A predator-prey model with SEIR and SEIRS epidemic in the prey

Abstract

In this paper, the author studies a predator-prey model based on epidemic disease. The epidemic disease is described by SEIR (Susceptible-Exposed-Infected-Recovered) and SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) models with a logistic growth function only in the susceptible prey population. The function of response in the predator-prey model is given by a Lotka-Volterra type. Notice, all epidemic models are characterized by the basic reproduction number R0. This number gives information whether the epidemic can be controlled. In this work, the author introduces two systems of five nonlinear ordinary differential equations which describe the eco-epidemiological model above using the following functions: susceptible prey S (t), exposed prey E(t), infected prey I(t), recovered prey R(t) and predator P(t). The solutions of the systems are studied by proving their positivity and boundedness. A set of equilibrium points is presented and conditions for their existence and stability are analyzed. A formula for the basic reproduction number R0 is found. Biological interpretation for different equilibria depending on R0 is discussed. The numerical experiments presented show some differences in the dynamic of all populations when the SEIR and SEIRS epidemic models are used.