A study of a prey-predator system with disease in prey

In this paper, the dynamical behavior of some eco-epidemiological models is investigated. Two types of prey-predator models involving infectious disease in prey population, which divided it into two compartments; namely susceptible population S and infected population I

lthough the dynamics of two species prey-predator model with Holling type-II functional response received a lot of attention in literature, it is well known that in nature there are different factors in any given environment, such as disease, refuge, switches, age structure, etc. effect the dynamics of such model. Anderson and May [1] were the first who formulated a prey-predator model involving disease in prey species. Later on many researchers, especially in the last two decades, have proposed and studied different types of prey-predator models in the presence of disease in one of the species, see for example Haque and Chattopadhyay [2] which studied the role of transmissible diseases in a prey dependent prey-predator system with prey infection; Li et al [3] proposed the SIS model with a limited resource for treatment. In most of the previous studies, the only way of transmission of disease is taken as the direct contact between the individuals. However, many diseases are transmitted to the susceptible individuals in the species not only through direct contact, but also indirectly from environment. Das et al [4] proposed on a prey-predator model with disease in prey that spread by contact and external sources, included Holling type II as a functional response. Dobson [5] studied the situations where the behavior of the infected host is modified by the action of a parasite. the infected prey may become weaker and less active so that they may be easily caught by the predator (Moore , [6]) In this chapter a prey-predator model involving SIS infectious disease in prey species is proposed and analyzed. It is assumed that the disease transmitted within the prey population by contact and an external sources .The existence, uniqueness and boundedness of the solution are discussed. The existence and the stability analysis of all possible equilibrium points are studied. Finally, the global dynamics of the model is carried out analytically as well as numerically.

The mathematical model:
In this section, a prey-predator system involving an SIS epidemic disease in prey population is proposed for study. In the presence of disease, the prey population is divided into two classes: the susceptible individuals represents the infected individuals at time t. The prey population grows logistically with intrinsic growth rate r and environmental carrying capacity . We impose the following assumptions: A 1. Only the susceptible prey can reproduce logistically, however the infected prey can't reproduce but still has a capability to compete with the other prey individuals for carrying capacity. 2. The susceptible prey becomes infected prey due to contact between both the species as well as an external sources for the infection with the contact infection rate constant 0  β and external infection rate 0  C . However, the infected prey recover and return to the susceptible prey with a recover rate constant 0  γ . According to the above hypothesis the dynamics of a prey-predator model involving an SIS epidemic disease in prey population can be describe by the following set of nonlinear differential equations: Obviously the interaction functions of the system (1) are continuous and have continuous partial derivatives on the region.
Therefore these functions are Lipschitzian on 3 + R , and hence the solution of the system (1) exists and is unique. Further, in the following theorem, the boundedness of the solutions of the system (1) in 3 + R is established.
it is easy to verify that ( )   if and only if the following conditions are hold.
Now the stability analysis of the above feasible equilibrium points of system (1) are studied analytical with help of Linearization method. Note that it is easy to verify that, the Jacobian matrix of system (1) at the trivial equilibrium point Thus the characteristic equation of ( ) can be written as Accordingly, the eigenvalues of ( ) 1 E J satisfy the following relations: 0 , ) ( . , E is locally asymptotically stable provided that the following condition holds r Cd d C r However, it is saddle point otherwise. The Jacobian matrix of system (1) at the predator free equilibrium point 2 E can be written as: Consequently, the characteristic equation can be written as ( ) Clearly the eigenvalues of this Jacobian matrix satisfy the following relationships: ,and Accordingly the equilibrium point 2 E is locally asymptotically stable provided that: However, it is a saddle point otherwise. The Jacobian matrix of system (1) at 3 E is given by ( ) ( ) 3 where: ,Hence straight forward computation show That equilibrium point 3 E is locally asymptotically stable provided that Proof: Consider the following positive definite function: is continuously differentiable function so that ( ) 0 . Therefore by differentiating this function with respect to the variable t we get that:  , , Therefore by differentiating this function with respect to the variable t we get: . R that satisfy the given conditions.

Numerical Simulation:
In this section the dynamical behavior of system (1) is studied numerically for different sets of parameters and different sets of initial points. The objectives of this study are: first investigate the effect of varying the value of each parameter on the dynamical behavior of system (1) and second confirm our obtained analytical results. Now for the following set of hypothetical parameters values: The trajectory of the system (1) is drawn in the Fig.(1)for different initial points. In the above figure, system (1) approaches asymptotically to the stable coexistence equilibrium point starting from different initial points. Note that in time series figures, we will use throughout this section that: blue color for describing the trajectory of S ; green color for describing the trajectory of I ; red color for describing the trajectory of Y . Now in order to discuss the effect of varying the intrinsic rate r on the dynamical behavior of system (1), the system (1) Eng. &Tech.Journal, Vol. 33,Part (B), No.3,2015 A study of a prey- Fig.(2a) ,(b) time series for the attractor in Fig.(2b), (c) time series for the attractor in Fig. (2c) ,(d) time series for the attractor in Fig. (2d) The effect of contact infection rate β on the dynamic of system (2.1)studied and the trajectories of system (1) are drawn in Fig. (4a)-( 4c) for the values β = 0.05, 0.5, respectively, keeping other parameters fixed as given in Eq.(10). Journal, Vol. 33,Part (B), No.3,2015 A study Fig. (4a) (b) time series for the attractor in Fig. (4b) , (c) time series for the attractor in Fig. (4c) The effect of varying recover rate γ on the dynamic behavior of system (1) is studied and the trajectories of system (1) are drawn in Fig. (6a)-( 6c) for the values γ = 0.01,0.1,0.8 respectively keeping other parameters fixed as given in Eq. (10). The effect of varying half-saturation constant m the dynamic behavior of system (1) is studied and the trajectories of system (1) are drawn in Fig. (10a)-(10c) for the values m = 10,60 respectively.

Discussion and conclusion:
In this chapter, we proposed and analyzed an eco-epidemiological model that describe the dynamical behavior of a prey-predator model with linear functional response. The model consisting of three non-linear differential equations that describe the dynamics of three different populations namely predator Y, susceptible prey S, infected prey I. The boundedness of the system (1) has been discussed. The conditions for existence and stability of each equilibrium points are obtained. To understand the effect of varying each parameter on the dynamical behavior of the system a numerically simulation has been used and the obtained results can be summarized as follow 1. Decreasing the intrinsic growth rate r in the range 02 . 0 ≤ r causes that extinction in all populations and the system (1) 3 E loses its stability and the system approaches asymptotically to the periodic dynamic in the 3 . + R Int