Numerical Study of Computer Virus Reaction Diffusion Epidemic Model

Reaction–diffusion systems are mathematical models which link to several physical phenomena. The most common is the change in space and time of the meditation of one or more materials. Reaction–diffusion modeling is a substantial role in the modeling of computer propagation like infectious diseases. We investigated the transmission dynamics of the computer virus in which connected to each other through network globally. The current study devoted to the structure-preserving analysis of the computer propagation model. This manuscript is devoted to finding the numerical investigation of the reaction–diffusion computer virus epidemic model with the help of a reliable technique. The designed technique is finite difference scheme which sustains the important physical behavior of continuous model like the positivity of the dependent variables, the stability of the equilibria. The theoretical analysis of the proposed method like the positivity of the approximation, stability, and consistency is discussed in detail. A numerical example of simulations yields the authentication of the theoretical results of the designed technique.


Introduction
Computer viruses are automated programs that, against the users' wish, make copies of themselves to spread to new targets and as a result infect the computers [1]. With the accelerated advancement of modern technologies, the internet has assimilated into each part of our life, which is a great help for us, as well as it also poses a serious problem to individual and corporate computer systems through cyberattacks [2]. A lot of effort has been dedicated to studying how to avoid harmful actions. To control effectively the diffusion of computer viruses, it is very crucial to figure out the ways that nasty codes with initial conditions and homogenous Neumann boundary conditions. Where S x; t ð Þ represents the class of uninfected computers, L x; t ð Þ represents latent computers and B x; t ð Þ represents the seizing computers at time t. The constant rate of connecting external computers with internet and disconnecting internal computers from internet is denoted by d: b is the constant rate of infection and the term bS L þ B ð Þ is the percentage of internal computers infected at time t. a is the constant rate of latent computer breakout. Latent computers are cured at the constant rate of c 1 , while breaking out computers are cured at the constant rate of c 2 . d 1 ; d 2 ; d 3 are constant rate of diffusion. All these parameters used in this model are positive. It is observed that the variables of interest of the proposed computer virus model are the computer populations. Now it is the basic need for the solution of system (Eqs. (1)-(3)) to be positive as values of unknown variables involved are taken as absolute [21]. In the present era, several positivity persevering numerical techniques are proposed by various authors because many dynamical continuous systems require the positive solution [22,23]. The current work is dedicated to designing and analyzing a finite difference algorithm which retains the positive solution of the state variables of the continuous system for the solution of computer virus epidemic model (Eqs. (1)- (3)). This manuscript is sectioned as follows. The reproductive number and equilibrium points of the system under study is explained in Section 2. In Section 3, a numerical algorithm is designed to solve the computer virus epidemic system. In the same section, the theoretical analysis of the proposed technique is performed. It is shown that the designed numerical technique is capable of retaining the positivity of the solution. The stability and consistency of the proposed algorithm are also authenticated in this section. Section 4 is devoted to the computation results. The simulations are justified the theoretical results of the designed method.

Equilibria of the Model
The model Eqs.

Consistency of the Proposed Scheme
This section is concerned about the verification of the proposed scheme to be consistent. For this, By putting all these definitions' in Eq. (7), we have Similarly we can check for Ļ L and Ļ B i.e., Hence our proposed scheme is consistent and first order accurate in time and second order accurate in space.

Stability of Proposed Scheme
In this section, we will use von Neumann stability criteria to show our proposed implicit scheme from Eqs. (4)-(6) is unconditionally stable. For this purpose, we will introduce the following terms, Put all these terms in Eq. (4), we have After proper calculation and rearranging terms, we have Take absolute value on both sides, we have the following inequality, By using similar process for Eq. (5), and Eq. (6) simultaneously we have, we have and Inequalities from Eqs. (11)- (13) are showing that our proposed implicit scheme is unconditionally stable by using von Neumann stability analysis.

Positivity
Theorem 1: For any positive k and h, S n ; L n and B n appertaining the to the Eqs. (4)-(6) are positive for all n ¼ 0; 1; . . .

Proof:
We will use m-matrix theory and mathematical induction to show our scheme preserve positivity. For this purpose rewrite Eqs. (4)- (6) in vector form, as GS nþ1 ¼ S n þ þdk þ c 1 kL n þ c 2 kB n ; (14) If Q represents any of G; H or I; where these are all real square matrices, then Q can be written in general form as Q ¼ The off-diagonal and diagonal entries of G are g 1 ¼ À2 1 , g 2 ¼ À 1 and The off-diagonal and diagonal entries of I are i 1 ¼ À2 3 , Suppose that S n ; L n ; B n > 0; ⇒ S n þ dk þ c 1 kL n þ c 2 kB n > 0; L n þ bkS n j L n j þ B n j > 0; and B n þ aL n > 0: Hence our proposed implicit scheme preserve positivity.

Numerical Example and Simulations
In this section, we demonstrate a numerical example and simulations for the application of proposed structure preserving technique. For this we consider the following initial conditions,

CVF Point
First we discuss the simulations of proposed structure preserving method at CVF point. For the CVF point we take the following values of parameters involved in the model so that the value of < 0 is less than one.

CVE Point
Now we present the simulations of proposed structure preserving method at CVE point. For the CVE point we use the following values of parameters involved in the model so that the value of < 0 is greater than one.

Conclusion
In this paper, we propose an extended reaction-diffusion epidemic model of computer virus dynamics for the numerical investigation. An efficient and reliable numerical technique is designed which preserves the stability of equilibria and positivity of the approximation. The stability, consistency, and positivity of the proposed algorithm are shown mathematically and are validated graphically with the help of a numerical example. The proposed algorithm can be used for the solution of reaction-diffusion models like predatorprey models, chemical reaction models and infectious disease models. In future work, we shall extend the modeling of a computer virus in the computer population in the well-known notations like fractional and stochastic fractional-order derivatives [24][25][26].