1550078481-Ordinary_Differential_Equations__Roberts_

Applications of the Initial Value Problem y' = f(x, y); y(c) = d 139

des Sciences in which a mathematical model was used for the first time to

study the population dynamics of infectious disease. Bernoulli was investigat-

ing mortality due to smallpox and trying to assess the risks and advantages of

preventive inoculation. In his mathematical model, Bernoulli formulated and

solved a relevant differential equation. He evaluated the results in terms of

the value of preventive inoculation. The modern mathematical theory of epi-

demics originated in the works of William Hamer and Sir Ronald Ross which

appeared early in the twentieth century.

In the simplest epidemic model, we assume the population size has the

constant value of N. Thus, we assume there are no births and no immigration

to increase the population size and we assume there are no deaths and no

emigration to decrease the population size. Since the time span of an epidemic

is short (usually a few weeks or months) in comparison to the life span of

a person, the assumption of a constant population size is fairly reasonable.

Next, we assume the population is divided into two mutually exclusive sets:

The infectives is the set of people who are infected with and capable of

transmitting the disease. The susceptibles is the set of people who do not

have the disease but may become infected later. We denote the number of

infectives at time t by I(t) and the number of susceptibles at time t by S(t).

Under the assumptions we have made

(1) I(t) + S(t) = N for all t.

An assumption which was first made by William Hamer in 1906, and which
has been included in every deterministic epidemic model ever since, is that
the rate of change of the number of susceptibles is proportional to the product
of the number of susceptibles and the number of infectives, which represents
the rate of contact between susceptibles and infectives. Thus, it is assumed
that

(2) ~~ = S' (t) = -/3S(t)I(t) for all t

where f3 > 0 is called the infection rate. Solving equation (1) for I(t) and

substituting into equation (2), we obtain the following differential equation
for the number of susceptibles

(3) S'(t) = - /3S(t)(N -S(t)).

Differentiating (1), we see that

I'(t) + S'(t) = 0.

So the number of infectives satisfies the differential equation

(4) I'(t) = -S'(t) = j3S(t)I(t) = /3(N -I(t))I(t).

since S ( t) = N - I ( t) from ( 1). The differential equations ( 3) and ( 4) can

both be solved expli citly by the technique of separation of variables, or one