Threshold dynamics in an SEIRS model with latency and temporary immunity.

A disease transmission model of SEIRS type with distributed delays in latent and temporary immune periods is discussed. With general/particular probability distributions in both of these periods, we address the threshold property of the basic reproduction number R0 and the dynamical properties of the disease-free/endemic equilibrium points present in the model. More specifically, we 1. show the dependence of R0 on the probability distribution in the latent period and the independence of R0 from the distribution of the temporary immunity, 2. prove that the disease free equilibrium is always globally asymptotically stable when R0, and 3. according to the choice of probability functions in the latent and temporary immune periods, establish that the disease always persists when R0 < 1 and an endemic equilibrium exists with different stability properties. In particular, the endemic steady state is at least locally asymptotically stable if the probability distribution in the temporary immunity is a decreasing exponential function when the duration of the latency stage is fixed or exponentially decreasing. It may become oscillatory under certain conditions when there exists a constant delay in the temporary immunity period. Numerical simulations are given to verify the theoretical predictions.