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 susceptible and carriers, respectively, in the population. Suppose that carriers are identified and removed from the population at a rateβ, so dy/dt=−βy.(i)Suppose also that the disease spreads at a rate proportional to the product of x and y; thus dx/dt=−αxy.(ii)

(a) Determine y at any time t by solving Eq. (i) subject to the initial condition y(0)=y0.

(b) Use the result of part (a) to find x at any time t by solving Eq. (ii) 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→∞.