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In this paper, we consider a leptospirosis epidemic model to implement optimal campaign by using multiple control variables. First, we show the existence of the control problem. Then we derive the conditions under which it is optimal to eradicate the leptospirosis infection and examine the impact of a possible educatioal/vaccinaction campaign using Pontryagin’s Maximum Principle. We completely characterize the optimal control problem and compute the numerical solution of the optimality system using an iterative method. The results obtained from the numerical simulations of the model show that a possible educational/vaccinaction combined with effective treatment regime would reduce the spread of the leptospirosis infection appreciably.

Leptospirosis disease is a globally zoonotic disease. The cause of the disease is bacteria which is called leptospira. Human as well as cattle are mostly infected from this disease. The human are infected by means of drinking the water in which a rat (dead) found, while cattle that drink this water are become infectious. The human whose urine is used by other animals and cattle are also infected, because the leptospirosis germs come out in urine. It is also reported that people belong to city are mostly infected from this disease and got liver infection. Leptospirosis is known by different names such Weil’s disease, canicola fever, canefield fever, 7-day fever, nanukayami fever [

The mathematical formulation and dynamical sketch of this infection has been studied by several authors see for example [8-12]. Pongsuumpun et al. [

In case of vector born diseases some authors focused on eradication of the disease, by targeting the vector population as a strategy for controlling the disease [15, 16] while some scientists studied the effect of vaccinetion on the dynamics of the disease [

In this paper, we consider the basic model studied in [

The structure of the paper is organized as follows. Section 2 is devoted to the formulation of the basic mathematic model. In Section 3, we present the control problem and develop reproductive number. In Section 4, we present the endemic equilibria for both systems with and without control and bifurcation analysis. In Section 5, we present the existence of the control problem and derive the necessary conditions for an optimal control and the corresponding state system by using Pontryagin’s Maximum Principle. Section 6 is devoted to numerical solution of the optimality system and finally, we conclude our work.

Basic epidemic models allow for variations in the different stages(classes) of the infection. Several researcher developed different mathematical models to identifying the stages which depends on the dynamics of the disease and the composition of the population. In these mathematical models an individual can be in any one of the stages of infection. Susceptible (S), the individual is able to contract the infection; exposed (E), the individual has contracted the disease but is not yet infectious or symptomatic; infectious (I), the individual is contagious and may or may not be showing symptoms; and removed (R), an individual can be removed from the population by recovering with immunity, being quarantined or by death. In this work, we present the basic model proposed by [

For each category, we assume the population changes over time. Thus, we write the number of humans in each category susceptible, exposed, infected and recovered human as functions of time t. The total human population is denoted by with. Similarly, we write the number of vector in each category: susceptible, exposed, and infected vector, respectively as functions of time t. The total vector class is denoted by N_{v}(t) with . The complete system of non-linear differential equation is given by:

With initials conditions

The parameters involved in the basic model are as under:

is the recruitment rate of human population,

is the transmission coefficient,

is the transmission coefficient,

is the Transmission coefficient,

is the natural mortality rate of human,

is the death rate of infected human,

is the recruitment rate of vector,

is the natural mortality rate of vector,

is the death rate of infected vector,

is the rate at which exposed vector move to exposed class,

is the rate at which exposed human move to exposed class.

Optimal control is one of the techniques to minimize (maximize) the infection in the human class of individuals. Several articles have been published on different population models by applying the optimal control techniques to reduce the infection at the human population using different control variables [17,19]. In this section, we present an optimal control technique by using multiple control variables to reduce the spread of leptospirosis infection in a community. Our educational/vaccination campaign consisting of the following control variables:

: represents (cover all cuts, water dry, full-cover boots, shoes and long sleeve shirts when handling animals),

: represents (wash hands thoroughly on a regular basis and shower after work),

: represents (clean up both work place and home).

Our control strategies by using the above three control variables can be easily implemented to eradicate the spread of this disease in the community.

The control set for the control variables is defined as,

The above control variables in the system (1) are adjusted in the following form

with the initials conditions given in (2).

Here represents the constant at which the rate of vector decreases at time t. The factor and, are used to reduce the force of infections.

Our aim is to decrease the number of susceptible, exposed human and total vector population and increase the recovered human population. In order to do this, we define the objective functional is given by

The objective functional includes the susceptible individuals, exposed individuals, and the class of vector population. The constants and for are weight/balance factors to keep the balanced of individuals in the objective functional. The Lagrange for the control problem (4) is given by

To do this, we define the Hamiltonian H for the control problem as follows:

In order to understand the dynamical behavior, we find the threshold quantity, also known as the basic reproductive number. This number is obtained by setting the right hand side of all equations equal to zero of the system (1) without control and the system (4) with control and do some rearrange to get the following two basic reproductive numbers. We obtain two reproductive numbers and form the above two systems without and with optimal control, respectively. The threshold quantity denoted by for the system (1) without optimal control variable is given by,

where,

The threshold quantity for control problem in the control system (4) is given by

where,

, , and are defined above for both threshold quantity and.

In this section, we find the endemic equilibria of the control system (4) and check that the backward bifurcation of the optimal control problem exists or not. For the endemic equilibria we set left hand side of the control system (4) equal to zero, to obtain

Here ,

, ,.

In order to find the backward bifurcation, we put the above endemic equilibria in the first equation of the system (4), with setting left hand side equal to zero to get

where,

Here the coefficient a is positive always and c depends upon the value of, if the value of, then c is positive, otherwise negative. The positive solution of the above equation depends upon the value of b and c. For the value of, the above equation leads to two different roots one positive and negative. If we substitute, then the equation has no positive solution. This is possible if and only if b < 0. For b < 0 and, the equilibria depends upon then there exists an open interval having two positive roots that is

and.

For either or, then the above have no positive solution. For backward bifurcation, we set, and solving for the critical value of, which is given by

The numerical simulation of the backward bifurcation is obtained by using MATLAB. First we find the numerical results represented in Figures 1-3 for control variable respectively.

In this section, we show the existence of the control system (4). Let and be the state variables with control variables and. We can write the system (4) in the following form:

where

where denotes the derivative with respect to time t. The system (8) is a non-linear system with bounded coefficients. We set

The second term on the right hand side of (9) satisfies

where the positive constant

is independent of the state variables. Also we have

where

So, it follows that the function G is uniformly Lipschitz continuous. From the definition of control variables and non-negative initial conditions we can see that a solution of the system (8) exists see [

Theorem 5.1: There exists an optimal control

such that

subject to the control system (4) with the initial conditions (2).

Proof: For the proof of this result, we use the same result presented in [

is convex in the control set U. Also we can easily see that, there exists a constant and positive numbers and such that

which shows the existence of an optimal control problem.

To find the optimal solution to the control problem (4), we using the necessary conditions presented in [23,24] are given by

Now we apply the necessary conditions to Hamiltonian (7), for our optimal solution.

Theorem 5.2: Suppose and be the optimal state solutions with associated optimal control variables for the optimal control problem (4), with the initial conditions (2). Then there exists adjoint variables, for satisfying

with transversality conditions (or boundary conditions)

Furthermore, optimal controls and are given by

Proof: To prove the above result, i.e. the adjoint equation and the transversallity conditions, we use the Hamiltonian (7). The adjoint system was obtained by Pontryagin’s Maximum Principle [23,24].

with To obtained the required characterization of the optimal control given by (13) to (15), solving equations,

in the interior of the control set and by the control space U, we derive Equations (13) to (15).

In this section, we present numerical simulations of the system (1) and the control system (4). We use forward Runge-Kutta order four schemes to solve both the system (1) and the control system (4). For the numerical solution of the adjoint system (11), we use backward Runge-Kutta order four schemes because of the transversality conditions or boundary conditions (12). For numerical simulation we consider parameters value presented in

In