*2.3. Delay Model*

Let us assume that disease duration is *τ*, and the individuals *J*(*t* − *τ*) infected at time *t* − *τ* recover or die at time *t* with certain probabilities. This assumption corresponds to the following choice of the functions *r*(*t* − *η*) and *d*(*t* − *η*):

$$r(t - \eta) = r\_0 \delta(t - \eta - \tau), \quad d(t - \eta) = d\_0 \delta(t - \eta - \tau),$$

where *r*0, *d*<sup>0</sup> are constants, *r*<sup>0</sup> + *d*<sup>0</sup> = 1, and *δ* is the Dirac delta-function. Then,

$$\frac{dR(t)}{dt} = \int\_0^t r(t-\eta)f(\eta)d\eta = r\_0f(t-\tau),$$

$$\frac{dD(t)}{dt} = \int\_0^t d(t-\eta)f(\eta)d\eta = d\_0f(t-\tau).$$

Note that *J*(*t*) is the number of newly infected individuals appearing at time *t*. If we assume that the first infected case was reported at time *t* = 0, then we can set *J*(*t*) = 0 for all *t* < 0. Now, integrating the above two equations from 0 to *t* and assuming that *R*(0) = *D*(0) = 0, we obtain

$$R(t) = r\_0 \int\_0^t I(s - \tau) ds = r\_0 \int\_{-\tau}^{t - \tau} I(y) dy \ = r\_0 \int\_0^{t - \tau} I(y) dy,$$

$$D(t) = d\_0 \int\_0^t I(s - \tau) ds = d\_0 \int\_{-\tau}^{t - \tau} I(y) dy \ = d\_0 \int\_0^{t - \tau} I(y) dy.$$

$$I(t) = 1.$$

Then, instead of (2), we have

$$I(t) = \int\_{t-\tau}^{t} J(s)ds,\tag{5}$$

and from (3b),

$$\frac{dI(t)}{dt} = \beta \frac{\mathcal{S}(t)}{N} \left[ I(t) - I(t-\tau) \right] \tag{6}$$

From (5) we obtain

$$J(t) = \beta \frac{S(t)}{N} \int\_{t-\tau}^{t} J(s)ds, \quad \frac{dS(t)}{dt} = -J(t). \tag{7}$$

Hence,

$$\frac{dS(t)}{dt} = -\beta \frac{S(t)}{N} \int\_{t-\tau}^{t} J(s)ds = \beta \frac{S(t)}{N} \int\_{t-\tau}^{t} \frac{dS(s)}{ds} ds = -\beta \frac{S(t)}{N} \left(S(t-\tau) - S(t)\right). \tag{8}$$

Once we obtain the solution *S*(*t*) from (8), then we can find *I*(*t*) using the following relation:

$$I(t) = \int\_{t-\tau}^{t} J(s)ds = -\int\_{t-\tau}^{t} \frac{dS(s)}{ds}ds = S(t-\tau) - S(t).$$

Hence, System (3) is reduced to the following delay model:

$$\frac{d\mathcal{S}(t)}{dt} = -J(t),\tag{9a}$$

$$\frac{dI(t)}{dt}\_{\cdot} = \left[J(t) - J(t-\tau)\_{\cdot}\right] \tag{9b}$$

$$\frac{dR(t)}{dt} \quad = \quad r\_0 I(t - \tau), \tag{9c}$$

$$\frac{dD(t)}{dt} \quad = \quad d\_0 I(t-\tau),\tag{9d}$$

$$J(t) \quad = \ \beta \frac{S(t)}{N} \ I(t),\tag{9e}$$

with *J*(*t*) = 0 for all *t* < 0. A similar model was proposed in [33] without derivation from the distributed model.
