DIRECT TRANSMISSION

This paper introduces the usage of three controls as a way to reduce the occurrence of vector-borne disease. The governing equation of the dynamical system used in this paper describes both direct and indirect transmission mode of vector-borne disease. This means that the disease can be transmitted in two different ways. First, it can be transmitted through mosquito bites and the other is through human blood transfusion. The three controls that are incorporated in the dynamical system include a measurement of basic practice for blood donation procedure, self-prevention effort and vector control strategy by health authority. The optimality system of the three controls is characterized using optimal control theory and the existence and uniqueness of the optimal control are established. Then, the effect of the incorporation of the three controls is investigated by performing numerical simulation.


INTRODUCTION
Vector-borne disease is a kind of disease which is transmitted by organisms that carry infectious pathogen from one host to another.Some of the organisms which serve as vectors include mosquitoes, fleas, biting flies, bugs, mites and ticks.There are several type of diseases that are caused by this kind of transmission including malaria, dengue fever, yellow fever, chikungunya and several others.According to the World Health Organization (WHO), 17% of the estimated global burden of all infectious diseases are caused by vector-borne diseases.
As far as mathematical modelling is concerned, vector-borne disease is often being modelled as an interaction between the hosts population, particularly between human and vector population.The population for each of them is then being classified into several distinct classes or also known as compartment with each portrays the host and the mosquito disease status.Some of the early studies on mathematical model for vector-borne disease include Feng and Velasco-Hernández [1], Esteva and Vargas [2], Ferguson et al. [3], Ngwa and Shu [4], and Esteva and Matias [5].
Though vector-borne diseases are primarily transmitted via vectors, there are several findings which indicated the occurrence of virus transmission through blood transfusion.This is reported in several articles including the one written by Kelly et al. [6], Kitchen and Chiodini [7], Tambyah et al. [8], Punzel et al. [9] and Stramer et al. [10].Furthermore, Harif et al. [11] did an experimentation to investigate the existence of viremia among blood donors, where a random serum samples from 360 donors were selected from blood donated during the period of December 2009 to January 2010, in which the period mentioned was the time of dengue fever outbreak.Their results indicated that 15 of the donors may be in the carrier stage of the dengue virus which may eventually lead to the possibility of transmitting the virus through blood transfusion.
In consideration of this fact, several studies had addressed the situation of vector-borne transmission through blood transfusion from the mathematical point of view.One of them is Wei et al. [12], who considered this situation and denoted it as direct transmission mode of vector-borne disease.A mathematical model for the transmission was formulated in [12], which include both populations, hosts and vectors, and also presented a differential-delay model with a discrete time delay which accounts for the incubation period of the vectors.Then, there is Cai and Li [13] who also addressed this issue and formulated two mathematical models.Lashari and Zaman [14] extended the model in [13] by introducing exposed classes to both host and vector populations.Whereas, Cai et al. [15] extended the study of [12] with modification on the incidence rate of the model.
On the formulation of optimal control problem, it is originally part of the basis in modelling infectious disease to understand transmission mechanism and investigate the appropriate control strategy [16].For example, one can determine how to optimally manage limited resources of vaccines, or treatment facilities when there is an outbreak of infectious disease.Specifically in the case of vector-borne disease, the main objectives can be to reduce the number of infected people as well as to eradicate, if possible, the vector population so that the virus cannot be further circulated.Hence, this paper will investigate the usage of optimal control in achieving the mentioned objectives.The optimal control problem is formulated by modifying the model in [15] with the incorporation of three controls.The first control is on the practice of blood screening procedure as a way to prevent the virus to be transmitted through blood transfusion.Second is on the self-prevention effort, which is to clean up house compound, to use insect repellent, and others, particularly the one who lives at the area of high disease occurrence.The third is on the effort by health authority, which is on the usage of adulticide, larvacide and others.This type of optimal control problem with the said controls had been addressed in [17] and in [18].However, the underlying dynamical system presented in this paper is slightly different from theirs, which then lead to a different results.In [17], the authors presented seven classes of disease status, in which they introduced an exposed class for both host and vector.Whereas, in [18], density dependent mortality rates are used for both vector and host populations.
This paper is organized as follows.In Section 2, the formulation of optimal control problem is presented, in which its existence and characterization are also shown.Then, in Section 3, discussion and the results from numerical simulation is presented.Finally, the conclusion is given in Section 4.

THE FORMULATION OF OPTIMAL CONTROL PROBLEM
In this section, a vector-host epidemic model which characterizes both direct transmission and the conventional way of transmission through vector is presented.The dynamical system for both host and vector population is governed by a model which is formulated in [15].The dynamical system is as presented in (1). denote the occurrence of new incidence through blood transfusion, infection from infected vector to susceptible host and infection from infected host to susceptible vector respectively.h  and v  represent the natural death rate for host and vector respectively.The host recovery rate is denoted by  and  represent the disease induced death rate.
Accordingly, in this paper the dynamical system of ( 1) is modified by the inclusion of three control variables.The three controls introduced are, cost of basic prevention practice in blood donation procedure,   1 ut, self-prevention effort (repellent, clean up possible vector breeding sites such as vase, pail and others),   2 ut , and the cost to reduce the vector population by health authority such as adulticide and larvacide,   3 ut.Also, the recruitment rate in both host and vector susceptible population is modified so that it will be density dependent [1,16].This is denoted as follows., which then, transformed the system of (1) to the one in (2).Note that  represents the proportionality constant showing the impact of density to host recruitment rate.If there is no new host recruitment, then  will be the per capita birth rate of host [19].
Nt denotes the size of total host population at time t, in which it is the sum of which represents the size of total vector population.
It should be noted that there is an addition of new rate constant, 0 r in (2) since the mortality rate of both susceptible and infected vectors may increase in proportion to the third control,   3 ut [19].The rate of 0 r is always positive.Then, the objective functional is defined as: which is subject to the system of (2).The parameters

The terms  
A N t denote the cost associated in reducing the infected host and vector population respectively.Also,   represent the cost associated with the basic practice of blood donation procedure, self-prevention effort by host, and vector control, respectively.The purpose is then to find an optimal control triplet *

U
Accordingly, the existence and characterization of its optimal control for the above formulation will be shown in the following section.The Hamiltonian function, H with respect to

Existence of Optimal Control
Theorem 2.1: Consider the objective functional of (3) with   ,, u u u U subject to the controlled system of (2)  .There exists [20], the following conditions should be satisfied as to ensure the existence of optimal control.i. the set of controls and corresponding state variables is nonempty ii.
the control set U is convex and closed iii. the right hand side of the state system is bounded by a linear function in the state and control variables iv. the integrand of the objective functional is convex on U v. the integrand of the objective functional is bounded below by , where To verify these properties, the result from Lukes [21] is used to give the existence of solutions for the state system of (2) with bounded coefficients, which gives Condition i.The control set is closed and convex by definition, hence satisfies Condition ii.The right hand side of system (2) satisfies Condition iii since the state solutions are bounded.The integrand of our objective functional is clearly convex on U , which then gives Condition iv.Also, there are 12 ,0 cc and because the state variables are bounded.□

Characterization of Optimal Control
Pontryagin's maximum principle [22] is used to derive the necessary conditions for the optimal control triplet.The Lagrangian, which is the Hamiltonian augmented with penalty terms for the control constraints is defined as follows.

t u t S t I t r u t S t S t I t r u t I t I t w t u t w t m u t w t u t w t m u t w t u t w m u t
with the transversality condition of (5) The optimal controls are given by, Proof: The form of the adjoint functions and transversality conditions are standard results from Pontryagin's Maximum Principle [22].The Lagrangian is differentiated with respect to states,

, , , and
I respectively, which resulted in the following adjoint functions.
 with zero transversality conditions.The characterization of the optimal control ( 6) is obtained by solving Then, to determine an explicit expression for * Combining these three cases, the optimal control of * 1 u can be characterized by  

NUMERICAL RESULTS AND DISCUSSION
In this section, the optimal solution to the vector host model ( 2) is solved numerically using forwardbackward sweep method introduced in [23].The optimality system, which consisted of the state equations of (2), the adjoint equations ( 4) and controls characterization (6), is solved using iterative method of fourth-order Runge-Kutta scheme.The algorithm started with the initial guess for the control variables.The state variables are solved forward in time using fourth-order Runge-Kutta method by considering the initial conditions.With the output of the state variables, and also considering the transversality conditions of ( 5) , the adjoint variables are solved backward in time also using the fourth-order Runge-Kutta method.The control variables were then being updated with the insertion of the current value of the state and adjoint variables.The process is repeated until they converge sufficiently.
The initial conditions, which represent the initial population size of each classes are assumed to be .These values of initial conditions are extracted from [24].Table 1 presents the value of parameters used in the numerical simulation.
Other parameters values are taken from [17,18,19] in which they are arbitrarily assumed.These are, .As for the weights in the objective functional of (3), they are assumed with the value of A are chosen in that particular way so as to show that the minimization of the infected host population is given more importance as compared to the reduction of vector population.Whereas, the weights of Considering the estimated value of parameters and initial conditions mentioned, the results from the numerical simulation is presented in Figure 1 -6. Figure 1 shows the dynamic of infected host population with and without controls.It can be seen from the figure, without the controls, the size of population will increase and eventually reduced to 0. On the other hand, with the controls, the population size of infected host never increase and eventually reached 0 in a time faster than the one without the controls.Whereas, the population dynamic of susceptible vector is shown in Figure 2. From the figure, it can be seen that the population size with the controls reached the size of 0 in a slower time as compared to the one without the controls.This means that the controls allow less vector to be infected with the virus and causing the size of susceptible vector population to be slowly reduced to 0. Figure 3 shows the population dynamic of infected vector with and without the controls.Rationally, Figure 2 and Figure 3 are directly related since once susceptible vector acquired the virus, it will move to the class of infected vectors.As can be seen from Figure 3, the population size without the controls increased higher than with the controls.Eventually, both populations, which are with and without the controls reached 0 but the one with the controls reached to 0 in a faster time.Also, the rate of each of the three controls over time are shown in Figure 4, Figure 5 and Figure 6.From Figure 4, during the initial outbreak of the disease, full rate of control for blood screening procedure need to be enforced.This enforcement need to be undertaken until it reaches the time between day 30-35, in which afterward the rate of control can gradually be reduced.In relation with the disease status, as what is presented in Figure 1, this is the time when the infected host population started to be zero.Full enforcement rate of control also need to be implemented for self-prevention effort and pesticide control during initial disease outbreak.This enforcement need to be implemented until it reaches the time between day 40-45.Afterward, both rate of control can steadily be reduced and eventually reached to the point of no controls are needed.

CONCLUSION
This paper has introduced an optimal control problem to a dynamical system of vector-borne disease with direct transmission.By direct transmission, it means that the disease can also be transmitted directly from one host to another apart from the conventional way of its transmission through vector.This particularly occurred through blood transfusion in which real life cases have been reported in several studies.As a way to reduce or if possible to eradicate the disease, three controls have been introduced in this study.One is to perform blood screening procedure during blood transfusion process.The other control is to do self-prevention effort, for example by cleaning up house compound and wearing insect repellent.The last control is the responsibility of the health authority to provide adulticide and lavarcide as a method to reduce the size of vector population.With the said controls, and its underlying dynamical system, the optimal control problem is formulated, with the objective to reduce the size of infected host population and also the size of vector population.Eventually, the optimality system is derived using the Pontryagin's maximum principle.Then, numerical simulation is performed to observe the impact of the controls to the population dynamic of both host and vector.Based on the results, it can be said that the introduction of the controls have caused the population dynamic of host and vector to reform in a positive way.It should be noted that several modification can be done to the dynamical system.Firstly, instead of using constant population for both host and vector, one can change it to variable population such as the one done by [4] and [27].Also, the incidence rate can be changed to bilinear and saturation incidence as were done by [5] and [28].This subject matter is also under consideration by the authors.

1 A , 2 A , 1 B , 2 B and 3 B
are positive weight constants.

1 c and 2 c
are positive constants and 1 b  .
are the susceptible, infected and recovered population for host respectively.On the other hand, for the population of vectors, they are assumed to be 

1 B , 2 B and 3 B 2 ut is lower than   3 ut and the cost of   3 3 ut
indicate the cost associated with the controls, considering the fact that the cost for   may vary from 0 to 1.

Table 1
Estimated value of parameters