7.2.5. An isolated island population of 100 individuals is exposed to a disease. The disease is particularly deadly; an infected individual remains contagious until overcome by death after 4 days. We want to predict the diseases's effect on the community on a daily basis. Suppose initially one individual is stricken with the disease.

a. What is the removal rate ? ?

b. For what values of the relative removal rate ? will an epidemic occur? Use this to determine for what values of the transmission coefficient ? an epidemic will occur.

c. Use a computer program such as sir to estimate the number of days until the epidemic peaks for the values of ? = .003, .005, .01, and .0125, presenting your data in a table. How does the magnitude of ? relate to the time until the peak?

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