Dynamics of disease spread in a predator-prey system

Maǧallaẗ al-handasaẗ wa-al-tiknūlūǧiyā, 2015

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, are proposed and analyzed. The proposed model deals with SIS infectious disease that transmitted directly from external sources, as well as, through direct contact between susceptible and infected individuals. The model are represented mathematically by the set of nonlinear differential equations .The existence, uniqueness and boundedness of this model are investigated. The local and global stability conditions of all possible equilibrium points are established. Finally, using numerical simulations to study the global dynamics of the model .

Journal of Mathematical and Computational Science, 2012

In this paper an eco-epidemiological model, consisting of a Crowley-Martin prey-predator with disease in prey ,is investigated analytically as well as numerically. The conditions for the existence and local stability of equilibrium points are obtained. The global dynamics is studied numerically for different sets of initial values and for different sets of parameters values.

2003

We propose a model to describe the interaction between a diseased sh population and their predators. Analysis of the system is performed to determine the stability of equilibrium points for a large range of parameter values. The existence and uniqueness of solutions is established and solutions are shown to be uniformly bounded for all nonnegative initial conditions. The model predicts

Mathematical Methods in the Applied Sciences, 2003

The present paper deals with the problem of a classical predator-prey system with infection of prey population. A classical predator-prey system is split into three groups, namely susceptible prey, infected prey and predator. The relative removal rate of the susceptible prey due to infection is worked out. We observe the dynamical behaviour of this system around each of the equilibria and point out the exchange of stability. It is shown that local asymptotic stability of the system around the positive interior equilibrium ensures its global asymptotic stability. We prove that there is always a Hopf bifurcation for increasing transmission rate. To substantiate the analytical ÿndings, numerical experiments have been carried out for hypothetical set of parameter values. Our analysis shows that there is a threshold level of infection below which all the three species will persist and above which the disease will be epidemic.

IOP Conference Series: Materials Science and Engineering, 2019

A predator-prey model with disease in both populations is proposed to illustrate the possibility of disease transmission between prey and predator through contact and predation. We used saturated incidence rate which takes behavioural changes of healthy population into consideration when disease spreads around them. The existence of eight non-negative equilibrium points is analysed and their local stability has been investigated. Numerical simulations are given to illustrate analytic results.

International Journal of Computing Science and Mathematics, 2016

This paper presents a nonlinear mathematical model of prey-predator interaction in which the prey is infected by an infectious disease while assuming that the disease is not transmitted to predator though the rates of predation can be different for the susceptible and infected preys. We also assume that only susceptible prey population contributes in the reproduction. The infective population competes with susceptible population to population growth towards the carrying capacity and here the disease transmission follows the standard incidence. The basic reproduction numbers both in absence and presence of the predator are computed and the equilibria of the mathematical model are obtained. Our results show that there is a possibility of two coexistence equilibria for some set of parameters but only one of them can be locally asymptotically stable. We also observed that the system undergoes 'Hopf-bifurcation' when the maximum predation rate β crosses a threshold value. Finally, the numerical simulation is performed and that supports the analytical findings.

The paper is done to study the Spread of disease in prey-predator population. For these problem a Dynamical system of differential Equations has been proposed. The Positivity, Boundedness and existence of model solutions of the Equation has been analyzed and proved. Existence of all possible Equilibrium has been checked and computed. Stability Analysis of all Equilibrium points of the model has been done. Moreover Local and global stability of disease free and endemic equilibrium points are established with concept of Jacobian matrix and Routh Hurwitz criterion respectively. Numerical simulations are presented to clarify analytical results.

Nonlinear Analysis: Theory, Methods & Applications, 1999

Parasitism and predation are two ecological interactions that can occur simultaneously in any system of species. Specially, predation becomes particularly interesting in host/prey – parasite systems because predation can significantly modify the abundance of parasites and their host populations. The combined effect of parasites and predator on host/prey population leads to a larger effect on the dynamics of the population sizes. In this paper a prey – predator system is considered. The host species or prey population is categorized into susceptible and infected due to the presence of parasites. Predators are assumed to consume both the susceptible and infected hosts/prey with some partial preference given to susceptible ones. Thus, a mathematical model is developed to describing the population dynamics of susceptible prey – Infected prey – Predator system. Positivity and boundedness of the model are verified. Disease free equilibrium is found and shown that it is locally and asymptotically stable. Interior equilibrium is also identified and shown that it is locally, asymptotically and globally stable. Simulation study is conducted so as to verify the results of mathematical analysis. Different simulation scenarios are presented by assigning varying values to the parameters of the system using mathematical soft ware. Lastly, conclusions of the results are forwarded

Journal of Theoretical Biology, 2009

We study the effects of a disease affecting a predator on the dynamics of a predator-prey system. We couple an SIRS model applied to the predator population, to a Lotka-Volterra model. The SIRS model describes the spread of the disease in a predator population subdivided into susceptible, infected and removed individuals. The Lotka-Volterra model describes the predator-prey interactions. We consider two time scales, a fast one for the disease and a comparatively slow one for predator-prey interactions and for predator mortality. We use the classical ''aggregation method'' in order to obtain a reduced equivalent model. We show that there are two possible asymptotic behaviors: either the predator population dies out and the prey tends to its carrying capacity, or the predator and prey coexist. In this latter case, the predator population tends either to a ''disease-free'' or to a ''disease-endemic'' state. Moreover, the total predator density in the disease-endemic state is greater than the predator density in the ''disease-free'' equilibrium (DFE).