A delayed computer virus model with nonlinear incidence rate

An Susceptible-Vaccinated-Exposed-Infectious-Recovered computer virus model with nonlinear incidence rate and two delays is proposed and its Hopf bifurcation is investigated. The existence of Hopf bifurcation at the viral equilibrium is established by analyzing its distribution of characteristic values. Moreover, the stability and the direction of the Hopf bifurcation are obtained by using the normal form theory and centre manifold theory. Finally, a numerical simulation is performed to validate the theoretical results obtained in the paper.


Introduction
The computer network has brought great convenience to our daily life. While enjoying the convenience from the computer network, we have to confront the threat of computer virus intrusions (Kumar, Mishra, & Panda, 2016;Ren & Xu, 2014). The file system is mostly attacked by the computer viruses which use system vulnerability to attack computers in networks. Based on Cybercrime-Report (2017), cybercriminal activity is one of the biggest challenges that humanity will face in the next two decades and Cybersecurity Ventures predicts that cybercrime will cost the world in excess of $6 trillion annually by 2021. Thus, conducting research on computer virus propagation dynamics is of considerable interest to elevate the computer network security. So far, a wide variety of models have been proposed to simulate the behaviour of computer virus throughout networks since the macroscopic model featuring the spread of computer viruses formulated by Kephart, White, and Chess (1993). In addition, researchers have paid attention to the combination of computer virus model and antivirus countermeasures such as quarantine and virus immunization, in order to analyse and counter against computer viruses. In Mishra, Srivasstava, and Mishra (2014), Khanh (2016), Nwokoye, Ozoegwu, and Ejiofor (2017), worm propagation models with quarantine in wireless sensor network to explore the spreading law of worms were investigated. Xiao et al. (2017) studied a worm propagation model with quarantine in mobile internet. Fatima, Ali, Ahmed, and Rafiq (2018) proposed a computer virus epidemic model with quarantine and infectivity in latent period.
CONTACT Wanjun Xia xiawj0982@sina.com Mishra and Keshri (2013), Mishra and Tyagi (2014) formulated the malicious code propagation models with immunization to describe the dynamics of malicious code propagation in wireless sensor network. Worm propagation models with immunization were proposed by Nwokoye and Umeh (2017), Singh, Awasthi, Singh, and Srivastava (2018) who considered the communication radius and node density of wireless sensor network. All the models mentioned above use the bilinear incidence rate to describe the transmission process of computer viruses, which is only suitable for the case that the proportion of the infected computers in a small network. To address the issue of bilinear incidence rate, Upadhyay, Kumari, and Misra (2017) proposed an Susceptible-Vaccinated-Exposed-Infectious-Recovered (SVEIR) computer virus model with nonlinear incidence rate and immunization. They also studied the stability and the persistence of the model. However, Upadhyay et al. (2017) neglected time delays in the process of computer virus propagation. Robustness of computer networks depends on their stability. If a computer network is stable then it will work properly. Time delays may lead to Hopf bifurcation and make the computer virus model become unstable, which may result in a crash of the entire computer network. Hence, it is important to know the critical point at which a computer virus model changes its stability. The analysis of Hopf bifurcation can ensure that the model is stable. To this end, Zhao, Zhang, and Upadhyay (2018) analysed effect of the time delay due to the period the antivirus software uses to clean the viruses on the SVEIR computer virus model with nonlinear incidence rate proposed by Upadhyay et al. (2017). It should be pointed out that  omitted the latent period delay and the temporary immunity period delay in the SVEIR computer virus model. As is known to all, there is usually a latent period for the exposed nodes to obtain infectious capacity. On the other hand, the vaccinated nodes will lose their immunity when the new computer viruses appear and it needs a period to develop the new computer viruses. In Zhang and Bi (2015), Zhang and Bi investigated a delayed computer virus model with the effect of external computers and studied the Hopf bifurcation of the model by choosing the latent period delay as the bifurcation parameter. In Ren, Yang, Yang, Xu, and Yang (2012), Ren et al. formulated a delayed SIRS (Susceptible-Infectious-Recovered-Susceptible) computer virus propagation model and analysed the existence of the Hopf bifurcation by choosing the different combinations of the latent period delay and the temporary immunity period delay as the bifurcation parameter. Subsequently, Muroya, Enatsu, and Li (2014) investigated global stability and permanence of the delayed SIRS computer virus propagation model formulated by Ren et al. (2012). Zhao and Bi (2017) studied the Hopf bifurcation of a delayed computer virus spreading model in the network with limited antivirus ability when the latent period delay equals the time delay due to the period that antivirus software uses to clean the viruses. In Zhao, Wei, and Bi (2018), Zhao et al. proposed a computer virus propagation model with the latent period delay and temporary immunity period delay. They also studied the properties of the Hopf bifurcation when the two delays are not equal. Motivated by the work about computer virus models with time delay in , Zhang and Bi (2015), Ren et al. (2012), Muroya et al. (2014), Zhao and Bi (2017), , we are concerned with another form of delayed SVEIR computer virus model including two delays based on the work in Upadhyay et al. (2017).
In this paper we consider a delayed SVEIR computer virus model with nonlinear incidence rate by incorporating the time delay due to the latent period and the time delay due to the temporary immunity period into the model considered in the literature (Upadhyay et al., 2017). Its main contributions are (i) local stability of viral equilibrium of the proposed model; (ii) the existence of local Hopf bifurcation. (iii) the properties of the Hopf bifurcation.
The rest of this paper is organized as follows: in Section 2, preliminaries are given and the SVEIR computer virus model with two delays is formulated. In Section 3, the existence of Hopf bifurcation is illustrated by analyzing the corresponding characteristic equation and the results of determining the direction and stability of the bifurcating periodic solutions are obtained. In Section 4, a numerical simulation is presented to support our theoretical results. Finally in Section 5, we end our paper with a conclusion.

i. Notations
The notations used in this paper are defined as follows if not otherwise stated. R and Z denote the set of real numbers and integers, respectively. R n denotes n-dimensional space. C 1 denotes the set of functions which have continuous first derivative. C([−1, 0], R n ) denotes the set of continuous functions.
ii. Hopf bifurcation theorem has a pair of purely imaginary roots ±iλ 0 with λ 0 = 0, and no other root that is an integer multiple of iλ 0 , and if the derivative of real part of characteristic root at μ 0 = 0 is not equal to 0, that is Reλ (0) = 0, then a Hopf bifurcation will occur around the equilibrium ofẋ(t) = F(μ 0 , x t ).

Model formulation
The SVEIR computer virus model with nonlinear incidence rate and immunization proposed by Upadhyay et al. (2017) is as follows: (1) in which the computers are divided into five subclasses: the susceptible computers (S), the exposed computers (E), the infectious computers (I), the recovered computers (R) and the vaccinated computers (V). S(t), E(t), I(t), R(t) and V(t) denote the numbers of S, E, I, R and V computers at time t, respectively. The detailed meanings of the parameters in system (1) are as follows. New computers are connected to the network at the constant recruitment rate A. Due to the contact with I computers, every S computer becomes an E computer at the constant rate α. Each computer is disconnected from the network at the constant rate δ 0 naturally and each I computer is disconnected from the network at the constant rate δ 3 due to the attack of the computer viruses. Every I computer is cured at the constant rate β. a and c are the half saturation constant for I computers and S computers, respectively. η, δ 1 , δ 2 and μ are state transition rates. The characterizing feature of computer viruses is their latency. Thus, there is usually a time delay before the exposed computers becomes infectious ones. On the other hand, the vaccinated computers will lose their immunity when the new computer viruses are developed and it usually needs a period for the new computer viruses to be developed. For this reason, we incorporate the latent period delay and the time delay due to the temporary immunity period into system (1) and investigate the following computer virus model with two delays: (2) where τ 1 is the time delay due to the temporary immunity period and τ 2 is the time delay due to the latent period.
To obtain the results about the Hopf bifurcation of system (2), some assumptions are listed in the following for clarity.
Proof: Throughout this section, we assume that τ 1 * < τ 2 * . Define the space of continuous real-valued functions and rescale the time delay by t → (t/τ 2 ). Letting τ 2 = τ 2 * + , ∈ R, then the Hopf bifurcation occurs at = 0. Thus, system (2) can be transformed intȯ and By the Riesz representation theorem, there exists a function η(θ, ) whose components are of bounded variation for θ ∈ [−1, 0] such that In fact, we choose with δ(θ) is the Dirac delta function.

Numerical simulation
In this section, we present a numerical simulation to validate the obtained main results in this paper. By extracting some values from Upadhyay et al. (2017) and considering the conditions for the existence of the Hopf bifurcation, we consider the following special case of system (2): from which we obtain the unique viral equilibrium P * (10. 8619, 8.3082, 17.1832, 38.8068, 0.1481).

Conclusion
A delayed SVEIR computer virus model with nonlinear incidence rate was investigated in this paper. First, conditions guaranteeing the local stability of the viral equilibrium and the existence of the Hopf bifurcation were obtained by choosing the different combination of the two delays. Then, the properties of the Hopf bifurcation are studied by applying a method based on the centre manifold theorem and normal form theory. Finally, numerical results have been presented to validate of the theoretical analysis. It has shown that the propagation of the viruses can be controlled when the value of the delay is below the corresponding critical value. However, a Hopf bifurcation occurs when the value of the delay passes through the corresponding critical value, which means that computers of the five classes in the model may coexist in an oscillatory mode under some conditions and the viruses will be out of control in this case. Through the simulation, we found that the time delay caused by the temporary immunity period and the time delay caused by the latent period play different roles. The effect of the time delay caused by the temporary immunity period is more significant because the corresponding critical value is smaller when we consider it in isolation.