Modelling epidemics with variable contact rates.

In this paper we look at models for epidemics where the contact rate is a monotone increasing function of the population density. The background death rate also depends on the population density. We first examine the case of a constant contact rate (motivated by AIDS) and here obtain some global stability results. We consider an SIR model where a typical individual starts off susceptible, at some stage catches the disease and after a short infectious period becomes permanently immune. We also look at the effects of vaccination. First we perform an equilibrium and local stability analysis. Next we reformulate the model in terms of the proportions of individuals susceptible, infected, and immune to obtain some global stability results. We find three possible equilibrium values: one where the population is extinct, one where the disease has died out but the population has not died out, and a unique equilibrium where disease is present. We determine conditions for global stability of these equilibria. For certain parameter values none of these equilibria are locally stable. In this case there is a formal proportional endemic equilibrium with a strictly positive proportion of infected individuals. We expect the population size to die out but the proportions of susceptible, infected, and immune individuals to tend to this endemic proportional equilibrium. We find two critical contact rates which help determine the behaviour of the system. Next we extend some of these results to the case where the contact rate depends on population density. Finally the paper examines these results further using numerical methods.