Hopf bifurcation and patterns in a modified SIR model

Abstract

Infectious diseases have constantly threatened human safety because the diffusion of the susceptible and infected may make more individuals infected and even die. In this paper, a modified SIR model with both external stimulus and diffusion is considered to illustrate the dynamical mechanism of the periodic outbreak and pattern formation. Firstly, we propose a modified SIR model based on the propagation behaviour of infectious diseases to show the effects of the different parameters and diffusion on the outbreak. The Hopf bifurcation and multiscale methods are performed to analyze the stability of this model, which explains the dynamical mechanism of the periodic outbreak. Then, the pattern formation and Turing instability are discussed through comparison principles to reveal the role of periodic disturbances and diffusion in selecting pattern formation. Also, we find rich patterns that may occur when the frequency modulation is close to the intrinsic frequency. Finally, our theoretical results are verified by numerical simulation.