2" . Consider the system ds = A - BS(t) I(t) dt N (t ) - US + 1 (1.12) di = BS(t) I(t) dt N (t ) - ( # + 7) I, where A denotes the total recruitment rate, assumed constant. (a) Look at dNV/dt = d(S + I)/dt and solve the resulting differential equation for N, obtaining N(t) = K + (K - N(0))e-#t. This shows that the system is equivalent to the solution of the single nonautonomous differential equation di I = P(N(t) - N(t) - ( #+ 7)1, (1.13) where N (t ) = K + (K - N(0))e-ut, (1.14) with K = A/J. (b) Show that N(t) - K as t -+ 00. (c) Choose K = 1,000, 1/u = 10 years, and two initial popula- tion sizes, N(0) = 1, 200 and NV(0) = 700. Using a differential equation solver find /(10), /(20), and /(50) using values of the parameters that give Ro > 1. (d) If we look at the right hand side of equation (1.13) and let t - oo and replace S(t) by K - I, then we arrive formally at the following "asymptotic" differential equation at = B(K - D) - (# + 4) I(t), (1.15) where K = _ = lim N(t). 1-+00 Here, without justification, S(t) has been replaced by K - / and hence, Equation (1.15) and Equation (1.11) are not the same. However recent work [Castillo-Chavez and Thieme (1995)] has shown that these equations have the same qualitative dynamics. Compare the values found in (c) with those found using the limiting equation (1.15) numerically