# Considering Quarantine in the SIRA Malware Propagation Model

Authors: José Roberto C. Piqueira; Cristiane M. Batistela
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

As the beginning of the21st century was marked by a strong development in data science and, consequently, in computer networks, models for designing preventive actions against intruding, data stealing, and destruction became mandatory. Following this line, several types of epidemiological models have been developed and improved, considering different operational approaches. The development of the research line using traditional SIR(Susceptible, Infected, Removed) model for data networks started in the 1990s. In 2005, an epidemiological compartmental model containing antidotal nodes, SIRA (Susceptible, Infected, Removed, Antidotal), was introduced to study how the antivirus policies affect the network reliability. The idea here is to study the consequence of quarantine actions in a network by modifying the SIRA model, introducing quarantine nodes generating the SIQRA (Susceptible, Infected, Quarantine, Removed, Antidotal) model. Analytical and numerical approaches result in parameter conditions for the existence and stability of disease-free and endemic equilibrium points for two different cases: saturation and nonsaturation of the quarantine population block. Based on these results, operational actions can be planned to improve the network reliability.


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## Body

## 1. Introduction

A substantial development occurred in data network technology, providing accurate and fast public and private facilities. This has changed the way of data transfer in society, creating a strong dependence on the communication services at all levels of relations. Computers, laptops, and smart phones are an integral part of human life, with a large universe of different types of connections emerging, facilitating daily tasks and improving personal relations.Unfortunately, there are people and groups using their intelligence to develop hate and bad intentions, creating programs, such as viruses and worms, to invade and to destroy processes, causing collective and individual harmful effects. The power of destruction and the capacity of propagation of the viruses and malwares have increased over the years representing important threats to society and to corporative networks [1–3].Consequently, there are two main concerns related to the propagation of computer virus and malwares: estimating the economic losses [4–8] and developing security and safety strategies [9]. Thus, a new important branch of computer science has been developed, cybernetic security, having as one of its approaches to build mathematical models, related to the dynamic operation of the systems [10].This mathematical approach to computer virus propagation was inspired by biology and was divided into two levels: microscopic and macroscopic [1, 11]. The microscopic level is related to the development of antivirus software. Antivirus software is a program capable of detecting threats and preventing damages in the machine. Besides, the spread of virus through the network is avoided as a consequence of the antivirus action [12, 13].The macroscopic level is based on the classical model of disease propagation [14–16], whose dynamics can indicate the possible infection process and show how to establish policies and strategies to reduce risk, combining parameters as strategies of control [17, 18]. The classical epidemiological macroscopic model SIR, proposed by Kermack and McKendrick, is the basis for the development of macroscopic models, providing a dynamic model dividing the individuals of a population into three compartments containing the susceptible individuals (S), the infected (I), and the removed (R) ones [14, 15, 19].Adapting the SIR model to computer populations, two different strategies were developed: deterministic [11, 13, 20–23] or stochastic [12, 24–26]. Regarding the mathematical treatment of time, discrete [27] and continuous [28, 29] models appeared.Based on the classical SIR model, the deterministic SIRA model was developed, considering constant population [2, 18, 30]. In this case, robust conditions for disease-free equilibrium stability, depending on the node recovery rate, were derived, considering that antidotal nodes are part of the network. Improving the SIRA model, mortality and variable population were considered in the descriptions, allowing new robust parameter relations for disease-free equilibrium that seem to be useful to design data networks [31].Models taking delays and quarantine into account in the classical SIR model [32] show interesting operational actions to be taken to improve reliability, with bifurcation diagrams indicating parameters values to be avoided [33, 34]. Recently, models including quarantine compartments seem to avoid virus and malware propagation in data networks [35, 36] and to prevent attacks in sensor networks [22, 23, 37, 38].The main idea to control the virus spread is to keep the nodes isolated for some time [38, 39]. The word “quarantine” means this interaction is interrupted. When a node is found to be infected, it can be quarantined by the detection virus program. It is then monitored for an interval corresponding to the inappropriate behavior indicated by the process. If during this action no wrong behavior is observed, it is released.In this work, quarantine concept is applied as an additional improvement of the SIRA model and the SIQRA (Susceptible, Infected, Quarantine, Removed, Antidotal) model is developed. Two different cases are studied: with and without interaction between infected and quarantine compartments allowing robust relations between parameters for disease-free equilibrium, providing ideas to improve the reliability of networks.It will be shown that, if the quarantine compartment is introduced with its population not saturated by the number of infected nodes, i.e., the antivirus detection rate is greater than the infection rate, the model does not contain nonlinear terms given byIQ. In this case, the network presents a very satisfactory behavior, with stable disease equilibrium point and no endemic possibility. If the saturation term IQ is present, i.e., the infection rate is greater than the virus detection capacity, in spite of the benefits given by the presence of the quarantine population, an endemic equilibrium appears.In the next section, hypothesis and equations are presented, followed by a section deriving equilibrium conditions and the parameters relations for the several equilibrium situations. Numerical results are shown next, simulating the network dynamics for some relevant real cases, followed by the conclusion section giving hints about the model improvements.


## 2. SIQRA Model: Hypothesis and Equations

The model to be studied is a modification to the compartmental SIRA [2, 18, 30], improved in [31], adding a quarantine computer compartment (Q), representing possible infected machines, which are separated to be evaluated concerning infection or recovery (Figure 1). Once entering in the quarantine compartment, after the evaluation, the machine can either return to the network or be removed. If a machine returns, it can become either susceptible or antidotal.Figure 1
SIQRA model.In Figure1, αIA represents the interaction coefficient between compartments I and A, related to the vaccination rate of the network; β, the interaction between S and I, i.e., measuring how the infected nodes change the operational systems of the susceptible ones; σ, the rate of transformation from R to S, i.e., the recovering capacity of the nodes; and ω, the rate of transformation from Q to S, giving the rate of quarantine liberation.ParametersαSA and αQA represent how the antidotal nodes actuates over susceptible and quarantine operational systems, respectively; α is the interaction coefficient between compartments Q and R, giving an idea of the rate of nonrecovered nodes subjected to quarantine.Two different quarantine strategies are presented: one without nonlinear interaction between blocksI and Q, i.e., without a saturation term, and other considering that blocks I and Q interact nonlinearly allowing the onset of endemic situations. Parameter δ represents either the rate of transformation from I to Q in the model without nonlinear interaction or the interaction between compartments Q and I in the model with nonlinear IQ term.


### 2.1. Model without NonlinearIQ Term

In this case, examining Figure1, the state equation representing the model is given by(1)S˙=−αSASA−βSI+σR+ωQ,I˙=βSI−αIAAI−δI,Q˙=δI−ωQ−αQAQ−αQ,R˙=αQ−σR,A˙=αSASA+αIAAI+αQAQ.The initial conditions are assumed to beS0 ≥ 0, I0 ≥ 0, Q0 ≥ 0, R0 ≥ 0, and A0 ≥ 0. It can be verified that, for equation (1), the total population T=S+I+Q+R+A is constant.As a consequence, the state space is 4-dimensional, and examining the dynamic equation forI˙, no endemic equilibrium point is verified, i.e., there is no equilibrium point with I≠0. However, two disease-free equilibrium points (I=0) are possible, described by using the following expressions:(2)P1=S,I,Q,R,A=0,0,0,0,A=0,0,0,0,T,P2=S,I,Q,R,A=S,0,0,0,0=T,0,0,0,0.To study the local stability of these equilibrium points, the Jacobian matrix, (J), must be calculated [40]:(3)J=−αSAA−βI−βSωσ−αSASβIβS−αIAA−δ00−αIAI0δ−ω−αQA−α0000α−σ0αSAAαIAAαQA0αSAS+αIAI.Considering theP1 equilibrium point, the Jacobian matrix, JP1, is given by(4)JP1=−αSAT0wσ00−αIAT+δ0000δ−w+αQA+α0000α−σ0αSATαIATαQA00.By using MATLAB R2013a [41], the eigenvalues of JP1 are λ1=0, λ2=−αIAT+δ, λ3=−σ, λ4=−αSAT, and λ5=−α+ω+αQA. The zero eigenvalue only indicates that the state space dimension is 4 in spite of the 5-equation model. The other eigenvalues are real and negative, for any parameter combination with physical meaning, implying that P1 is asymptotically stable.Repeating the procedure for the equilibrium pointP2, the Jacobian matrix, JP2, is given by(5)JP2=0−βTwσ−αSAT0βT−δ0000δ−w+αQA+α0000α−σ000αQA0αSAT,whose eigenvalues are λ1=0, λ2=αSAT+δ, λ3=βT−δ, λ4=−σ, and λ5=−α+ω+αQA.Again, the zero eigenvalue only indicates that the state space dimension is 4 in spite of the 5-equation model. For any parameter combination with physical meaning, one of the eigenvalues,λ2, is real and positive, indicating that P2 is unstable.


### 2.2. Model with NonlinearIQ Term

Including the possibility of saturation for theIQ term is important mainly for small networks, where the increase of this product could provoke a time-out situation for the whose system.In this case, examining Figure1, the state equation representing the model is given by(6)S˙=−αSASA−βSI+σR+ωQ,I˙=βSI−αIAAI−δQI,Q˙=δQI−ωQ−αQAQ−αQ,R˙=αQ−σR,A˙=αSASA+αIAAI+αQAQ.The initial conditions are assumed to beS0 ≥ 0, I0 ≥ 0, Q0 ≥ 0, R0 ≥ 0, and A0 ≥ 0. It can be verified that, for equation (6), the total population T=S+I+Q+R+A is constant. As a consequence, the state space is 4-dimensional.Examining the model equations, it can be concluded that the system presents three equilibrium points: two disease-free (I=0), given by P1 and P2, and one endemic (I≠0), given by P3, with(7)P1=S,I,Q,R,A=0,0,0,0,A=0,0,0,0,T,P2=S,I,Q,R,A=S,0,0,0,0=T,0,0,0,0,P3=S,I,Q,R,A=0,I,0,0,0=0,T,0,0,0.To study the local stability of these equilibrium points, the Jacobian matrix, (J), must be calculated [40]:(8)J=−αSAA−βI−βSωσ−αSASβIβS−αIAA−δQδI0−αIAI0δQδI−ω−αQA−α0000α−σ0αSAAαIAAαQA0αSAS+αIAI.Considering theP1 equilibrium point, the Jacobian matrix, JP1, is given by(9)JP1=−αSAT0ωσ00−αIAT00000−ω−αQA−α0000α−σ0αSATαIATαQA00.By using MATLAB R2013a [41], the eigenvalues of JP1 are λ1=0, λ2=−σ, λ3=−αIAT, λ4=−αSAT, and λ5=−α+ω+αQA.The zero eigenvalue only indicates that the state space dimension is 4 in spite of the 5-equation model. The other eigenvalues are real and negative, for any parameter combination with physical meaning, implying thatP1 is asymptotically stable.Repeating the procedure for the equilibrium pointP2, the Jacobian matrix, JP2, is given by(10)JP2=0−βTωσ−αSAT0βT00000−ω−αQA−α0000α−σ000αQA0αSAT,whose eigenvalues are λ1=0, λ2=αSAT, λ3=βT, λ4=−σ, and λ5=−α+ω+αQA.Again, the zero eigenvalue only indicates that the state space dimension is 4, in spite of the 5-equation model. For any parameter combination with physical meaning, one of the eigenvalues,λ2, is real and positive, indicating that P2 is unstable.The endemic equilibrium point,P3, is characterized by the presence of infected individuals (I≠0) and to perform calculations, it is supposed that there is no antidotal node in the network A=0.ForP3, the Jacobian matrix, JP3, is given as follows:(11)JP3=−βT0ωσ0βT0δT0−αIAT00δT−ω−αQA−α0000α−σ000αQA0αIAT,whose eigenvalues are λ1=−βT, λ2=0, λ3=δT−α−ω−αQA, λ4=−σ, and λ5=αIAT.As in the other cases, the zero eigenvalue only indicates that the state space dimension is 4, in spite of the 5-equation model. For any parameter combination with physical meaning, one of the eigenvalues,λ5, is real and positive, indicating that P3 is unstable.


## 2.1. Model without NonlinearIQ Term

In this case, examining Figure1, the state equation representing the model is given by(1)S˙=−αSASA−βSI+σR+ωQ,I˙=βSI−αIAAI−δI,Q˙=δI−ωQ−αQAQ−αQ,R˙=αQ−σR,A˙=αSASA+αIAAI+αQAQ.The initial conditions are assumed to beS0 ≥ 0, I0 ≥ 0, Q0 ≥ 0, R0 ≥ 0, and A0 ≥ 0. It can be verified that, for equation (1), the total population T=S+I+Q+R+A is constant.As a consequence, the state space is 4-dimensional, and examining the dynamic equation forI˙, no endemic equilibrium point is verified, i.e., there is no equilibrium point with I≠0. However, two disease-free equilibrium points (I=0) are possible, described by using the following expressions:(2)P1=S,I,Q,R,A=0,0,0,0,A=0,0,0,0,T,P2=S,I,Q,R,A=S,0,0,0,0=T,0,0,0,0.To study the local stability of these equilibrium points, the Jacobian matrix, (J), must be calculated [40]:(3)J=−αSAA−βI−βSωσ−αSASβIβS−αIAA−δ00−αIAI0δ−ω−αQA−α0000α−σ0αSAAαIAAαQA0αSAS+αIAI.Considering theP1 equilibrium point, the Jacobian matrix, JP1, is given by(4)JP1=−αSAT0wσ00−αIAT+δ0000δ−w+αQA+α0000α−σ0αSATαIATαQA00.By using MATLAB R2013a [41], the eigenvalues of JP1 are λ1=0, λ2=−αIAT+δ, λ3=−σ, λ4=−αSAT, and λ5=−α+ω+αQA. The zero eigenvalue only indicates that the state space dimension is 4 in spite of the 5-equation model. The other eigenvalues are real and negative, for any parameter combination with physical meaning, implying that P1 is asymptotically stable.Repeating the procedure for the equilibrium pointP2, the Jacobian matrix, JP2, is given by(5)JP2=0−βTwσ−αSAT0βT−δ0000δ−w+αQA+α0000α−σ000αQA0αSAT,whose eigenvalues are λ1=0, λ2=αSAT+δ, λ3=βT−δ, λ4=−σ, and λ5=−α+ω+αQA.Again, the zero eigenvalue only indicates that the state space dimension is 4 in spite of the 5-equation model. For any parameter combination with physical meaning, one of the eigenvalues,λ2, is real and positive, indicating that P2 is unstable.


## 2.2. Model with NonlinearIQ Term

Including the possibility of saturation for theIQ term is important mainly for small networks, where the increase of this product could provoke a time-out situation for the whose system.In this case, examining Figure1, the state equation representing the model is given by(6)S˙=−αSASA−βSI+σR+ωQ,I˙=βSI−αIAAI−δQI,Q˙=δQI−ωQ−αQAQ−αQ,R˙=αQ−σR,A˙=αSASA+αIAAI+αQAQ.The initial conditions are assumed to beS0 ≥ 0, I0 ≥ 0, Q0 ≥ 0, R0 ≥ 0, and A0 ≥ 0. It can be verified that, for equation (6), the total population T=S+I+Q+R+A is constant. As a consequence, the state space is 4-dimensional.Examining the model equations, it can be concluded that the system presents three equilibrium points: two disease-free (I=0), given by P1 and P2, and one endemic (I≠0), given by P3, with(7)P1=S,I,Q,R,A=0,0,0,0,A=0,0,0,0,T,P2=S,I,Q,R,A=S,0,0,0,0=T,0,0,0,0,P3=S,I,Q,R,A=0,I,0,0,0=0,T,0,0,0.To study the local stability of these equilibrium points, the Jacobian matrix, (J), must be calculated [40]:(8)J=−αSAA−βI−βSωσ−αSASβIβS−αIAA−δQδI0−αIAI0δQδI−ω−αQA−α0000α−σ0αSAAαIAAαQA0αSAS+αIAI.Considering theP1 equilibrium point, the Jacobian matrix, JP1, is given by(9)JP1=−αSAT0ωσ00−αIAT00000−ω−αQA−α0000α−σ0αSATαIATαQA00.By using MATLAB R2013a [41], the eigenvalues of JP1 are λ1=0, λ2=−σ, λ3=−αIAT, λ4=−αSAT, and λ5=−α+ω+αQA.The zero eigenvalue only indicates that the state space dimension is 4 in spite of the 5-equation model. The other eigenvalues are real and negative, for any parameter combination with physical meaning, implying thatP1 is asymptotically stable.Repeating the procedure for the equilibrium pointP2, the Jacobian matrix, JP2, is given by(10)JP2=0−βTωσ−αSAT0βT00000−ω−αQA−α0000α−σ000αQA0αSAT,whose eigenvalues are λ1=0, λ2=αSAT, λ3=βT, λ4=−σ, and λ5=−α+ω+αQA.Again, the zero eigenvalue only indicates that the state space dimension is 4, in spite of the 5-equation model. For any parameter combination with physical meaning, one of the eigenvalues,λ2, is real and positive, indicating that P2 is unstable.The endemic equilibrium point,P3, is characterized by the presence of infected individuals (I≠0) and to perform calculations, it is supposed that there is no antidotal node in the network A=0.ForP3, the Jacobian matrix, JP3, is given as follows:(11)JP3=−βT0ωσ0βT0δT0−αIAT00δT−ω−αQA−α0000α−σ000αQA0αIAT,whose eigenvalues are λ1=−βT, λ2=0, λ3=δT−α−ω−αQA, λ4=−σ, and λ5=αIAT.As in the other cases, the zero eigenvalue only indicates that the state space dimension is 4, in spite of the 5-equation model. For any parameter combination with physical meaning, one of the eigenvalues,λ5, is real and positive, indicating that P3 is unstable.


## 3. Numerical Experiments

By using the Simulink tool from MATLAB R2013a [41], model simulations were performed to confirm the analytical results, showing the possible behaviors of the dynamical system when quarantine and antidotal compartments are present. The total population T=S+I+Q+R+A is set equal to N = 100, to express the results in percentage, and the time unit is normalized, according to the normalized rates of interactions.


### 3.1. Numerical Experiments: Model without theIQ Saturation Term

To simulate this situation, the parameters and population were chosen to show the stability of the equilibrium points. ConsideringαSA=0.025, β=0.1, ω=2, σ=0.8, αIA=0.25, δ=9, αQA=0.5, and α=0.5 for all simulations.If an infected node is introduced into the network, the asymptotically stable equilibrium,P1, is reached, as Figure 2 shows, with initial conditions S=99, I=1, A=0, R=0, and Q=0.Figure 2
SIQRA without theIQ term: time evolution reaching P1.If the infected population is high, the introduction of either an antidotal node (Figure3(a)) or a quarantine node (Figure 3(b)) drives the populations to stable equilibrium.Figure 3
SIQRA without theIQ term: time evolution reaching P1, starting with high I: (a) Introducing one antidotal, (b) Introducing one quarantine.
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(b)When the network consists only of susceptible and removed computers, it is possible to verify that even the disease-free unstable equilibrium point,P2, is reachable for any parameter combination, because the unstable eigenvector direction I has been avoided. Figure 4 shows this situation starting with S=90 and R=10 and reaching P2.Figure 4
SIQRA without theIQ term: time evolution reaching P2.


### 3.2. Numerical Experiments: Model with theIQ Saturation Term

As analytically shown, depending on the initial condition, the temporal evolution of network ends at three different points: stable disease-free, unstable disease-free, and unstable endemic. To show these three cases, the parameters are set atαSA=0.025, β=0.1, ω=2, σ=0.8, αIA=0.25, δ=9, αQA=0.5, and α=0.5.If the network starts with only one antidotal machine, the stable disease-freeP1 is always reached, as Figure 5 shows for the initial condition S,I,Q,R,A=85,9,0,5,1.Figure 5
SIQRA with theIQ term: time evolution reaching P1 starting with an antidotal machine.Another way to capture the disease-free stable equilibriumP1 is considering the same parameter values and starting with a condition without antidotal machines but with at least a quarantine one. Figure 6 shows the temporal evolution of the dynamic model for the initial conditions S,I,Q,R,A=85,9,1,5,0.Figure 6
SIQRA with theIQ term: time evolution reaching P1 starting with a quarantine machine.When the network consists only of susceptible and removed computers, it is possible to verify that even the disease-free unstable equilibrium point,P2, is reachable for any parameter combination, because the unstable eigenvector direction (I) has been avoided. Figure 7 shows a temporal evolution starting at S=90 and R=10 and reaching the disease-free equilibrium P2.Figure 7
SIQRA with theIQ term: time evolution reaching P2.Starting without antidotal or quarantine machines, even one infected node drives the network to collapse at an endemic equilibrium point. Figure8 shows a temporal evolution initial condition S,I,Q,R,A=99,1,0,0,0, reaching the endemic equilibrium point P3.Figure 8
SIQRA with theIQ term: time evolution reaching P3.


### 3.3. Simulations with Real Data

In [42], an infection control model was developed and tested by using data from cybernetic crimes in Japan networks. The population data S and I given in [42] was taken to test the SIQRA models, identifying β with the infection rate and σ with the recovering rate.The normalization of these practical data to be compatible with the time and population scales used here givesS0=23.01, I0=76.99, A0=0, Q0=0, R0=0, αSA=0.025, β=7.85.10−11, ω=2, σ=8.890729.10−4, αIA=0.25, δ=9, αQA=0.5, and α=0.5. Simulating the SIQRA model without the IQ term, the results are shown in Figure 9, with the final state presenting about 10% of removed machines.Figure 9
SIQRA without theIQ term: simulation with real data.To avoid removing machines, the SIQRA model withIQ term can be used but demands at least an initial antidotal machine. Without changing parameters, Figure 10 shows the results adopting S0=22.01, I0=76.99, A0=1, Q0=0, and R0=0.Figure 10
SIQRA with theIQ term: simulation with real data.


## 3.1. Numerical Experiments: Model without theIQ Saturation Term

To simulate this situation, the parameters and population were chosen to show the stability of the equilibrium points. ConsideringαSA=0.025, β=0.1, ω=2, σ=0.8, αIA=0.25, δ=9, αQA=0.5, and α=0.5 for all simulations.If an infected node is introduced into the network, the asymptotically stable equilibrium,P1, is reached, as Figure 2 shows, with initial conditions S=99, I=1, A=0, R=0, and Q=0.Figure 2
SIQRA without theIQ term: time evolution reaching P1.If the infected population is high, the introduction of either an antidotal node (Figure3(a)) or a quarantine node (Figure 3(b)) drives the populations to stable equilibrium.Figure 3
SIQRA without theIQ term: time evolution reaching P1, starting with high I: (a) Introducing one antidotal, (b) Introducing one quarantine.
(a)
(b)When the network consists only of susceptible and removed computers, it is possible to verify that even the disease-free unstable equilibrium point,P2, is reachable for any parameter combination, because the unstable eigenvector direction I has been avoided. Figure 4 shows this situation starting with S=90 and R=10 and reaching P2.Figure 4
SIQRA without theIQ term: time evolution reaching P2.


## 3.2. Numerical Experiments: Model with theIQ Saturation Term

As analytically shown, depending on the initial condition, the temporal evolution of network ends at three different points: stable disease-free, unstable disease-free, and unstable endemic. To show these three cases, the parameters are set atαSA=0.025, β=0.1, ω=2, σ=0.8, αIA=0.25, δ=9, αQA=0.5, and α=0.5.If the network starts with only one antidotal machine, the stable disease-freeP1 is always reached, as Figure 5 shows for the initial condition S,I,Q,R,A=85,9,0,5,1.Figure 5
SIQRA with theIQ term: time evolution reaching P1 starting with an antidotal machine.Another way to capture the disease-free stable equilibriumP1 is considering the same parameter values and starting with a condition without antidotal machines but with at least a quarantine one. Figure 6 shows the temporal evolution of the dynamic model for the initial conditions S,I,Q,R,A=85,9,1,5,0.Figure 6
SIQRA with theIQ term: time evolution reaching P1 starting with a quarantine machine.When the network consists only of susceptible and removed computers, it is possible to verify that even the disease-free unstable equilibrium point,P2, is reachable for any parameter combination, because the unstable eigenvector direction (I) has been avoided. Figure 7 shows a temporal evolution starting at S=90 and R=10 and reaching the disease-free equilibrium P2.Figure 7
SIQRA with theIQ term: time evolution reaching P2.Starting without antidotal or quarantine machines, even one infected node drives the network to collapse at an endemic equilibrium point. Figure8 shows a temporal evolution initial condition S,I,Q,R,A=99,1,0,0,0, reaching the endemic equilibrium point P3.Figure 8
SIQRA with theIQ term: time evolution reaching P3.


## 3.3. Simulations with Real Data

In [42], an infection control model was developed and tested by using data from cybernetic crimes in Japan networks. The population data S and I given in [42] was taken to test the SIQRA models, identifying β with the infection rate and σ with the recovering rate.The normalization of these practical data to be compatible with the time and population scales used here givesS0=23.01, I0=76.99, A0=0, Q0=0, R0=0, αSA=0.025, β=7.85.10−11, ω=2, σ=8.890729.10−4, αIA=0.25, δ=9, αQA=0.5, and α=0.5. Simulating the SIQRA model without the IQ term, the results are shown in Figure 9, with the final state presenting about 10% of removed machines.Figure 9
SIQRA without theIQ term: simulation with real data.To avoid removing machines, the SIQRA model withIQ term can be used but demands at least an initial antidotal machine. Without changing parameters, Figure 10 shows the results adopting S0=22.01, I0=76.99, A0=1, Q0=0, and R0=0.Figure 10
SIQRA with theIQ term: simulation with real data.


## 4. Conclusions

Classical SIR models applied to data communication systems inspired the first SIRA model, introducing antidotal nodes that supposedly improve the operational network robustness [2]. Bifurcations of equilibrium points and conditions for existence and stability of disease-free equilibrium points for the SIRA model were derived [18, 30], suggesting network design strategies.Recently, the SIRA model showed improvement related to the practical situation considering variable populations, deriving network robustness conditions [31]. Here, the SIRA model was complemented with a quarantine strategy (SIQRA model), and the simulations indicate that network performance can be increased in the great majority of the cases and hypothesis that must be tested in real situations.The main results regarding the SIQRA model are divided into two cases: with and without saturation condition. Saturation is about the behavior of theQ term that depends on the antivirus program capacity and the infection rate. If the antivirus detection rate is greater than the infection rate, there is no saturation. If the infection rate surpasses the antivirus detection capacity, saturation occurs.If theQ compartment is not subjected to the possibility of saturation (no IQ term), the stable disease-free equilibrium point exists and is always reached. However, if there is any possibility of saturation (IQ term present), an initial antidotal or quarantine node is necessary. If there is neither initial antidotal nor quarantine node, even a single infected machine collapses the network.Testing the models with real data shows that quarantine is effective to combat infection in data networks. The noninclusion ofIQ terms, in spite of eliminating infected nodes, implies the removal of a certain number of machines. The inclusion of the term IQ avoids machine removal but demands the inclusion of either a quarantine or an antidotal node.



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*Source: 6467104-2019-11-21.xml*