Chapter 3.3, Problem 17E

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 that everyone is initially 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 infected with the disease, and the number of people who have recovered from the disease. Explain why the system of differential equations

$\begin{array}{l}\frac{ds}{dt}=-k_{1}si\\ \frac{di}{dt}=-k_{2}i+k_{1}si\\ \frac{dr}{dt}=k_{2}i,\end{array}$

where k₁ (called the infection rate) and k₂ (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.