7.2.1. The SIR model has many equilibria.

a. To find the equilibria, why is it not necessary to find when R = 0, if we find points where S = 0 and I = 0?

b. Algebraically, find the equilibrium points S?, I ? for the SIR model. Give a common-sense explanation of why the values you find are equilibria.

c. Are these equilibria stable? Explain intuitively why they should or should not be

400 350 300 250 Infectives 200 150 100 50 O 0 50 100 150 200 250 300 350 400 450 500 Susceptibles Figure 7.2. S/ phase plane for the S/ R model. epidemic, you can tell that the number of infectives increases rapidly at the onset of the epidemic. In fact, just before the epidemic peaks, /, is increasing by approximately 80 individuals per time step. In a population of only 500. this is extreme growth. Of course, a transmission coefficient of a = .002 is quite large for a population of that size. Note that you can approximate the relative removal rate p from the graphs of the three orbits, since we know p is the value of S, when /, begins to dec- line. Because So is also easily read from the graph, once we know p, we can find Ro Determine from the graph of the orbits approximate values for p and Re. Do these values match what you would calculate from the values of the parameters y and o? We can make intelligent guesses about the equilibrium points of the S/ R model from the phase plane, too: Each epidemic follows a wave, progressing toward a point on the horizontal axis. You will see in the problems that the S / R model has a set of equilibria, including all the points along that axis