*4.1. Basic Reproduction Number (R*0*)*

In epidemiology modeling, a basic reproduction number is defined as the advent of a new infection in an entirely susceptible population due to an infected individual, and is usually represented by *R*0. The value of *R*<sup>0</sup> determines the spread of infection; for *R*<sup>0</sup> > 1 infection will spread in the population, and for *R*<sup>0</sup> < 1 infection will soon end [51].

To simplify the derivation process, a reduced model (14) has been utilized for further investigation of *R*0. The calculation of *R*<sup>0</sup> is based on the value of *α* = 1. The necessary condition of a disease epidemic is based on the increase in the infected individuals, with the supposition that, initially, the entire population is susceptible.

For the case of *D<sup>α</sup> I* > 0, we have *D<sup>α</sup>UI* > 0

$$\frac{\beta\_1 (N^\* - I - M)I}{2^{32}} + \frac{\beta\_2 (N^\* - I - M) \mathcal{U}\_I}{N^\*} - \rho I - r\_1 I > 0,\tag{15}$$

and, accordingly, in case of *D<sup>α</sup>UI* > 0, we have

$$\frac{\beta\_2 (\mathcal{U}^\* - \mathcal{U}\_I) I}{N^\*} - r\_2 \mathcal{U}\_I > 0. \tag{16}$$

With the assumption that all the population is susceptible at the start, the above expressions may be written as:

$$\frac{\beta\_1 N^\* I}{2^{32}} + \frac{\beta\_2 N^\* U\_I}{N^\*} - \rho I - r\_1 I > 0,\tag{17}$$

$$\frac{\beta\_2 \mathcal{U}^\* I}{N^\*} - r\_2 \mathcal{U}\_I > 0. \tag{18}$$

Simplifying the above relations, we have

$$\frac{\beta\_1 N^\*}{(\rho + r\_1)2^{32}} + \frac{\beta\_2^2 \mathcal{U}^\*}{r\_2 N^\*(\rho + r\_1)} > 1. \tag{19}$$

Accordingly,

$$R\_0 = \frac{\beta\_1 N^\*}{2^{32} (\rho + r\_1)} + \frac{\beta\_2^2 U^\*}{r\_2 N^\* (\rho + r\_1)}.\tag{20}$$

Equation (20) represents the basic reproduction number derived for the model.

#### *4.2. Equilibria Studies*

In this subsection, we study the equilibrium points of FO-SVM model Equation (14). The FO-SVM model has virus-free equilibrium and endemic equilibrium points. In the endemic equilibrium point, the spread of infection is observed.

For equilibria studies, we have

$$D^{\alpha}I = 0,\\ D^{\alpha}M = 0,\\ D^{\alpha}UI\_{I} = 0,$$

equilibrium points of system (14) for virus-free and endemic are as: *K*<sup>0</sup> = (*I*, *M*, *UI*) = (0, 0, 0) and *K*∗ = (*I*∗, *M*∗, *U*∗ *<sup>I</sup>* ) for *R*<sup>0</sup> > 1.

The analysis for the endemic equilibria of model (14) is written as:

$$\begin{aligned} \frac{\beta\_1 (N^\* - I - M) I}{2^{32}} + \frac{\beta\_2 (N^\* - I - M) \mathcal{U}\_I}{N^\*} - \rho I - r\_1 I &= 0, \\ \rho I - r\_1 M &= 0, \\ \frac{\beta\_2 (\mathcal{U}^\* - \mathcal{U}\_I) I}{N^\*} - r\_2 \mathcal{U}\_I &= 0. \end{aligned} \tag{21}$$

Solving the equations in set (21), we obtain the expressions for the endemic equilibrium point (*I*∗, *M*∗, *U*∗ *<sup>I</sup>* ) as:

$$I^\* = \frac{\sqrt{b^2 - 4ac} - b}{2a},\tag{22}$$

$$M^\* = \frac{\rho}{r\_1} I^\*,$$
 
$$M^\* = \frac{\rho}{r\_1} I^\*,$$

$$
\Delta I\_I^\* = \frac{\beta\_2 \mathcal{U}^\*}{\beta\_2 I^\* + r\_2 \mathcal{N}^\*} I^\*,\tag{24}
$$

where

$$\begin{aligned} a &= \frac{(\rho + r1)\beta 1 \beta 2}{2^{32} r\_1 N^\*}, \\ b &= \frac{\beta\_2 (\rho + r\_1)(1 - R\_0)}{N^\*} + \frac{\beta\_2^3 L I^\*}{N^\* r\_2} + \frac{\beta\_1 (r\_2) \beta\_2^2 L^\*}{2^{32} r\_1} (\rho + r\_1), \\ c &= (\rho + r\_1)(1 - R\_0) r\_2. \end{aligned}$$

It is evident from Equation (22) that the possibility of infection spread, i.e., *I*∗ > 0, is only verified for the value of *R*<sup>0</sup> > 1.
