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

Authors: Muhammad Ozair; Abid Ali Lashari; Il Hyo Jung; Kazeem Oare Okosun
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## 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 numberR0 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.


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## Body

## 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 rateβSI 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 Section2. 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 byNh(t), is split into susceptible individuals (Sh(t)) and infected individuals (Ih(t)) so that Nh(t)=Sh(t)+Ih(t). Whereas, the total vector population, denoted by Nv(t), is subdivided into susceptible vectors (Sv(t)) and infectious vectors (Iv(t)). Thus Nv(t)=Sv(t)+Iv(t).The dynamics of the disease are described by the following system of differential equations:(2.1)dShdt=Λh-bβ1ShIv1+α1Iv-β3ShIh1+α2Ih-μhSh+γhIh,dIhdt=bβ1ShIv1+α1Iv+β3ShIh1+α2Ih-μhIh-γhIh,dSvdt=Λv-bβ2IhSv1+α3Ih-μvSv,dIvdt=bβ2IhSv1+α3Ih-μvIv.Susceptible humans are recruited at a rateΛh, whereas susceptible vectors are generated by Λv. 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. μh is natural death rate of human, μv is death rate of vectors, respectively. We assume that infectious individuals do not acquire permanent immunity and become susceptible again by the rate γh. Further we assume that incidence terms for human population and vector population that transmit disease are saturation interactions and are given by bβ1ShIv/(1+α1Iv), β3ShIh/(1+α2Ih), and bβ2IhSv/(1+α3Ih), where α1, α2, and α3 determine the level at which the force of infection saturates.Obviously,Δ={(Sh,Ih,Sv,Iv)∈R4:Sh+Ih=Λh/μh,Sv+Iv=Λv/μv} is positively invariant, system (2.1) is dissipative, and the global attractor is contained in Δ.The total dynamics of vector population aredNv/dt=Λv-μvNv. Thus we can assume without loss of generality that Nv=Λv/μv for all t≥0 provided that Sv(0)+Iv(0)=Λv/μv. On Δ, Sv=Λv/μv-Iv. Therefore, we attack system (2.1) by studying the subsystem
(2.2)dShdt=Λh-bβ1ShIv1+α1Iv-β3ShIh1+α2Ih-μhSh+γhIh,dIhdt=bβ1ShIv1+α1Iv+β3ShIh1+α2Ih-μhIh-γhIh,dIvdt=bβ2μv(Λv-μvIv)Ih1+α3Ih-μvIv.From biological considerations, we study system (2.2) in the closed set Ω={(Sh,Ih,Iv)∈R+3:Sh+Ih=Λh/μh,Iv≤Λv/μv}, where R+3 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 numberR0. 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
(3.1)R0=β3Λhμh(μh+γh)+b2β1β2ΛhΛvμv2μh(μh+γh).
Direct calculation shows that system (2.2) has two equilibrium states. For R0≤1, the only equilibrium is disease-free equilibrium E0=(Λh/μh,0,0). For R0>1, there is an additional equilibrium E*(Sh*,Ih*,Iv*) which is called endemic equilibrium, where
(3.2)Sh*=Λh-μhIh*μh,Iv*=bβ2ΛvIh*μv2+(α3μv2+bβ2μv)Ih*,
and Ih* is the root of the following quadratic equation. 						(3.3)a1Ih2*+a2Ih*+a3=0,
with
(3.4)a1=α2μhb2β1β2Λv+(α3μv2+bβ2μv+α1bβ2Λv)[β3μh+α2μh(μh+γh)],a2=μh(b2β1β2Λv+β3μv2)+α2(μv2μh(μh+γh)-b2β1β2ΛhΛv)+(α3μv2+bβ2μv+α1bβ2Λv)(μh(μh+γh)-Λhβ3),a3=μh(μh+γh)μv2(1-R0).
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*=(Sh*,Ih*,Iv*) 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.
IfR0≤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:(3.5)L(t)=Ih(t)+bβ1ΛhμhμvIv(t).
Calculating the time derivative of L along the solutions of system (2.2), we obtain(3.6)L′(t)=Ih′(t)+bβ1ΛhμhμvIv′(t)=bβ1ShIv1+α1Iv+β3ShIh1+α2Ih-(μh+γh)Ih+bβ1Λhμhμv{bβ2Λvμv(1+α3Ih)Ih-bβ2IvIh1+α3Ih-μvIv}≤bβ1ΛhμhIv+β3ΛhμhIh-(μh+γh)Ih+bβ1Λhμhμv{bβ2ΛvμvIh-bβ2IvIh1+α3Ih-μvIv}=-(μh+γh)Ih(1-R0)-b2β1β2ΛhμhμvIvIh1+α3Ih.
Thus L′(t) is negative if R0≤1. When R0<1, the derivative L′=0 if and only if Ih=0, while in the case R0=1, the derivative L′=0 if and only if Ih=0 or Iv=0. Consequently, the largest compact invariant set in {(Sh,Ih,Iv)∈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 equilibriumE* 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 aC1 map f:x↦f(x) from an open set D⊂Rn to Rn such that each solution x(t,x0) to the differential equation
(3.7)x′=f(x)
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 setK⊂D;(H3)
Equation (3.7) has unique equilibrium x- in D.LetP:x↦P(x) be a nonsingular (n2)×(n2) matrix-valued function which is C1 in D and a vector norm |·| on RN, where N=(n2).Letμ be the Lozinskiĭ measure with respect to the |·|. Define a quantity q-2 as
(3.8)q-2=limsupt→∞supx0∈K1t∫0t‍μ(B(x(s,x0)))ds,
where B=PfP-1+PJ[2]P-1, the matrix Pf is obtained by replacing each entry p of P by its derivative in the direction of f, (pij)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 q-2<0.ObviouslyΩ 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.
IfR0>1, then the system (2.2) is uniformly persistent; that is, there exists c>0   (independent of initial conditions), such that liminft→∞Sh≥c, liminft→∞Ih≥c, and liminft→∞Iv≥c,Proof.
LetΦ be semidynamical system (2.2) in (R0+)3, let Ϝ be a locally compact metric space, and let Γ0={(Sh,Ih,Iv)∈Ω:Iv=0}. Γ0 is a compact subset of Ω and Ω/Γ0 is positively invariant set of system (2.2). Let P:Ϝ→R0+ be defined by P(Sh,Ih,Iv)=Iv and set S={(Sh,Ih,Iv)∈Ω:P(Sh,Ih,Iv)<ϕ}, where ϕ is sufficiently small so that
(3.9)β3Λhμh(μh+γh)(1+α2ϕ)+b2β1β2ΛhΛv(1-(μv/Λv)ϕ)μv2μh(μh+γh)(1+α3ϕ)>1.
Assume that there is a solution x∈S such that for each t>0P(Φ(x,t))<P(x)<ϕ. Let us consider
(3.10)L(t)=bβ1Λhμhμv(1-δ*)Iv+Ih,
where δ* is sufficiently small so that
(3.11)β3Λhμh(μh+γh)(1+α2ϕ)+b2β1β2ΛhΛv(1-(μv/Λv)ϕ)(1-δ*)μv2μh(μh+γh)(1+α3ϕ)>1.
By direct calculation we have(3.12)L′(t)≥(μh+γh)(β3Λhμh(μh+γh)(1+α2ϕ)+b2β1β2ΛhΛv(1-(μv/Λv)ϕ)(1-δ*)μv2μh(μh+γh)(1+α3ϕ)-1)Ih+bβ1Λhμhδ*Iv,L′(t)≥αL(t),
where
(3.13)α=min{(μh+γh)(β3Λhμh(μh+γh)(1+α2ϕ)+b2β1β2ΛhΛv(1-(μv/Λv)ϕ)(1-δ*)μv2μh(μh+γh)(1+α3ϕ)-1),μvδ*1-δ*}.
This implies thatL(t)→∞ as t→∞. However L(t) is bounded on Ω. According to [21, Theorem  1] the proof is completed.The boundedness ofΩ and the above lemma imply that (2.2) has a compact absorbing set K⊂Ω [22]. Now we shall prove that the quantity q-2<0. We choose a suitable vector norm |·| in R3 and a 3×3 matrix valued function
(3.14)P(x)=(1000IhIv000IhIv).
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. 								(3.15)J(E*)=(-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh-β3Sh(1+α2Ih)2+γh-bβ1Sh(1+α1Iv)2bβ1Iv1+α1Iv+β3Ih1+α2Ihβ3Sh(1+α2Ih)2-(μh+γh)bβ1Sh(1+α1Iv)20bβ2μvΛv-μvIv(1+α3Ih)2-bβ2Ih(1+α3Ih)-μv).
The second additive compound matrix of J(E*) is given by
(3.16)J[2]=(M11bβ1Sh(1+α1Iv)2bβ1Sh(1+α1Iv)2bβ2μvΛv-μvIv(1+α3Ih)2M22-β3Sh(1+α2Ih)2+γh0bβ1Iv1+α1Iv+β3Ih1+α2IhM33),
where
(3.17)M11=-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh+β3Sh(1+α2Ih)2-(μh+γh),M22=-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh-bβ2Ih(1+α3Ih)-μv,M33=β3Sh(1+α2Ih)2-(μh+γh)-bβ2Ih(1+α3Ih)-μv.
The matrix B=PfP-1+PJ[2]P-1 can be written in block form as
(3.18)B=(B11B12B21B22),
with
(3.19)B11=-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh+β3Sh(1+α2Ih)2-(μh+γh),B12=(bβ1Sh(1+α1Iv)2IvIh,bβ1Sh(1+α1Iv)2IvIh),B21=((IhIv)bβ2μvΛv-μvIv(1+α3Ih)20),B22=(Q11Q12Q21Q22),
where
(3.20)Q11=IvIh(IhIv)f-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh-bβ2Ih(1+α3Ih)-μv,Q12=-β3Sh(1+α2Ih)2+γh,Q21=bβ1Iv1+α1Iv+β3Ih1+α2Ih,Q22=IvIh(IhIv)f+β3Sh(1+α2Ih)2-(μh+γh)-bβ2Ih(1+α3Ih)-μv,IvIh(IhIv)f=Ih′Ih-Iv′Iv.
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
(3.21)g1=μ1(B11)+|B12|,g2=μ1(B22)+|B21|.
From system (2.2) we can write
(3.22)Ih′Ih=bβ1Sh1+α1IvIvIh+β3Sh1+α2Ih-(μh+γh),Iv′Iv=bβ2μv(Λv-μvIv)1+αIhIhIv-μv.
Since B11 is a scalar, its Lozinskiĭ measure with respect to any vector norm in R1 will be equal to B11. Thus
(3.23)B11=-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh+β3Sh(1+α2Ih)2-(μh+γh),|B12|=bβ1Sh(1+α1Iv)2IvIh,
and g1 will become
(3.24)g1=-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh+β3Sh(1+α2Ih)2-(μh+γh)+bβ1Sh(1+α1Iv)2IvIh≤-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh+β3Sh(1+α2Ih)-(μh+γh)+bβ1Sh(1+α1Iv)IvIh≤Ih′Ih-μh-bβ1Iv1+α1Iv-β3Ih1+α2Ih.
Also |B21|=(Ih/Iv)(bβ2/μv)((Λv-μvIv)/(1+α3Ih)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. 								(3.25)μ(B22)=Sup{IvIh(IhIv)f-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh-bβ2Ih(1+α3Ih)-μv+bβ1Iv1+α1Iv+β3Ih1+α2Ih,IvIh(IhIv)f+β3Sh(1+α2Ih)2-(μh+γh)-bβ2Ih(1+α3Ih)-μv-β3Sh(1+α2Ih)2+γh}=IvIh(IhIv)f-μh-bβ2Ih(1+α3Ih)-μv.
Hence
(3.26)g2=Ih′Ih-Iv′Iv+(IhIv)bβ2μvΛv-μvIv(1+α3Ih)2-μh-bβ2Ih(1+αIh)-μv≤Ih′Ih-Iv′Iv+(IhIv)bβ2μvΛv-μvIv(1+α3Ih)-μh-bβ2Ih(1+α3Ih)-μv≤Ih′Ih-Iv′Iv+Iv′Iv-μh-bβ2Ih(1+α3Ih)≤Ih′Ih-μh-bβ2Ih(1+α3Ih).
Thus,
(3.27)μ(B)=Sup{g1,g2}≤Ih′Ih-μh.
Since (2.2) is uniformly persistent when R0>1, so for T>0 such that t>T implies Ih(t)≥c, Iv(t)≥c and (1/t)logIh(t)<μ/2 for all (Sh(0),Ih(0),Iv(0))∈K. Thus
(3.28)1t∫0t‍μ(B)dt<logIh(t)t-μ<-μ2
for all (Sh(0),Ih(0),Iv(0))∈K, which further implies that q-2<0. Therefore all the conditions of Theorem 3.3 are satisfied. Hence unique endemic equilibrium E* is globally stable in Ω.


## 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.
IfR0≤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:(3.5)L(t)=Ih(t)+bβ1ΛhμhμvIv(t).
Calculating the time derivative of L along the solutions of system (2.2), we obtain(3.6)L′(t)=Ih′(t)+bβ1ΛhμhμvIv′(t)=bβ1ShIv1+α1Iv+β3ShIh1+α2Ih-(μh+γh)Ih+bβ1Λhμhμv{bβ2Λvμv(1+α3Ih)Ih-bβ2IvIh1+α3Ih-μvIv}≤bβ1ΛhμhIv+β3ΛhμhIh-(μh+γh)Ih+bβ1Λhμhμv{bβ2ΛvμvIh-bβ2IvIh1+α3Ih-μvIv}=-(μh+γh)Ih(1-R0)-b2β1β2ΛhμhμvIvIh1+α3Ih.
Thus L′(t) is negative if R0≤1. When R0<1, the derivative L′=0 if and only if Ih=0, while in the case R0=1, the derivative L′=0 if and only if Ih=0 or Iv=0. Consequently, the largest compact invariant set in {(Sh,Ih,Iv)∈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 equilibriumE* 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 aC1 map f:x↦f(x) from an open set D⊂Rn to Rn such that each solution x(t,x0) to the differential equation
(3.7)x′=f(x)
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 setK⊂D;(H3)
Equation (3.7) has unique equilibrium x- in D.LetP:x↦P(x) be a nonsingular (n2)×(n2) matrix-valued function which is C1 in D and a vector norm |·| on RN, where N=(n2).Letμ be the Lozinskiĭ measure with respect to the |·|. Define a quantity q-2 as
(3.8)q-2=limsupt→∞supx0∈K1t∫0t‍μ(B(x(s,x0)))ds,
where B=PfP-1+PJ[2]P-1, the matrix Pf is obtained by replacing each entry p of P by its derivative in the direction of f, (pij)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 q-2<0.ObviouslyΩ 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.
IfR0>1, then the system (2.2) is uniformly persistent; that is, there exists c>0   (independent of initial conditions), such that liminft→∞Sh≥c, liminft→∞Ih≥c, and liminft→∞Iv≥c,Proof.
LetΦ be semidynamical system (2.2) in (R0+)3, let Ϝ be a locally compact metric space, and let Γ0={(Sh,Ih,Iv)∈Ω:Iv=0}. Γ0 is a compact subset of Ω and Ω/Γ0 is positively invariant set of system (2.2). Let P:Ϝ→R0+ be defined by P(Sh,Ih,Iv)=Iv and set S={(Sh,Ih,Iv)∈Ω:P(Sh,Ih,Iv)<ϕ}, where ϕ is sufficiently small so that
(3.9)β3Λhμh(μh+γh)(1+α2ϕ)+b2β1β2ΛhΛv(1-(μv/Λv)ϕ)μv2μh(μh+γh)(1+α3ϕ)>1.
Assume that there is a solution x∈S such that for each t>0P(Φ(x,t))<P(x)<ϕ. Let us consider
(3.10)L(t)=bβ1Λhμhμv(1-δ*)Iv+Ih,
where δ* is sufficiently small so that
(3.11)β3Λhμh(μh+γh)(1+α2ϕ)+b2β1β2ΛhΛv(1-(μv/Λv)ϕ)(1-δ*)μv2μh(μh+γh)(1+α3ϕ)>1.
By direct calculation we have(3.12)L′(t)≥(μh+γh)(β3Λhμh(μh+γh)(1+α2ϕ)+b2β1β2ΛhΛv(1-(μv/Λv)ϕ)(1-δ*)μv2μh(μh+γh)(1+α3ϕ)-1)Ih+bβ1Λhμhδ*Iv,L′(t)≥αL(t),
where
(3.13)α=min{(μh+γh)(β3Λhμh(μh+γh)(1+α2ϕ)+b2β1β2ΛhΛv(1-(μv/Λv)ϕ)(1-δ*)μv2μh(μh+γh)(1+α3ϕ)-1),μvδ*1-δ*}.
This implies thatL(t)→∞ as t→∞. However L(t) is bounded on Ω. According to [21, Theorem  1] the proof is completed.The boundedness ofΩ and the above lemma imply that (2.2) has a compact absorbing set K⊂Ω [22]. Now we shall prove that the quantity q-2<0. We choose a suitable vector norm |·| in R3 and a 3×3 matrix valued function
(3.14)P(x)=(1000IhIv000IhIv).
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. 								(3.15)J(E*)=(-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh-β3Sh(1+α2Ih)2+γh-bβ1Sh(1+α1Iv)2bβ1Iv1+α1Iv+β3Ih1+α2Ihβ3Sh(1+α2Ih)2-(μh+γh)bβ1Sh(1+α1Iv)20bβ2μvΛv-μvIv(1+α3Ih)2-bβ2Ih(1+α3Ih)-μv).
The second additive compound matrix of J(E*) is given by
(3.16)J[2]=(M11bβ1Sh(1+α1Iv)2bβ1Sh(1+α1Iv)2bβ2μvΛv-μvIv(1+α3Ih)2M22-β3Sh(1+α2Ih)2+γh0bβ1Iv1+α1Iv+β3Ih1+α2IhM33),
where
(3.17)M11=-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh+β3Sh(1+α2Ih)2-(μh+γh),M22=-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh-bβ2Ih(1+α3Ih)-μv,M33=β3Sh(1+α2Ih)2-(μh+γh)-bβ2Ih(1+α3Ih)-μv.
The matrix B=PfP-1+PJ[2]P-1 can be written in block form as
(3.18)B=(B11B12B21B22),
with
(3.19)B11=-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh+β3Sh(1+α2Ih)2-(μh+γh),B12=(bβ1Sh(1+α1Iv)2IvIh,bβ1Sh(1+α1Iv)2IvIh),B21=((IhIv)bβ2μvΛv-μvIv(1+α3Ih)20),B22=(Q11Q12Q21Q22),
where
(3.20)Q11=IvIh(IhIv)f-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh-bβ2Ih(1+α3Ih)-μv,Q12=-β3Sh(1+α2Ih)2+γh,Q21=bβ1Iv1+α1Iv+β3Ih1+α2Ih,Q22=IvIh(IhIv)f+β3Sh(1+α2Ih)2-(μh+γh)-bβ2Ih(1+α3Ih)-μv,IvIh(IhIv)f=Ih′Ih-Iv′Iv.
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
(3.21)g1=μ1(B11)+|B12|,g2=μ1(B22)+|B21|.
From system (2.2) we can write
(3.22)Ih′Ih=bβ1Sh1+α1IvIvIh+β3Sh1+α2Ih-(μh+γh),Iv′Iv=bβ2μv(Λv-μvIv)1+αIhIhIv-μv.
Since B11 is a scalar, its Lozinskiĭ measure with respect to any vector norm in R1 will be equal to B11. Thus
(3.23)B11=-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh+β3Sh(1+α2Ih)2-(μh+γh),|B12|=bβ1Sh(1+α1Iv)2IvIh,
and g1 will become
(3.24)g1=-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh+β3Sh(1+α2Ih)2-(μh+γh)+bβ1Sh(1+α1Iv)2IvIh≤-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh+β3Sh(1+α2Ih)-(μh+γh)+bβ1Sh(1+α1Iv)IvIh≤Ih′Ih-μh-bβ1Iv1+α1Iv-β3Ih1+α2Ih.
Also |B21|=(Ih/Iv)(bβ2/μv)((Λv-μvIv)/(1+α3Ih)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. 								(3.25)μ(B22)=Sup{IvIh(IhIv)f-bβ1Iv1+α1Iv-β3Ih1+α2Ih-μh-bβ2Ih(1+α3Ih)-μv+bβ1Iv1+α1Iv+β3Ih1+α2Ih,IvIh(IhIv)f+β3Sh(1+α2Ih)2-(μh+γh)-bβ2Ih(1+α3Ih)-μv-β3Sh(1+α2Ih)2+γh}=IvIh(IhIv)f-μh-bβ2Ih(1+α3Ih)-μv.
Hence
(3.26)g2=Ih′Ih-Iv′Iv+(IhIv)bβ2μvΛv-μvIv(1+α3Ih)2-μh-bβ2Ih(1+αIh)-μv≤Ih′Ih-Iv′Iv+(IhIv)bβ2μvΛv-μvIv(1+α3Ih)-μh-bβ2Ih(1+α3Ih)-μv≤Ih′Ih-Iv′Iv+Iv′Iv-μh-bβ2Ih(1+α3Ih)≤Ih′Ih-μh-bβ2Ih(1+α3Ih).
Thus,
(3.27)μ(B)=Sup{g1,g2}≤Ih′Ih-μh.
Since (2.2) is uniformly persistent when R0>1, so for T>0 such that t>T implies Ih(t)≥c, Iv(t)≥c and (1/t)logIh(t)<μ/2 for all (Sh(0),Ih(0),Iv(0))∈K. Thus
(3.28)1t∫0t‍μ(B)dt<logIh(t)t-μ<-μ2
for all (Sh(0),Ih(0),Iv(0))∈K, which further implies that q-2<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.
Parameter
Value
Reference
Λ
h
0.00011
[23]
Λ
v
0.13
[23]
b
0.5
[23]
γ
h
0.7
Assumed
β
1
0.022
[23]
β
2
0.48
[23]
β
3
0.004
Assumed
α
5
Assumed
μ
h
0.000016
[23]
μ
v
0.033
[23]Definition 4.1.
The normalized forward sensitivity index of a variable,h, that depends differentiably on a parameter, l, is defined as γlh=(∂h/∂l)×(l/h).Table2 represents sensitivity indices of model parameters to R0.Table 2
Sensitivity indices ofR0 to parameters for the model, evaluated at the parameter values given in Table 1.
Parameter
Description
Sensitivity index
b
Rate of biting of a host by mosquito
1.97493
γ
h
Loss of immunity
−0.999977
β
1
Probability of transmission from mosquitoes to host
0.987467
β
2
Probability of transmission from host to mosquitoes
0.987467
β
3
Probability of transmission from infectious human to susceptible human
0.0125332
Λ
h
Recruitment rate of susceptible hosts
1
Λ
v
Recruitment rate of susceptible mosquitoes
0.987467
μ
v
Death rate of mosquitoes
−1.97493
μ
h
Death rate of hosts
−1.00002By analyzing sensitivity indices we observe that the most sensitive parameters are biting rate of mosquitoesb and death rate of mosquitoes μv. 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 Table1. The most sensitive parameter for Ih* is mosquito biting rate. Change in mosquito biting rate is directly related to change in Ih* and inversely related to change in γh. This suggests  that personal protection and human treatment strategies can lead to marvelous decrease in Ih*. The most sensitive parameter for Iv* is mosquito death rate μv, followed by mosquito biting rate. We observe that Iv* can be reduced by personal protection, larvcide adulticide, and so forth.The analysis of the sensitivity indices ofR0, Ih*, and Iv* 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 Sh*, Ih*, and Iv* with respect to all parameters are given in Table 3.Table 3
The sensitivity indices of the state variables at the endemic equilibrium,xi, to the parameters, pj, for parameter values given in Table 1.
S
h
*
I
h
*
I
v
*
Λ
h
0.998946
1.50275
0.00011
Λ
v
−0.00108296
0.516688
1.45019
b
−0.00401088
1.91363
2.57621
γ
h
0.00314305
−1.49958
−1.30657
β
1
−0.00302661
1.44402
1.25817
β
2
−0.000984278
0.469608
1.31805
β
3
−0.000116517
0.0555912
0.0484363
α
1
0.00194365
−0.927333
−0.80798
α
2
6.33734 × 10−6
−0.0030236
−0.00263445
α
3
0.0000407045
−0.0194205
−0.0545075
μ
h
−0.998946
−1.50279
−1.30937
μ
v
0.00206723
−0.986295
−2.76823


## 4.1. Sensitivity Indices of Endemic Equilibrium

We have numerically calculated the sensitivity indices at the parameter values given in Table1. The most sensitive parameter for Ih* is mosquito biting rate. Change in mosquito biting rate is directly related to change in Ih* and inversely related to change in γh. This suggests  that personal protection and human treatment strategies can lead to marvelous decrease in Ih*. The most sensitive parameter for Iv* is mosquito death rate μv, followed by mosquito biting rate. We observe that Iv* can be reduced by personal protection, larvcide adulticide, and so forth.The analysis of the sensitivity indices ofR0, Ih*, and Iv* 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 Sh*, Ih*, and Iv* with respect to all parameters are given in Table 3.Table 3
The sensitivity indices of the state variables at the endemic equilibrium,xi, to the parameters, pj, for parameter values given in Table 1.
S
h
*
I
h
*
I
v
*
Λ
h
0.998946
1.50275
0.00011
Λ
v
−0.00108296
0.516688
1.45019
b
−0.00401088
1.91363
2.57621
γ
h
0.00314305
−1.49958
−1.30657
β
1
−0.00302661
1.44402
1.25817
β
2
−0.000984278
0.469608
1.31805
β
3
−0.000116517
0.0555912
0.0484363
α
1
0.00194365
−0.927333
−0.80798
α
2
6.33734 × 10−6
−0.0030236
−0.00263445
α
3
0.0000407045
−0.0194205
−0.0545075
μ
h
−0.998946
−1.50279
−1.30937
μ
v
0.00206723
−0.986295
−2.76823


## 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
(5.1)dShdt=Λh-(1-u1)bβ1ShIv1+α1Iv-β3ShIh1+α2Ih-μhSh+γhIh,dIhdt=(1-u1)bβ1ShIv1+α1Iv+β3ShIh1+α2Ih-(μh+γh+ru2)Ih,dSvdt=Λv(1-u3)-(1-u1)bβ2IhSv1+α3Iv-μvSv-cu3Sv,dIvdt=(1-u1)bβ2IhSv1+α3Iv-μvIv-cu3Iv.
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 functionalJ, which is to minimize the number of infected human and the cost of applying the control u1,u2,u3. One has
(5.2)J=∫0tf‍(A1Ih+A2Iv+A3Sv+12(B1u12+B2u22+B3u32))dt,
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 u1*,u2*,u3* such that
(5.3)J(u1*,u2*,u3*)=min{J(u1,u2,u3)∣u1,u2,u3∈U},
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 u3(5.4)H=A1Ih+A2Iv+A3Sv+12(B1u12+B2u22+B3u32)+λ1(Λh-(1-u1)bβ1ShIv1+α1Iv-β3ShIh1+α2Ih-μhSh+γhIh)+λ2((1-u1)bβ1ShIv1+α1Iv+β3ShIh1+α2Ih-(μh+γh+ru2)Ih)+λ3(Λv(1-u3)-(1-u1)bβ2IhSv1+α3Iv-μvSv-cu3Sv)+λ4((1-u1)bβ2IhSv1+α3Iv-μvIv-cu3Iv),
where λi 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(u1*,u2*,u3*) that minimize J over U subject to the control system (5.1). Further, for the system (5.1), there exists adjoint variables λi,i=1,…,4 satisfying
(5.5)dλ1dt=(λ1-λ2)bβ1(1-u1)Iv1+α1Iv+(λ1-λ2)β3Ih1+α2Ih+λ1μh,dλ2dt=(μh+γh+ru2)λ2-λ1γh+(λ1-λ2)β3Sh(1+α2Ih)2+(λ3-λ4)(1-u1)bβ2Sv1+α3Iv-A1,dλ3dt=(λ3-λ4)(1-u1)bβ2Ih1+α3Iv+(μv+cu3)λ3-A3,dλ4dt=(λ1-λ2)bβ1(1-u1)Sh(1+α1Iv)2+(λ4-λ3)(1-u1)αbβ2IhSv(1+α3Iv)2+(μv+cu3)λ4-A2,
with transversality conditions λi(tf)=0,i=1,…,4. The optimal controls are given by
(5.6)u1*=min{max{0,1B1[(λ2-λ1)bβ1ShIv1+α1Iv+(λ4-λ3)bβ2IhSv1+α3Iv]},1},u2*=min{max{0,1B2[λ2rIh]},1},u3*=min{max{0,1B3[λ3(Λv+cSv)+λ4cIv]},1}.Proof.
The integrand of the objective functionalJ 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 Sh, Ih, Sv, and Iv, respectively, evaluated at the optimal control. To get the characterization of the optimal control given by (5.6), solving the equations
(5.7)∂H∂u1=0,∂H∂u2=0,∂H∂u3=0,
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 controlu1 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 Iv between the case with control and the case without while there is a significant difference in the number of infected humans Ih 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.Simulations of the malaria model showing the effect of control strategies on malaria transmission.
(a)
(b)
(c)
(d)


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

With this strategy, we set the controlu2 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 Ih and infected mosquitoes Iv, respectively, with control compared to the situation where there is no control. More specifically, we observe a decrease in Ih and Iv 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).Simulations of the malaria model showing the effect of control strategies on malaria transmission.
(a)
(b)
(c)
(d)


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

With this strategy, we set controlu1 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 (Ih), infected mosquitoes (Iv), and susceptible humans (Sh) while an increase is observed in the number of infected humans (Ih) and infected mosquitoes (Iv) 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.Simulations of the malaria model showing the effect of control strategies on malaria transmission.
(a)
(b)
(c)
(d)


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

With this strategy, the controlu1, 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 (Ih), infected mosquitoes, (Iv) and susceptible humans (Sh) while an increase is observed in the number of infected humans (Ih) and infected mosquitoes (Iv) 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).Simulations of the malaria model showing the effect of control strategies on malaria transmission.
(a)
(b)
(c)
(d)


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

With this strategy, only the preventive controlu1 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 Iv between the case with control and the case without while there is a significant difference in the number of infected humans Ih 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.Simulations of the malaria model showing the effect of control strategies on malaria transmission.
(a)
(b)
(c)
(d)


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

With this strategy, we set the controlu2 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 Ih and infected mosquitoes Iv, respectively, with control compared to the situation where there is no control. More specifically, we observe a decrease in Ih and Iv 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).Simulations of the malaria model showing the effect of control strategies on malaria transmission.
(a)
(b)
(c)
(d)


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

With this strategy, we set controlu1 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 (Ih), infected mosquitoes (Iv), and susceptible humans (Sh) while an increase is observed in the number of infected humans (Ih) and infected mosquitoes (Iv) 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.Simulations of the malaria model showing the effect of control strategies on malaria transmission.
(a)
(b)
(c)
(d)


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

With this strategy, the controlu1, 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 (Ih), infected mosquitoes, (Iv) and susceptible humans (Sh) while an increase is observed in the number of infected humans (Ih) and infected mosquitoes (Iv) 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).Simulations of the malaria model showing the effect of control strategies on malaria transmission.
(a)
(b)
(c)
(d)


## 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 numberR0 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.



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*Source: 595487-2012-11-01.xml*