**4. SBG for Modeling the Spread Trend of COVID-***n*

SBG can be utilized to model the spread trend of COVID-*n* by travelers in different countries and on a large scale, involved countries. In this pattern, the vertices represent individuals/countries and edges appoint the relationship among individuals/countries which are based on a fundamental relation.

#### *4.1. Application 1*

Let *H* be the number of individuals. Consider *H* = {Michael, Robert, Emma, Olivia}. Then, the SBG of *G* = &*H*, *E*' is determined in the following way:


Define a binary relation "◦" on *H* as follows:

*a* ◦ *b* = {*x* |*x* get infected to COVID − *n* by person *a* or person *b*}

In Table 2, the pair (*H*, ◦) is a hypergroup.

**Table 2.** Hypergroup (*H*, ◦).


The following statements are attained from Table 2:


Consider the relation *γ<sup>n</sup>* as edges for two arbitrary vertices *x* and *y* as:

$$\forall x \gamma\_n y \iff \exists (a\_1, \dots, a\_n) \in H^n, \exists \sigma \in S\_n : \ x \in \prod\_{i=1}^n a\_{i\prime} \ y \in \prod\_{i=1}^n a\_{\sigma(i)}.$$

Note that ∏*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *ai* is regarded as a hyperproduct of distinct elements *ai* for *i* ∈ {1, 2, . . . , *n*}, that is *a*<sup>1</sup> ◦ *a*<sup>2</sup> ◦ ... *an*. We follow the procedure for all components, i.e.,

> *γ*22 ⇐⇒ 1 ∈ 2 ◦ 2, 2 ∈ 2 ◦ 2 *γ*24 ⇐⇒ 3 ∈ 2 ◦ 3, 4 ∈ 3 ◦ 2 *γ*23 ⇐⇒ 2 ∈ 3 ◦ 4, 3 ∈ 4 ◦ 3 *γ*24 ⇐⇒ 1 ∈ 4 ◦ 4, 4 ∈ 4 ◦ 4 *γ*24 ⇐⇒ 2 ∈ 3 ◦ 3, 4 ∈ 3 ◦ 3 *γ*23 ⇐⇒ 1 ∈ 3 ◦ 3, 3 ∈ 3 ◦ 3

This means that (1, 2) ∈ *e*1,(3, 4) ∈ *e*2,(2, 3) ∈ *e*3,(1, 4) ∈ *e*4,(2, 4) ∈ *e*5,(1, 3) ∈ *e*<sup>6</sup> where, *E* = {*e*1,*e*2,*e*3,*e*4,*e*5,*e*6} are the edges of *SBG*. The corresponding SBG of *G* is depicted in Figure 6a and Table 3.

**Figure 6.** SBGs of *G* corresponding to (**a**) Application 1 and (**b**) Application 2.


**Table 3.** SBGs of *G*.

Furthermore, the equivalence class of [*x*] is considered as the individuals who transmit viral disease COVID to specific person *x*, that is [*x*] = {*y* |*xγ*∗*y*}, where *γ*<sup>∗</sup> is the transitive closure of *γ* and *γ* = *n*≥1 *γn*. Therefore, the class [Michael] = {Robert, Emma, Olivia}, and so on. By applying Proposition 4, the degree of Michael is |*γ*∗(Michael) | = 3 and by Corollary 2, the SBG is 3-regular.
