Nonlinear Analysis and Differential Equations, Vol. 4, 216, no. 12, 583-595 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/nade.216.687 A Fractional-Order Model for Computer Viruses Propagation with Saturated Treatment Rate Moustafa El-Shahed and Fatima Alissa Department of Mathematics, Faculty of Arts and Sciences Qassim University, P.O. Box 3771, Qassim Unizah 51911, Saudi Arabia Copyright c 216 Moustafa El-Shahed and Fatima Alissa. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, a fractional-order model for the spread of computer viruses with saturated treatment rate is presented. This model consists of three components: Susceptible, Infectious, and Recovered. The SIR epidemic model with saturated treatment rate is considered. The stability of the disease free and endemic equilibrium points are studied. The global stability of the disease free equilibrium point is also investigated. In addition, fractional Hopf bifurcation conditions for the model are proposed. The generalized Adams-Bashforth-Moulton method is used to solve and simulate the system of fractional differential equations. Keywords: viruses Fractional order; Stability; Numerical method; Computer 1 Introducation A computer virus is a small program that is inserted on a computer without the user s knowledge in order to destroy some or all of the software and hardware components of the Computer. Virus has begun to spread in the mid-eighties of the last century and since that time has evolved and appeared more aggressive, especially speed types with the end of the nineties and the known number of known viruses and modified them copies of hundreds of thousands of viruses.
584 Moustafa El-Shahed and Fatima Alissa Because of their potential applications in science in engineering, interest in fractional calculus and fractional differential equations has grown dramatically in recent decades [15, 24]. Pinto and Machado[23] proposed a fractional order model for computer virus propagation. The model includes the interaction between computers and removable devices. Ansari et.al.[5] have studied the dynamical behavior of delay-varying computer virus propagation (CVP) model with fractional order derivative. The spread of computer virus in a computer network can be seen analogous to the spread of an infectious disease in a population. The susceptible and infective individuals in the spread of epidemic diseases correspond to non-infected and infected computer nodes in the dynamics of virus infected computer network[22, 25]. Zhang and suo [26] formulated a new SIR epidemic model by considering the saturated removed rate and supposing the removal individuals perpetual immunity, is formulated. In this paper, we consider the fractional-order model for computer viruses propagation with saturated treatment rate. We give a detailed analysis of the asymptotic and global stability of the model. To solve and simulate the system of fractional differential equations, we use the Adams-Bashforth-Moulton algorithm. We simulate numerically the model for different values of the order of the fractional derivatives. 2 Model Formulation Following [25, 26], the mathematical model for computer viruses propagation can be written as a set of three coupled nonlinear ordinary differential equations as follows: ds dt di dt dr dt = A S βsi + ωr, = βsi ( + γ + δ)i ɛi 1 + qi, (2.1) = γi + ɛi ( + ω)r. 1 + qi The above model has three populations, the susceptible nodes (S), infectious nodes (I ) and recovered nodes (R). A denotes the recruitment rate of new nodes, is the natural crashing rate of nodes all classes, β is the rate of transfer of virus from an infectious node to the susceptible node. Due to the loss of immunity, at any time a recovered computer becomes susceptible with constant probability ω. Due to treatment, every infectious node becomes an
A fractional-order model for computer viruses propagation 585 recovered node with constant probability γ.δ is the virus induced crashing rate, ɛ is the maximal treatment resources supplied per unit time, and q measures the extent of the effect of the infective node being delayed for treatment. Here all the parameters A,, β, γ, δ, ω, ɛ and q are positive. Fractional order models are more accurate than integer-order models as fractional order models allow more degrees of freedom. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. The presence of memory term in such models not only takes into account the history of the process involved but also carries its impact to present and future development of the process. Fractional differential equations are also regarded as an alternative model to nonlinear differential equations. In consequence, the subject of fractional differential equations is gaining much importance and attention. For some recent work on fractional differential equations, see [15, 24]. Now we introduce fractional order in to the ODE model. The new system is described by the following set of fractional order differential equations: D α t S = A S βsi + ωr, D α t I = βsi ( + γ + δ)i ɛi 1 + qi, (2.2) D α t R = γi + ɛi 1 + qi ( + ω)r. where D α t is the Caputo fractional derivative. Because model (2.2) monitors the dynamics of human populations, all the parameters are assumed to be nonnegative. Furthermore, it can be shown that all state variables of the model are non-negative for all time t. Lemma 2.1. The Region Ω = {(S, I, R) R+ 3 : S + I + R A }is positivelyinvariant for the model (2.2) with non-negative initial conditions. Proof. Let N(t) = S(t) + I(t) + R(t).Adding the equations of the model (2.2) gives D α t N(t) = A N(t) δi(t), (2.3) then we have D α t N(t) A N(t). Lemma 2.3[13] can then be used to show that N(t) At α E α,α+1 ( t α ) + N()E α,1 ( t α ),
586 Moustafa El-Shahed and Fatima Alissa where E α,β () is the Mittag-Leffler function. Based on Lemma (2.4)[13], one can observe that N(t) A, thus the Region Ω is positively-invariant for the model (2.2) with non-negative initial conditions. In the following, we will study the dynamics of system (2.2). 3 Equilibrium Points and Stability To evaluate the equilibrium points, let D α t S =, D α t I =, D α t R =. The disease free equilibrium point is E = ( A,, ). The point E always exists. Theorem 3.1. For the system (2.2), the basic reproduction number is R = βa (Ψ + ɛ). Proof We use the next generation method to find the basic reproduction number for the system (2.2). One can rewrite the system (2.2) as follows: Dt α I = βsi ( + γ + δ)i ɛi 1 + qi, Dt α S = A S βsi + ωr, (3.1) Dt α R = γi + ɛi ( + ω)r. 1 + qi We make matrices f, v, such that the system (3.1) in the form dx dt = f(x) v(x), where f(x) = f 1 f 2 f 3 = βsi, v(x) = v 1 v 2 v 3 = ΨI + ɛi 1+qI A + S + βsi ωr γi ɛi + qr 1+qI, where Ψ = + γ + δ and θ = + ω. Now the Jacobian matrix of f(x) and v(x) at E are respectively given by F (x) = [ βa ] [ Ψ + ɛ, V (x) = βa ].
A fractional-order model for computer viruses propagation 587 The eigenvalues of F V 1 is the solution of the equation F V 1 λi =, (3.2) where λ is the eigenvalues and I is the identity matrix. Then λ 1 = βa, (Ψ+ɛ) λ 2 =. It then follows that the spectral radius of the matrix F V 1 is ρ (F V 1 ) = max (λ i ), i = 1, 2. Then R = βa. (H+ɛ) Theorem 3.2. If R < 1,disease free equilibrium point E of the system (2.2) is a asymptotically stable, while if R > 1, the disease free equilibrium point is E is unstable. Proof. The Jacobian matrix J for the system given in (2.2) is given by J = βi βs ω βi βs Ψ ɛ (1+qI) 2. γ + ɛ (1+qI) 2 ( + ω) The Jacobian matrix J(E ) for the system given in (2.2), evaluated at E, is as follows: βa ω J(E ) = βa Ψ ɛ. γ + ɛ ( + ω) The disease free equilibrium point E is locally asymptotically stable if all the eigenvalues λ i, i = 1, 2, 3 of J(E ) satisfy the following condition [1, 2, 3, 8, 11, 12]: arg (λ 1i ) > απ 2. (3.3) The eigenvalues corresponding to the equilibrium E are λ 1 =, λ 2 = ( + ω)and λ 3 = βa (Ψ + ɛ) = (Ψ + ɛ)(r 1). Then we have λ 1 <, λ 2 < and λ 3 < when R < 1.It follows that the disease free equilibrium point of system (2.2) is unstable point, when R > 1. Theorem 3.3. The disease free equilibrium point E is globally asymptotically stable. Proof. Let us consider the following Lyapunov candidate function, which is positive definite. V = 1 2 I2,
588 Moustafa El-Shahed and Fatima Alissa Applying Lemma 1[6, 7], one get D α t V = 1 2 I2, I.D α t I, ( βa Ψ ɛ + qa )I2, [R (Ψ + ɛ) Ψ ɛ + qa ]I2. Thus E is globally asymptotically stable if R < ɛ+ψ(ψ+ɛ). (+qa)(ψ+ɛ) The endemic equilibrium point is E 1 = (S, I, R ),where, R = I (γ + θ ɛ ), S = Ψ + ɛ and 1+qI β β(1+qi ) I satisfying the following equation I 2 + c 1 (1 R 1 )I + c (1 R ) =, c = θ(ψ+ɛ) qβ(ψ+ωθ), c 1 = β(ψ+ɛ)+βω(+δ)+qθψ, qβ(ψ+ωθ) qβθa. β(ψ+ɛ)+βω(+δ)+qθψ R 1 = When R > 1, there exists a unique endemic equilibrium point E 1 = (S, I, R ).The Jacobian matrix J(E 1 )for the system given in (2.2) at E 1 is given by βi βs ω ɛqi J(E 1 ) = βi (1+qI ). 2 ɛ γ + θ (1+qI ) 2 The characteristic equation of J(E 1 ) is as follows: λ 3 +a 1 λ 2 +a 2 λ+a 3 =,where a 1 = θ + + I (β(1 + qi ) 2 + q((θ + )(2 + qi ) ɛ)) (1 + qi ) 2, a 2 = S I (β + βqi ) 2 + β( q)i 2 ɛ + βθi (1 + qi ) 2 + θ(1 + qi ) 2 qi ɛ( + θ) (1 + qi ) 2, a 3 = I (θs (β + βqi ) 2 βω (γ(1 + qi ) 2 + ɛ) θqɛ( + βi )) (1 + qi ) 2. Let D(Φ) denote the discriminant of a polynomial Φ (λ) = λ 3 +a 1 λ 2 +a 2 λ+a 3, then D(Φ) = 18a 1 a 3 a 3 + (a 1 a 2 ) 2 4a 3 a 3 1 4a 3 2 27a 2 3. Following [1, 2, 3, 18], one have the proposition.
A fractional-order model for computer viruses propagation 589 Proposition 3.4. One assumes that E 1 exists in R 3 +. 1. If the discriminant of Φ(λ), D(Φ)is positive and Routh-Hurwitz are satisfied, that is, D(Φ) >, a 1 >, a 3 >, a 1 a 2 > a 3,then E 1 is locally asymptotically stable. 2. If D(Φ) <, a 1 >, a 2 >, a 1 a 2 = a 3, α [, 1) then E 1 is locally asymptotically stable. 3. If D(Φ) <, a 1 <, a 2 <, α > 2 3, then E 1 is unstable. 4. The necessary condition for the equilibrium point E 1, to be locally asymptotically stable, is a 3 >. 4 Numerical Methods and Simulation Because most fractional-order differential equations lack exact analytic solutions, approximation and numerical techniques must be used. There exist several different analytical and numerical methods for solving fractional-order differential equations. For numerical solutions of system (2.2), one can use the generalized Adams-Bashforth-Moulton method. To give the approximate solution by means of this algorithm, consider the following nonlinear fractionalorder differential equation[8, 9, 14, 17]: D α t y(t) = f (t, y (t)), t T, y (k) () = y k, k =, 1, 2,..., m 1. This equation is equivalent to the Volterra integral equation: y(t) = m 1 k= y (k) t k k! + 1 Γ (α) t (t s) α 1 f (s, y (s)) ds. (4.1) Diethelm et al. used the predictor-corrector scheme [8, 9], based on the Adams- Bashforth-Moulton algorithm, to integrate Eq. (4.1). By applying this scheme to the fractional-order model for a tritrophic model consisting of tea plant, pest, and predator, and setting h = T, t N n = nh, n =, 1, 2,,..., N Z +, one
59 Moustafa El-Shahed and Fatima Alissa can discretize Eq. (4.1) as follows: where h α [ ] S n+1 = S + A S p n+1 βs p Γ (α + 2) n+1i p n+1 + ωr p n+1 n + hα a j,n+1 [A S j βs j I j + ωr j ], Γ (α + 2) j=1 [ h α I n+1 = I + βsn+1i p p n+1 ( + γ + δ)i p n+1 ɛip n+1 Γ (α + 2) n + hα Γ (α + 2) h α j=1 R n+1 = R + Γ (α + 2) n + hα Γ (α + 2) S p n+1 = S + 1 Γ (α) I p n+1 = I + 1 Γ (α) R p n+1) = R + 1 Γ (α) j=1 a j,n+1 [ βs j I j ( + γ + δ)i j [ γi p n+1 + ɛip n+1 1 + qi p ( + ω)r p n+1 n+1 a j,n+1 [ γi j + ɛi j 1 + qi j ( + ω)r j n b j,n+1 [A S j βs j I j + ωr j ], j= n j= n j= b j,n+1 [ βs j I j ( + γ + δ)i j b j,n+1 [ γi j + 1 + qi p n+1 ɛi ] j, 1 + qi j ] ], ɛi j 1 + qi j ( + ω)r j ɛi j 1 + qi j ], n α 1 (n α) (n + 1), j =, a j,n+1 = (n j 2) α+1 + (n j) α+1 2 (n j + 1) α+1 1 j n, 1 j = n + 1, 5 Conclusion b j,n+1 = hα α [(n j + 1)α (n j) α ], j n. ], ] In this paper, we consider the fractional-order model for computer virus propagation. We obtained a stability condition for equilibrium points, provided a numerical example, and verified our results. Figure.1 show that the process of transforming a classical model into a fractional one is very sensitive to the order of differentiation α; for instance, a small change in α may result in a
A fractional-order model for computer viruses propagation 591 substantial change in the final result. From the numerical, it is clear that the approximate solutions depend continuously on the fractional derivative α. We used some documented data for several parameters, such as A = 22, =.6, β =.8, q =.3, ɛ = 1, γ = 5, δ =.2 and ω =.2. The approximate solutions S(t), I(t) and R(t) are displayed in Figure. 1 for the order of the fractional derivative α =.99,.95,.9. An important feature of the fractionalorder model is that it controls the speed at which the solution to equilibrium is reached. It follows from (3.3), that α =.92799. When α < α, the trajectories converge to the equilibrium point, as shown in Figs. 2 (α =.9); whereas when α is increased to exceed α, the origin loses its stability, and a Hopftype bifurcation occurs, as shown in Fig. 2 for the following values:a = 4, =.6, β =.4, q =.3, ɛ = 1, γ = 5, δ =.2 and ω =.2. Furthermore, Fig. 3 illustrates that the system (2.2) has two endemic equilibrium points. Unlike the integer order differentiation, the fractional order changes the point of equilibrium from (38.5382,.21976, 1.92386) when α =.99 to the point ( 32.381, 1.632, 7.15582), when α =.9, for the following values: A = 4, =.98, β =.4, q =.6, ɛ = 1, γ = 5, δ =.4 and ω =.6. S(t),I(t),R(t) S(t),I(t),R(t) S(t),I(t),R(t) 4 2 α =.99 2 4 6 8 1 t α =.95 4 S(t) 2 I(t) R(t) 2 4 6 8 1 t α =.9 4 2 2 4 6 8 1 t Figure 1: When α =.99,.95,.9 the trajectory of system (2.2) converges to the endemic equilibrium point E 1 = (16.4254, 1.2781, 18.639).
592 Moustafa El-Shahed and Fatima Alissa 12 α =.99 6.5 α =.9 6 1 5.5 8 5 4.5 I(t) 6 I(t) 4 3.5 4 3 2 2.5 2 1 2 3 4 5 S(t) 1.5 2 25 3 35 S(t) Figure 2: When α =.99 the system (2.2) converges to limit cycle, when α =.9 the system (2.2) converges to equilibrium point. 1 α =.99 1 α =.9 9 9 8 8 7 7 6 6 I(t) 5 I(t) 5 4 4 3 3 2 2 1 1 1 2 3 4 5 S(t) 1 2 3 4 S(t) Figure 3: Phase portraits corresponding to the system (2.2) for the following values: A = 4, =.98, β =.4, q =.6, ɛ = 1, γ = 5, δ =.4 and ω =.6.
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