Citation: Hu Ge-JiLe, Saima Rashid, Muhammad Aslam Noor, Arshiya Suhail, Yu-Ming Chu. Some unified bounds for exponentially tgs-convex functions governed by conformable fractional operators[J]. AIMS Mathematics, 2020, 5(6): 6108-6123. doi: 10.3934/math.2020392
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In the past decade, information technology has gotten a rapid development, and is playing an important role in industry, commerce and humans' daily life. Computers, the most primary carriers of the advanced technique, are the requisite hardware of the modern science. However, computers have to face the threaten of worm and virus which are the biggest menace to the normal operation[16,14]. Generally, the operators do not know the existence of the computer worm, least of all how and when their computers have been infected. Worm and virus are always attached to a junk mail, masqueraded as innocent files for downloading, even ensconced in a code spreading through the Internet[5]. The dissemination of the worm and virus only needs someone to open the host file, then the worm and virus can transmit automatically without any additional operation[30]. Up to now, computer worm and virus can averagely cause billions of economic damage every year. To the operators, the direct results caused by infectious, such as causing system collapse, occupying computer RAM and demolishing existence data etc, always make them pay a heavy price. In response, updating antivirus software and patching operating system are two primary protections.
Recently, some researchers described the transmission and prevention of computer worm and virus with the state feedback compulsive model, and they have got some interesting results[7,12,27]. In their models, when the number of infected computers reaches a certain level the impulsive control will be triggered. Then new antivirus software will be published and installed, operating system will be patched to the latest version. All those activities happened instantaneously, and they are depicted as an impulse in the model.
However, the published researches all assume the diffusion of worm and virus is linear with the number of infected and susceptible computers which goes against the truth. In fact, at the initial period of a certain transmission, among the huge number of healthy computers connected with the Internet, only few of them have the opportunity to touch the virus[9,22,31]. And following the development, the infectious rate can not keep being linear with the infected and susceptible ones if the infected computers are numerically superior. So it is proper to describe the infectious with saturated transmission, and we will study the impulsive control of computer virus under this spreading fashion.
The remaining of this paper is arranged as follows: we set up the model and introduce some preliminaries in section 2; the existence, uniqueness and stability of order-1 limit cycle are analyzed in section 3; finally, some numerical simulation are showed to confirm the result and some deeper discussion is carried out in section 4.
Enlightened by the development in the field of impulsive control and the progress of infectious disease study [19,29,25,26], all the computers connected with the Internet are divided into two disjoint groups:
Considering objectively, we make the following assumptions:
(1) all the new emerging computers are susceptible ones, and the rate is
(2) rejection rate is not affected by the computer worm and virus, and both suceptibles and infectives are sifted out from the Internet at rate
(3) the transmission incidence is
(4)
When infectives is less than
{dSdt=K−βSI1+αS−μS,dIdt=βSI1+αS−μI,I<ˉI, ΔS=−σ1S, ΔI=−σ2I,I=ˉI. | (1) |
Considering the practical significance, all the parameters in this paper are positive,
In the reminder part of this section, we introduce some definitions and lemmas to support the proof of the existence and stability of the order-1 limit cycle of system (1).
Definition 2.1 (19-25) A typical state feedback impulsive differential model can be defined as
{dxdt=P(x,y), dydt=Q(x,y), (x,y)∉M{x,y},Δx=α(x,y), Δy=β(x,y), (x,y)∈M{x,y}, | (2) |
and can be denoted by
Definitions of successor function and order-1 periodic solution of state feedback impulsive dynamic system are also vitally important in the research.
Definition 2.2. [2,21,15] Simply suppose the impulse set
Remark 1. The necessary and sufficient condition that the solution passing point
Lemma 2.3. [2,21,15] To a state feedback impulsive dynamic system
Without counting the impulsive control measure, system (1) can be simplified as
{dSdt=K−βSI1+αS−μS,dIdt=βSI1+αS−μI, | (3) |
system (3) is called uncontrolled system of (1). Change the variables with linear transform
{dSdt=−βSI+(K−μS)(1+αS),dIdt=βSI−μI(1+αS). | (4) |
It is obvious that system (4) has two equilibria
S∗=μβ−μα and I∗=(K−μS∗)(1+αS∗)βS∗=Kμ−μβ−μα. |
β>μ2K+μα. | (5) |
In the following discussions, we assume condition (5) is satisfied consistently. The Jacobian matrix of
JE1=(−μ−αK−βKμ0βKμ−μ−αK), |
JE2=(−Kμ(β−μα)−αμ2β−μα−βμβ−μαKμ(β−μα)−μ0). |
Assume
λ1+λ2=−Kμ(β−μα)−αμ2β−μα<0 and λ1⋅λ2=βμβ−μα(Kμ(β−μα)−μ)>0, |
so
In this part, we will consider the uniform boundedness in the first quadrant. From system (4), we can find that
I=−μαβS+Kβ1S+Kα−μβ |
is vertical isoclinic line, and
S=μβ−μα and I=0 |
are horizontal isoclinic lines.
Following the succeeded statement, we select a piece of bounded region (see Fig. 3). Designate a line
Assume
dφdt=dIdt−adSdt=(1+a)aβ⋅x2+[bβ(1+a)−α(aK+bμ)]⋅x−(aK+bμ). | (6) |
Set
Assume Dulac function
{dSdt=−βSI+(K−μS)(1+αS)Δ=P(S,I),dIdt=βSI−μI(1+αS)Δ=Q(S,I), | (7) |
then
{dSdt=u(S,I)P(S,I)=−βSI+(K−μS)(1+αS)Δ=P1(S,I),dIdt=u(S,I)Q(S,I)=βSI−μI(1+αS)Δ=Q1(S,I). | (8) |
It is obvious that
Basing on the above discussion, if condition (5) is satisfied, then there exists only one positive equilibrium
In this section, we will certificate the existence and uniqueness of order-1 limit cycle of system (1). In reality, it is unreasonable that a single kind of computer virus lasts for long time without being detected. So we assume
The existence of order-1 limit cycle is studied in this subsection.
Theorem 4.1. If condition (5) holds, for any
Proof. In system (1), to any
Assume the intersection of imagine set
Case of
Case of
Basing on the above discussion, to any
Since the order-1 periodic solution is isolated, its trajectory and the proper impulsive line can form order-1 limit cycle of system(1).
In this part we prove the uniqueness of the order-1 limit cycle with the homogeneous of successor function.
Theorem 4.2. The order-1 limit cycle of system (1) is unique.
Proof. Select two points
The above content has proved the existence and uniqueness of order-1 limit cycle. In this subsection, we discuss the stability of the order-1 limit cycle of system (1). Unlike the case of continuous dynamic system, the stability proof of impulsive dynamic system is quite complex. Some definition and lemma which are significant in the proof will be illustrated first.
Definition 4.3. [24] The order-1 limit cycle
Basing on the theorem of subsection 4.1 and 4.2, to any
S0,S1,⋯,Sk,Sk+1,⋯, |
where
sk={−dk,Sk is on left side of A,dk,Sk is on right side of A, |
where
Lemma 4.4. [24] Denote the mapping of the general impulsive dynamic system (2) as
|dˉsds|s=0<1(>1). |
It is not easy to prove the stability of order-1 limit cycle of system (2) only with the previous lemma, so we set up an orthogonal coordinate
Without loss of generality, we assume that functions
To any point in that region, there exists a certain point in the closed orbit
x=ϕ(s),y=ψ(s), | (9) |
where
Then the rectangular coordinate
x=ϕ(s)−nψ′(s),y=ψ(s)+nϕ′(s), | (10) |
Following (2) and (10), we have
dnds=Qϕ′−Pψ′−n(Pϕ″+Qψ″)Pϕ′+Qψ′Δ=F(s,n). | (11) |
It is apparent that
dnds=Fn′(s,n)|n=0⋅n+o(n). | (12) |
Following (11), we can calculate
Fn′(s,n)|n=0=P20Qy0−P0Q0(Py0+Qx0)+Q20Px0(P20+Q20)32Δ=H(s), |
where
dnds=H(s)n, | (13) |
and its solution is
n=n0e∫T0H(s′)ds′,n0=n(0). | (14) |
Theorem 4.5. Assume
∫h0H(s)ds<0(>0). |
Theorem 4.6. Suppose the region closed by order-1 periodic solution
The proofs of Theorem 4.5 and Theorem 4.6 are similar with that in [27], and omitted here.
Then we can draw the corollary naturally.
Corollary 1. Suppose the area closed by order-1 limit cycle
In this section, we present and discuss a special example to check the theoretical results in section 3 and section 4.
The uncontrolled part of system (1) is numerically calculated with fourth-order Runge-Kutta of Matlab software package. In the following, we discuss the solution of system (1), considering the parameters
From Fig. (10), it is obvious that system (1) has an order-1 limit cycle with the parameters mentioned above. We can also find that the limit cycle forms soon after the trajectory initiate from the original point which means that the impulsive patching and updating of antivirus software is quite effective. During the first period of the transmission, the worm or the virus spreads freely, the number represents susceptible computers decreases rapidly and the number stands for the infectious ones increased sharply. The developing trend of
We would like to sincerely thank the reviewers for their careful reading and constructive opinions of the original manuscript. This work is supported by NSFC(No.11671346, No.61751317, No.11701026), the Twin Tower Excellent Teacher supporting program (BUCEA2016) and the Fundamental Research Funds for Beijing University of Civil Engineering and Architecture (X18225, X18080) for M. Zhang. M. Zhang would like to thank the China Scholarship Council for financial support of her overseas study(No.201808110071).
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