Stability and Hopf bifurcation analysis of a networked SIR epidemic model with two delays

Population migration within spatial structures significantly influences disease spread. Given the typically uneven environment, individuals randomly interact with various others, facilitating disease transmission over time. Complex networks are integrated into infectious disease models and effectively capture these contact dynamics. In this paper, we propose a two-delay networked SIR epidemic model featuring a Crowley-Martin type incidence rate and Holling III type treatment rate. The stability of three equilibria of the model is proved by analyzing the distribution of characteristic roots. When two delays change at the same time, the stable region of the equilibrium on delays plane and the existence of Hopf bifurcation are obtained by the method of stability switching curves. Furthermore, we calculate the normal form of Hopf bifurcation to obtain the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulation on a small-world Watts-Strogatz network is performed to illustrate the theoretical results.