The steady state for the infected human population implies that m — (r/ab)(HT/MT)(h/(1 — h)) and this curve is shown in Figure 5.11a. Note that the slope of the tangent line to this curve at the origin (or, alternatively, the slope of the linear approximation to this curve) is (r/ab)(HT/MT). The steady state for the infected mosquito population implies that m — ach/(p + ach) and this curve is shown in Figure 5.11b. The slope of this curve at the origin is ac/p. We understand the dynamics of the disease by putting the isoclines together, which I have done in three ways in Figures 5.11c-e. When the steady state determined by the intersection of the two isoclines is at a relatively high level of infection, MacDonald called the malaria "stable" (Anderson and May 1991, p. 397). When the steady state is at a lower level of infection, he called it "unstable" and it is possible for malaria to become extinct: if the mosquito isocline starts off below the human isocline, then the only steady state is the origin.
Malaria persists if the mosquito isocline rises faster than the human isocline at the origin. We can derive the condition for this to be true in terms of the slopes; in particular we must have ac/p > (r/ab)(HT/MT) and combining these terms we conclude that malaria will persist if (a2bc/pr)(MT/HT) > 1. Compare this with the computation that we did for the basic reproductive rate and you will see that they are the same: in
Figure 5.11. Analysis of the dynamics of malaria. (a) The isocline for infected humans, found by setting dh/dt = 0. (b) The isocline for infected mosquitoes, found by setting dm/dt = 0. Panels (c, d and e) show three ways that the isoclines can intersect.
(c) stable malaria (d) unstable malaria
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