Some results for an SEIR epidemic model with density dependence in the death rate

This paper deals with an SEIR epidemic model for an infectious disease where the death rate depends on the number of individuals in the population. It is also assumed that there is an additional death rate suffered by infected individuals. It is found that there are three steady-state values: one where the population is extinct, one where the population maintains itself at a constant level and the disease is extinct, and one where there is a unique equilibrium with disease present. An interesting and unusual feature is that it is possible for this third equilibrium to exist and be locally unstable. Numerical work and simulation show that we can have cycles of disease incidence with increasing amplitude, a constant amplitude, and a decreasing amplitude, depending on the parameter values of the model.