Ecoepidemiological Model and Analysis of Prey-Predator System

Abstract

In this paper, the prey-predator model of five compartments is constructed with treatment 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 analyses of trivial equilibrium, axial equilibrium, and disease-free equilibrium points are performed with the method of variation matrix and the Routh-Hurwitz criterion. It is found that the trivial equilibrium point $E_o$ is always unstable, and axial equilibrium point $E_A$ is locally asymptotically stable if $\beta k-\left(t_1+d_2\right)<0,q{p}_{1}k-d_3\left(s+k\right)<0$ and $q{p}_{3}k-\left(t_2+d_4\right)\left(s+k\right)<0$ conditions hold true. Global stability analysis of an endemic equilibrium point of the model has been proven by considering the appropriate Lyapunov function. The basic reproduction number of infected prey and infected predators are obtained as $R_{01}={\left(q{p}_1-d_3\right)}^{2}k\beta d_3{s}^2/\left(q{p}_1-d_3\right)\left\{{\left(q{p}_1-d_3\right)}^{2}ks\left(t_1+d_2\right)+rsq{p}_2\left(kq{p}_1-k{d}_3-d_{3}s\right)\right\}$ and $R_{02}=\left(q{p}_1-d_3\right)\left(q{p}_3{d}_3\right)k+\alpha rsq\left(kq{p}_1-k{d}_3-d_{3}s\right)/{\left(q{p}_1-d_3\right)}^2\left(t_2+d_4\right)k$ , respectively. If the basic reproduction number is greater than one, then the disease will persist in the prey-predator system. If the basic reproduction number is one, then the disease is stable, and if the basic reproduction number is less than one, then the disease dies out from the prey-predator system. Finally, simulations are done with the help of DEDiscover software to clarify results.