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

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.


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.

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:

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 β 1 be the transmission rate from vector to human, and let β 2 be the transmission rate from human to vector.β 3 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 bβ 1 S h I v / 1 α 1 I v , β 3 S h I h / 1 α 2 I h , and bβ 2 I h S v / 1 α 3 I h , where α 1 , α 2 , and α 3 determine the level at which the force of infection saturates.

Obviously, Δ { S h , I h , S v , I
} 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 v N 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

2.2
From biological considerations, we study system 2.2 in the closed set where R 3 denotes the nonnegative cone of R 3 including its lower dimensional faces.It can be easily verified that Ω is positively invariant with respect to 2.2 .

Mathematical Analysis of the Model
The dynamics of the disease are described by the basic reproduction number R 0 .The threshold quantity R 0 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
Direct calculation shows that system 2.2 has two equilibrium states.For R 0 ≤ 1, the only equilibrium is disease-free equilibrium and I * h is the root of the following quadratic equation.

3.4
From 3.3 , we see that R 0 > 1 if and only if a 3 < 0. Since a 1 > 0, 3.3 has a unique positive root in feasible region.If R 0 < 1, then a 3 > 0. Also, it can be easily seen that a 2 > 0 for R 0 < 1.Thus, by considering the shape of the graph of 3.3 and noting that a 3 > 0 , we have that there will be zero positive endemic equilibrium in this case.Therefore, we can conclude that if R 0 < 1, 3.3 has no positive root in the feasible region.
If, R 0 > 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 E 0 .If R 0 > 1, system 2.2 has a unique endemic equilibrium E * S * h , I * h , I * v defined by 3.2 and 3.3 .

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 E 0 of the system.
Theorem 3.2.If R 0 ≤ 1, then the infection-free equilibrium E 0 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: Calculating the time derivative of L along the solutions of system 2.2 , we obtain

3.6
Thus L t is negative if R 0 ≤ 1.When R 0 < 1, the derivative L 0 if and only if I h 0, while in the case R 0 1, the derivative L 0 if and only if Hence, LaSalle's invariance principle 19 implies that E 0 is globally asymptotically stable in Ω.This completes the proof.

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 C 1 map f : x → f x from an open set D ⊂ R n to R n such that each solution x t, x 0 to the differential equation is uniquely determined by the initial value x 0, x 0 .We have following assumptions: H 2 there exists a compact absorbing set K ⊂ D; H 3 Equation 3.7 has unique equilibrium x in D.
Let μ be the Lozinskiȋ measure with respect to the | • |.Define a quantity q 2 as where B P f P −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 H 1 , H 2 and H 3 hold, the unique endemic equilibrium E * is globally stable in Ω if q 2 < 0.
Obviously Ω is simply connected and E * is unique endemic equilibrium for R 0 > 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.
Proof.Let Φ be semidynamical system 2.2 in R 0 3 , let be a locally compact metric space, and let Γ 0 { S h , I h , I v ∈ Ω : I v 0}.Γ 0 is a compact subset of Ω and Ω/Γ 0 is positively invariant set of system 2.2 .Let P : → 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 3.9 Assume that there is a solution x ∈ S such that for each t > 0 P Φ x, t < P x < φ.Let us consider where δ * is sufficiently small so that

3.11
By direct calculation we have

3.12
where 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 q 2 < 0. We choose a suitable vector norm | • | in R 3 and a 3 × 3 matrix valued function

3.14
Obviously P is C 1 and nonsingular in the interior of Ω. Linearizing system 2.2 about an endemic equilibrium E * gives the following Jacobian matrix.

3.15
The second additive compound matrix of J E * is given by where 3.17 The matrix B P f P −1 PJ 2 P −1 can be written in block form as

3.20
Consider the norm in R 3 as | u, v, w | max |u|, |v| |w| where u, v, w denotes the vector in R 3 .The Lozinskiȋ measure with respect to this norm is defined as μ B ≤ sup g 1 , g 2 , where From system 2.2 we can write

3.22
Since B 11 is a scalar, its Lozinskiȋ measure with respect to any vector norm in R 1 will be equal to B 11 .Thus

3.23
and g 1 will become

3.24
Also

3.27
Since 2.2 is uniformly persistent when R 0 > 1, so for T > 0 such that t > T implies I h t ≥ c, I v t ≥ c and 1/t log I h t < μ/2 for all S h 0 , I h 0 , I v 0 ∈ K. Thus for all S h 0 , I h 0 , I v 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.

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, R 0 , 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.
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 R 0 .
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 R 0 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 R 0 by 20%.Similarly increase or decrease in death rate of mosquitoes by 10% decreases or increases R 0 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.

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 R 0 , 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.

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 − u 1 and the reproduction rate of the mosquito population is reduced by a factor of 1 − u 3 .It is assumed that under the successful control efforts the mortality rate of mosquito population increases at a rate proportional to u 3 , where c > 0 is a rate constant.The per capita recovery rate is proportional to u 2 , where r > 0 is a rate constant.One has

5.1
The control variable u 1 represents the use of drugs or vaccine which are preventive measures to minimize vector human contacts.The control function u 2 represents the treatment supplied to the infected humans.The control function u 3 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 u 1 , u 2 , u 3 .One has where A 1 , A 2 , and A 3 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 u 1 , u 2 , u 3 , our goal is to minimize the number of infected humans, while minimizing the cost of control u 1 t , u 2 t , and u 3 t .We seek an optimal control u * where 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 u 1 , u 2 , and u 3 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 .

5.6
Proof.The integrand of the objective functional J given by 5.2 is a convex function of u 1 , u 2 , u 3 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 on the interior of the control set and using the property of the control space U, we can derive the desired characterization 5.6 .

Numerical Results and Discussions
Here, we investigate numerically the effect of the optimal control strategies on the spread of the disease in a population.With this strategy, only the preventive control u 1 on the vector biting rate and treatment u 2 is used to optimize the objective function J, while the control u 3 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 u 1 rise to the upper bound after 400 days , while control u 2 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.With this strategy, we set the control u 2 to zero and use only preventive control u 1 and larvacide u 3 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 u 3 is at the upper bound for t 450 days .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 u 3 is at the upper bound for t 450 days and the control u 2 is at the upper bound for t 440 days and then dropped to the lower bound at the final time.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 u 1 is at the upper bound for t 440 days before dropping gradually to the lower bound, the control u 2 is at the upper bound for t 440 days and then dropped to the lower bound at the final time, and the control u 3 is maintained at the upper bound for t 450 days .

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 R 0 is less than or equal unity and has a unique positive globally asymptotically stable endemic equilibrium whenever R 0 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

Treatment u 2 / 1 3 Time
0 and Larvacide u 3 / 0 With this strategy, we set control u 1 to zero, while the control u 2 and the control u 3 are both used to optimize the objective function J.In Figures 3 a and 3 b , we observed that = 0, u 2 = 0, u 3 = 0 = 0 u 1 / = 0, u 2 / = 0, u

u 1 =Figure 1 :
Figure 1: Simulations of the malaria model showing the effect of control strategies on malaria transmission.

Measures u 1
/ 0 , Treatment u 2 / 0 , and Larvacide u 3 / 0 With this strategy, the control u 1 , control u 2 , and the control u 3 are all used to optimize the objective function J.In Figures 4 a and 4 b , we observed that the control strategies

Figure 2 :
Figure 2: Simulations of the malaria model showing the effect of control strategies on malaria transmission.
l 1 norm.μ 1 B 22 is the Lozinskiȋ measure of 2 × 2 matrix B 22 with respect to l 1 norm in R 2 .
|B 12 | and |B 21 | are the operator norms of B 12 and B 21 which are mapping from R 2 to R and from R to R 2 , respectively, and R 2 is endowed with the

Table 1 :
Values for parameters used for sensitivity analysis.

Table 2 :
Sensitivity indices of R 0 to parameters for the model, evaluated at the parameter values given in Table1.

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 Table1.