*3.3. Maximum Number of Infected Individuals*

We now derive an approximate formula for the maximal number of infected individuals for the delay model (9). We have *I*(*t*) = *S*(*t* − *τ*) − *S*(*t*). Suppose that *I*(*t*) attains its maximum at *t* = *tm*. Set *Im* = *I*(*tm*), *Sm* = *S*(*tm*). From the equality *I* (*tm*) = 0, we obtain *S* (*tm* − *τ*) = *S* (*tm*). This implies

$$\frac{\beta S(t\_m - \tau)I(t\_m - \tau)}{N} = \frac{\beta S(t\_m)I(t\_m)}{N}.\tag{17}$$

Substituting the relation *I*(*tm*) = *S*(*tm* − *τ*) − *S*(*tm*) in (17), we obtain

$$I(t\_m - \tau) = \frac{\mathcal{S}\_m I\_m}{\mathcal{S}\_m + I\_m}.\tag{18}$$

From (8), we have

$$\frac{dS}{dt} = -\frac{\beta S(t)}{N} \left( S(t-\tau) - S(t) \right). \tag{19}$$

Integrating (19) from 0 to *tm* and changing the variable inside the first integral of the right-hand side, we obtain:

$$\int\_{0}^{t\_{m}} \frac{dS}{S} \quad = \quad -\frac{\beta}{N} \left( \int\_{-\tau}^{0} S(t)dt - \int\_{t\_{m}-\tau}^{t\_{m}} S(t)dt \right). \tag{20}$$

We assume that *S*(*t*) = *S*<sup>0</sup> for all *t* ∈ [−*τ*, 0] and use the approximation (Figure 2):

$$\int\_{t\_m-\tau}^{t\_m} S(t)dt \approx \frac{\tau}{2}I\_m + \tau S\_m.$$

Then from (20) we have

$$\ln \frac{S\_m}{S\_0} = -\frac{\beta}{N} \left( \tau S\_0 - \frac{\pi}{2} I\_m - \tau S\_m \right).$$

**Figure 2.** The red curve represents *S*(*t*). The integral *tm tm*−*<sup>τ</sup> <sup>S</sup>*(*t*)*dt*, i.e., the area under the red curve (yellow color region + cyan color region), is approximated by the sum of the areas of the cyan color region, the yellow color region, and the green color region.

Let *x* = *Im S*0 , *y* = *Sm <sup>S</sup>*<sup>0</sup> and *<sup>S</sup>*<sup>0</sup> *<sup>N</sup>* ≈ 1. Then we have

$$
\ln y = -\beta \tau \left( 1 - \frac{1}{2} x - y \right). \tag{21}
$$

Again, integrating (19) from *tm* − *τ* to *tm*, we obtain:

$$\int\_{t\_m-\tau}^{t\_m} \frac{dS}{S} = -\frac{\beta}{N} \left( \int\_{t\_m-\tau}^{t\_m} S(t-\tau)dt - \int\_{t\_m-\tau}^{t\_m} S(t)dt \right).$$

Changing the variable inside the first integral of the right-hand side, we get:

$$\ln \frac{S\_m}{S(t\_m - \tau)} = -\frac{\beta}{N} \left( \int\_{t\_m - 2\tau}^{t\_m - \tau} S(t)dt - \int\_{t\_m - \tau}^{t\_m} S(t)dt \right).$$

Now, using the approximation described in Figure 2, we conclude that

$$\ln \frac{S\_m}{S(t\_m - \tau)} = -\frac{\beta}{N} \left( \tau S(t\_m - \tau) + \frac{\tau}{2} I(t\_m - \tau) - \tau S\_m - \frac{\tau}{2} I\_m \right). \tag{22}$$

Using the relation *I*(*tm*) = *S*(*tm* − *τ*) − *S*(*tm*), after some transformations, (22) can be written as:

$$\ln \frac{S\_m}{S\_m + I\_m} = -\frac{\beta}{N} \frac{\tau}{2} \left( I\_m + I(t\_m - \tau) \right).$$

Using (18), we obtain:

$$\ln \frac{S\_m}{S\_m + I\_m} = -\frac{\beta}{N} \frac{\tau}{2} \left( I\_m + \frac{S\_m I\_m}{S\_m + I\_m} \right).$$

Substituting *x* = *Im S*0 , *y* = *Sm <sup>S</sup>*<sup>0</sup> and *<sup>S</sup>*<sup>0</sup> *<sup>N</sup>* ≈ 1, we have

$$
\ln \frac{y}{x+y} = -\frac{\beta \pi}{2} \left( x + \frac{xy}{x+y} \right). \tag{23}
$$

Solving (21) and (23), we can find *x*, *y* and, consequently, *Im*, *Sm*.

In Figure 3, we show a comparison between the maximum number of infected obtained by Equations (21) and (23) and the maximum number of infected obtained by direct numerical simulation of the delay model (9). From Figure 3, we can observe that the approximation gives a very close upper bound to the maximum number of infected.

**Figure 3.** The red curve and the blue curve show the maximum number of infected using the direct numerical simulation of the delay model and using Equations (21) and (23) respectively. Parameter values: *<sup>N</sup>* <sup>=</sup> 105, *<sup>τ</sup>* <sup>=</sup> 4, *<sup>S</sup>*<sup>0</sup> <sup>=</sup> *<sup>N</sup>* <sup>−</sup> 1, *<sup>I</sup>*<sup>0</sup> <sup>=</sup> 1.
