Global Stability of a Computer Virus Propagation Model with Two Kinds of Generic Nonlinear Probabilities

and Applied Analysis 3 Proof. All viral equilibria of system (1) are determined by the following system of equations: (1 − p) b − μS − βSI h (I) − α 1 f (I) S + γ 2 I + α 2 R = 0, βSI h (I) − μI − γ 1 I − γ 2 I = 0, pb − μR − α 2 R + α 1 f (I) S + γ1I = 0, (4) where I ̸ = 0. Simplifying, one can get (1 − p) b − μS − (μ + γ 1 ) I − α 1 f (I) S + α 2 R = 0,

Vaccination (i.e., the measure that an uninfected computer has the newest-version antivirus software installed) plays an important role in repressing computer virus, by which a susceptible computer would have temporary immunity.The fact that a large number of susceptible computers are infected would enhance the probability that the user of a susceptible computer has his/her computer vaccinated, implying that vaccination probability is related to the number of infected computers.Indeed, Gan et al. [9,11] recently investigated two SIRS models by incorporating a linear or nonlinear vaccination probability (i.e., the probability that a susceptible computer gets vaccinated is linear or nonlinear in the number of currently infected computers).Unfortunately, the bilinear incidence rate assumption for these two models, which is a good first approximation of the general incidence rate, is inconsistent with the actual conditions [10].In reality, overcrowded infected computers and active protection measures would render this approximation to fail terribly.As a result, it is worthwhile to explore a dynamical model with generic nonlinear vaccination probability under more reasonable assumptions.
Having this idea in mind, in this paper, a new dynamical model of computer virus with generic nonlinear vaccination probability and nonlinear incidence rate is proposed.A detailed study of the model is provided.Specifically, the basic reproduction number (i.e., the average number of secondary infections produced by a single infected computer during its life time),  0 , is determined, and the virus-free (or viral) equilibrium is shown to be globally asymptotically stable if  0 ≤ 1 (or  0 > 1), implying that computer virus would tend to extinction or persist according to the value of  0 .The related analysis of  0 is also conducted.Additionally, some numerical examples are examined to illustrate the main results, from which it can be seen that the generic nonlinear vaccination is helpful to suppress computer virus diffusion.
The organization of the rest of the paper is as follows.Section 2 formulates the new model.Section 3 proves the global stabilities of the virus-free and viral equilibria.A parameter analysis of the basic reproduction number is performed in Section 4. Finally, Section 5 summarizes this work.

Model Formulation
As usual, a computer is either internal (i.e., on the Internet) or external (i.e., outside the Internet).Moreover, an internal computer is assumed to be in one of three possible states: susceptible (i.e., uninfected but not immune), infected, and recovered (i.e., uninfected and immune).Now, let us introduce some notations as follows, which will be adopted in the sequel: (): the average number of susceptible internal computers at time , (): the average number of infected internal computers at time , (): the average number of recovered internal computers at time , (): the average number of internal computers at time ; that is, () = () + () + ().
The following fundamental assumptions of the new model are made.This collection of assumptions can be schematically shown in Figure 1, from which one can derive the differential system with initial condition ((0), (0), (0)) ∈ R 3 + .

Model Analysis
This section is devoted to study model (1) theoretically.The analysis of this model comprises the basic reproduction number, the existence of equilibria, and their global stabilities.

Basic Reproduction Number.
Employing the next generation method (see [26]) to model (1), the basic reproduction number can be derived as 3.2.Equilibria.Obviously, system (1) always has a unique virus-free equilibrium  0 = ( 0 , 0,  0 ), where Next, let us examine the existence of viral equilibria.The following result is obtained.

Global Stability of the Virus-Free Equilibrium
Theorem 2.  0 is globally asymptotically stable if  0 ≤ 1.
Remark 3. Theorem 2 implies that computer virus on the Internet would tend to extinction when the basic reproduction number is less than or equal to unity.

Global Stability of the Viral Equilibrium.
Firstly, let us consider the following lemma.
Now, let us explore the global stability of the viral equilibrium.Theorem 6.  * is globally asymptotically stable if  0 > 1.
Remark 7. Theorem 6 implies that computer virus on the Internet would tend to persist when the basic reproduction number is greater than unity.
Proof.It is easy to see that the first four inequalities are true.Consider The proof is complete.
In addition, the following example indicates the effect of different incidence rates and vaccination probabilities on  (see Figure 7).

Conclusions
This paper has studied the long-term behavior of computer virus in terms of a new propagation model with generic nonlinear incidence rate and nonlinear vaccination probability.An elaborate analysis of the model including the basic reproduction number, the existence of virus-free and viral equilibria, and their global stabilities has been conducted, from which it is found that computer virus on the Internet would tend to extinction or persist according to the value of the basic reproduction number.To illustrate the obtained main results, some numerical examples have been examined, from which it can be seen that the generic nonlinear vaccination is useful for the inhibition of viral spread.

(Figure 1 :
Figure 1: The transfer diagram of the new model.

Example 4 .RFigure 2 :
Figure 2: An illustration of the dynamics of system (1) given in Example 4.

Figure 4 :Figure 5 :Figure 6 :
Figure 4: An illustration of the impact of  1 and  2 on  0 .