5. Nonlinear systems often find use in mathematical epidemiology, which is concerned with the study of disease transmission within a population. Let s represent the susceptible fraction of the population and i represent the infected fraction of the population. One simple model is given by ds 1 di = -Bsi + =yi dt 2 dt Bsi yi where quantifies the transmission rate and corresponds to a recovery rate. This model assumes that half of the recovered individuals will again be susceptible to infection, while the other half will gain immunity. (a) Show that solution curves starting within the triangular region, A = {(s, i): s 0, i 0,i + s 1} must remain there for all t 0. Hint: Look at the quantity s(t) + i(t) and its time derivative. (b) Sketch the phase portrait of the nonlinear system within A. (c) Find a conserved quantity H(s, i) for the system via separation of variables. (d) Use Dulac's criterion with g(s, i) = to show that there are no closed orbits within A.