Basic Epidemic Models

In this chapter we follow up on our discussion from Chapter 1 and model the spread of a disease through a population, gradually adding new features. Epidemics, such as the one modeled here, are of great concern to human societies. The complex interrelationships of biological, social, economic and geographic relationships that drive or constrain an epidemic make dynamic models an invaluable tool for the analysis of particular diseases. The model developed here is fairly idealized but can be applied easily to real populations affected by a disease.

Assume that an initial population of 1,000,000 (per 100 square miles) is not immune to a contagious disease. The rate at which they become sick is assumed to be a function of the product of the nonimmune population times the contagious plus sick population. This equation for contagion is the simplest form that meets the obvious requirements that the contagion rate must be zero if either the immune or the contagious populations are zero. The contagious population is assumed to become the sick population for a week and then, with a survival rate of 0.9, the survivors join the immune population. The nonimmune population is augmented with a constant birth rate of 5000/week. The people in this model do not die of other causes.