The Preventive Control of a Dengue Disease Using Pontryagin Minimum Principal

Behaviour analysis for host-vector model without control of dengue disease is based on the value of basic reproduction number obtained using next generation matrices. Furthermore, the model is further developed involving a preventive control to minimize the contact between host and vector. The purpose is to obtain an optimal preventive strategy with minimal cost. The Pontryagin Minimum Principal is used to find the optimal control analytically. The derived optimality model is then solved numerically to investigate control effort to reduce infected class.


Introduction
In the host-vector epidemic model, the population is divided into the host, in this case humans, and the vector, the Aedes aegypti mosquito as the carrier of the dengue virus. Research on the dynamic behaviour of the spread of dengue fever using a host-vector model by assuming that there is immunity after recovery has been performed by Esteva [1]. But in reality, dengue fever is caused by the dengue virus that has four serotypes, and a person can be infected by dengue fever again with a different serotype. The mathematical model that addresses this has been done by Soewono and Apriatna [2]. Their model has assumed of becoming susceptible again after recovering from a dengue fever, but has not considered the possibility of infected hosts that cannot spread the dengue fever.
Therefore, this paper discusses the extension of a dengue fever model where after recovery a host can be susceptible again. In addition, it is also assumed that there are some infected hosts but cannot spread the dengue fever. The proposed model (i.e. the first proposed model) and then will be analysed on its behaviour based on the value of the basic reproduction number ( [3], [4], [5]).
Furthermore, the first model will be expanded by adding a control variable. Research on the optimal control using control insecticide has been carried out by Rodrigues [6]. The results show that the control must be done repeatedly for the best result. But this leads to the increasing of the cost control. Dumont's research result [7] shows that using a combination treatment with chemicals, larvae control and adult mosquitoes can decrease the epidemic. Of course it is not easy to be done as it requires more data.
On the other hand, WHO [8] brings out some strategies to resolve the epidemic by prevention and control. Therefore, this paper also discusses the behaviour of our first proposed model if we give a preventive control variable to the model (i.e. second proposed model). This second model then will be 2 1234567890 analysed analytically by Pontryagin Minimum Principle. The behaviour of the first and second model (before and after being given controls) will be done numerically. At the end, we obtain the most effective control recommendations to minimize the dengue epidemic.

Mathematical Model of Dengue Disease without Control
The host population is divided into susceptible class, infectious class and recoverable (immune) class. Vector population are divided into susceptible class and infectious class. There is no recoverable (immune) class in the vector population as mosquitoes will always carry a dengue virus in their body for all their life. The spread of the dengue fever starts from the infected host is bitten by the vulnerable mosquito, resulting in the transmission of dengue virus to the mosquito, so the mosquito becomes infected. If the infected mosquito then bites the vulnerable host, then there will be a transmission which result on the host becomes infected. And so on. Assume The following Fig.1 is the transmission diagram for the spread of the dengue fever by taking all the above assumptions. The mathematics model for the dengue fever transmission is given as follow.

( )
As V µ denotes the mortality transmission for the vector, thus the total of mortality for vector population is denotes by V V N µ . In the other hand, the rate of change of the vector population is denoted by , which is the recruitment rate subtracted by the total of mortality, means that Let the proportion for each class is denoted by 1 while for (1b), we obtain ( ) Using the same way, we derive (1e) as follow.
( ) ( ) The equilibrium point of System (2) is achieved when Using this condition, we have two equilibrium points that are ( ) 1, 0, 0 , which is called as disease free, and ( ) which is called as the endemic equilibrium point. Then, we will discuss about the value of secondary infections as a result of the first infection. This value is called the basic reproduction number. To determine this value, we use the next generation matrix method [9].
Defined matrix ‫ܨ‬ as a matrix whose entries are the first derivative of all terms that "enters" the infectious class at ( ) 1,0,0 1 5 2 International Conference We also define matrix V whose entries are the first derivatives from the terms that are "out" from the infectious class at ( ) 1,0,0 2 2 2 5 5 1,0,0 The value of the basic reproduction number is obtained from the biggest eigen value from matrix G, which is , , x x x is stable if 0 1 R > .

Optimal Control
This section discusses about the extension of the transmission model of dengue fever (1) controlled by the prevention control variable so that the contact between the host and the mosquito can be reduced, and denoted by 1 If the prevention control is given, then the infection rate will become ( ) ( ) The prevention efforts in this case could be a counselling about the dengue fever, the provision of the abate powder or the fogging readiness if necessary.
To investigate the optimality of those efforts, given the objective function J, namely to minimize the infected host class and the cost to implement the 1 u control, where B is the weights for the infected hosts class, while C is the weights for prevention efforts. It means that we will find the solution from (5), which minimizes the number of the infected hosts with a minimal ( ) 1 u t control cost. In other words, will be sought so an * 1 u optimal control such that with system (4) as a constraint and Further, we define Hamiltonian as follow.
Proof. The differential equations developing the costate variables are obtained by differentiation of the Hamiltonian function that evaluated at the optimal control. Then the costate system can be written as follow

Numerical Simulation
In this section, we present numerical simulation to describe the optimal solution of the hostvector model. The parameter values are given as below.
As shown in Figure 2, for susceptible hosts' class, the population rose initially, but dropped later on. This is because the population will go to the infectious class. While with the presence of the control, the population of the susceptible class increased over time.  Figure 3 explains that without control, the population of the infected hosts' increases. Conversely when the control is given, the population decreases sharply. In Figure 4, it can be seen that the vector population increases, both with and without controls. But, the population with control rose faster and higher than without control.  It can be seen that with preventive control can reduce the number of infectious host significantly.

Conclusion
This paper has discussed the development of a mathematical model for the dengue fever spread in the form of host-vector. The model then is extended by giving a prevention control for the spread. We described analytically that using Pontryagin Minimum Principle there is an optimal control. We also proved numerically that with a control strategy, we can decrease the population of infectious class both for the host and the vector.