2. (50 pts) Let S, I and R represent the number of individuals subject to a disease who are susceptible, infective and revovered respectively. Assume that the total population size N(t) is a constant for all t 2 0. Suppose that recovered individuals can become susceptible again after some time. The model equations with frequency incidence are s follows: dS dt dI dt dR dt where N = S+ 1 + R, and 3, γ and v are the infection, loss of immunity, and recovery rates, respectively. (a) Reduce this model to a two-dimensional system of equations in S and I. (b) Normalize the two dimensional system by converting to proportions s =-and 2- F (Convert the ODEs in S, l to ODEs in s, 2). (c) Find the equilibrium points in the s- i phase plane. Is there a disease-free equilibrium (an equilibrium with i = 0)? Is there an endemic equilibrium (an equilibrium with i >0)? (d) Determine the local stability of each of the equilibrium points found above using inearization (e) Plot equilibria, nullclines and the flow of solutions in the phase plane (f) What biological conclusions can you make from the analysis of this model?
