In the theory of the spread of contagious disease (see [Bal] or [Ba2]), a relatively

Chapter 5, Problem 5

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In the theory of the spread of contagious disease (see [Bal] or [Ba2]), a relatively elementary differential equation can be used to predict the number ofinfective 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, to remain in that state. Suppose x(/) denotes the number ofsusceptible individuals at time t and y(t) denotes the number of infectives. It is reasonable to assume that the rate at which the number of infectives changes is proportional to the product of x{t) and y{t) because the rate depends on both the number ofinfectives and the number ofsusceptibles present atthattime. Ifthe population is large enough to assume that x{t) and y(t) are continuous variables, the problem can be expressed as y\t) = kx{t)y{t), where ^ 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, that k 2 x 10-6 , and that time is measured in days, find an approximation to the number of infective individuals at the end of 30 days. b. The differential equation in part (a) is called a Bernoulli equation and it can be transformed into a linear differential equation in u(t) = (y(0)_l - Use this technique to find the exact solution to the equation, under the same assumptions as in part (a), and compare the true value of y{t) to the approximation given there. What is lim^ooyfr)? Does this agree with your intuition?