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Equation. (40) then leads us to the following differential equation dI 7]-;--- ——otI. This means that I (t) = I (0) e‘°°* and, hence, R(t)=N—S(t)——I(t) = I (O) [1 — e‘°°*]. Figure 9 provides diagrams that illustrate the changes with the passage of time in the 48 Differential Equations in Applications number of individuals in each of the three groups. Case 2. I (0) > I*. In this case there must exist a time interval OQ t < T in which I (t) > I* for all values of t, since by its very meaning the function I = I (t) is continuous.

Let us establish the relationship between u and v. To this end we divide the first equation in system (36) by the second and integrate the resulting differential equation. We get ow+u—-lnv°°u =ow0—i-un-—lnvg°u0 EH, where H is a constant determined by the initial conditions (37) and parameter oc. Figure 7 depicts the dependence of u on v for different values of H. We see that the (u, v)—plane contains only closed curves. Let us now assume that the initial values uo and vt, are specified by point A on the trajectory that corresponds to the value H = H3.

1 (O)< 1*. With the passage of time the individuals in the population will not become infected because in this case dS/dt = O, and, hence, in accordance with Eq. (38) and the condition R (O) == O, we have an equation valid for all values Ch. 1. Construction of Differential Models 47 N . (¢) IO5) 0 it Fig. 9 of tz S(t)==S(O)=N——I(0). The case considered here corresponds to the situation when a fairly large number of infected individuals are placed in quarantine. Equation. (40) then leads us to the following differential equation dI 7]-;--- ——otI.