*2.1. Model with Distributed Parameters*

The number of newly infected individuals *J*(*t*) is determined by the rate of decrease of the number of susceptible individuals, *J*(*t*) = −*S* (*t*). Assuming that

$$N = S(t) + I(t) + R(t) + D(t) \tag{1}$$

is constant, where *I*(*t*) is the total number of infected at time *t* and *R*(*t*) and *D*(*t*) denote, respectively, recovered and dead, we can write

$$I(t) = \int\_0^t J(\eta)d\eta - R(t) - D(t). \tag{2}$$

Following conventional epidemiological models, we set

$$\frac{dS(t)}{dt} = -\beta \frac{S(t)}{N} \,\, I(t),$$

where *β* is the disease transmission rate. Let *r*(*η*) and *d*(*η*) be the recovery and death rates depending on the time-since-infection *η*. Then, the number of infected individuals who will recover at time *t* is given by the expression:

$$\frac{d\mathcal{R}(t)}{dt} = \int\_0^t r(t-\eta)f(\eta)d\eta.$$

and the number of infected individuals who will die at time *t*:

$$\frac{dD(t)}{dt} = \int\_0^t d(t-\eta)f(\eta)d\eta.$$

Differentiating Equality (1), we obtain

$$\frac{dI(t)}{dt} = \beta \frac{S(t)}{N} \left. I(t) - \int\_0^t r(t - \eta) I(\eta) d\eta - \int\_0^t d(t - \eta) I(\eta) d\eta. \right|$$

Thus, we obtain the following integro-differential equation model:

$$\frac{dS(t)}{dt}\_{\text{in}} = -\beta \frac{S(t)}{N} I(t) \quad (= -I(t)), \tag{3a}$$

$$\frac{dI(t)}{dt}\_{\text{min}} = \beta \frac{S(t)}{N} I(t) - \int\_0^t r(t-\eta)f(\eta)d\eta - \int\_0^t d(t-\eta)f(\eta)d\eta,\tag{3b}$$

$$\frac{d\mathcal{R}(t)}{dt} = -\int\_0^t r(t-\eta)f(\eta)d\eta,\tag{3c}$$

$$\frac{dD(t)}{dt} = -\int\_0^t d(t-\eta)f(\eta)d\eta,\tag{3d}$$

with the initial condition *S*(0) = *N*, *I*(0) = *I*<sup>0</sup> > 0 (*I*<sup>0</sup> is sufficiently small as compared to *N*), *R*(0) = 0, *D*(0) = 0.
