Dynamics of an infectious diseases with media/psychology induced non-smooth incidence

This paper proposes and analyzes a mathematical model on an infectious disease system with a piecewise smooth incidence rate concerning media/psychological effect. The proposed models extend the classic models with media coverage by including a piecewise smooth incidence rate to represent that the reduction factor because of media coverage depends on both the number of cases and the rate of changes in case number. On the basis of properties of Lambert W function the implicitly defined model has been converted into a piecewise smooth system with explicit definition, and the global dynamic behavior is theoretically examined. The disease-free is globally asymptotically stable when a certain threshold is less than unity, while the endemic equilibrium is globally asymptotically stable for otherwise. The media/psychological impact although does not affect the epidemic threshold, delays the epidemic peak and results in a lower size of outbreak (or equilibrium level of infected individuals).