Traditional models of respiratory diseases overlook the spatial heterogeneity of populations and \(\hbox {PM}_{2.5}\) concentration in disease transmission. To capture the variations in the spatial distribution of populations, we propose a delay-induced susceptible-infected-susceptible- \(\hbox {PM}_{2.5}\) concentration (SISP) respiratory diseases model on networks. According to the Hurwitz criterion and the Hopf bifurcation theorem, the local asymptotic stability at the equilibria and the existence of Hopf bifurcation are given. Moreover, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are discussed by using the normal form and center manifold theory. In addition, by constructing an appropriate Lyapunov functional and employing LaSalle’s invariance principle, we study the global asymptotic stability of the equilibria.
