*3.2. Final Size of the Epidemic*

Next, we determine the final size of the susceptible compartment *Sf* = lim*t*→<sup>∞</sup> *S*(*t*). From (3a), we obtain:

$$\frac{dS(t)}{dt} = -\beta \frac{S(t)}{N} \int\_{t-\tau}^{t} J(\eta) d\eta.$$

Then, integrating from 0 to ∞, we obtain

$$\int\_0^\infty \frac{dS}{S} = -\frac{\beta}{N} \int\_0^\infty \left( \int\_{t-\tau}^t J(\eta) d\eta \right) dt.$$

Changing the order of integration, we get

$$\begin{split} \ln \frac{S\_0}{S\_f} &= \frac{\beta}{N} \left[ \int\_{-\tau}^0 \left( \int\_0^{\eta+\tau} J(\eta) dt \right) d\eta + \int\_0^{\infty} \left( \int\_{\eta}^{\eta+\tau} J(\eta) dt \right) d\eta \right] \\ &= \frac{\beta}{N} \left[ \int\_{-\tau}^0 (\eta+\tau) J(\eta) d\eta + \int\_0^{\infty} \tau J(\eta) d\eta \right]. \end{split}$$

Now, integrating (9a) from 0 to ∞, we have:

$$\int\_0^\infty J(\eta)d\eta = S\_0 - S\_f. \tag{13}$$

Since *J*(*t*) = 0 for all *t* ∈ [−*τ*, 0], then

$$\ln \frac{S\_0}{S\_f} \quad = \, \frac{\beta}{N} \left[ \int\_0^\infty \tau J(\eta) d\eta \right] = \frac{\beta}{N} \tau (S\_0 - S\_f) . \tag{14}$$

Thus, the final size can be obtained from the equation:

$$\ln \frac{S\_0}{S\_f} = \frac{\beta}{N} \text{tr} \left( S\_0 - S\_f \right). \tag{15}$$

Integrating (9c) and (9d) from 0 to ∞ and using (13), we obtain:

$$R\_f \equiv \lim\_{t \to \infty} R(t) = r\_0 \int\_0^\infty J(s - \tau) ds = r\_0 \int\_0^\infty J(\eta) d\eta \, = r\_0 (S\_0 - S\_f),$$

$$D\_f \equiv \lim\_{t \to \infty} D(t) = d\_0 \int\_0^\infty J(s - \tau) ds = d\_0 \int\_0^\infty J(\eta) d\eta \, = d\_0 (S\_0 - S\_f).$$

Final size *Sf* obtained from the numerical simulation and using the Formula (15) is verified for three different values of *τ* and a range of values of *β* (Figure 1). Observe that Equation (15) can be written as:

$$\ln \frac{S\_0}{S\_f} = \mathcal{R}\_0 \left( 1 - \frac{S\_f}{S\_0} \right). \tag{16}$$

Note that the SIR model (4) gives the same equation for the final size, and also in the SIR model (4), if we assume that *r*<sup>0</sup> + *d*<sup>0</sup> = 1/*τ*, then the expression of R<sup>0</sup> is equivalent for both the SIR model (4) and the delay model (9).

**Figure 1.** Dependence of *Sf* on *β* found analytically by Formula (15) and in numerical simulations of (8) for *τ* = 3 (upper curves), *τ* = 4 (middle curves), and *τ* = 5 (lower curves). The analytical and numerical solutions coincide.
