Consider the following model for the spread of an illness such as measles in a city:

dS/dt = -0.00002 SI

dI/dt = 0.00002 SI - 0.08 I

dR/dt = 0.08 I

where t is measured in days, S is the susceptible population, I is the infected (and infectious) population, and R is the recovered (and hence immune) population.S, I, and R are all expressed in units of number of individuals in the total population (= S + I + R).

We assume the disease comes and goes in a time period short enough so that we can neglect any overall change in the total population due to births and deaths.

Roughly how long does someone who catches this disease remain infectious?Suppose 200 people come to the city, each infected, and thus bring the first cases of the disease.How large does the population of the city have to be for the disease to spread (meaning that the number of infected will increase beyond 200 spread)?