A periodic SEIRS epidemic model with a time-dependent latent period.

Many infectious diseases have seasonal trends and exhibit variable periods of peak seasonality. Understanding the population dynamics due to seasonal changes becomes very important for predicting and controlling disease transmission risks. In order to investigate the impact of time-dependent delays on disease control, we propose an SEIRS epidemic model with a periodic latent period. We introduce the basic reproduction ratio [Formula: see text] for this model and establish a threshold type result on its global dynamics in terms of [Formula: see text]. More precisely, we show that the disease-free periodic solution is globally attractive if [Formula: see text]; while the system admits a positive periodic solution and the disease is uniformly persistent if [Formula: see text]. Numerical simulations are also carried out to illustrate the analytic results. In addition, we find that the use of the temporal average of the periodic delay may underestimate or overestimate the real value of [Formula: see text].