QUALITATIVE AND BIFURCATION ANALYSIS USING A COMPUTER VIRUS MODEL WITH A SATURATED RECOVERY FUNCTION

In this paper, we introduce a saturated treatment function into the computer virus propagation model, where the treatment function is limited for increasing number of infected computers. By carrying out global qualitative and bifurcation analysis, it is shown that the system exhibits some new and complicated behaviors: if the basic reproduction number is larger than unity, the number of infected computers will show persistent behavior, either converging to some positive constant or oscillating; and if the basic reproduction number is below unity, the model may exhibit complicated behaviors including: (i) backward bifurcation; (ii) almost sure virus eradication where the number of infective computers tends to zero for all initial positions except the interior equilibria; (iii) oscillating backward bifurcation where either the number of infective computers oscillates persistently, if the initial position lies in a region covering the stable virus equilibrium, or virus eradication, if the initial position lies outside this region; (iv) virus eradication for all initial positions if the basic reproduction number is less than a turning point value.


Introduction
Mathematical modeling has been playing an ever more important role in the study of epidemiology. Various epidemic models have been proposed and explored extensively and great progress has been achieved in the studies of disease control and prevention [3,4,6,7,8,11] and the references therein.
Due to the high similarity between computer viruses and biological viruses [2], the classical SIR (Susceptible-Infected-Recovered) computer virus propagation model was proposed [5,9,12], which is formulated as the following system of differential equations: Here it is assumed that all the computers connected to the network in concern are classified into three categories: susceptible, infected and recovered computers. Let S(t), I(t) and R(t) denote their corresponding numbers at time t, respectively. This model involves four positive parameters: b denotes the rate at which external computers are connected to the network, ε denotes the recovery rate of infected computers due to the anti-virus ability of the network, d denotes the rate at which one computer is removed from the network, λ denotes the rate at which, when having a connection to one infected computer, one susceptible computer can become infected.
Recently, Jianguo Ren et. all in [10] introduced a new recovery function where ε is the recovery rate when the anti-virus ability is not fully utilized, m = εI 0 to characterize the saturation phenomenon of the limited anti-virus ability of a network and carefully investigated the dynamics of the following computer virus propagation model Similar to the argument in [13], we take the saturated recovery function as where µ is positive and α is nonnegative, to show that the network condition is limited for increasing number of infected computers. Then the model to be studied takes the following form: Before going into any detail, we simplify the model. Since the first two equations of (1.3) are independent of the third one and its dynamic behavior is trivial when I(t 0 ) = 0 for some t 0 > 0, it suffices to consider the first two equations with I > 0. Thus, we restrict our attention to the following reduced model: (1.4) To our knowledge, this is the first time the effect of anti-virus ability is taken into account this way. First, we give the threshold value determining whether the virus dies out completely. Second, we study the existence of equilibria, and investigate their local asymptotic stability. Next, we find that, depending on the anti-virus ability, the system may undergo a backward bifurcation, which is instructive for us when choosing an appropriate virus-controlling strategy. Finally, we prove that, under appropriate conditions, the system may admit bistable states: a stable virusfree equilibrium and a stable virus equilibrium, or two stable virus equilibria.
The organization of this paper is as follows. In the next section, we present preliminary results for our model including the boundedness and existence of equilibria and backward bifurcation. In Section 3, we present a stability on virus-free equilibrium.

Backward bifurcation
In this section, we will give some preliminary results from (1.4), including the boundedness and existence of equilibria and backward bifurcation.
One of the key concepts in dealing with computer virus models is the basic reproduction number, usually denote it by R 0 , which plays an important role in computer virus propagation. One often observes the threshold property that the virus removed out if R 0 ≤ 1 and invade the susceptible host if R 0 > 1. In this case, the bifurcation leading from a virus-free equilibrium to an virus equilibrium is called forward bifurcation.
However, more and more studies reveal that, under appropriate conditions, many computer virus and epidemic models admit backward bifurcation, i.e., both the virus-free equilibrium and the virus equilibrium coexist when R 0 ≤ 1 and the basic reproduction number cannot be the necessary threshold for the virus eradication. It is very important to identify the backward bifurcations to obtain some necessary thresholds for the control of virus.
We can define the basic reproduction number as It is obvious that (1.4) always has a trivial equilibrium E = (0, 0) and unique virus- , 0). The virus equilibria of (1.4) can be obtained by solving the following algebraic equations First, from the first equation of (2.1), we obtaining I = k(r−d)−rS kλ . We substitute this into the second equation, which yields Then it is easy to see that Therefore, we have the following simple statements which describe the number and location of equilibria of system (1.4). Now, we know that ∆ ≥ 0 is equivalent to where S 2 < S 1 . So one can get then E 1 = (S 1 , I 1 ) and E 2 = (S 2 , I 2 ) are the candidates of the virus equilibria of (1.4). Then To facilitate the discussion below, define This implies that It follows that By arguments similar to those above, we see that Note that λ < √ µαr k is equivalent to ϱ 1 < 1. The proof is complete. Note that ϱ 0 < 1. If λ < √ µαr k , then ϱ 1 < 1. Then, from (i) of Theorem 2.1, we have the following corollary for giving conditions for such a backward bifurcation to occur.

Corollary 2.1. System (1.4) has a backward bifurcation with virus equilibria when
R 0 < 1 and λ < √ µαr k . Note that a backward bifurcation with virus equilibria when R 0 < 1 is very interesting in applications. The basic reproduction number does not provide a description of the necessary elimination effort; rather the description of the effort is provided through the value of the critical parameter at the turning point. Thus, it is important to identify backward bifurcation to obtain thresholds for the control of virus.

Stability of equilibria
In this section, we deal with the global dynamics of (1.4). First, we examine the local stability of the equilibria by analyzing the eigenvalues of the Jacobian matrices of (1.4) at the equilibria.
The Jacobian matrix of (1.4) at the virus-free equilibrium E is which implies that E is always a saddle. The Jacobian matrix evaluated at E 0 is which has negative eigenvalues, implying asymptotic stability of the disease-free equilibrium if and only if which is equivalent to R 0 < 1. So the virus-free equilibrium E 0 is locally asymptotically stable if R 0 <1, and is unstable when R 0 > 1. Then, from the above discussions, we get  Consider model (1.4). The following assertions hold.
(i) E is a saddle.
(ii) If 0 < R 0 < 1, then the virus-free equilibrium E 0 is locally asymptotically stable while it is unstable if R 0 > 1.
Let J i be the Jacobian matrix of (1.4) at E i = (S i , I i ), i = 1, 2; then we get .
Thus, one can get .
Note that R 0 > ϱ 0 and It follows from the above two conditions that then, det(J 1 ) < 0. It follows that E 1 = (S 1 , I 1 ) is a saddle point. By the same argument, we obtain det(J 2 ) > 0. Thus, E 2 = (S 2 , I 2 ) is a focus, a node, or a center. Further, we have Since µI2 1+αI2 = λS 2 I 2 − dI 2 , we see that the trace of J 2 is Thus, the trJ 2 is negative if λ ≤ r k . Suppose that λ > r k . Let us find the conditions under which trJ 2 = 0. Since S 2 = k(r−d) r − λk r I 2 , it follows from (3.2) that trJ 2 = 0 is equivalent to Thus, by (3.3), we can get that trJ 2 ̸ = 0 if It follows from that we obtain In view of I 2 < r−d λ , we have Hence, after a long and tedious calculation and using (3.7) can be reduced to where (3.9) As a consequence, we see that (3.6), (3.5) and (3.8) are the necessary and sufficient conditions for trJ 2 = 0. The previous discussion show that the stability of E 2 does not change if (3.4) holds. It follows from the definition of trJ 2 = 0 that (3.4) implies that trJ 2 < 0. Therefore, E 2 is stable if (3.4) holds. It follows from (3.1) and (3.9) that trJ 2 < 0 if , (3.10) and that trJ 2 > 0 if . (3.11) From the above discussions, we get

Analysis at
In this section, we consider the system (1.3), when α = 0. To use the center manifold theory, as described in [1] (Theorem 4.1). To apply this method, the following simplification and change of variables are made first. (3.12) with R 0 = 1 corresponding to λ = λ * = r(d+µ) k(r−d) . The virus-free equilibrium is ( . The linearization matrix of system (3.12) around the infection-free equilibrium when λ = λ * is The matrix J x f has eigenvalues (0, −d, −(r−d)) T , which meets the requirement of a simple zero eigenvalue and others having negative real part. A right eigenvector ω corresponding to the zero eigenvalue is ω = (− d+µ r−d , 1, µ d ) and the left eigenvector satisfying ν · ω = 1 is ν = (0, 1, 0). For te system (3.12) we can get Thus, a < 0, b > 0, by item (iv) of Theorem 4.1 in [1], we can give the following result: