A CAREFUL STUDY OF THE EFFECT OF THE INFECTIOUS DISEASES AND REFUGE ON THE DYNAMICAL BEHAVIOR OF PREY-SCAVENGER MODELING

In this paper, the dynamics of scavenger species predation of both susceptible and infected prey at different rates with prey refuge is mathematically proposed and studied. It is supposed that the disease was spread by direct contact between susceptible prey with infected prey described by Holling type-II infection function. The existence, uniqueness, and boundedness of the solution are investigated. The stability constraints of all equilibrium points are determined. In addition to establishing some sufficient conditions for global stability of them by using suitable Lyapunov functions. Finally, these theoretical results are shown and verified with numerical simulations.


INTRODUCTION
The terms mathematical model is one of the most important subjects for study and has a wider scope. The first mathematical model in the field of ecology that involves the interactions between 3

THE MATHEMATICAL MODEL FORMULATION
In this section an eco-epidemiological model consisting of a prey-scavenger model incorporating prey refuge with infectious disease in the prey is proposed for study. In order to construct our model the following hypotheses considered: 1. In the absence of disease, the prey population grows logistically with carrying capacity and intrinsic birth rate .
2. In the existence of SIS infectious disease, the prey population is divided into two groups, namely susceptible prey denoted by ( ) and infected prey denoted by ( ). Therefore at time t, the total population is ( ) = ( ) + ( ).
3. Disease spreads among the prey population and it transmitted between the prey individuals (but not the scavenger) by contact, according to Holling type-II infection function with maximum incidence rate and half saturation constant . Further the disease disappears and the infected prey becomes susceptible prey again at a recover rate 1 .
4. The susceptible prey is capable of reproducing only and the infected prey is removed by death at a natural rate 1 .
5. The susceptible prey species are assumed to take a refuge. That is (1 − ) , is a prey refuge constant, of the susceptible prey is available for feeding by scavenger. 6. According to nature of scavenger, we assume scavenger feeds upon susceptible prey killed by other animals or dead naturally according to ratio-dependent functional response with maximum attack rate and half saturation constant or linearly with maximum attack rate 1 , respectively. The consumed susceptible prey, which killed by other animals, is converted into scavenger with efficiency 1 . Also, we assume scavenger feeds upon infected prey killed by other animals or dead naturally by linear functional response with maximum attack rates or 2 , respectively. The consumed infected prey, which killed by other animals, is converted into scavenger with efficiency 2 .
7. Finally in the absence of the prey the scavenger decay exponentially with natural death rate 2 and intra-specific competition rate 3 .
According to the above assumptions the prey-scavenger model (1) can be modified to the following set of differential equations.

BOUNDEDNESS OF THE MODEL
Theorem (1): All the solutions of system (1), which initiate in ℜ + 3 are uniformly bounded provided that the following condition holds Proof: note that the prey population is ( ) = ( ) + ( ), so when = 0 the first equation of system (1) can be rewritten as: The right handside must be positive that implies (1 − ) > 0.

EXISTENCE OF EQUILIBRIUM POINTS
The system (1) has at most five non negative equilibrium points, namely = ( , , ) where = 0, ⋯ ,4. The existence conditions for each of these equilibrium points are established in the following: 1. The vanishing equilibrium point 0 = (0,0,0) always exists.

STABILITY OF THE MODEL
At equilibrium points ; = 1, ⋯ ,4 the Jacobian matrix of the system (1)  Here: In fact, they are Lipschizian on ℜ + 3 . Accordingly, the solution of the system (1) with nonnegative initial condition exists and is unique. Thus, the int.ℜ + 3 is invariant for system (1). Clearly, the system (1) Functions ( , , ); = 1,2,3 are continuous and have second order derivatives on ℜ + 3 .
Accordingly, the solution of the system (8) with nonnegative initial condition exist and is unique.
The Jacobian matrix ≡ ℋ ( , , ) for system (8)  where: Then, the Jacobian matrix of system (8) at the equilibrium point 0 is: And the characteristic equation is since we have positive and negative eigenvalues then 0 is saddle point.

DYNAMICS OF THE SYSTEM AROUND EQUILIBRIUM POINT
The Jacobian matrix at equilibrium point 1 is: And the characteristic equation is: Then, the equilibrium point 2 is asymptotically stable if the conditions hold Otherwise the equilibrium point 2 is saddle point.

DYNAMICS OF THE SYSTEM AROUND EQUILIBRIUM POINT
The Jacobian matrix at equilibrium point 2 is: Implies equilibrium point 2 is asymptotically stable and it's Saddle point otherwise.

DYNAMICS OF THE SYSTEM AROUND EQUILIBRIUM POINT
The Jacobian matrix at equilibrium point 3 is: Obviously, the equilibrium point 3 is asymptotically stable if the following conditions hold, and 3 Saddle point otherwise.

Theorem (3):
The equilibrium point 1 is a globally asymptotically stable provided that the following conditions hold And the derivative of 1 ( ) with respect to time can be written as 1 = ( − 1 ) + + So, by using system (1) with some algebraic manipulations we get Clearly, 1 is negative definite function under the conditions (17.a-17.c). Moreover it's clear that the function 1 ( ) is radially unbounded; then according to the Lyapunov first theorem the equilibrium point 1 is a globally asymptotically stable point.

Theorem (4):
The equilibrium point 2 is globally asymptotically stable provided that the following sufficient conditions hold Now, by using conditions (18.a-18.c) guarantees that 2 < 0. It's clear that the equilibrium point 2 is a globally asymptotically stable point.

Theorem (5):
The equilibrium point 3 is globally asymptotically stable that satisfied the following conditions Consequently, by using conditions (20.a-20.c) we get that in theorem (6), which determined the sufficient condition (20) for globally stable positive equilibrium point 4 .

DISCUSSION AND RESULTS
In this paper, the interaction dynamics of prey and scavenger proposed and analyzed. 5. According to the above discussion, it's observed that system (1) is sensitive to varying in many of its parameters and hence there is higher possibility to control.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.