Stability and bifurcation on a delayed multi-strain patch epidemic model with varying incidence and migration induced by media coverage

In this paper, a delayed multi-strain patch SIS epidemic model with varying incidence and migration of susceptible individuals induced by media coverage of infected individuals is proposed. For the subsystem without migration, if there is no delay, criteria on the globally asymptotical stability of (disease-free, dominant and coexistent) equilibria are obtained. Otherwise, criteria on the local Hopf bifurcations at the dominant and coexistent equilibria are obtained, meanwhile, the global Hopf bifurcation is unbounded. For the model with migration, the basic reproduction number R0 is derived, by which criteria on the locally and globally asymptotic stability of disease-free equilibrium are derived. The uniform persistence of (strain 1 or 2 dominant and coinfection) disease are obtained respectively. The theoretical results are illustrated by numerical simulation, from which we find that R0 is non-monotonic with the migration rates between patches. Ignoring the multi-strain factor will greatly underestimate the scale of the disease, and even obtain the opposite conclusion (disease extinction or persistence). Delay could cause periodic oscillation, and the multi-strain and migration between patches could further produce chaos. Both the media effect with delay and migration could greatly influence the transmission dynamics of disease from stability to unstability until chaos, which further increase the difficulty of disease control.