Question: some diseases such as typhoid fever are spread largely by...
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 rate β, so
dy/dt = −βy
Suppose also that the disease spreads at a rate proportional to the product of x and y; thus
dx/dt = −αxy
(a) Determine y at any time t by solving the first equation subject to the initial condition y(0) = y0.
(b) Use the result of part (a) to find x at any time by solving the second equation 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 tø∞.