The classic endemic model For an endemic disease, births and deaths need to be taken into account, and the SIR model becomes as =UN - us - as I N dI = -ul + as N - BI, (7.5) dt dR = -UR + BI, dt with initial conditions S(0) = So 2 0, I(0) = 10 > 0, and R(0) 2 0. Here the new parameters are the per capita death rate / and per capita birth rate v of the pop- ulation. By choosing u = v, the total population N = S + I + R is constant. In this case, using the same dimensionless variables as for the classic SIR model, we are left with a two-dimensional dynamical system, which reads, in dimensionless form, ds = n - ns - si, (7.6) di = - (nto)i+ si, where n = v/ a = u/Q. As before, it is easy to check that trajectories starting in T remain in T (see exercises). The fixed points of (7.6) in the (s, i) plane are P1 = (1, 0) and ( 17 + 8 , " ( 1 - 1 - 8 ) nts wheren = 11/0; = p/a. As before, it is easy to check that trajectories starting in T remain in T (see exercises). The fixed points of (E) in the (s, i) plane are _ _ 77(117-6) P1 (1,0) and P2 (71+le . The Jacobian of (E) is 5,: (\"17' 3671) and J 1, P1 is the only fixed point in T and since the eigenvalues of J(P1) are 1] and 1 (11+ 6), P1 is a stable node. All of the trajectories starting in T must converge to this xed point, which means that in the long run there are only susceptible individuals in the population. This is because those who have recovered from the disease eventually die and are replaced by new borns, who are susceptible. This is illustrated in Figure 7.2. which shows the phase por trait of (E), obtained with PPLAN E, with 6 = 0.2 and 17 = 1. s'=n-ns-si n = 1 i' = - (nto)itsi 8= 0.2 TTTTTTTTTTTTTTT TTTTTTTTTTTTTTTT .~ 0.5 1 1 1 1 1 1 T T T T T T T T TT TT 0 0 0.5 S Figure 7.2. Phase plane of system (7.6), with o = 0.2 and n = 1, plotted with the software PPLANE. Only the dynamics inside T (not shaded) is relevant. If on the other hand n + 8