**7. Mathematical modeling for healthcare and cybersecurity**

Mathematics is one of the key components for cybersecurity data analysis. Mathematics has a direct impact on the advancement of the science of cybersecurity. Considering the complexity and dynamics of cyberspace it is essential to have a formal scientific basis for the field of cybersecurity. Mathematics plays a critical role in the construction of the science of cybersecurity.

There have been many research studies for modeling of dynamics and spread of COVID-19. Most of them are based on the Susceptible (Si)-Exposed (Ei)-Infected (Ii)-Removed (Ri) and susceptible-infected-recovered (SIR) model as shown in **Figure 5**. Susceptible individuals might acquire the infection at a given rate when they are in contact with an infectious individual and enter the exposed disease state before they become infectious and later either recover or die.

For a given age group i, epidemic transitions can be described as,

$$\mathbf{S}\_{i,t+1} = \mathbf{S}\_{i,t} - \boldsymbol{\beta} \cdot \mathbf{S}\_{i,t} \sum\_{j=1}^{n} \mathbf{C}\_{i,j} \mathbf{I}\_{j,t}^{\mathbf{c}} - \boldsymbol{a} \,\boldsymbol{\beta}\_{i,t} \sum\_{j=1}^{n} \mathbf{C}\_{i,j} \mathbf{I}\_{j,t}^{\mathbf{c}c} \tag{1}$$

$$E\_{i,t+1} = (\mathbf{1} - k)E\_{i,t} + \boldsymbol{\mathcal{J}} \cdot \mathbf{S}\_{i,t} \sum\_{j=1}^{n} \mathbf{C}\_{i,j} I\_{j,t}^{\epsilon} + a \boldsymbol{\mathcal{J}} \cdot \mathbf{S}\_{i,t} \sum\_{j=1}^{n} \mathbf{C}\_{i,j} I\_{j,t}^{\epsilon c} \tag{2}$$

$$I\_{j,t+1} = \rho\_i k \to\_{i,t} + (1-\gamma)I\_{j,t}^{\epsilon} \tag{3}$$

$$I\_{j,t+1} = (\mathbf{1} - \rho\_i)\mathbb{k}\,E\_{i,t} + (\mathbf{1} - \gamma)I\_{j,t}^{\epsilon} \tag{4}$$

$$
\begin{matrix}
\boxed{\mathcal{S}\_{\bar{t}}} \xrightarrow{\phi\_{\bar{t}}} \begin{matrix} \boxed{\mathcal{E}\_{\bar{t}}}
\end{matrix} \xrightarrow{\begin{bmatrix} \rho\_{\bar{t}}k \\ \end{bmatrix}} \begin{matrix} \boxed{I\_{\bar{t}}^{c}} \xrightarrow{\mathcal{Y}} \begin{matrix} \mathcal{Y} \\ \end{matrix} \xrightarrow{\mathcal{Y}} \begin{matrix} \mathcal{S}\_{\bar{t}} \\ \end{matrix}
\end{matrix}
$$

**Figure 5** *SEIR model for Dynamics and Spread Prediction of Covid-19 [32].*

$$R\_{i,t+1} = R\_{i,t} + \gamma I\_{j,t+1}^c + \gamma I\_{j,t+1}^{sc} \tag{5}$$

Where,

*β* ¼ Transmission rate.

*Ci*,*<sup>j</sup>* ¼ Contact of age group j made by age group i.

*<sup>k</sup>* <sup>¼</sup> **<sup>1</sup>** � *<sup>e</sup>*� **<sup>1</sup>** *dL* � � <sup>¼</sup> the daily probability of an exposed individual becoming infectious.

*<sup>γ</sup>* <sup>¼</sup> **<sup>1</sup>** � *<sup>e</sup>*� **<sup>1</sup>** *dI* � � <sup>¼</sup> the daily probability that an infected individual recovers when the average duration of infection is *dI*.

*dL* ¼ average incubation period.

*dI* ¼ average duration of infection.

*α* ¼ infection acquired from subclinical individual.

*ρ<sup>i</sup>* ¼ the probability that an individual is symptomatic or clinical.

**1** � *ρ<sup>i</sup>* ¼ probability of an infected case being asymptomatic or subclinical.

*I <sup>c</sup>* <sup>¼</sup> an infected individual can be clinical.

*I sc* <sup>¼</sup> an infected individual can be subclinical.

∅*i*,*<sup>t</sup>* ¼ *β* P *j Ci*,*jI c <sup>j</sup>*,*<sup>t</sup>* <sup>þ</sup> *αβ*<sup>P</sup> *j Ci*,*jI sc <sup>j</sup>*,*<sup>t</sup>* ¼The force of infection.

Primarily, these models were used in the past for the research of epidemic spreading with various forms of networks of transmission. The principle of AI techniques, like, Neural Networks (NN) are based on the collection of artificial neurons, without any prior knowledge, this AI technique automatically generates identification characteristics for cybersecurity.
