Some diseases (such as typhoid fever) are spread largely by carriers, individuals who

Chapter 2, Problem 22

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Some diseases (such as typhoid fever) are spread largely by carriers, individuals who can transmit the disease but who exhibit no overt symptoms. Let x and y denote the proportions of susceptibles and carriers, respectively, in the population. Suppose that carriers are identified and removed from the population at a rate , so dy dt = y. (23) Suppose also that the disease spreads at a rate proportional to the product of x and y; thus dx dt = xy. (24) a. Determine y at any time t by solving equation (23) subject to the initial condition y(0) = y0. b. Use the result of part a to find x at any time t by solving equation (24) subject to the initial condition x(0) = x0. c. Find the proportion of the population that escapes the epidemic by finding the limiting value of x as t .