*2.2. Reduction to SIR Model*

In a particular case, if we assume the uniform distribution of recovery and death rates:

$$r(t - \eta) = \begin{cases} \begin{array}{c} r\_0 \\ 0 \end{array}, \quad t - \tau < \eta \le t \\\ 0, \quad \eta < t - \tau \end{array}, \quad d(t - \eta) = \begin{cases} \begin{array}{c} d\_0 \\ 0 \end{array}, \quad t - \tau < \eta \le t \\\ 0, \quad \eta < t - \tau \end{array},$$

where *τ* > 0 is disease duration, *r*<sup>0</sup> and *d*<sup>0</sup> are some constants, and if *r*<sup>0</sup> and *d*<sup>0</sup> are small enough, then the model (3) can be reduced to the conventional SIR model (see [31]):

$$\frac{dS(t)}{dt} \quad = \quad -\beta \frac{S(t)}{N} \ I(t),\tag{4a}$$

$$\frac{dI(t)}{dt} = -\beta \frac{S(t)}{N} \left[ I(t) - (r\_0 + d\_0)I(t) \right] \tag{4b}$$

$$\frac{dR(t)}{dt} \quad = \quad r\_0 I(t), \quad \frac{dD(t)}{dt} \quad = \; d\_0 I(t). \tag{4c}$$

However, the approximation of a uniform distribution of recovery and death rates may not be precise since infected individuals are unlikely to recover or die at the beginning of the disease. In the following subsection, we consider another choice of recovery and death rates, which will approximate the real scenario of recovery and death more accurately.
