DYNAMICAL BEHAVIOR OF SOME ECO-EPIDEMIOLOGICAL MODELS

Abstract

A large number of recent studies by researchers have led to an increased interest in the concept of predator-prey system in epidemiology, which leads to new fields of studies popularly known as eco-epidemiology. Predator-prey interactions provide a regulatory mechanism for evolution of biological species in natural ecosystem. It has been observed that not only the disease in the system affects the dynamics of prey predator species, but the prey predator interactions also affect the dynamics of disease. The presence of infectious disease in one or both of the populations leads to reduced population sizes and destabilization of equilibrium into oscillations. The aim of the present research work is to study the dynamical behavior of some eco-epidemiological models. This is an attempt to analyze temporal and spatio- temporal dynamics of some eco-epidemiological systems. Some mathematical models for eco-epidemiology have been proposed and investigated. The effects of migration and diffusion on such models have been studied. Stability of dynamical models has been investigated and bifurcation analysis has been carried out. Numerical simulations have been carried out to explore complexity in nonlinear models. Efforts have also been made to interpret the mathematical results. Further, biological relevance of the mathematical results has also been explored. This thesis is comprised of seven chapters. Chapter wise summary of the thesis is given below: Chapter 1 gives a brief introduction to the dynamics of ecological, epidemiological and eco-epidemiological systems. The related concepts are overviewed in this chapter. Brief discussion on tools / techniques used in the study is also included. It also gives the review of work done in this area. In chapter 2, an SIS eco-epidemiological model for disease transmission in a predator-prey system in which disease is spreading to susceptible prey and consequently to predator species is proposed and studied. It is assumed that only susceptible prey is capable of reproducing logistically. The infected prey are weakened due to disease and become easier to catch, while susceptible prey can easily escape the predation. The infected prey and predator do not reproduce but still continue to consume resources. The susceptible predator gets food from infected prey for its survival. It may become infected due to interaction with infected prey. Mortality rate of infected predator is higher than that of susceptible predator but hunting ability of infected predator is not affected. Further, all the predators who predate the infected prey becomes infected. The model is analyzed mathematically. The next generation approach is used to obtain the epidemiological threshold quantities for the model system. Conditions for the existence and stability of disease free prey predator system are obtained. Conditions for endemic disease in prey species are discussed. Global stability of all the locally asymptotically stable equilibrium is established. Numerical Simulations confirm the results obtained mathematically. The model in Chapter 2 is modified in Chapter 3. In this chapter it is assumed that the infected predators are weaker due to disease, so they cannot hunt with the same rate as that of susceptible predators. It is also assumed that all predators feeding on infected prey do not become infected. Accordingly, a fraction (g) of intake from infected prey contributes towards growth of susceptible predator. Differential predation rates for susceptible and infected predators and parameter g affects the dynamics of the system. Analytical results show the possibility of bi-stability in ecosystem. The nonzero vi equilibrium point may become stable in this case. It is also noticed that for lower predation rate of infected predator, disease may remain endemic in the system. In chapter 4, an eco-epidemic model with migration is studied. The susceptible preys are migrating into a disease infested habitat where the disease is spreading from infected prey to predators through predation. The SI model has been used for both prey and predator species. The epidemiological threshold quantities have been obtained for the model system using next generation approach. Due to migration, the carrying capacity is enhanced. The local and global stability analysis of various equilibrium points have been carried out. It is observed that immigration of susceptible prey at a sufficiently high constant rate stabilizes the system to disease free equilibrium state. The disease may be endemic in both the prey and predator species. The impact of disease in prey on Jhe extinction of the predator is also investigated. In chapter 5, an eco-epidemic model in which whole prey species is infected with disease and the disease is spreading from prey to predator species. The prey dynamics includes logistic growth with Rolling type II functional response while modified Leslie-Gower type dynamics is considered for predator. The predator species - is compartmentalized into susceptible and infected classes. The disease does not cause immunity in the predator species therefore, the SIS model is considered for predator species. Several threshold parameters from local analysis of various equilibria of the proposed system as well as coupled conditions on these thresholds which determine the stability of these equilibria are obtained. Existence of Hopf bifurcation is established. By considering infection in both populations, the present model yields more complex dynamics that includes bi-stability and periodic oscillation. The supercritical as well as subcritical Hopf bifurcations are observed through numerical simulations vii In chapter 6, the emergence of spatial patterns is investigated in an eco-epidemic predator—prey system in a heterogeneous habitat. It is assumed that the infected preys are more vulnerable to predation. Simple law of mass action is assumed for the interaction of susceptible and infected prey. The predators eat both healthy and infected prey at different rates, as the healthy prey may easily escape attacking predator. The conditions for Turing bifurcation on the spatial domain are derived for SIS model. It is observed numerically that infection in prey species may give rise to variety of spatio-patterns. Chapter 7, lists the achievements of the present work and presents a discussion on the future scope of work. viii