It was observed in London during the early 1900's, that large outbreaks of measles epidemics recurred about once every two years. The mathematical biologist H. E. Soper tried to explain this phenomenon by assuming that the stock of susceptibles is constantly replenished by new recruits to the population. Thus, he assumed that
dtdS=−rSI+μ,dtdI=rSI−γI(∗)
for some positive constants r, γ, and μ.
(a) Show that S=γ/r, I=μ/γ is the only equilibrium solution of (∗).
(b) Show that every solution S(t), I(t) of (∗) which starts sufficiently close to this equilibrium point must ultimately approach it as t approaches infinity.
(c) It can be shown that every solution S(t), I(t) of (∗) approaches the equilibrium solution S=γ/r, I=μ/γ as t approaches infinity. Conclude, therefore, that the system (∗) does not predict recurrent outbreaks of measles epidemics. Rather, it predicts that the disease will ultimately approach a steady state.