# Differential Equations in Applications (Science for by V. V. Amel'kin

By V. V. Amel'kin

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38) Let us also assume that when the number of infected individuals exceeds a certain fixed number I *, the rate of change in the number of individuals susceptible to the disease is proportional to the number of such individuals. As for the rate of change in the number of infected individuals that eventually recover, we assume that it is proportional to the number of infected individuals. Clearly, these assumptions simplify matters and in a number of cases they reflect the real situation. Because of the first assumption, we suppose that when the number of infected individuals I (t) is greater than I *, they can infect the individuals susceptible to the disease.

Construction of Differential Models 49 We devote our further investigations to the problem of finding the specific value of T and the moment of time tmax at which the number of infected individuals proves maximal. The answer to the first question is important because starting from T the individuals susceptible to the disease cease to become infected. If we turn to Eq. (43), the aforesaid implies that at t= T its right-hand side assumes the value 1*, that is, I* = [I (O) -1- ocS (O) Tle‘°°T. (44) But S (T) = 1imS (t) =S (oo) f->00 is the number of individuals that are susceptible to the disease and yet avoid falling ill; for such individuals the following chain of equalities holds true: S (T) = S (oo) = S (0) e‘°°T.

And although the latter controls this territory by firepower, it cannot know the effectiveness of its actions. lt is also highly probable that the losses suffered by the guerilla forces x are, on the one hand, proportional to the number of personnel sc (t) on R and, on the other, to the number y (t) of personnel of the opposing side. Hence, the term corresponding to the losses suffered by the guerilla forces x has the form —ga; (t) y (t), where the coefficient reflecting the effectiveness of combat operations of side y is, in general, more difficult to estimate than the coeffiient b in the first relationship in (48).

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