Endemic models with arbitrarily distributed periods of infection I: Fundamental properties of the model

A model is developed for the spread of an infectious disease in a population with constant recruitment of new susceptibles and the fundamental properties of its solutions are analyzed. The model allows for arbitrarily many stages of infection all of which have general length distributions and disease mortalities. Existence and uniqueness of solutions to the model equations are established. A basic reproduction ratio is derived and related to the existence of an endemic equilibrium, to the stability of the disease-free equilibrium, and to weak and strong endemicity (persistence) of the disease. A characteristic equation is found, the zeros of which determine the local stability of the endemic equilibrium, and sufficient stability conditions are given for the case that infected individuals do not return into the susceptible class. In a subsequent paper, explicit sufficient and necessary stability conditions will be derived for the case that the disease dynamics are much faster than the demographics.