Epidemic spreading on multi-layer networks with active nodes

Investigations on spreading dynamics based on complex networks have received widespread attention these years due to the COVID-19 epidemic, which are conducive to corresponding prevention policies. As for the COVID-19 epidemic itself, the latent time and mobile crowds are two important and inescapable factors that contribute to the significant prevalence. Focusing on these two factors, this paper systematically investigates the epidemic spreading in multiple spaces with mobile crowds. Specifically, we propose a SEIS (Susceptible-Exposed-Infected-Susceptible) model that considers the latent time based on a multi-layer network with active nodes which indicate the mobile crowds. The steady-state equations and epidemic threshold of the SEIS model are deduced and discussed. And by comprehensively discussing the key model parameters, we find that (1) due to the latent time, there is a "cumulative effect" on the infected, leading to the "peaks" or "shoulders" of the curves of the infected individuals, and the system can switch among three states with the relative parameter combinations changing; (2) the minimal mobile crowds can also cause the significant prevalence of the epidemic at the steady state, which is suggested by the zero-point phase change in the proportional curves of infected individuals. These results can provide a theoretical basis for formulating epidemic prevention policies.