Database Reference
In-Depth Information
Chapter 2
Basic Epidemic Models
2.1 Basic Model
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
1
.
Assume that an initial population of 1,000,000 (per 100 square miles) is not im-
mune 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 con-
tagious 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.
Setting the contagion rate proportional to the product of the nonimmune and
the contagious and sick population is arbitrary. The form is suspiciously similar
1
See Spain, J.D. 1982.
BASIC Microcomputer Models in Biology
, Addison-Wesley, Reading,
Massachusetts, p. 118. For some realism, see the data on the Black Death in 1300s Italy (Curtis H.
and N. Barnes. 1985.
Invitation to Biology
, Worth Publishers, New York.) These data show a
declining peak as people became aware of the vector, or those most likely exposed to the vec-
tor died off, or the naturally immune were selected for and that immunity was inheritable. The
four occurrences of the plague in that century had a period of about eleven years. For chaotic epi-
demics, see: Schaffer, W. 1985. Can Nonlinear Dynamics Elucidate Mechanisms in Ecology and
Epidemiology,
IMA Journal of Mathematics Applied in Medicine and Biology
, Vol. 2, pp. 221-252.
Schaffer shows how a cyclic contact coefficient can produce chaos in this form of epidemic model.
