A new SEIRS disease transmission model is formulated for a variable size population that is divided into susceptible, exposed, infective, and recovered classes. The model has recruitment-death demographics and includes a disease-related death rate. Standard incidence, a constant latent period, and a constant period of immunity are assumed. The model is formulated as an integro-differential

... [Show full abstract] equation system, which is shown to be equivalent to a delay differential equation system with two discrete delays. This latter system is analysed. A reproduction number R 0 is identified; if R 0 <1 the disease dies out of the population, and a unique endemic equilibrium exists if and only if R 0 >1. Periodic solutions about the endemic equilibrium are possible for a range of parameter values when R 0 >1. The limiting case for diseases with no latent period, resulting in an SIRS model, is analysed in detail. Periodic solutions are found for a range of parameter values, which is algorithmically determined and related to data on Pasteurella muris.