**3. Cluster SIRA model: Hypothesis and equations**

There is a lot of compartimental models indicated for epidemiology [43] and their origin is Kermack and Mckendrick SIR (susceptible—infected—removed) models [9, 43, 44].

The population is considered constant and is divided into three compartments: Susceptible computers are uninfected and subject to infection (S); infected computers are represented by (I), those removed by infection or not (R), as shown in **Figure 1**.

**Figure 1.** *SIR model.*

As reported by [21] the dynamic equation for the populations *S*, *I* and *R* are:

$$\begin{cases} \dot{\mathcal{S}} &= -a\mathcal{S}I; \\ \dot{I} &= a\mathcal{S}I - \beta I; \\ \dot{\mathcal{R}} &= \beta I. \end{cases} \tag{1}$$

The susceptible population *S* is infected with a rate, that is, related to the probability of susceptible individuals to establish effective communications with infected ones. Therefore, this rate is proportional to the product *SI*, with proportion factor represented by *α* and infected individual can become removed with a rate controlled by *β*.

Considering initial conditions *S*ð Þ 0 ≥ 0, *I*ð Þ 0 ≥0 and *R*ð Þ 0 ≥0, in model such as SIR the interest is to investigate the dynamics of virus propagation indicates whether the virus will remain in the network or if it will naturally be eradicated. One of the ways to evaluate this behavior is to study the basal reproduction rate ð Þ *R*<sup>0</sup> . This number indicates whether the virus will continue to be propagated and will be considered a situation analogous to the endemic one, or if it will become extinct in the network.

Based on a model described by (1), a model with a modification, including an antidotal population compartment (A) representing nodes of the network equipped with fully effective antivirus programs, is studied and considering constant population with four compartments: susceptible computers are uninfected and subject to infection (S); infected computers are represented by (I), and those removed by infection or not (R) and (A) are uninfected computers equipped with antivirus, as shown in **Figure 2**.

As reported by [21] the dynamic equation for the populations *S*, *I*, *R* and *A* are:

$$\begin{cases} \dot{\mathcal{S}} &= N - \sigma\_{\rm SA} \mathcal{S} \mathcal{A} - \beta \mathcal{S} I - \mu \mathcal{S} + \sigma R; \\ \dot{I} &= \beta \mathcal{S} I - a\_{\rm IA} A I - \delta I - \mu I; \\ \dot{R} &= \delta I - \sigma R - \mu R; \\ \dot{A} &= \sigma\_{\rm SA} \mathcal{S} A + a\_{\rm IA} A I - \mu A. \end{cases} \tag{2}$$

**Figure 2.** *SIRA model.*

## *A Review of Mathematical Model Based in Clustered Computer Network DOI: http://dx.doi.org/10.5772/intechopen.108891*

The influx rate is considered to be *N* ¼ 0 because during the propagation of the considered virus, there is no incorporation of new computers in the network. The choice of *μ* ¼ 0 is justified that the machines obsolescence time is larger than the time of the virus action.

The model represents the spread of a known virus and the conversion of the antidoto to the infected is not considered. In this model, a vaccination strategy can be defined implying a control strategy associated with the economic use of antivirus programs.

The analysis of the SIRA model shows that it is possible to reach an disease-free equilibrium, guaranteeing a good operational performance of the network and that even in a situation of endemic equilibrium, the introduction of at least one machine equipped with antivirus guarantees a good performance of the network, tending to a disease-free equilibrium.

Furthermore, considering a constant total number of machines, the main control parameters are associated with the infection rate and how quickly infected machines are removed for formatting procedure. The other network parameters are associated with the transient response of the network in some small disturbance.

The other variation of the SIRA model is improved by considering that, when the machines pass to the removed condition, a fraction of these machines is recovered and the complement is considered dead. The introduction of the mortality rate results in an increase of the robustness of the disease-free equilibrium point of a computer network [38].

The study of the SIRA model was complemented by considering another control strategy by adding a quarantined compartment. The new compartment can be evaluated for the presence or absence of saturation and both situations indicate robustness in control strategies.

Based on a model described by [20], the virus propagation in a cluster is studied [45]. The model proposed is an association of two networks constituted by the SIRA model that interacts as shown in **Figure 3**.

Considering this hypothesis, adding another compartimental model, and associating a new infection rate, representing the infection capacity to the network, the cluster SIRA model for viruses propagation was proposed has the following dynamical eqs. (3):

$$\begin{cases} \dot{S}\_{1} &= -a\_{\rm SA1}S\_{1}A\_{1} - \beta\_{1}S\_{1}I\_{1} - \rho\_{2}I\_{2}S\_{1} + \theta \mathbf{1}R\_{1}; \\ \dot{I}\_{1} &= \beta\_{1}S\_{1}I\_{1} - \delta\_{1}I\_{1} - a\_{\rm IA}I\_{1}A\_{1} + \rho\_{2}I\_{2}S\_{1}; \\ \dot{R}\_{1} &= \delta\_{1}I\_{1} - \theta\_{1}R\_{1}; \\ \dot{A}\_{1} &= a\_{\rm SA1}S\_{1}A\_{1} + a\_{\rm IA}I\_{1}A\_{1}; \\ \dot{S}\_{2} &= a\_{\rm SA2}S\_{2}A\_{2} - \beta\_{2}S\_{2}I\_{2} - \rho\_{1}I\_{1}S\_{2} + \theta 2R\_{2}; \\ \dot{I}\_{2} &= \beta\_{2}S\_{2}I\_{2} - \delta\_{2}I\_{2} - a\_{\rm IA}I\_{2}A\_{2} + \rho\_{1}I\_{1}S\_{2}; \\ \dot{R}\_{2} &= \delta\_{2}I\_{2} - \theta\_{2}R\_{2}; \\ \dot{A}\_{2} &= a\_{\rm SA2}S\_{2}A\_{2} + a\_{\rm A2}I\_{2}A\_{2}. \end{cases} \tag{3}$$

For the cluster SIRA model, the susceptible population *S* is infected with a rate, that is, related to the probability of susceptible elements to establish effective communications with infected ones and this rate is proportional to the product *SI*, with proportion factor represented by *α* or if infectivity occurs between network, by rate *ρ* that is related to the probability of infected elements to establish effective communications with susceptible computer in another network.

#### **Figure 3.** *SIRA cluster model.*

Clustered sets, represented by subscripts 1 and 2, are divided into four groups, as shown in **Figure 3** and populations are considered constant in each cluster.

The SIRA cluster study, despite presenting a simple model composed of two connected grouped networks, points out the main control strategies associated with the control of parameters in order to avoid new forms of attacks.

Among the possibilities for adjustments, we consider infection rates in susceptible populations, due to contact with infected populations from the same cluster (*β*); conversion rates are removed by infected (*δ*) and infection rates in susceptible populations due to contact with the infected population of the other cluster (*ρ*).

The choice of network topology and connection strategies is an effective strategy to reduce the spread of viruses, since infection rates are not known in advance.

Another way to prevent the spread of viruses is to maintain the removal rates of damaged machines, which plays an important role in controlling the spread of networks.

If the virus is known, the best strategy to prevent its spread is to introduce antidote nodes containing programs that can be propagated throughout the network, immunizing the other nodes.
