In the theory of the spread of contagious disease (see [Ba1] or [Ba2]), a relatively elementary differential equation can be used to predict the number of infective individuals in the population at any time, provided appropriate simplification assumptions are made. In particular, let us assume that all individuals in a fixed population have an equally likely chance of being infected and once infected remain in that state. Suppose x(t) denotes the number of susceptible individuals at time t and y(t) denotes the number of infective. It is reasonable to assume that the rate at which the number of infective changes is proportional to the product of x(t) and y(t) because the rate depends on both the number of infective and the number of susceptible present at that time. If the population is large enough to assume that x(t) and y(t) are continuous variables, the problem can be expressed
y' (t) = kx(t)y(t),
Where k is a constant and x(t) + y(t) = m, the total population. This equation can be rewritten involving only y(t) as
y' (t) = k(m − y(t))y(t).
a. Assuming that m = 100,000, y(0) = 1000, k = 2 × 10−6, and that time is measured in days, find an approximation to the number of infective individuals at the end of 30 days.