SIR Model A communicable disease is spread throughout a small community, with a fixed

Chapter 2, Problem 17

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SIR Model A communicable disease is spread throughout a small community, with a fixed population of n people, by contact between infected individuals and people who are susceptible to the disease. Suppose initially that everyone is susceptible to the disease and that no one leaves the community while the epidemic is spreading. At time t, let s (t), i(t), and r (t) denote, in turn, the number of people in the community (measured in hundreds) who are susceptible to the disease but not yet infected with it, the number of people who are irifected with the disease, and the number of people who have rec overed from the disease. Explain why the system of differential equations ds - = -ksi dt 1 di dt = -k2i + k1si dr dt = k1i, where k1 (called the irifection rate) and ki (called the removal rate) are positive constants, is a reasonable mathematical model, commonly called a SIR model, for the spread of the epidemic throughout the community. Give plausible initial conditions associated with this system of equations. Show that the system implies that d dt (s + i + r) = 0. Why is this consistent with the assumptions?