Eco-Epidemiological Modelling and Analysis of Prey-Predator Population

In this paper, prey-predator model of five Compartments are constructed with treatment is given to infected prey and infected predator. We took predation incidence rates as functional response type II and disease transmission incidence rates follow simple kinetic mass action function. The positivity, boundedness, and existence of the solution of the model are established and checked. Equilibrium points of the models are identified and Local stability analysis of Trivial Equilibrium point, Axial Equilibrium point, and Disease-free Equilibrium points are performed with the Method of Variation Matrix and Routh Hourwith Criterion. It is found that the Trivial equilibrium point is always unstable, and Axial equilibrium point is locally asymptotically stable if βk (t1+d2) < 0, qp1k d3(s+k) < 0, & qp3k (t2+d4)(s+k) < 0 conditions hold true. Global Stability analysis of endemic equilibrium point of the model has been proved by Considering appropriate Liapunove function. In this study, the basic reproduction number of infected prey is obtained to be the following general formula R01=[(qp1-d3) 2 kβd3s 2 ]⁄[(qp1-d3){(qp1-d3) 2 ks(t1+d2 )+rsqp2 (kqp1-kd3-d3s)}] and the basic reproduction number of infected predator population is computed and results are written as the general formula of the form as R02=[(qp1-d3 )(qp3 d3 )k+αrsq(kqp1-kd3-d3s)]⁄[(qp1-d3) 2 (t2+d4)k]. If the basic reproduction number is greater than one, then the disease will persist in prey-predator system. If the basic reproduction number is one, then the disease is stable, and if basic reproduction number less than one, then the disease is dies out from the prey-predator system. Finally, simulations are done with the help of DEDiscover software to clarify results.


Introduction
Mathematical Modeling of prey-predator systems of interaction of species have a long history since original remarkable work was done by Lotka-Volterra Model in 1920 [1,3,5,6], and SIR model Compartment of systems of population is another vital area of research after pioneering work of Kermack and Mckendrick [1][2][3][5][6][7][8][9][10]. Anderson and May where the first who combined these two modeling systems, while Chattopadhyay and Arino were the first who used the term ''eco-epidemiology'' for such models [3,5,7]. The dynamics of disease in prey-predator systems now become an interesting area of research due to the fact that prey-predator interaction is rich and complex in nature [4,6,7,[11][12][13]. Several mathematical models have been proposed and studied on prey-predator systems [1][2][3][4][5][6][7][9][10][11][12]. Many studies focused on the study of disease in a prey only [1-5, 7, 12], other researchers were interested in the study of disease within the predator population only [14], and there are also some studies on diseases in both prey and predators [6,9,11] In this paper, we proposed and studied infectious disease on both prey and predator interaction of species with treatment given to infected prey and infected predator.

Model Formulation and Assumptions
In this paper, the prey-predator population divided into five compartments. let us denote X(t)-Susceptible prey, infected prey, -Susceptible predator, -infected predator, both infected prey and infected predator population under treatment. In the absence of infectious disease, the susceptible prey population grows logistically with intrinsic growth rate and environmental carrying capacity and only susceptible prey can reproduce. In the presence of infectious disease, susceptible predator become infected predator when they come into contact with infected predator, susceptible prey become infected prey when they come into contact with infected prey and the contact process assumed to follow bilinear functional with convolution rate , respectively. The predation functional response of predator towards the prey assumed to follow a different holling type II functional response form with , respective predation coefficients of , due to susceptible predator, and , respective predation coefficient of , due to infected predator. suppose Consumed prey converted into predator with efficiency and also half saturated constant . It is also assumed that Infected prey and infected predator can only recover through treatment, and treated at treatment rate of , respectively. The prey-predator population , , , and suffer from infectious disease with death rate , , , and respectively. Moreover, Assume that all variables and parameters used in the model are non negative. According to the above assumptions, we have the following Model flow diagram From the Model flow diagram in Figure 1 we have the following set of differential equations with initial conditions X 0 ≥ 0, W 0 ≥ 0, Y 0 ≥ 0, Z 0 ≥ 0, H 0 ≥ 0, p 5 > 0, ! = 1,2,3,4, & 0 < q ≤ 1 Depending on the assumptions of per capita growth of function * , for susceptible prey, and different type II functional responses ' , ! = 1,2,3,4 .We have more feasible model (6)-(10) emanated from model (1) with initial conditionsX 0 ≥ 0, W 0 ≥ 0, Y 0 ≥ 0, Z 0 ≥ 0, H 0 ≥ 0, p , p , p , p > 0 & 0 < ≤ 1

Mathematical Analysis of the Model
In this section, positivity, boundedness, and existence of the solution of the model is checked. This mathematical analysis of the model could be considered as primarly results.  ; Without loss of generality, After removing all the positive terms from the right hand side of the differential equation, we have the following differential inequality; dX dt ⁄ ≥ : rX + rXW k ⁄ ; + βXW + : P XY + P XZ S + X ⁄ ; divide both sides by negative yields: :dX dt ⁄ ; ≤ : rX + rXW k ⁄ ; + βXW + : P XY + P XZ S + X ⁄ ;, But It is also clear that the following inequality holds : rX + rXW k ⁄ ; + βXW + : P XY + P XZ S + X ⁄ ; ≤ rX + rXW + βXW + p XY + p XZ = X rX + rW + βW + p Y + p Z Assume that rW + βW + p Y + p Z = C, Then the differential inequality reduced to :dX dt ⁄ ; ≤ X rX + C .This inequality can be arranged for integration by partial fraction asX:  (6) -(10) together with the initial conditions 0 > 0, 0 ≥ 0, 0 ≥ 0, 0 ≥ 0, 0 ≥ 0 exist inℝ C D i.e., the model variables , , , and exist for all and remain in ℝ C D . Proof: From the system of differential equation (6)- (10) given as have partial derivatives in the following Table 3 According to Derrick and Groosman theorem, let Ω denote the region Ω = P , , , , ∈ ℝ C D ; N ≤ μ Λ ⁄ S . Then model (2)-(10) have a unique solution if all partial derivatives of the above functions are continuous and bounded in Ω.
Here, The continuity and the Boundedness can be shown as follows: Thus, all the partial derivatives of these functions exist, continuous, and bounded in a regionΩ forallpositive values ofmodel variable and model parameter. Hence, by Derrick and Groosman theorem, a solution for the model (6)-(10) exists and unique.

Stability Analysis
Stability analysis in the absence of predators in model, That is when y(t) and Z(t) are Zero, model (6) The system (12) with Jacobian matrix is given by , ofmodel (2) that satisfiesdX dt ⁄ = dW dt ⁄ = dY dt ⁄ = dZ dt ⁄ = dH dt ⁄ = 0,provided that each variable is non-negative. In Model(2) Five steady state points are identified and listed here: trivial steady state E § 0, 0, 0, 0, 0 , Axial steady stateE¨ k, 0, 0, 0, 0 ,Disease-free steady state © © , 0, © , 0, 0 and endemic steady state E * X * , W * , Y * , Z * , H * . computation of disease free and endemic equilibrium points are presented as follows: Disease free equilibrium points[DFEP] of model(2)-(10) are steady state Solutions when there is no infectious disease in the population. In the absence of infectious disease in prey-predator system the variables = = = 0 anddX dt ⁄ = dW dt ⁄ = dY dt ⁄ = dZ dt ⁄ = dH dt ⁄ = 0,Then model (2) To study the Stability analysis of equilibrium points of model (6)-(10), it is better to linearize mode (6)-(10) using Variation matrix. Then the Variation Matrix of these functions (6)-(10)is given by Where each element of the matrix represent partial derivatives of functions (6)-(10) with respect to model variables, and Computations of each element of the variation matrix given as: Eigen value of variation matrix can be computed from the characteristic polynomial det V E § ˆ‰ D = 0 The eigen values are: λ = r > 0, λ = t < 0, λ = < 0, λ = t < 0, λ D = r r < 0, Thus the trivial equilibrium point is a saddle point with locally asymptotically unstable manifold in X-direction, and locally asymptotically stable manifold in , , , directions. The Axial equilibrium point E is locally asymptotically stable, if βk t < 0, : qp k s + k ⁄ ; < 0, &: qp k s + k ⁄ ; t < 0 otherwise E is locally asymptotically unstable.  },Otherwise the disease free equilibrium point is asymptotically unstable. Now let see again, the Global stability analysis of model(2) around the endemic equilibrium point or positive equilibrium point * * , * , * , * , * which showsco-existence. For that let us state following theorem and prove by taking appropriate Liapunove function L.

Conclusions and Recommendation
In this paper, It can be concluded that the formulated model is Mathematically meaningful, valid, and biologically well posed by proving the boundedness, positivity and existence of the solutions of the model. Trivial, Axial, Disease-free and endemic Equilibrium points are investigated. Moreover, It is observed that in our model trivial equilibrium point is always locally asymptotically unstable. Axial equilibrium point is locally asymptotically stable if and only if the variables satisfy the following three conditions: i βk t + d 8 0, ii qp k d s + k 8 0, & !!! p k t + d s + k 8 0. Treatment is helpful tool to minimize or eradicate infection in prey-predator system. Therefore Providing treatment in infected prey-predator system creates opportunity to recover from illness and the prey-predator population can be saved and exist in stable situation. Thus, it is recommended to apply treatment on infected prey-predator to make the whole prey-predator population safe and abundant in nature. One can extend this paper by Assuming the predator grows logistically or by adding parameter like death rate on the prey or by including other variables like vaccination, immigration, migration on prey-predator system, and these things can be considered as limitation of this paper.