Global Dynamics and Optimal Control of a Viral Infection Model with Generic Nonlinear Infection Rate

This paper is devoted to exploring the combined impact of a generic nonlinear infection rate and infected removable storage media on viral spread. For that purpose, a novel dynamical model with an external compartment is proposed, and the explanations of the main model assumptions (especially the generic nonlinear infection rate) are also examined. The existence and global stability of the unique equilibrium of the model are fully investigated, from which it can be seen that computer virus would persist. On this basis, a next-best approach to controlling the level of infected computers is suggested, and the theoretical analysis of optimal control of the model is also performed. Additionally, some numerical examples are given to illustrate the main results.


Introduction
In the wake of developments in computer and network technologies, computer virus has become more capable of conquering computer system.In the meantime, the study of fighting against computer virus has in the past few decades been paid more attention.In reality, there is no doubt that antivirus software and firewall are the most effective prevention measures.However, they are incompetent to inhibit computer virus diffusion over the Internet [1].To deal with this problem, a wide variety of mathematical models have been widely studied (for the related references, see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15]).
The infection rate is an important and essential system parameter in computer virus propagation models.However, the dominating majority of previous models assume a bilinear incidence rate (for the related references, see, e.g., [16][17][18][19][20]) or a nonlinear increasing incidence rate (for the related references, see, e.g., [21]).The former assumption is suitable for the case where the proportion of infected computers is small.The latter assumption neglects the fact that, due to active protection measures taken during viral spread, the infection rate would be decreasing, while the infected computers may be increasing.In order to depict the case where the infection rate could be decreasing with the infected computers and inspired by the previous work (e.g., [22,23]), the proposed model of this paper adopts a nonlinear function () = /(), where  > 0 and function  ∈  2 [0, +∞) with   ≥ 0,   < 0, and (0) = 1 (see also the model assumption (A3) in the next section).
External computers (i.e., computers outside the Internet) and infected removable storage media play an important role in viral spread (for the related references, see, e.g., [24,25]).In [24], the impact of infected removable storage media is considered, but the influence of external computers is insufficient.In [25], a dynamical model, in which all external computers are regarded as a separate compartment, was proposed.Unfortunately, this model ignores the effects of generic nonlinear infection rate and infected removable storage media.Consequently, it is necessary to consider the combined impact of a generic nonlinear infection rate and infected removable storage media on viral spread.
Combining the above discussions, a novel SIES model with two kinds of incidence rates, which are caused by infected computers and infected removable storage media, is established in this paper.A systematic analysis of the proposed model shows that the unique (viral) equilibrium is globally asymptotically stable.This result indicates that any effort in eradicating computer virus cannot succeed.In this regard, theoretical analysis of a next-best approach and optimal control of the model is performed.Numerical analysis of the model is also included.
The remaining materials of this paper are organized as follows: Section 2 formulates the model.Section 3 considers the viral equilibrium and its global stability.In Section 4, a theoretical analysis of optimal control of the model is performed.Finally, Section 5 concludes the contributions of this work and points out some further works that are worth doing.

Model Characterization
In this paper, the proposed model consists of three compartments (see (i)-(iii)), and the assumptions of it are also made (see (A1)-(A8)): (i) -compartment: the set of all -computers (susceptible internal computers, i.e., computers in the Internet) (ii) -compartment: the set of all -computers (infected internal computers) (iii) -compartment: the set of all -computers (external computers, i.e., computers outside the Internet) (A1) Every computer is out of use with probability per unit time  > 0 (A2) Every or -computer leaves the Internet with probability per unit time  1 > 0 (A3) Due to possible communication with -computers over the Internet, every -computer is infected with probability per unit time () = /(), where  > 0 and function  ∈  2 [0, +∞) with   ≥ 0,   < 0, and (0) = 1 (A4) Due to possible effect of infected removable storage media, every -computer is infected with probability per unit time  ≥ 0 (A5) Due to treatment, every -computer is cured with probability per unit time  2 > 0 (A6) The rate of all newly accessed -computers is  > 0 (A7) Every -computer is either susceptible or infected when it enters the Internet (A8) Every susceptible (or infected) -computer enters the Internet with probability per unit time  2 > 0 (or  1 > 0) For convenience, let , , and  represent the average number of computers in -compartment, -compartment, and -compartment at time , respectively.Collecting the foregoing hypotheses, the proposed model can be depicted by Figure 1 or the differential system with initial condition ((0), (0), (0)) ∈  3 + .
Remark 2. It follows from the above proof that 0 <  * < .

Global Stability
Theorem 3.  * is globally asymptotically stable with respect to Ω.

Proof. Consider the Lyapunov function
Then, Here, we proceed by treating two cases. Then, Next, we need to further distinguish two subcases.
Remark 5. Theorem 3 reveals that () tends to a positive constant  * .From an epidemiological standpoint, it indicates that computer virus would persist in network.Thus, one can conclude that any effort in eradicating computer virus is doomed to failure.Thus, the best achievable goal is to make the number of infected computers below an acceptable level (i.e., as low as possible).Note that  * cannot be expressed in a specific formula, and it follows from Remark 2 that 0 <  * < .Then, one could keep the value of  below an acceptable threshold.To this end, the following result is made.Theorem 6.From Equation ( 3),  = ( 1 + 2 )/(+ 1 + 1 +  2 ).Then / < 0, / 1 < 0, / > 0, / 1 > 0, and / 2 > 0.
Remark 7. Theorem 6 implies the effects of some system parameters on the value of .

Optimal Control of the Model
To make a tradeoff between control cost and control effect, the control variable () meaning the control strategy is applied in the proposed model (system (1)).Then, one can get the controlled dynamical system with initial condition ((0), (0), (0))  ∈ Ω  , where The admissible control set is where  2 and  2 are positive constants and 0 <  2 <  2 < 1.
Let x() = ((), (), ())  .Then system (15) can be written in matrix notation as with initial condition x(0) ∈ Ω  .Now, the objective is to find a control variable (⋅) so as to minimize both the number of infected computers and the total budget for treatment and vaccination during the time period [0, ].That is, it suffices to solve the optimal control problem.
subject to system (18), where is the Lagrangian and  > 0 is a tradeoff factor based on the control cost and control effect.
Theorem 10.The optimal control problem ( 19) has an optimal control.
Thus, the claimed result follows directly from [31].
Figures 5 and 6 exhibit the evolution of  and  with different control strategies, respectively.From these two figures, it is natural to see that the optimal control ũ is superior to others.Figure 7 shows the corresponding optimal control strategy.
Table 1 lists the values of infected computers  and objective function  under different control strategies.It is easy to conclude that ũ is the best choice.

Conclusions and Prospects
This paper has investigated an SIES model with generic nonlinear infection rate.A thorough analysis shows that the unique (viral) equilibrium is globally asymptotically stable.This result implies that any effort in eradicating computer virus is inoperative.As a result, a countermeasure, which mainly aims to maintain the number of infected computers at an acceptable level, and optimal control analysis of the proposed model have been posed.Some numerical examples are also included.
The study of this model not only implies some new practical measures but also gives a theoretical support to the usefulness of some existing antivirus strategies and provides the basis to developing many other more elaborated models.
The study can be continued in several directions.First of all, it would be interesting for complex network (e.g., scale-free network) because our model is a homogeneous model.Next, delays (or pulses) could be incorporated in our model so as to characterize the delay in the development of new vaccine (or the emergency of new virus).Finally,  our model involves a relatively large number of parameters, which are very difficult to establish with accuracy.Therefore, it is appropriate to modify our model by considering the parameters to be random variables.

Figure 1 :
Figure 1: The transfer diagram of the proposed model.

Figure 3 :
Figure 3: An illustration of the impact of () on () for system (1) given in Example 8.

Figure 4 :
Figure 4: An illustration of the impact of  on () for system (1) given in Example 9.

Figure 5 :
Figure 5: Evolution of number of susceptible computers with different control strategies.

Figure 6 :
Figure 6: Evolution of number of infected computers with different control strategies.

Table 1 :
The values of  and objective function  under different control strategies .