Mean-field dispersal induced synchrony and stability in an epidemic model under patchy environment

Abstract

In our highly connected modern world, the dispersion of population is very important at the time when disease spread to evaluate its nature i.e., whether the disease will die-out or grow. Therefore, in this paper, SIR (Susceptible–Infected–Recovered) model is framed to describe the dispersal of population for different patches (communities) which are interconnected using mean field diffusive coupling. The effect of coupling on the dynamics of the epidemiology classes among different patches has been investigated. The dynamics of the epidemiology classes is explored under the environment of symmetric coupling and asymmetric coupling. It is shown that the synchronization and stability of Disease Free Equilibrium (or Endemic Equilibrium) can be attained simultaneously through the mechanism of suppression of oscillations namely amplitude death (AD) and oscillation death (OD). Irrespective of coupling environment i.e., symmetric and asymmetric, we have observed the tri-stable state and its transition from different states through Hopf bifurcation and transcritical bifurcation. The basic reproduction number is also established for the stability of Disease Free Equilibrium and Endemic Equilibrium. Through bifurcation analysis, it has been found that the disease can become stable or it can die out through dispersal of population among patches.