Hopf Bifurcation of Nonresident Computer Virus Model with Age Structure and Two Delays Effects ()
1. Introduction
With the rapid development of computer network technology, computer risks such as hackers, viruses, phishing emails, and other threats to information security are becoming increasingly serious. Computer viruses may damage computer data, occupy computer space and memory resources, damage computer hardware and software, and are known as the “biggest hidden danger of the 21st century”. Understand the essence and characteristics of computer virus transmission, strengthen research on network risk prevention strategies, and ensure information security. Therefore, understanding the nature and characteristics of the spread of computer viruses and strengthening research on network risk prevention strategies can help ensure network information security.
There is a high similarity between the transmission process of viruses in the network and the spread of diseases within the population [1] . It leads to more and more scholars constructing computer virus transmission dynamics models based on epidemic compartment models such as SIR models [2] , SIRS models [3] [4] [5] [6] , SLBS models [7] [8] , and then meanwhile a variety of computer virus models have also been built [9] - [15] . In recent years, more experts and scholars have begun to focus their attention on nonresident computer viruses that do not execute themselves from or in computer memory [16] [17] [18] . In [16] , the authors constructed the following nonresident computer virus SLAS epidemiological model in the network:
(1)
where
represents the numbers of the uninfected computer having no immunity in S class at time t,
represents the numbers of infected computers in which viruses are not yet loaded in their memory (latent computers, for short) in L class at time t,
represents the numbers of infected computers in which viruses are located in memory (infectious computers, for short) in A class at time t, and
is the rate of the uninfected computers disconnects from the network. And then, give the following assumptions. First, all newly accessed computers are virus-free. Second, all viruses staying in computers are nonresident. Third, users of latent computers cannot perceive the existence of virus, so latent computers cannot get cured.
The model (1) divides the overall process of virus transmission in a computer into three stages. However, combined with the actual situation, we found that the virus in the latent class into the memory at different times, will lead to its transformation into the infection rate is also different. At the same time, the time of virus infection is different, and the time of computer cure is different. Therefore, based on the above analysis, we make improvements to the model and carry out a specific analysis of the model below.
In fact, nonresident viruses in the latent class of computers are loaded into memory after a period of time before they become infected. Therefore, we use the piecewise function
to characterize the rate at which a nonresident virus in a latent computer is loaded into memory to become an infection class computer, that is,
(2)
where
,
,
is the length of time the computer stays in the latent compartment, and
is the minimum time when the latent computers in L class enters in A class. Correspondingly, the outflow of the latent computers in Equation (1) can be rewritten by the partial differential equation as
(3)
with
is the density of the latent computers with the latent age
at time t,
is the rate of the latent computers disconnects from the network, and
is the recovery rate of latent computers returns to S class.
In addition, it takes time for the infected computers in A class to be cured and become the uninfected computers in S class. Therefore, we use the piecewise function
to characterize the recovery rate of the infected computers, that is,
(4)
where
,
, b refers to the length of time the nonresident virus is in the infection compartment, and
the maximum time for an infected computer in A class to be cured of being the uninfected computers in S class. Correspondingly, the outflow of the infected computers in the Equation (1) also can be rewritten by the partial differential equation as
(5)
where
is the density of the infected computers with the immunity age b at time t,
is the rate of the infected computers disconnects from the network,
is the rate of return of infected computers in A class to latent computers in L class.
Therefore, we have constructed the computer virus epidemic model with age structure and two delays effects as follows. Where infection age and immunity age refers to the time the computer spent in the infected class compartment and the recovery class compartment, respectively, and is a class age rather than the actual age.
(6)
where the initial condition
Here,
is the external computers are accessed to the Internet at positive constant number,
,
are the nonresident virus transmission rate, It’s worth noting that
.
The overall construction of this paper is as follows. In section 2, we study the non-negativity and the boundedness of the solution of the system. In section 3, we investigate the existence of all the feasible equilibria, including the virus-free equilibrium
and the computer virus equilibrium
. In section 4, we explore the local stability of the virus-free equilibrium
, and the local stability of the computer virus equilibrium
when
. In section 5, we study the existence of Hopf bifurcation of the system under three different cases, which are 1)
and
, 2)
and
, and 3)
, respectively. In section 6, we present some numerical examples and conclusions.
2. Preliminaries
In this section, we focus on the non-negativity and consistent boundedness of the system(6) solution under any non-negative initial value condition.
Theorem 1 If
, then the solution of the system (6) is non-negative for all
, and it is ultimately bounded for t large enough.
Proof. For any
, assume that
,
remains non-negative for
. Suppose the assumption does not hold, then the continuity of the solution of the system (6) concerning the initial value shows that the following two cases may occur:
1) There exists a
such that
and
for
,
,
and
;
2) There exists a
such that
and
for
,
,
and
;
For case 1), by using of the second and fourth equation of system (6), we can get
Since
, and
, we can obtain
. And then, the first equation of system (6) implies that
which contradicts with
.
Similarly, for case 2), from the third equation of the system (6) it follows that
which also contradicts with
. So the assumption is valid.
Based on the above analysis, for any nonnegative initial values
,
, the solution of the system (6) is guaranteed to be
,
, always for all
. We directly integrate the second equation in system (6) along the characteristic line yields that
It is clear that
remains nonnegative for any
when
,
and
.
Next, we prove the boundedness of solution of system (6). To this end, we denote
, and
, which represents the total number of the latent computers and the infectious computers at time t, respectively. Suppose that
, and
. It is well known these assumptions are in line with practical biological significance, as the lifespan of computers is limited. Then,
It implies that
. That is, the set
is positively invariant with respect to system (6).
3. The Existence of the Equilibria
In this section can be divided into two main parts, in the first part, we prove the existence of the virus-free equilibrium of the system (6); in the second part, we prove the existence of the computer virus equilibrium and also give an explicit expression for the basic reproduction number of the system (6).
First, a direct calculation shows that there always exists virus-free equilibrium
of the system (6), where
.
Next, in the second part, in order to obtain the existence of the system (6) computer virus equilibrium
, we need to determine the existence of nonnegative solutions to the following system of equations.
(7)
The second of Equation (7) implies that
(8)
Further from the combination of the third and fifth equations of the system of Equation (7) we can obtain
(9)
Substituting Equation (8) and Equation (9) into the forth equation in Equation (7), we get
(10)
with
Substituting Equation (8), Equation (9) and Equation (10), into the first equation in Equation (7), we get
with
It is clear that
and
can ensure
. That is, system (6) exists the computer virus equilibrium
, where
Theorem 2. For the system (6), there always exists the virus-free equilibrium
, and when
and
, it exists the computer virus equilibrium
.
4. The Stability of the Equilibria
In this section, we focus on the stability of the virus-free equilibrium
and the computer virus equilibrium
of the system (6).
4.1. The Stability of the Virus-Free Equilibrium
Theorem 3. When
, the virus-free equilibrium
of the system (6) is locally asymptotically stable, and
is unstable when
.
Proof. By linearizing system (6) at virus-free equilibrium
, we can obtain the characteristic equation as follows:
(11)
Obviously,
is a root of Equation (11). And, the remaining characteristic roots of the characteristic Equation (11) must satisfy the following equation:
(12)
When
, it is clear that
is a continuous strictly monotonically increasing real-valued function on
, and satisfies
It implies that
has at least one positive real root when
, that is,
is unstable. Obviously,
does not have a positive real root when
. We assert that
also does not have the complex roots with the real part greater than 0. Otherwise, suppose
is an arbitrary complex root of
. Therefore,
which is clearly contradictory. That is, the real parts of the roots of
all are less than 0.
In summary, when
, the virus-free equilibrium
is locally asymptotically stable.
4.2. The Stability of the Computer Virus Equilibrium
In this subsection, we discuss stability of the computer virus equilibrium
of system (6) in the case where
and
.
By direct calculation, we can obtain the characteristic equation of system (6) at
as follows:
(13)
where
Theorem 4. Suppose
, and
. Then the computer virus equilibrium
of the system (6) is locally asymptotically stable.
Proof. When
, Equation (13) can be rewritten as
At this time,
It is clear that
.
Next,
where
Furthermore,
The Rausch-Helwitz criterion implies that the real part of the roots of
are all negative. It implies that the computer virus equilibrium state
of the system (6) is locally asymptotically stable.
5. Hopf Bifurcation
In this section, we will explore the dynamic behaviors of system (6) in three different cases, including 1)
,
; 2)
,
; and 3)
, respectively. When an unstable computer virus equilibrium occurs in the system (6), at this point
bifurcates and thus changes from unstable to stable. In other words, the system transitions from one stable state to another with periodic oscillations.
Case.1 In the case where
, and
.
We will use the method in Section 2 of [19] to discuss the existence of Hopf bifurcation. To this end, Equation (13) can be rewritten as
(14)
where
and the following hypotheses need to be justified:
1)
;
2)
;
3)
,
;
4)
;
5) Each positive roots
of
is continuous and differentiable in
whenever it exists.
The rigorous calculations can help us obtain
It is clear that the above conditions 1), 2), and 3) are satisfied. Since
we can get
(15)
It means that condition 4) holds. Furthermore, since both
,
and
are continuous differentiable functions with respect to
the implicit function theorem ensures condition 5) also satisfied.
Let
be a pure imaginary root of Equation (14), then
(16)
And then, let
, then Equation (15) can be rewritten as
(17)
where
and
ensures
holds. Since the existence of pure imaginary root of Equation (15) is equivalent to the existence of the positive root of Equation (17), we first discuss the existence of the positive root of
.
Let
. On the one hand, if
, then
has no real root; on the other hand, if
, then
at least has one real root, in which
is a bigger root. Therefore, the following lemma gives the existence of the positive root of
.
Lemma 5. 1) If
, then
has no positive root; 2) if
, and
, then
has no positive root; 3) if
,
, and
, then
has no positive root; 4) if
,
, and
, then
has the positive roots.
If Equation (17) has no positive roots, then the stability of the computer virus equilibrium
does not change as
increases. Conversely, if there exists the positive root in Equation (17), the stability of the computer virus equilibrium
may change when
reaches some critical value
. At this point, Hopf bifurcation may appear in the system. In summary, we have the following conclusions:
Theorem 6. Suppose than
,
, and
hold. 1) If
, then the computer virus equilibrium
of system (6) is locally asymptotically stable; 2) if
, and
, then the computer virus equilibrium
of system (6) is locally asymptotically stable; 3) if
,
, and
, then the computer virus equilibrium
of system (6) is locally asymptotically stable.
Next, we judge the Hopf bifurcation of the system. If
has a positive root, then the stability of the computer virus equilibrium
may change when
passes through some specific values. Let us consider whether the stability of the positive equilibrium changes when
has one positive root. In the case where
has two positive roots, the analysis is similar.
Let
be the positive root of
, that is,
is the positive real root of
. Then, we define a set by
That is, for
, there exists
such that
.
Separating the real part and the imaginary part from Equation (16)
Let
(
) be a solution of the following equations:
We conclude that
. Hence,
(
) is a purely imaginary root of Equation (14) if and only if
is a zero of
for some
, which is defined by
Theorem 2.2 in [19] implies that the following lemma is true.
Lemma 7. ( [19] ): Assume that
is a positive real root of
for
, and at some
,
Then a pair of simple conjugate pure imaginary roots
, and
of the characteristic Equation (14) exists at
which crosses the imaginary axis from left to right if
and crosses the imaginary axis from right to left if
, where
Therefore
that means the transversality condition holds. So a Hopf bifurcation occurs when
and
, and the following conclusion is obtained according to the Hopf bifurcation theorem.
Theorem 8. Suppose
,
, and
. If
,
, and
, then the network virus equilibrium
of system (6) is locally asymptotically stable for
, and the system (6) undergoes a Hopf bifurcation at the network virus equilibrium
when
.
Case.2 In the case where
,
.
When
and
, using the same approach to prove the stability analysis of the computer virus equilibrium
and the occurrence of Hopf bifurcation as
increases. To this end, Equation (13) can be rewritten as
(18)
where
and then
Unlike in case 1, there exists
which means that
has a positive real root, and the system (6) also generates a Hopf bifurcation. In summary, we have the following conclusions:
Theorem 9. Suppose than
,
, and
hold. 1) If
, then the computer virus equilibrium
of system (6) is locally asymptotically stable; 2) if
, and
, then the computer virus equilibrium
of system (6) is locally asymptotically stable; 3) if
,
, and
, then the computer virus equilibrium
of system (6) is locally asymptotically stable.
And the following conclusion is about the Hopf bifurcation.
Theorem 10. Suppose
,
and
holds. If
, or
,
,
, and
, then the computer virus equilibrium
of system (6) is locally asymptotically stable for
, and the system (6) undergoes a Hopf bifurcation at the computer virus equilibrium
when
.
Case.3 In the case where
.
The characteristic Equation (13) is rewritten as the transcendental equation
(19)
where
, and
When
are independent of the delay
. Following Section 2 in [20] , we need to justify the following hypotheses:
1)
;
2)
;
3) The polynomials
have no common factor;
4)
;
5)
,
;
6) For any
, at least one of
,
tends to
as
.
The conditions 1) and 3) apparently established. Through a tedious manipulation, we can derive that
which implies that the above conditions 2), 4) and 5) are satisfied. Noting that
When
, we can get
. Obviously, 6) also satisfies.
In the following, we search the points
such that
is a zero of Equation (19). Let
Then,
is zero of Equation (19), if and only if
(20)
Suppose that
is the zero of Equation (19), Then the 3 parts 1,
, and
on the right side of the above equation must be connected and form a triangle in the complex plane, as shown in Figure 1. Therefore, we can obtain the feasible region of
in the following lemma.
Lemma 11. For
, and
holds, the feasible region
for
, such that 1,
and
create a triangle, is
where
at the same time
Proof. We let
.
First,
is equivalent to
Figure 1. Triangle formed by 1,
,
.
Second, for
. The result of inequality is consistent with the above equation. Third,
is equivalent to
Therefore, we can get the feasible region
.
Once again, we consider two possible scenarios:
1) If
, we can get
where
Then, we can get
and
2) If
, then we can get the triangular formed by 1,
, and
is the mirror image of the one in Figure 1 about the real axis. Therefore, we obtain
and
We denote by
the interval of
for the feasible region
. Now, for fixed
and
, we can introduce the functions of
, say
, as
(21)
For the above equation, if zero exists, it can be denoted as
. So, the corresponding value of
. When
takes the values throughout the interval
, then we get the curve
on
, which will later determine the shape of the crossing curves
on
-plane.
Lemma 12. The characteristic Equation (19) admits a pair of conjugate roots
, for
. Denote by
the pair of conjugate complex roots of (19) in some neighborhood of
, such that
and
. If
, then
cross the imaginary axis from left to right, as
passes through the crossing curve to the region on the right. While if
, then
cross the imaginary axis from left to right, as
passes through the crossing curve to the region on the left, where
with
,
,
.
Comprehensive analysis of the above, then we have the following result.
Theorem 13. Suppose that
and
holds. If
, then the computer virus equilibrium
of the system (6) is locally asymptotically stable, and the system (6) undergoes a Hopf bifurcation at the computer virus equilibrium
when
.
6. The Numerical Simulations and Conclusions
A computer virus epidemic model with age structure and two delays effects are constructed and studied in this paper. In contrast to traditional computer virus models, we take into account not only the spreading ability of latent computers, but also the healing ability of infected computers. That is, the models considered in this paper are more realistic. There are fewer studies on computer virus models for time lag and age systems, and in contrast to the existing ones, this paper proposes specific defenses through basic regenerative number analysis.
Theoretical analysis shows that the solution of the system (6) is non negative and bounded, the system only has the virus-free equilibrium
when
, which is locally asymptotically stable, and except the virus-free equilibrium, there is also an the computer virus equilibrium
when
and
, which is locally asymptotically stable for
. We also have discussed the existence of Hopf bifurcation under three different cases, which are 1)
and
, 2)
and
, and 3)
, respectively. In the following, we will use Matlab to verify the dynamic behaviors of the system (6), which includes the stability of the computer virus equilibrium
and the Hopf bifurcations under some cases.
In the cases where
and
. Firstly, let
,
,
,
,
,
,
,
,
,
, and
. then we can obtain
,
,
. Figure 2 displays that the solution of the system (6) will converge to the computer virus equilibrium
as t tends to infinity for different initial value conditions.
And then, we take
,
,
,
,
,
,
,
,
,
, and
. then
,
,
,
. Figure 3(a) displays the computer virus equilibrium
gradually tends to stabilize as t tends to infinity. However, when we take
,
,
,
,
,
,
,
,
,
, and
, then
,
,
,
, and
. Similarly, we can see the computer equilibrium
keeps stability in Figure 3(b).
Therefore, under these conditions of Theorem 6, the system also does not have periodic behavior regardless of the change in the delay
. Further, the distribution of infected individuals at the computer virus equilibrium
,
is shown in Figure 4(a), corresponding to the situation in Figure 3(b), and the distribution of immunization age and time, and the distribution of infection age and infection time
is shown in Figure 4(b).
Figure 2. The stability of the computer virus equilibrium
when
.
Figure 3. The stability of the computer virus equilibrium
when
.
Figure 4. The distributions of latent individuals when
is asymptotically stable under
.
Finally, choosing
,
,
,
,
,
,
,
,
,
, and
, we can get
,
,
,
, and
. Figure 5 shows that the system (6) experiences the Hopf bifurcation.
In the cases where
and
. Firstly, let
,
,
,
,
,
,
,
,
,
, and
. then we can obtain
,
,
. Figure 6 displays that the solution of the system (6) will converge to the computer virus equilibrium
as t tends to infinity for different initial value conditions.
And then, we take
,
,
,
,
,
,
,
,
,
, and
. then
,
,
,
. Figure 7(a) displays the computer virus equilibrium
gradually tends to stabilize as t tends to infinity. However, when we take
,
,
,
,
,
,
,
,
,
, and
. then
,
,
,
,
. Similarly, we can see the computer equilibrium
keeps stability in Figure 7(b).
Next, choosing
,
,
,
,
,
,
,
,
,
, and
. we can get
,
, and
. Figure 8 shows that the system (6) experiences the Hopf bifurcation. Finally, choosing
,
,
,
,
,
,
,
,
,
, and
, we can get
,
,
,
, and
. Figure 9 shows that the system (6) experiences the Hopf bifurcation.
Figure 5. The Hopf bifurcation around the computer virus equilibrium
for
and
.
Figure 6. The stability of the computer virus equilibrium
when
.
Figure 7. The stability of the computer virus equilibrium
when
.
Figure 8. The Hopf bifurcation around the computer virus equilibrium
for
.
Figure 9. The Hopf bifurcation around the computer virus equilibrium
for
.
In the cases where
. Firstly, we illustrate the stability of the computer virus equilibrium
when
. Let
,
,
,
,
,
,
,
,
, and
. Then we can obtain
,
when
. Figure 10 illustrates the computer virus equilibrium
gradually tends to stabilize as t increases. Secondly, we discuss the stability of the network virus equilibrium
when
. Let
,
,
,
,
,
,
,
,
,
, and
, then we can obtain
, and
. Figure 11 illustrates the system (6) experiences the Hopf bifurcation.
In the case where
,
, and
. Let
, and
. Taking
, and
, then we can obtain
, and
. Figure 12 displays that the system (6) still undergoes the Hopf bifurcation.
Figure 10. The stability of the computer virus equilibrium
for
.
Figure 11. The Hopf bifurcation around the computer virus equilibrium
for
.
Figure 12. The Hopf bifurcation around the computer virus equilibrium
for
.
In order to completely control the spread of computer viruses in the network, it is necessary to control the basic regeneration number
when it is less than 1, which means that the spread of computer viruses can be controlled when virus-free equilibrium exists. Based on the basic regeneration number
in the previous theoretical analysis, it is known that it is related to both delays
and
. That
is a strictly monotonically increasing with respect to
. This indicates that the increase of
will have an impact on the stability of the system, so in order to control the spread of computer viruses, the range of values of
should be as small as possible, which means that the impact on the system is small at the same time. Therefore, we can effectively control the spread of computer viruses by reducing the recovery time of computer systems after infection. By adding age structure and two delays to the system, it is more suitable for the actual virus propagation in the computer in reality, which also provides effective measures for computer virus prevention.
The emergence of the Hopf bifurcation of the system (6) implies that when both factors, age structure and delay, are introduced together into the nonresident computer virus SLAS model, it leads to destabilization of the system, followed by the phenomenon of stability switching in the system. In other words, this would disrupt the threshold of dynamic behavior, while allowing the spread of the virus to get out of control.
The limitation of this paper is that it does not discuss the global stability of virus-free equilibrium
when
. In addition, this paper does not determine the dynamical behavior of the system as the two delays
and
increases. We will continue to discuss these aspects in the future. At the same time, the spread of viruses in the computer and the infection process is more detailed, which is conducive to the subsequent study of computer virus propagation model, in which case the proposed control strategy will be more effective.
Acknowledgements
This research was partially supported by National Natural Science Foundation of China through Grants No. 12071268 and No. 11971281.