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Question: some diseases such as typhoid fever are spread largely by...

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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 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∞.

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