Hopf bifurcation in a time-delayed multi-group SIR epidemic model for population behavior change

In this study, we construct a time-delayed multi-group SIR epidemic model to discuss the impact of population behavior change on the occurrence of recurrent epidemic waves. We obtain the basic reproduction number $R_0$ and show that if $R_0\le 1$, then the disease-free equilibrium is globally asymptotically stable, whereas if $R_0>1$, then the disease-free equilibrium is unstable and an endemic equilibrium exists. In a special two-group case, we show sufficient conditions for Hopf bifurcation and obtain index values that determine the direction, stability and period of bifurcated periodic solutions. By numerical simulation, we investigate the occurrence of periodic solutions in two groups representing an urban area and a non-urban area. We conclude that the epidemic size, response intensity of population behavior change and heterogeneity in two different groups can be the key factors of the occurrence of recurrent epidemic waves.