Nonstandard finite difference schemes for solving a modified epidemiological model for computer viruses
In this paper we construct two families of nonstandard finite difference (NSFD) schemes preserving the essential properties of a computer virus propagation model, such as positivity,
boundedness and stability. The first family of NSFD schemes is constructed based on the nonlocal
discretization and has first order of accuracy, while the second one is based on the combination of
a classical Runge-Kutta method and selection of a nonstandard denominator function and it is of
fourth order of accuracy. The theoretical study of these families of NSFD schemes is performed with
support of numerical simulations. The numerical simulations confirm the accuracy and the efficiency
of the fourth order NSFD schemes. They hint that the disease-free equilibrium point is not only
locally stable but also globally stable, and then this fact is proved theoretically. The experimental
results also show that the global stability of the continuous model is preserved.
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Tóm tắt nội dung tài liệu: Nonstandard finite difference schemes for solving a modified epidemiological model for computer viruses
we have the eigenvalue µ∗ of J∗. Obviously, µ∗ > 1 because λ∗ > 0. Hence, the point P3 is unstable. Now we consider the stability of the equilibrium point P1. Recall that σ(J(P1)) = {−αSAT,−αIAT − δ,−σ, 0}. Corresponding to the eigenvalue λ = 0 of J we have the eigenvalue µ = 1 of J∗. Nevertheless, as in the continuous case, this eigenvalue does not imply bifurcation or central manifold for the model [25], representing only the fact that one equation can be expressed as a linear combination of the other three. Corresponding to the eigenvalues λ1 = −αSAT, λ2 = −αIAT − δ, λ3 = −σ of J we have the following eigenvalues of J∗ P(z) = a5z5 + 1 24 z4 + 1 6 z3 + 1 2 z2 + z + 1, z = ϕλ1, P(z) = a5z5 + 1 24 z4 + 1 6 z3 + 1 2 z2 + z + 1, z = ϕλ2, P(z) = a5z5 + 1 24 z4 + 1 6 z3 + 1 2 z2 + z + 1, z = ϕλ3. By Lyapunov theorem [10, 16], the necessary and sufficient condition for P1 to be locally stable is |P(λi)| < 1, i = 1, 2, 3. This is equivalent to the system a5ϕ 5(λi) 5 + 1 24 ϕ4(λi) 4 + 1 6 ϕ3(λi) 3 + 1 2 ϕ2(λi) 2 + ϕλi < 0, (9) Pi(ϕ) := a5ϕ5(λi)5 + 1 24 ϕ4(λi) 4 + 1 6 ϕ3(λi) 3 + 1 2 ϕ2(λi) 2 + ϕλi > −2, (10) for i = 1, 2, 3. We also see that (9) is equivalent to Qi := a5ϕ4(λi)5 + 1 24 ϕ3(λi) 4 + 1 6 ϕ2(λi) 3 + 1 2 ϕ(λi) 2 + λi < 0. NSFD SCHEMES FOR SOLVING A MODIFIED EPIDEMIOLOGICAL MODEL 179 It is easy to see that Pi(ϕ) → 0 as ϕ → 0 and Qi(ϕ) → λi < 0 as ϕ → 0. Therefore, from the definition of limit of a function it follows that there exists a number τ∗ > 0 such that Pi(ϕ) > −2 and Qi(ϕ) < 0 for any ϕ < τ∗, or in other words, (9) and (10) are satisfied if ϕ < τ∗. Thus, the theorem is proved. Remark 1. In Theorem 6, the number τ∗ can be determined as τ∗ = mini=1,2,3{pi, qi}, where pi and qi are minimal root of the polynomials Pi(ϕ) and Qi(ϕ), respectively. Summarizing the results in this section we obtain Theorem 7. NSRK (5) preserves the properties (P1) and (P2) of (1) if the denominator function satisfies the condition ϕ(h) < min { r (αSA + β)T , r (αIA + δ)T , r σ , τ∗ } , ∀h > 0. (11) Clearly, the denominator function ϕ(h) = h does not satisfy (11). Therefore, we should select the denominator function satisfying (11) and not influencing on the accuracy order of the original Runge-Kutta methods. For doing this we need the following Corollary 2. NSRK (c, A, bT , ϕ) is of fourth order of accuracy if the denominator function satisfies the condition ϕ(h) = h+O(h5). (12) Thus, we have to select denominator functions satisfying simultaneously (11) and (12). The selection of such functions is an interesting and important problem. Analogously as in the recent work [6], we select denominator functions of the form ϕ(h) = (1− θ(h))1− e −τh τ + θ(h)he−µh m , θ(h) = 1 +O(hp). 5. NUMERICAL SIMULATIONS In this section we report the results of some numerical simulations in order to confirm the validity of obtained theoretical results and to demonstrate the efficiency of designed NSFD schemes. It should be emphasized that all numerical simulations in [2, 4, 5, 6, 7, 8, 19, 20, 21, 22] showed that standard difference schemes do not preserve essential properties of the corresponding continuous models. Example 1. Consider the model (1) with the parameters β = 0.1, δ = 20, σ = 0.8, αSA = 0.25, αIA = 0.25. In this case we take T = 100. The numerical solutions obtained by NSFD schemes (3) for the model (2) is depicted in Figure 1. From the figure it is seen that P1 is globally asymptotically stable, P2 is unstable and P3 does not exist. Moreover, the properties of the continuous model are preserved. Example 2. Consider the model (1) with the parameters β = 0.1, δ = 9, σ = 0.8, αSA = 0.25, αIA = 0.25. For this case we take T = 100. The numerical solutions of the model (2) obtained by the NSFD schemes (3) are depicted in Figure 2. Obviously, P1 is globally asymptotically stable, P2 and P3 are unstable. Moreover, the properties of the continuous model are preserved. 180 DANG QUANG A, HOANG MANH TUAN, DANG QUANG LONG P1 P2 100 800 5 60 10 60 15 50 S 20R 40 25 40 30 I 30 35 40 20 20 10 00 Figure 1. Numerical solutions obtained by NSFD schemes (3) with ϕ(h) = 1−e−h and h = 1 Example 3. Accuracy and computation time of schemes Consider the model (1) with the parameters β = 0.01, δ = 0.02, σ = 0.5, αSA = 0.025, αIA = 0.02 and the initial values (20, 30, 20, 30). We report the errors of the NSFD and NSRK (c, A, bT , ϕ). For comparison we also consider the method proposed by Wood and Koruharov in [27], which preserves the positivity and stability of dynamical systems based on nonlocal discretization. The denominator functions used for NSFD scheme (3), NSRK scheme and Wood and Kor- uharov’s scheme, respectively are ϕ(h) = 1− e−0.6h 0.6 , ϕ(h) = e−h 8 he−0.0001h 6 + (1− e−h8)(1− e−1.1h) 1.1 , ϕ(h) = 1− e−0.6h 0.6h . Since it is impossible to find the exact solution of the model, as a benchmark we use the numerical solution obtained by 11-stage Runge-Kutta method [9]. The benchmark solution is depicted in Figure 3. From the figure it is seen that the components of the solution change very quickly in a short time from the starting points, after that they become stable. The differential problem in this case is stiff. Table 1 provides the errors and the rates of the NSFD NSFD SCHEMES FOR SOLVING A MODIFIED EPIDEMIOLOGICAL MODEL 181 P2 P1 P3 1000 8050 10 20 6040 S 30R 30 40 I 40 50 20 60 2010 00 Figure 2. Numerical solutions obtained by NSFD schemes (3) with ϕ(h) = e−h8he−0.0001h6 + (1− e−h8)(1− e−1.1h)/1.1 and h = 1 scheme, Wood and Koruharov’s (W-K’s) scheme and NSRK scheme for different stepsizes. There errors are computed by the formula err = max k {|Sk − S(tk)|+ |Ik − I(tk)|+ |Rk −R(tk)|+ |Ak −A(tk)|}, where Uk and U(tk) are the solutions obtained by a scheme and the benchmark solution, respectively. Besides, rate := log h1 h2 (err1/err2) (see [1, Example 4.1]) is an approximation for accuracy order of the schemes. In the last column of Table 1 (rate of NSRK scheme), we see an unexpected phenomenon, when h is small the rates decrease. A similar phenomenon also was indicated in [1, Example 4.1]) when studying explicit standard Runge-Kutta methods. The reason of this is that the rounding errors generally increase as h decreases. The computation time is given in Table 2. From the tables we see that NSFD (3) has better accuracy and faster then the Wood and Koruharov’s scheme. The reason is that at each step Wood and Koruharov’s scheme needs to consider the sign of the right- hand sides for choosing appropriate discretization. NSRK scheme has the best accuracy but the computation time is largest because at each step it requires computation of values of stages Ki. However, to obtain the solution of high order of accuracy the use of NSRK is more efficient than the use of extrapolation, which combines the solutions of first order of 182 DANG QUANG A, HOANG MANH TUAN, DANG QUANG LONG 0 10 20 30 40 50 t 0 5 10 15 20 S 0 10 20 30 40 50 t 0 5 10 15 I 0 10 20 30 40 50 t 0 2 4 6 8 10 R 0 10 20 30 40 50 t 0 10 20 30 40 A Figure 3. Benchmark solution obtained by RK8 with h = 10−5 Table 1. Errors and rates of the schemes Stepsize NSFD scheme Rate W-K’s scheme Rate NSRK scheme Rate 0.25 6.463466701595602 7.713254298410573 1.2780e-04 0.2 5.243459924068896 0.9374 6.056566836376636 1.0836 2.7012e-05 6.9650 0.15 3.989310828795733 0.9502 4.459865753797996 1.0638 5.5149e-06 5.5229 0.1 2.698912044130053 0.9638 2.920085782158711 1.0445 9.2956e-07 4.3913 0.05 1.369838435912756 0.9784 1.434358023863455 1.0256 5.6487e-08 4.0405 0.001 0.027798934145437 0.9963 0.028200224634866 1.0044 1.7708e-12 2.6509 0.00001 0.000278071734776 0.9999 0.000281905383956 1.0001 9.3258e-15 1.1392 accuracy. The advantage of NSRK is that it has high order of accuracy for small stepsizes and preserves the properties of the model for large stepsizes. Moreover, it is explicit, easy to be programmed. 6. CONCLUSIONS In this paper we have constructed two families of NSFD schemes preserving the essential properties of a computer virus spread model. They are positivity, boundedness and local stability. Besides, the first NSFD schemes are globally stable, the second NSFD schemes are NSFD SCHEMES FOR SOLVING A MODIFIED EPIDEMIOLOGICAL MODEL 183 Table 2. Computation times of the schemes in seconds Stepsize NSFD scheme (3) W-Ks scheme NSRK scheme 0.2 0.001897 0.027662 0.048200 0.1 0.001942 0.012523 0.043975 0.05 0.002224 0.016025 0.064037 0.01 0.006202 0.066508 0.306687 0.005 0.006793 0.118272 0.632497 0.001 0.028386 0.616064 2.859388 0.0001 0.210997 5.866315 27.817605 0.00001 2.069937 57.343785 273.413824 of fourth order of accuracy. The numerical simulations confirm the validity of obtained the- oretical results. Among these constructed schemes, NSFD schemes have advantage in order of accuracy for small stepsizes and preserves the properties of the corresponding continuous model for large stepsizes. The method for designing high order, dynamically consistent sche- mes are applicable to some other applied models. This is the subject of our research in the future. 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