Dynamics of a non-smooth epidemic model with three thresholds

A non-smooth epidemic model with piecewise incidence rate dependent on the derivative of the case number is proposed for the transmission dynamics of an infectious disease with media coverage, enhanced vaccination and treatment policy. This is an implicitly defined system, which is converted into an explicit system with three thresholds by employing the properties of the Lambert W function. We first analyze the dynamics of the proposed model for the limiting case, which induces two non-smooth but continuous models. The dynamic analysis of the model demonstrates that either one of the two generalized equilibria or the pseudo-equilibrium is globally asymptotically stable if the disease does not die out. This suggests that the case number can be contained either at an a priori level or at a high/low level, depending on the threshold, which governs whether the enhanced vaccination and treatment policies are implemented. Media coverage cannot help eradicate the disease, but it significantly delays the epidemic peak and lowers the peak case number. Hence, a good threshold policy and continuously updating the awareness of case numbers are required to combat the disease successfully.