#### Abstract

SEIQR (Susceptible, Exposed, Infectious, Quarantined, and Recovered) models for the transmission of malicious objects with simple mass action incidence and standard incidence rate in computer network are formulated. Threshold, equilibrium, and their stability are discussed for the simple mass action incidence and standard incidence rate. Global stability and asymptotic stability of endemic equilibrium for simple mass action incidence have been shown. With the help of Poincare Bendixson Property, asymptotic stability of endemic equilibrium for standard incidence rate has been shown. Numerical methods have been used to solve and simulate the system of differential equations. The effect of quarantine on recovered nodes is analyzed. We have also analyzed the behavior of the susceptible, exposed, infected, quarantine, and recovered nodes in the computer network.

#### 1. Introduction

It is a well-known fact that cyber world brought massive changes in the society. But nowadays cyber world is being threatened by the attack of malicious objects. Electronic mails and use of secondary devices are the major sources for the transmission of malicious objects in the computer network these days [1]. In accordance with their propagating behavior and characteristic, malicious objects spread in different way to each other. To curb the spread and impact of these malicious objects, it is important to study about their feature propagating methods, means, and limitation. Isolation may also be a very important and easy way to curb the transmission of these malicious objects. The word quarantine has evolved, meaning a forced isolation or stoppage of interactions with others. In biological world, quarantine has been adopted to reduce the transmission of human diseases, such as Leprosy, Plague, and Smallpox. Same concept has been adopted in the cyber world; the most infected nodes are isolated from the computer network till they get recovered. Anderson and May [2, 3] discussed the spreading nature of biological viruses, parasite, and so forth. leading to infectious diseases in human population through several epidemic models. The action of malicious objects throughout a network can be studied by using epidemiological models for disease propagation [4–9]. Richard and Mark [10] proposed an improved SEI (Susceptible-Exposed-Infected) model to simulate virus propagation. However, they do not show the length of latency and take into account the impact of antivirus software. Mishra and Saini [11, 12] presented a SEIRS model with latent and temporary immune periods to overcome limitation, which can reveal common worm propagation. Feng and Thieme [13–15] considered very general endemic models that include SEIQR model, with arbitrarily distributed periods of infection including quarantine and with a general form for the incidence term that includes the three forms. Wa and Feng [16] showed that an epidemic approximation near threshold number () can have a homoclinic bifurcation, so that some perturbation of the original model might also have a homoclinic bifurcation. Several authors studied the global stability of several epidemiological models [17–24]. Wang et al. studied the robustness of filtering on nonlinearities in packet losses, sensors, and so forth, [25–30].

In the SEIQR models for infection that confers immunity, susceptible nodes go to latent period, that is, nodes become infected but not infectious called exposed nodes, thereafter some nodes go to infectious class. Some infected nodes remain in the infected class while they are infectious and then move to the recovered class after the run of antimalicious software. Other most infected nodes are transferred into the quarantine class while they are infectious and then move to the recovered class after their recovery. The models here have a variable total population size, because they have recruitment into the susceptible class by inclusion of some new nodes and they have both crashing of nodes dueto reason otherthan the attack of malicious codes and crashing of nodes due to the attack of malicious codes. We have developed two models and have taken simple mass action incidence and standard incidence rate, because standard incidence rate is more realistic than the simple mass action incidence [31].

#### 2. Model 1: Mathematical Formulation for the SEIQR Model with Simple Mass Action Incidence

Let be the number of susceptible at time , be the number of exposed, be the number of infected nodes, be the number of quarantined nodes, be the recovered nodes after the run of antimalicious software, and be the total population size in time . The schematic diagram for the flow of malicious objects is depicted in Figure 1.

As per our assumption, we have the following system of equations: where is the recruitment rate of susceptible nodes, is the per capita natural mortality rate, the death rate in infective compartment due to malicious objects, and the death rate in quarantine class due to malicious objects. The per capita contact rate is average number of effective contacts with other nodes per unit time, is the rate constant which leaves the exposed compartment for infective class, is the rate constant which leaves the infective class for quarantine class, is the rate by which the nodes go from quarantine class into recovered class, and is the rate of vertical transmission into the infective class.

Lemma 2.1. *Consider the following two systems:
**
where , belong to , and are continuous function which satisfy a local Lipschitz condition in any compact set which belongs to , and as tends to infinity so that the second system is the limit system. Let and be the solutions of these systems, respectively. Suppose that is a locally asymptotically stable equilibrium of the limit system and its attractive region is
**
Let be the omega limit set of . If , then . *

* Lipschitz’s condition*

If for some function *,* the following condition is satisfied:
where and are any points in the domain and is a constant, this condition is called the Lipschitz’s condition.

Now, the total population size satisfies the equation
When , then from the above equation .

Let us define the solution region by .

The system (2.1) always has the malicious oject-free equilibrium .

Here the quarantine reproduction number is
If , then also contains a unique positive, endemic equilibrium .

Now from (2.1), on simplification, we have
We have , Hence,

Theorem 2.2. *Consider the system (2.1). If , then solution set is locally asymptotically stable for disease-free equilibrium . If , then the region , is an asymptotically stable region for the endemic equilibrium .*

*Proof. *For local stability, Jacobian of system (2.1) at equilibrium is
The eigenvalues of are . Since all the roots are real and negative, system is locally asymptotically stable.

In order to prove the global stability when , consider the Liapunov function . Liapunov derivative

As we know the Liapunov Lasallel theorem [17] implies that solutions in approach the largest positively invariant subset of the set where , which is the set where *. *

In this set,

We have
when
then
**

We have from (2.1),

this implies

Thus,
when

Thus, all solutions in the set go to the disease, free equilibrium . By Lemma 2.1, the system is globally asymptotically stable, when .

From the fourth equation in system (2.1), we can solve to obtain
Now, . This implies ;

The similarly, solving for by using, fifth equation in (2.1), we obtain
An application of Lemma 2.1 shows that the endemic equilibrium of model (2.1) is globally asymptotically stable in the region .

#### 3. Model 2: The SEIQR Model with the Standard Incidence Rate

The flow chart for the SEIQR model will be the same as depicted in Figure 1, but instead of simple mass action incidence , we take standard incidence rate , where .

The system of differential equations for this model is where the parameters are the same as in the previous model,

When then *.* When there are malicious objects free equilibrium *.* For this model basic reproduction number is
If , then also contains a unique positive, endemic equilibrium where

Theorem 3.1. *Consider the system (3.1). If , then solution set is locally asymptotic stable for disease-free equilibrium . If , then the region is an asymptotically stable region for the endemic equilibrium _{.}*

*Proof. *For the local stability, Jacobian of the system (3.1) at equilibrium is
The eigenvalues of are . Since all the roots are real and negative, system is locally asymptotically stable.

With a view to prove the global stability when , we use the Liapunov function by putting , we get the same equation as we have found in the model 1; therefore, the proof will be analogous with the proofs in the previous section.

In order to prove the global stability when and , first we get , this implies when tends to infinity.

The limit system for (3.1) is
where *.* The first three equations are independent of and . In the three dimensional first octant region with , the equilibrium (0, 0, 0) is saddle, that is, attractive along and has a repulsive direction into the region. The other equilibrium in the region is locally asymptotically stable.

Using Dulac’s criteria with multiplier , we have
so that there are no periodic solutions in the region. Thus, by the Poincare-Bendixson theory, all solutions starting in the first octant region with and approach as tends infinity.

In this case, the differential equation for has the limiting equation
so that tends to by Lemma 2.1.

Similarly, the differential equation for has the limiting equation
so that tends to by Lemma 2.1. Thus is a globally asymptotically stable equilibrium for the limit system (3.6). Hence, by Lemma 2.1, all solutions starting in the region of the system (3.1) approach the endemic equilibrium as tends to infinity. **

#### 4. Conclusion

Inspired by the biological compartmental epidemic model, we made an attempt to develop two SEIQR models, one using simple mass action incidence and the other using standard incidence rate. Vertical transmission has been included into infectious compartment. Runge Kutta Fehlberg fourth-fifth order method is used to solve and simulate the system (2.1) and (3.1) by using parametrical values mentioned in Table 1. The model has a constant recruitment of the nodes and exponential natural and infection-related death (crashing) of the nodes. Global stability of the unique endemic equilibrium for the epidemic model has been established. We observe that the behavior of the Susceptible, Exposed, Infected, and Quarantined nodes with respect to time is asymptotically stable, which is depicted in Figures 2 and 4. The effect of on is depicted in Figure 3. When the nodes are highly infected by different kinds of malicious objects, quarantine is one of the remedy. We run antimalicious software of latest signature against quarantined nodes, and these nodes are kept under observation. The more we quarantine the most infected nodes, the more the recovery is; the lesser we quarantine, the lesser the recovery is. Also at a very specific short interval of time, the recovery of the nodes is constant when the quarantine node decreases. These can be observed in Figure 3. Simulation result agrees with the real life situation. The basic reproduction number is obtained and has been identified as a threshold parameter. If , the disease-free equilibrium is globally stable in the feasible region and the disease always dies out. If , a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and once the disease appears, it eventually persists at the unique endemic equilibrium level. Lyapunov function is used to prove the global stability of when . In our model, the number of contacts is influenced by the size of the quarantine class . The quarantine process is an alternative method for reducing the average infectious period by isolating some infectives, so that they do not transmit the malicious objects in the computer network. We have observed that both the effective infectious period and decrease as the quarantine rate increases.

The analysis of quarantine reproduction number by Feng and Thieme [14, 15] andHethcote et al. [31] agrees with our model. Feng and Theieme [14, 15] in their SIQR model observed that the quarantine reproduction number was independent of the mean residence time in the quarantine class . Hethcote et al. [31] also had their same observation regarding the independence of the mean residence time in the class for all of their endemic models. We also have the same observation for our SEIQRS model. The mean residence time in the class for our model SEIQRS is . The expression for the threshold does not involve the parameter . This comes from our assumption that the nodes in the quarantine class do not infect other nodes and nodes are not infectious when they move out of the quarantine class. The quarantine reproduction number , depends on parameter . For example, if , then transfer out of infectious class to quarantine class is n times as frequent as transfer to the removed class . A positive rate constant to transfer out of infectious class by quarantine does decrease the quarantine reproduction number , so that it is less than its value without quarantine. Hence the use of quarantine to control a disease not only decreases the endemic infective class size when remains above 1, but also makes it easier to obtain leading to disease extinction.

The future work will involve in taking time delay constraints in various compartments which may lead to more interesting result.

#### Acknowledgment

The authors are thankful to the editor and anonymous referees for their comments which substantially improved the quality of the paper.