝_𝑖𝑗)_𝑓, and 𝐽^[2] is the second additive compound matrix of the Jacobian matrix 𝐽 of (3.7). The following result has been established in Li and Muldowney [20].

Theorem 3.3. Suppose that (𝐻₁), (𝐻₂) and (𝐻₃) hold, the unique endemic equilibrium 𝐸^∗ is globally stable in Ω if 𝑞₂<0.

Obviously Ω is simply connected and 𝐸^∗ is unique endemic equilibrium for 𝑅₀>1 in Ω. To apply the result of the above theorem for global stability of endemic equilibrium 𝐸^∗, we first state and prove the following result.

Lemma 3.4. If 𝑅₀>1, then the system (2.2) is uniformly persistent; that is, there exists 𝑐>0   (independent of initial conditions), such that liminf_𝑡→∞𝑆_ℎ≥𝑐, liminf_𝑡→∞𝐼_ℎ≥𝑐, and liminf_𝑡→∞𝐼_𝑣≥𝑐,

Proof. Let Φ be semidynamical system (2.2) in (𝑅⁺₀)³, let ϝ be a locally compact metric space, and let Γ₀={(𝑆_ℎ,𝐼_ℎ,𝐼_𝑣)∈Ω∶𝐼_𝑣=0}. Γ₀ is a compact subset of Ω and Ω/Γ₀ is positively invariant set of system (2.2). Let 𝑃∶ϝ→𝑅⁺₀ be defined by 𝑃(𝑆_ℎ,𝐼_ℎ,𝐼_𝑣)=𝐼_𝑣 and set 𝑆={(𝑆_ℎ,𝐼_ℎ,𝐼_𝑣)∈Ω∶𝑃(𝑆_ℎ,𝐼_ℎ,𝐼_𝑣)<𝜙}, where 𝜙 is sufficiently small so that 𝛽₃Λ_ℎ𝜇_ℎ𝜇_ℎ+𝛾_ℎ1+𝛼₂𝜙+𝑏²𝛽₁𝛽₂Λ_ℎΛ_𝑣1−𝜇_𝑣/Λ_𝑣𝜙𝜇²_𝑣𝜇_ℎ𝜇_ℎ+𝛾_ℎ1+𝛼₃𝜙>1. (3.9) Assume that there is a solution 𝑥∈𝑆 such that for each 𝑡>0𝑃(Φ(𝑥,𝑡))<𝑃(𝑥)<𝜙. Let us consider 𝐿(𝑡)=𝑏𝛽₁Λ_ℎ𝜇_ℎ𝜇_𝑣1−𝛿^∗𝐼_𝑣+𝐼_ℎ, (3.10) where 𝛿^∗ is sufficiently small so that 𝛽₃Λ_ℎ𝜇_ℎ𝜇_ℎ+𝛾_ℎ1+𝛼₂𝜙+𝑏²𝛽₁𝛽₂Λ_ℎΛ_𝑣1−𝜇_𝑣/Λ_𝑣𝜙1−𝛿^∗𝜇²_𝑣𝜇_ℎ𝜇_ℎ+𝛾_ℎ1+𝛼₃𝜙>1. (3.11)
By direct calculation we have 𝐿^′(𝑡)≥𝜇_ℎ+𝛾_ℎ𝛽₃Λ_ℎ𝜇_ℎ𝜇_ℎ+𝛾_ℎ1+𝛼₂𝜙+𝑏²𝛽₁𝛽₂Λ_ℎΛ_𝑣1−𝜇_𝑣/Λ_𝑣𝜙1−𝛿^∗𝜇²_𝑣𝜇_ℎ𝜇_ℎ+𝛾_ℎ1+𝛼₃𝜙−1𝐼_ℎ+𝑏𝛽₁Λ_ℎ𝜇_ℎ𝛿^∗𝐼_𝑣,𝐿^′(𝑡)≥𝛼𝐿(𝑡), (3.12) where 𝛼=min𝜇_ℎ+𝛾_ℎ𝛽₃Λ_ℎ𝜇_ℎ𝜇_ℎ+𝛾_ℎ1+𝛼₂𝜙+𝑏²𝛽₁𝛽₂Λ_ℎΛ_𝑣1−𝜇_𝑣/Λ_𝑣𝜙1−𝛿^∗𝜇²_𝑣𝜇_ℎ𝜇_ℎ+𝛾_ℎ1+𝛼₃𝜙,𝜇−1^𝑣𝛿^∗1−𝛿^∗. (3.13)
This implies that 𝐿(𝑡)→∞ as 𝑡→∞. However 𝐿(𝑡) is bounded on Ω. According to [21, Theorem  1] the proof is completed.

The boundedness of Ω and the above lemma imply that (2.2) has a compact absorbing set 𝐾⊂Ω [22]. Now we shall prove that the quantity 𝑞₂<0. We choose a suitable vector norm |⋅| in 𝑅³ and a 3×3 matrix valued function 𝑃(𝑥)=0𝐼100_ℎ𝐼_𝑣0𝐼00^ℎ𝐼_𝑣. (3.14) Obviously 𝑃 is 𝐶¹ and nonsingular in the interior of Ω. Linearizing system (2.2) about an endemic equilibrium 𝐸^∗ gives the following Jacobian matrix. 𝐽𝐸^∗=−𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ−𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²+𝛾_ℎ−𝑏𝛽₁𝑆_ℎ1+𝛼₁𝐼_𝑣²𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣+𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²−𝜇_ℎ+𝛾_ℎ𝑏𝛽₁𝑆_ℎ1+𝛼₁𝐼_𝑣²0𝑏𝛽₂𝜇_𝑣Λ_𝑣−𝜇_𝑣𝐼_𝑣1+𝛼₃𝐼_ℎ²−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣. (3.15) The second additive compound matrix of 𝐽(𝐸^∗) is given by 𝐽^[2]=𝑀₁₁𝑏𝛽₁𝑆_ℎ1+𝛼₁𝐼_𝑣²𝑏𝛽₁𝑆_ℎ1+𝛼₁𝐼_𝑣²𝑏𝛽₂𝜇_𝑣Λ_𝑣−𝜇_𝑣𝐼_𝑣1+𝛼₃𝐼_ℎ²𝑀₂₂−𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²+𝛾_ℎ0𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣+𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ𝑀₃₃, (3.16) where 𝑀₁₁=−𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ+𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²−𝜇_ℎ+𝛾_ℎ,𝑀₂₂=−𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣,𝑀₃₃=𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²−𝜇_ℎ+𝛾_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣. (3.17) The matrix 𝐵=𝑃_𝑓𝑃⁻¹+𝑃𝐽^[2]𝑃⁻¹ can be written in block form as 𝐵𝐵=¹¹𝐵₁₂𝐵₂₁𝐵₂₂, (3.18) with 𝐵₁₁=−𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ+𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²−𝜇_ℎ+𝛾_ℎ,𝐵₁₂=𝑏𝛽₁𝑆_ℎ1+𝛼₁𝐼_𝑣²𝐼_𝑣𝐼_ℎ,𝑏𝛽₁𝑆_ℎ1+𝛼₁𝐼_𝑣²𝐼_𝑣𝐼_ℎ,𝐵₂₁=𝐼_ℎ𝐼_𝑣𝑏𝛽₂𝜇_𝑣Λ_𝑣−𝜇_𝑣𝐼_𝑣1+𝛼₃𝐼_ℎ²0,𝐵₂₂=𝑄₁₁𝑄₁₂𝑄₂₁𝑄₂₂, (3.19) where 𝑄₁₁=𝐼_𝑣𝐼_ℎ𝐼_ℎ𝐼_𝑣𝑓−𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣,𝑄₁₂𝛽=−³𝑆_ℎ1+𝛼₂𝐼_ℎ²+𝛾_ℎ,𝑄₂₁=𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣+𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ,𝑄₂₂=𝐼_𝑣𝐼_ℎ𝐼_ℎ𝐼_𝑣𝑓+𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²−𝜇_ℎ+𝛾_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣,𝐼_𝑣𝐼_ℎ𝐼_ℎ𝐼_𝑣𝑓=𝐼^′_ℎ𝐼_ℎ−𝐼^′_𝑣𝐼_𝑣. (3.20) Consider the norm in 𝑅³ as |(𝑢,𝑣,𝑤)|=max(|𝑢|,|𝑣|+|𝑤|) where (𝑢,𝑣,𝑤) denotes the vector in 𝑅³. The Lozinskiĭ measure with respect to this norm is defined as 𝜇(𝐵)≤sup(𝑔₁,𝑔₂), where 𝑔₁=𝜇₁𝐵₁₁+𝐵₁₂,𝑔₂=𝜇₁𝐵₂₂+𝐵₂₁. (3.21) From system (2.2) we can write 𝐼^′_ℎ𝐼_ℎ=𝑏𝛽₁𝑆_ℎ1+𝛼₁𝐼_𝑣𝐼_𝑣𝐼_ℎ+𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ+𝛾_ℎ,𝐼^′_𝑣𝐼_𝑣=𝑏𝛽₂𝜇_𝑣Λ_𝑣−𝜇_𝑣𝐼_𝑣1+𝛼𝐼_ℎ𝐼_ℎ𝐼_𝑣−𝜇_𝑣. (3.22) Since 𝐵₁₁ is a scalar, its Lozinskiĭ measure with respect to any vector norm in 𝑅¹ will be equal to 𝐵₁₁. Thus 𝐵₁₁=−𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ+𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²−𝜇_ℎ+𝛾_ℎ,𝐵₁₂=𝑏𝛽₁𝑆_ℎ1+𝛼₁𝐼_𝑣²𝐼_𝑣𝐼_ℎ, (3.23) and 𝑔₁ will become 𝑔₁=−𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ+𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²−𝜇_ℎ+𝛾_ℎ+𝑏𝛽₁𝑆_ℎ1+𝛼₁𝐼_𝑣²𝐼_𝑣𝐼_ℎ≤−𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ+𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ+𝛾_ℎ+𝑏𝛽₁𝑆_ℎ1+𝛼₁𝐼_𝑣𝐼_𝑣𝐼_ℎ≤𝐼^′_ℎ𝐼_ℎ−𝜇_ℎ−𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ. (3.24) Also |𝐵₂₁|=(𝐼_ℎ/𝐼_𝑣)(𝑏𝛽₂/𝜇_𝑣)((Λ_𝑣−𝜇_𝑣𝐼_𝑣)/(1+𝛼₃𝐼_ℎ)²), |𝐵₁₂| and |𝐵₂₁| are the operator norms of 𝐵₁₂ and 𝐵₂₁ which are mapping from 𝑅² to 𝑅 and from 𝑅 to 𝑅², respectively, and 𝑅² is endowed with the 𝑙₁ norm. 𝜇₁(𝐵₂₂) is the Lozinskiĭ measure of 2×2 matrix 𝐵₂₂ with respect to 𝑙₁ norm in 𝑅². 𝜇𝐵₂₂𝐼=Sup^𝑣𝐼_ℎ𝐼_ℎ𝐼_𝑣𝑓−𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣+𝑏𝛽₁𝐼_𝑣1+𝛼₁𝐼_𝑣+𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ,𝐼_𝑣𝐼_ℎ𝐼_ℎ𝐼_𝑣𝑓+𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²−𝜇_ℎ+𝛾_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣−𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²+𝛾_ℎ=𝐼_𝑣𝐼_ℎ𝐼_ℎ𝐼_𝑣𝑓−𝜇_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣. (3.25) Hence 𝑔₂=𝐼^′_ℎ𝐼_ℎ−𝐼^′_𝑣𝐼_𝑣+𝐼_ℎ𝐼_𝑣𝑏𝛽₂𝜇_𝑣Λ_𝑣−𝜇_𝑣𝐼_𝑣1+𝛼₃𝐼_ℎ²−𝜇_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼𝐼_ℎ−𝜇_𝑣≤𝐼^′_ℎ𝐼_ℎ−𝐼^′_𝑣𝐼_𝑣+𝐼_ℎ𝐼_𝑣𝑏𝛽₂𝜇_𝑣Λ_𝑣−𝜇_𝑣𝐼_𝑣1+𝛼₃𝐼_ℎ−𝜇_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ−𝜇_𝑣≤𝐼^′_ℎ𝐼_ℎ−𝐼^′_𝑣𝐼_𝑣+𝐼^′_𝑣𝐼_𝑣−𝜇_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ≤𝐼^′_ℎ𝐼_ℎ−𝜇_ℎ−𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_ℎ. (3.26) Thus, 𝜇(𝐵)=Sup𝑔₁,𝑔₂≤𝐼^′_ℎ𝐼_ℎ−𝜇_ℎ. (3.27) Since (2.2) is uniformly persistent when 𝑅₀>1, so for 𝑇>0 such that 𝑡>𝑇 implies 𝐼_ℎ(𝑡)≥𝑐, 𝐼_𝑣(𝑡)≥𝑐 and (1/𝑡)log𝐼_ℎ(𝑡)<𝜇/2 for all (𝑆_ℎ(0),𝐼_ℎ(0),𝐼_𝑣(0))∈𝐾. Thus 1𝑡^𝑡₀𝜇(𝐵)𝑑𝑡<log𝐼_ℎ(𝑡)𝑡−𝜇<−𝜇2 (3.28) for all (𝑆_ℎ(0),𝐼_ℎ(0),𝐼_𝑣(0))∈𝐾, which further implies that 𝑞₂<0. Therefore all the conditions of Theorem 3.3 are satisfied. Hence unique endemic equilibrium 𝐸^∗ is globally stable in Ω.

4. Sensitivity Analysis

We would like to know different factors that are responsible for the disease transmission and prevalence. In this way we can try to reduce human mortality and morbidity due to disease. Initial disease transmission depends upon the reproductive number whereas disease prevalence is directly related to the endemic equilibrium point. The class of infectious humans is the most important class because it represents the persons who may be clinically ill and is directly related to the disease induced deaths. We will calculate the sensitivity indices of the reproductive number, 𝑅₀, and the endemic equilibrium point with respect to the parameters given in Table 1 for the model. By the analysis of these indices we could determine which parameter is more crucial for disease transmission and prevalence.

Table 1 

Values for parameters used for sensitivity analysis.

Definition 4.1. The normalized forward sensitivity index of a variable, ℎ, that depends differentiably on a parameter, 𝑙, is defined as 𝛾^ℎ_𝑙=(𝜕ℎ/𝜕𝑙)×(𝑙/ℎ).

Table 2 represents sensitivity indices of model parameters to 𝑅₀.

Table 2 

Sensitivity indices of 𝑅₀ to parameters for the model, evaluated at the parameter values given in Table 1.

By analyzing sensitivity indices we observe that the most sensitive parameters are biting rate of mosquitoes 𝑏 and death rate of mosquitoes 𝜇_𝑣. The reproductive number (𝑅₀) is directly related to the biting rate of mosquitoes and inversely related to the death rate of mosquitoes. We can say that an increase (or decrease) in biting rate 𝑏 by 10% increases (or decreases) 𝑅₀ by 20%. Similarly increase (or decrease) in death rate of mosquitoes by 10% decreases (or increases) 𝑅₀ by 20%. This suggests that strategies that can be applied in controlling the disease are to target the mosquito biting rate and death rate such as the use of insecticide-treated bed nets and indoor residual spray.

4.1. Sensitivity Indices of Endemic Equilibrium

We have numerically calculated the sensitivity indices at the parameter values given in Table 1. The most sensitive parameter for 𝐼^∗_ℎ is mosquito biting rate. Change in mosquito biting rate is directly related to change in 𝐼^∗_ℎ and inversely related to change in 𝛾_ℎ. This suggests that personal protection and human treatment strategies can lead to marvelous decrease in 𝐼^∗_ℎ. The most sensitive parameter for 𝐼^∗_𝑣 is mosquito death rate 𝜇_𝑣, followed by mosquito biting rate. We observe that 𝐼^∗_𝑣 can be reduced by personal protection, larvcide adulticide, and so forth.

The analysis of the sensitivity indices of 𝑅₀, 𝐼^∗_ℎ, and 𝐼^∗_𝑣 suggests us that three controls, personal protection, larvacide, and adulticide and treatment of infectious humans, can play an effective role to control the disease. The sensitivity indices for 𝑆^∗_ℎ, 𝐼^∗_ℎ, and 𝐼^∗_𝑣 with respect to all parameters are given in Table 3.

Table 3 

The sensitivity indices of the state variables at the endemic equilibrium, 𝑥_𝑖, to the parameters, 𝑝_𝑗, for parameter values given in Table 1.

5. Analysis of Optimal Control

In this section, model (2.1) is extended to assess the impact of some control measures, namely, prevention, treatment, and spray of insecticide against vector. In the human population, the associated force of infection is reduced by a factor of (1−𝑢₁) and the reproduction rate of the mosquito population is reduced by a factor of (1−𝑢₃). It is assumed that under the successful control efforts the mortality rate of mosquito population increases at a rate proportional to 𝑢₃, where 𝑐>0 is a rate constant. The per capita recovery rate is proportional to 𝑢₂, where 𝑟>0 is a rate constant. One has 𝑑𝑆_ℎ𝑑𝑡=Λ_ℎ−1−𝑢₁𝑏𝛽₁𝑆_ℎ𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝑆_ℎ𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ𝑆_ℎ+𝛾_ℎ𝐼_ℎ,𝑑𝐼_ℎ=𝑑𝑡1−𝑢₁𝑏𝛽₁𝑆_ℎ𝐼_𝑣1+𝛼₁𝐼_𝑣+𝛽₃𝑆_ℎ𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ+𝛾_ℎ+𝑟𝑢₂𝐼_ℎ,𝑑𝑆_𝑣𝑑𝑡=Λ_𝑣1−𝑢₃−1−𝑢₁𝑏𝛽₂𝐼_ℎ𝑆_𝑣1+𝛼₃𝐼_𝑣−𝜇_𝑣𝑆_𝑣−𝑐𝑢₃𝑆_𝑣,𝑑𝐼_𝑣=𝑑𝑡1−𝑢₁𝑏𝛽₂𝐼_ℎ𝑆_𝑣1+𝛼₃𝐼_𝑣−𝜇_𝑣𝐼_𝑣−𝑐𝑢₃𝐼_𝑣. (5.1) The control variable 𝑢₁ represents the use of drugs or vaccine which are preventive measures to minimize vector human contacts. The control function 𝑢₂ represents the treatment supplied to the infected humans. The control function 𝑢₃ represents the level of larvacide and adulticide used for vector control applied at those places at which vector breeding occurs.

To investigate the optimal level of efforts that would be needed to control the disease, we give the objective functional 𝐽, which is to minimize the number of infected human and the cost of applying the control 𝑢₁,𝑢₂,𝑢₃. One has 𝐽=^𝑡𝑓₀𝐴₁𝐼_ℎ+𝐴₂𝐼_𝑣+𝐴₃𝑆_𝑣+12𝐵₁𝑢²₁+𝐵₂𝑢²₂+𝐵₃𝑢²₃𝑑𝑡, (5.2) where 𝐴₁, 𝐴₂, and 𝐴₃ are positive weights. We choose a quadratic cost on the controls and this is similar with what is in other literature on epidemic controls [15]. With the given objective function 𝐽(𝑢₁,𝑢₂,𝑢₃), our goal is to minimize the number of infected humans, while minimizing the cost of control 𝑢₁(𝑡), 𝑢₂(𝑡), and 𝑢₃(𝑡). We seek an optimal control 𝑢^∗₁,𝑢^∗₂,𝑢^∗₃ such that 𝐽𝑢^∗₁,𝑢^∗₂,𝑢^∗₃=min𝐽𝑢₁,𝑢₂,𝑢₃∣𝑢₁,𝑢₂,𝑢₃∈𝑈, (5.3) where 𝑈={(𝑢₁,𝑢₂,𝑢₃) such that 𝑢₁,𝑢₂,𝑢₃  measurable with 0≤𝑢₁≤1,0≤𝑢₂≤1,0≤𝑢₃≤1} is the control set. The necessary conditions that an optimal must satisfy come from the Pontryagin’s Maximum Principle [24]. This principle converts (5.1) and (5.2) into a problem of minimizing pointwise a Hamiltonian 𝐻, with respect to 𝑢₁, 𝑢₂, and 𝑢₃𝐻=𝐴₁𝐼_ℎ+𝐴₂𝐼_𝑣+𝐴₃𝑆_𝑣+12𝐵₁𝑢²₁+𝐵₂𝑢²₂+𝐵₃𝑢²₃+𝜆₁Λ_ℎ−1−𝑢₁𝑏𝛽₁𝑆_ℎ𝐼_𝑣1+𝛼₁𝐼_𝑣−𝛽₃𝑆_ℎ𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ𝑆_ℎ+𝛾_ℎ𝐼_ℎ+𝜆₂1−𝑢₁𝑏𝛽₁𝑆_ℎ𝐼_𝑣1+𝛼₁𝐼_𝑣+𝛽₃𝑆_ℎ𝐼_ℎ1+𝛼₂𝐼_ℎ−𝜇_ℎ+𝛾_ℎ+𝑟𝑢₂𝐼_ℎ+𝜆₃Λ_𝑣1−𝑢₃−1−𝑢₁𝑏𝛽₂𝐼_ℎ𝑆_𝑣1+𝛼₃𝐼_𝑣−𝜇_𝑣𝑆_𝑣−𝑐𝑢₃𝑆_𝑣+𝜆₄1−𝑢₁𝑏𝛽₂𝐼_ℎ𝑆_𝑣1+𝛼₃𝐼_𝑣−𝜇_𝑣𝐼_𝑣−𝑐𝑢₃𝐼_𝑣, (5.4) where 𝜆_𝑖 for 𝑖=1,…,4 are adjoint variables. In the following we will state and prove the existence of the optimal control by using the result by Fleming and Rishel [25].

Theorem 5.1. There exists an optimal control (𝑢^∗₁,𝑢^∗₂,𝑢^∗₃) that minimize 𝐽 over 𝑈 subject to the control system (5.1). Further, for the system (5.1), there exists adjoint variables 𝜆_𝑖,𝑖=1,…,4 satisfying 𝑑𝜆₁=𝜆𝑑𝑡¹−𝜆₂𝑏𝛽₁1−𝑢₁𝐼_𝑣1+𝛼₁𝐼_𝑣+𝜆₁−𝜆₂𝛽₃𝐼_ℎ1+𝛼₂𝐼_ℎ+𝜆₁𝜇_ℎ,𝑑𝜆₂=𝜇𝑑𝑡^ℎ+𝛾_ℎ+𝑟𝑢₂𝜆₂−𝜆₁𝛾_ℎ+𝜆₁−𝜆₂𝛽₃𝑆_ℎ1+𝛼₂𝐼_ℎ²+𝜆₃−𝜆₄1−𝑢₁𝑏𝛽₂𝑆_𝑣1+𝛼₃𝐼_𝑣−𝐴₁,𝑑𝜆₃=𝜆𝑑𝑡³−𝜆₄1−𝑢₁𝑏𝛽₂𝐼_ℎ1+𝛼₃𝐼_𝑣+𝜇_𝑣+𝑐𝑢₃𝜆₃−𝐴₃,𝑑𝜆₄=𝜆𝑑𝑡¹−𝜆₂𝑏𝛽₁1−𝑢₁𝑆_ℎ1+𝛼₁𝐼_𝑣²+𝜆₄−𝜆₃1−𝑢₁𝛼𝑏𝛽₂𝐼_ℎ𝑆_𝑣1+𝛼₃𝐼_𝑣²+𝜇_𝑣+𝑐𝑢₃𝜆₄−𝐴₂, (5.5) with transversality conditions 𝜆_𝑖(𝑡_𝑓)=0,𝑖=1,…,4. The optimal controls are given by 𝑢^∗₁1=minmax0,𝐵₁𝜆₂−𝜆₁𝑏𝛽₁𝑆_ℎ𝐼_𝑣1+𝛼₁𝐼_𝑣+𝜆₄−𝜆₃𝑏𝛽₂𝐼_ℎ𝑆_𝑣1+𝛼₃𝐼_𝑣,𝑢,1_∗21=minmax0,𝐵₂𝜆₂𝑟𝐼_ℎ,𝑢,1_∗31=minmax0,𝐵₃𝜆₃Λ_𝑣+𝑐𝑆_𝑣+𝜆₄𝑐𝐼_𝑣,1. (5.6)

Proof. The integrand of the objective functional 𝐽 given by (5.2) is a convex function of 𝑢₁,𝑢₂,𝑢₃ and the state system satisfies the Lipschitz property with respect to the state variables since state solutions are bounded. The existence of an optimal control follows [25]. The equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function with respect to 𝑆_ℎ, 𝐼_ℎ, 𝑆_𝑣, and 𝐼_𝑣, respectively, evaluated at the optimal control. To get the characterization of the optimal control given by (5.6), solving the equations 𝜕𝐻𝜕𝑢₁=0,𝜕𝐻𝜕𝑢₂=0,𝜕𝐻𝜕𝑢₃=0, (5.7) on the interior of the control set and using the property of the control space 𝑈, we can derive the desired characterization (5.6).

6. Numerical Results and Discussions

Here, we investigate numerically the effect of the optimal control strategies on the spread of the disease in a population.

6.1. Use of Preventive Measures (𝑢₁≠0) and Treatment (𝑢₂≠0)

With this strategy, only the preventive control 𝑢₁ on the vector biting rate and treatment 𝑢₂ is used to optimize the objective function 𝐽, while the control 𝑢₃ on reducing the vector population is set to zero. Figure 1 shows no significant difference in the number of infected mosquitoes 𝐼_𝑣 between the case with control and the case without while there is a significant difference in the number of infected humans 𝐼_ℎ with control and the case without control. The control profile is shown in Figure 1(d); the control 𝑢₁ rise to the upper bound after 400 (days), while control 𝑢₂ rise to the upper bound after 50 (days). This shows that an effective and optimal use of preventive measure in the population without the use of larvacide against the vector will not be beneficial if total elimination of the disease is desirable in the community.

(a)

(b)

(c)

(d)

Figure 1 

Simulations of the malaria model showing the effect of control strategies on malaria transmission.

6.2. Use of Preventive Measures (𝑢₁≠0) and Larvacide Only (𝑢₃≠0)

With this strategy, we set the control 𝑢₂ to zero and use only preventive control 𝑢₁ and larvacide 𝑢₃ to optimize the objective function 𝐽. In Figures 2(a) and 2(b) show a significant difference in the infected humans 𝐼_ℎ and infected mosquitoes 𝐼_𝑣, respectively, with control compared to the situation where there is no control. More specifically, we observe a decrease in 𝐼_ℎ and 𝐼_𝑣 while an increase was observed in the uncontrolled cases. The control profile is shown in Figure 2(d), where we see that the control 𝑢₃ is at the upper bound for 𝑡=450 (days).

(a)

(b)

(c)

(d)

Figure 2 

Simulations of the malaria model showing the effect of control strategies on malaria transmission.

6.3. Use of Treatment (𝑢₂≠0) and Larvacide (𝑢₃≠0)

With this strategy, we set control 𝑢₁ to zero, while the control 𝑢₂ and the control 𝑢₃ are both used to optimize the objective function 𝐽. In Figures 3(a) and 3(b), we observed that the control strategies resulted in a decrease in the number of infected humans (𝐼_ℎ), infected mosquitoes (𝐼_𝑣), and susceptible humans (𝑆_ℎ) while an increase is observed in the number of infected humans (𝐼_ℎ) and infected mosquitoes (𝐼_𝑣) in strategy without control. The control profile is shown in Figure 3(d), where we see that the control 𝑢₃ is at the upper bound for 𝑡=450 (days) and the control 𝑢₂ is at the upper bound for 𝑡=440 days and then dropped to the lower bound at the final time.

(a)

(b)

(c)

(d)

Figure 3 

Simulations of the malaria model showing the effect of control strategies on malaria transmission.

6.4. Use of Preventive Measures (𝑢₁≠0), Treatment (𝑢₂≠0), and Larvacide (𝑢₃≠0)

With this strategy, the control 𝑢₁, control 𝑢₂, and the control 𝑢₃ are all used to optimize the objective function 𝐽. In Figures 4(a) and 4(b), we observed that the control strategies resulted in a decrease in the number of infected humans (𝐼_ℎ), infected mosquitoes, (𝐼_𝑣) and susceptible humans (𝑆_ℎ) while an increase is observed in the number of infected humans (𝐼_ℎ) and infected mosquitoes (𝐼_𝑣) in strategy without control. The control profile is shown in Figure 4(d), where we see that the control 𝑢₁ is at the upper bound for 𝑡=440 (days) before dropping gradually to the lower bound, the control 𝑢₂ is at the upper bound for 𝑡=440 (days) and then dropped to the lower bound at the final time, and the control 𝑢₃ is maintained at the upper bound for 𝑡=450 (days).

(a)

(b)

(c)

(d)

Figure 4 

Simulations of the malaria model showing the effect of control strategies on malaria transmission.

7. Conclusion

In this paper, we have studied a vector host epidemic model with saturated incidence rate. The global stability of the disease-free steady state is established by direct Lyapunov method and a geometric approach is used for the global stability of the endemic steady state. The model has a globally asymptotically stable disease-free solution whenever the basic reproduction number 𝑅₀ is less than or equal unity and has a unique positive globally asymptotically stable endemic equilibrium whenever 𝑅₀ exceeds unity. We found also from the sensitivity indices analysis that the most sensitive parameters are mosquito biting and death rates. The paper was also extended to asses the impact of some control measures. By the application of optimal control theory, we derived and analyzed the conditions for optimal control of the disease with personal protection, treatment and spray of insecticides. The optimal control has a very desirable effect for reducing the number of infected individuals and comparison between optimal control and without control is shown in figures. From our numerical results we found that an effective and optimal use of preventive measure in the population without the use of larvacide against the vector will not be beneficial if total elimination of the disease is desirable in the community. Control programs that follow these strategies can effectively reduce the spread of a vector-borne disease in the community.

Acknowledgments

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (2012-000599).