Biology Reference
In-Depth Information
The same reasoning applies if we double the number of susceptibles
while keeping the number of infectives fixed. At the end of this
chapter, we present a more rigorous mathematical derivation of
this idea.
Before we begin to analyze what happens when we have a group of
infective people in a population, we need to know several things about
the disease and the environment. For example, do those recovered from
the disease have immunity, or are they again susceptible? Does the
disease have an incubation period? Will the infectives be isolated, by
quarantine or natural separation, or will they be spread throughout the
population? We consider in detail two models, the SIS model (S stands
for susceptibles, I for infectives) and the SIR model (R stands for
recovered), and some variations of both.
B. The SIS Model
1. Description of the SIS Model
Our first model will be quite simple. It assumes that the population is
divided into two nonintersecting groups—the group of those who have
the disease and can infect others (I) and the group of those who do not
have the disease and can be infected (S). Our goal is to build a model
that describes the change in the sizes of the susceptible and infective
groups with time. We make the following assumptions:
1. The population is fixed and consists of N individuals. There are no
births or deaths, and no one migrates into or out of the population.
2. There is no incubation period for the disease.
3. The two groups—susceptibles and infectives—are uniformly mixed
within the population.
4. Once recovered from the disease, an individual is susceptible again;
that is, there is no immunity.
These assumptions may seem unjustifiably restrictive, but our goal is to
begin with the simplest model that will allow us to examine some
important questions. Also, there are real situations in which these
assumptions would be appropriate. Assumption 1, for example, would
apply to the SARS epidemic that developed in an apartment building in
Hong Kong, when the building was quickly sealed. For us, the SIS model
will provide a start, and we shall then proceed to models that remove or
relax its restrictive assumptions.
Assume that in a fixed population of size N, a small number I(0) of
individuals have somehow contracted an infectious disease. As time
progresses, the infection may spread in the population, and we want to
examine those changes with time. We denote:
