Sensitivity and Chaotic Dynamics of an Eco-Epidemiological System with Vaccination and Migration in Prey

A four-dimensional eco-epidemiological system is formulated consisting of susceptible prey, infected prey, vaccinated prey and predator. Mathematically, boundedness of the solutions of the proposed system, and the conditions on existence and stability of equilibrium points are discussed. The basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0$$\end{document} for the system is computed. The effect of death rate of infected prey is analyzed by considering it as bifurcation parameter in transcritical bifurcation. Adoption of vaccination helps to control future disease, and emigration leads for reduction in number of infected prey. Some numerical examples are carried out to substantiate our theoretical findings. The effect of migration is discussed by considering the cases of emigration and immigration. The sensitivity analysis is performed with respect to the migrating terms. The system is found to be chaotic (hyperchaotic) by evaluating Lyapunov exponents and Lyapunov dimension. Our main objective of this work is to balance the prey–predator relationship in the presence of vaccination, infection and migration in prey.