Summary of doctoral thesis: Development of nonstandard finite difference methods for some classes of differential equations

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Summary of doctoral thesis: Development of nonstandard finite difference methods for some classes of differential equations

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Although the research direction on NSFD schemes for differential equations have achieved a lot ofresults shown by both quantity and quality of existing research works, real-world situations have always posednew complex problems in both qualitative study and numerical simulation aspects. On the other hand, there aremany differential models that have been established completely in the qualitative aspect but their correspondingdynamically consistent discrete models have not yet been proposed and studied

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1. MINISTRY OF EDUCATION AND VIETNAM ACADEMY TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY Hoang Manh Tuan DEVELOPMENT OF NONSTANDARD FINITE DIFFERENCE METHODS FOR SOME CLASSES OF DIFFERENTIAL EQUATIONS Major: Applied Mathematics Code: 9 46 01 12 SUMMARY OF DOCTORAL THESIS HANOI - 2021
2. This thesis has been completed at: Graduate University of Science and Technology – Vietnam Academy of Science and Technology. Supervisor 1: Prof. Dr. Dang Quang A Supervisor 2: Assoc. Prof. Dr. Habil. Vu Hoang Linh Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be defended at the Board of Examiners of Graduate University of Science and Technology – Vietnam Academy of Science and Technology at ............................ on.............................. The thesis can be explored at: - Library of Graduate University of Science and Technology - National Library of Vietnam
3. INTRODUCTION 1. Overview of research situation Many essential phenomena and processes arising in fields of science and technology are mathematically modeled by ODEs of the form: dy(t) y(t0 ) = y0 ∈ Rn ,  = f y(t) , (0.0.1) dt  T where y(t) denotes the vector-function y1 (t), y2 (t), . . . , yn (t) , and the function f satisfies appropriate condi- tions which guarantee that solutions of the problem (0.0.1) exist and are unique. The problem (0.0.1) is called an initial value problem (IVP) or also a Cauchy problem. The problem (0.0.1) has always been playing an essential role in both theory and practice. Theoretically, it is not difficult to prove the existence, uniqueness and continuous dependence on initial data of the solutions of the problem (0.0.1) thanks to the standard methods of mathematical analysis. However, it is very challenging, even impossible, to solve the problem (0.0.1) exactly in general. In common real-world situations, the problem of finding approximate solutions is almost inevitable. Consequently, the study of numerical methods for solving ODEs has become one of the fundamental and practically important research challenges (see, for example, Ascher and Petzold 1998; Burden and Faires 2011; Hairer, Nørsset and Wanner 1993, Hairer and Wanner 1996, Stuart and Humphries 1998). Due to requirements of practice as well as the development of mathematical theory, many numerical methods, typically finite difference methods have been constructed and developed. It is safe to say that the general theory of the finite difference methods for the problem (0.0.1) has been developed thoroughly in many monographs. These methods will be called the standard finite difference (SFD) methods to distinguish them from the nonstandard finite difference (NSFD) schemes that will be presented in the remaining parts. Except for key requirements such as the convergence and stability, numerical schemes must correctly preserve essential properties of corresponding differential equations. In other words, differential models must be transformed into discrete models with the preservation of essential properties. However, in many problems, the SFD schemes revealed a serious drawback called ”numerical instabilities”. To describe this, Mickens, the creator of the concept of NSFD methods, wrote: ”numerical instabilities are an indication that the discrete models are not able to model the correct mathematical properties of the solutions to the differential equations of interest” (Mickens 1994, 2000, 2005, 2012). In a large number of works, Mickens discovered and analyzed numerous examples related to the numerical instabilities occurring when using SFD methods. In 1980, Mickens proposed the concept of NSFD schemes to overcome numerical instabilities. According to the Mickens’ methodology, NSFD schemes are those constructed following a set of basic rules derived from the analysis of the numerical instabilities that occur when using SFD schemes (Mickens 1994, 2000, 2005, 2012). Over the past four decades, the research direction on NSFD schemes has attracted the attention of many researchers in many different aspects and gained a great number of interesting and significant results. All of the works confirmed the usefulness and advantages of NSFD schemes. An advantage of NSFD schemes 1
4. over standard ones is that they can correctly preserve essential properties (positivity, boundedness, asymptotic stability, periodicity, etc.). In major surveys Mickens (2012) and Patidar (2005, 2016) and several monographs Mickens (1994, 2000, 2005), Mickens and Padidar systematically presented results on NSFD methods in recent decades as well as directions of the development in the future. Nowadays, NSFD methods have been and will continue to be widely used as a powerful and effective approach to solve ODEs, PDEs, delay differential equations (DDEs) and fractional differential equations (FDEs) (see, for instance, Arenas, Gonzalez-Parra and Chen-Charpentier 2016; Garba et al. 2015; Ehrardt and Mickens 2013; Mickens 1994, 2000, 2005, 2012; Modday, Hashim and Momani 2011; Patidar 2005, 2016). 2. The necessity of the research Although the research direction on NSFD schemes for differential equations have achieved a lot of results shown by both quantity and quality of existing research works, real-world situations have always posed new complex problems in both qualitative study and numerical simulation aspects. On the other hand, there are many differential models that have been established completely in the qualitative aspect but their corresponding dynamically consistent discrete models have not yet been proposed and studied. Therefore, the construction of discrete models that correctly preserve essential properties of differential models is truly necessary, has scientific significance and needs to be studied. Importantly, the construction of NSFD schemes for ODE models still faces many difficulties and has not been completely resolved, especially for models with at least one of the following characteristics: (i) Having higher dimensions. (ii) Having non-hyberbolic equilibrium points. (iii) Having global asymptotic stability (GAS) property. Generally, most of existing results only focus on differential models having hyperbolic equilibrium points with the local asymptotic stability (LAS), and there are no effective approaches for problems possessing non- hyperbolic equilibrium points and/or having the GAS property. On the other hand, the study of the LAS of NSFD schemes for models having higher dimensions is still a big challenge, and hence, effective approaches are needed for these models. Furthermore, the improvement of the accuracy of NSFD schemes and the construction of exact finite difference (EFD) schemes for ODE models are also essential with many important applications. From the above reasons, we believe that the continuation of study on NSFD schemes for differential equations is timely and really necessary and has great scientific and practical significance, and therefore, need to be studied. That is why we set the aim of developing NSFD schemes for important mathematical models modeled by ODEs, which arise in fields of science and technology. 3. Objectives and contents of the thesis The aim of the thesis is to develop Mickens’ methodology to construct NSFD methods for solving some important classes of differential equations arising in fields of science and technology. The thesis intends to study the following contents: Content 1: NSFD schemes for some classes of ODEs arising in science and technology. 2
5. Content 2: EFD schemes for some ODEs and their applications. Content 3: High order NSFD schemes for general autonomous dynamical systems and their applications. 4. Approach and research method We will approach to the proposed contents of the thesis from both theoretical and practical points of view. Differential models under consideration will be completely established on the qualitative aspect before proposing and studying NSFD schemes. Numerical simulations will be also performed to confirm the validity of theoretical results. In order to perform the above research, we will use a combination of tools including qualitative theory of discrete and continuous dynamical systems, the Lyapunov stability theory, the Mickens’ NSFD methodology, theory of numerical methods and finite difference schemes for differential equations. In addition, the experimental methods will be also used, especially when proofs for theoretical results are not finalized. 5. The new contributions of the thesis 1. Proposing and analyzing NSFD schemes for some important classes of differential equations, which are mathematical models of processes and phenomena arising in science and technology. The proposed NSFD schemes are not only dynamically consistent with the differential equation models, but also easy to be implemented; furthermore, they can be used to solve a large class of mathematical problems in both theory and practice. 2. Proposing novel efficient approaches and techniques to study asymptotic stability of the constructed NSFD schemes. 3. Constructing high-order NSFD methods for some classes of general dynamical systems; consequently, the contradiction between dynamic consistency and high order of accuracy of NSFD methods has been resolved. 4. Proposing exact finite difference schemes for linear systems of differential equations with constant coefficients. This result not only resolves some open questions related to exact schemes but also generalizes some existing works. 5. Performing many numerical experiments to confirm the theoretical results and to demonstrate the advantages and superiority of the proposed NSFD schemes over the standard numerical schemes. 6. Structure of the thesis In addition to ”Introduction”, ”General conclusions” and ”References”, the contents of this thesis are presented in three chapters, among which the main results are in Chapters 2 and 3. 3
6. CHAPTER 1. PRELIMINARIES In this chapter, we recall essential preliminaries related to continuous-time and discrete-time continuous dynamical systems, numerical methods for solving ODEs, exact finite difference schemes and nonstandard finite difference schemes for differential equations (Agarwal 2000; Allen 2007; Ascher and Petzold 1998; Brauer and Castillo-Chavez 2001; Burden 2011; Horv´ath 1998, 2005; Iggidr and Bensoubaya 1998; Edelstein-Keshet 1998; Khalil 2002; Kraaijevanger 1991; La Salle and Lefschetz 1961; Manning and Margrave 2006; Martcheva 2015; Mattheij and Molenaar 2002; Mickens 1994, 2000, 2005, 2012; Patidar 2005, 2016; Seibert and Suarez 1990; Smith and Waltman 1995; Stuart and Humphries 1998), which will be used in the next chapters. The contents of this chapter include: (i) Continuous-time dynamical systems. (ii) Discrete-time continuous systems. (iii) Runge-Kutta methods for ODEs. (iv) Positivity of Runge-Kutta methods. (v) Exact finite difference schemes. (vi) Nonstandard finite difference methods. 4
7. CHAPTER 2. NONSTANDARD FINITE DIFFERENCE SCHEMES FOR SOME CLASSES OF ORDINARY DIFFERENTIAL EQUATIONS In this chapter, we propose and study NSFD schemes for some classes of ODEs that describe essential phenomena and processes in fields of science and technology. The ODE models under consideration include: (i) Two metapopulation models. (ii) One predator-prey model. (iii) Two computer virus propagation models. It should be emphasized that all of these models have at least one of the following characteristics: (i) Having large dimensions. (ii) Having non-hyberbolic equilibrium points. (iii) Having the GAS property. Consequently, the stability analysis of the proposed NSFD schemes is a great challenge. To overcome this challenge, we propose novel approaches and techniques to study the stability of the constructed NSFD schemes. The main result is that we obtain NSFD schemes preserving essential properties of the continuous models for all finite step sizes. The validity of the theoretical results and the advantages of the constructed NSFD schemes over standard ones are supported by many numerical experiments. This chapter is written based on seven papers [A1]-[A7] in ”The list of works of the author related to the thesis”, p. 24. 2.1. Dynamically consistent NSFD schemes for a metapopulation model In this section, we construct NSFD schemes preserving essential properties of a metapopulation model formulated in Keymer et al. (2000). These properties include monotone convergence, boundedness, stability and non-periodicity. By standard techniques of mathematical analysis, we show that the constructed NSFD schemes are dynamically consistent with the continuous model. 2.1.1. Dynamical properties of the metapopulation model We consider a metapopulation model (or patch model) formulated in Keymer et al. (2000) d p0 = e(p1 + p2 ) − λ p0 , dt d p1 = λ p0 − β p1 p2 + δ p2 − ep1 , (2.1.1) dt d p2 = β p1 p2 − (δ + e)p2 . dt 5
8. We refer the readers to Keymer et al. (2000) for the details of the model. Because p0 + p1 + p2 = 1, it is sufficient to consider the following sub-model d p1 = λ (1 − p1 − p2 ) − β p1 p2 + δ p2 − ep1 , dt (2.1.2) d p2 = p2 (β p1 − δ − e). dt From the biological meaning of the model, we only consider the initial conditions p1 (0), p2 (0) satisfying p1 (0), p2 (0) ≥ 0, p1 (0) + p2 (0) ≤ 1. (2.1.3) The mathematical analyses in Allen (2007) and Keymer et al. (2000) show that (2.1.1) possesses the following properties: (P1 ) The monotone convergence of the sum s(t) = p1 (t) + p2 (t): With the initial conditions satisfying (2.1.3), the sum s(t) = p1 (t) + p2 (t) monotonically converges to s∗ = λ /(λ + e). (P2 ) Boundedness: All of the solutions of (2.1.1) with the initial conditions subjected to (2.1.3) satisfy p1 (t), p2 (t) ≥ 0, and p1 (t) + p2 (t) ≤ 1. (P3 ) The LAS: The model (2.1.1) has two equilibria     ∗ λ ∗ ∗ ∗ δ +e λ δ +e p1 = , 0 , p2 = (x , y ) = , − . λ +e β λ +e β βλ We define R0 := . Then, the first equilibrium p∗1 is locally asymptotically stable (LAS) if (λ + e)(δ + e) R0 < 1 and the second equilibrium p∗2 is LAS if R0 > 1. (P4 ) The GAS: If R0 < 1 then the first equilibrium is globally asymptotically stable (GAS), whereas, if R0 > 1, then the second equilibrium point is GAS.  (P5 ) Non-periodic solution: The model (2.1.2) does not have periodic solutions in the domain D = (p1 , p2 )|0 < p1 + p2 < 1 . 2.1.2. The construction of NSFD schemes For convenience of presentation, as usual, we denote the step-size by h, and adopt the notations x(t) ≡ p1 (t), y(t) ≡ p2 (t). We propose the following family of NSFD schemes xk+1 − yk+1 = −c1 (λ + e)xk − c2 (λ + e)xk+1 + c3 (δ − λ )yk + c4 (δ − λ )yk+1 ϕ(h) − c5 β xk yk − c6 β xk+1 yk − c7 β xk yk+1 − c8 β xk+1 yk+1 + λ , yk+1 − yk (2.1.4) = −c1 (λ + e)yk − c2 (λ + e)yk+1 + c3 (λ − δ )yk + c4 (λ − δ )yk+1 ϕ(h) + c5 β xk yk + c6 β xk+1 yk + c7 β xk yk+1 + c8 β xk+1 yk+1 , 6
9. where c1 + c2 = c3 + c4 = c5 + c6 + c7 + c8 = 1, ϕ(h) = h + O(h2 ). (2.1.5) We now consider some particular cases of (2.1.4)-(2.1.5) as follows. Scheme (2.1.4)-(i): Scheme (2.1.4) with c1 + c2 = 1, c3 = 1, c4 = 0, c5 + c6 = 1, c7 = c8 = 0, ϕ(h) = h + O(h2 ). (2.1.6) Scheme (2.1.4)-(ii): Scheme (2.1.4) with c1 + c2 = 1, c3 = 1, c4 = 0, c5 = 1, c6 = c7 = c8 = 0, ϕ(h) = h + O(h2 ). (2.1.7) Scheme (2.1.4)-(iii): Explicit nonstandard Euler’s scheme with c1 = 1, c2 = 0, c3 = 1, c4 = 0, c5 = 1, c6 = c7 = c8 = 0. (2.1.8) In [A1], we have obtained the following results. Theorem 2.1. The NSFD scheme (2.1.4)-(i) preserves Properties (P1 ) − (P3 ) of the model (2.1.2) if δ δ +e c1 ≤ − , 2c2 > , c5 ≤ 0, c2 ≥ c6 ≥ 0. (2.1.9) λ +e λ +e Theorem 2.2. The scheme (2.1.4)-(ii) preserves Properties (P1 ) − (P5 ) of the model (2.1.2) if β |y∗ |     −δ −β δ +e β c1 ≤ min , , c2 > max , , . (2.1.10) λ +e λ +e 2(λ + e) λ +e λ +e Theorem 2.3. Let q be a real number satisfying |λ |2     ∗ q > max max , λ +e+β, δ + e, β |y | , (2.1.11) Ω 2|Re(λ )| where Ω = S ∗ )), e∗ ∈{p∗1 ,p∗2 } σ (J(e and ϕ(h) be a function satisfying ϕ(h) < q, ∀h > 0. (2.1.12) Then, the explicit nonstandard Euler scheme (2.1.4)-(iii) preserves Properties (P1 ) − (P5 ) of the model (2.1.2). 2.2. A novel approach for studying asymptotic stability of NSFD schemes for two metapopulation models In this section, we consider a metapopulation model proposed in Amarasekare and Possingham (2001). We establish the complete GAS and construct semi-implicit and explicit NSFD schemes preserving essential properties for the model. It should be emphasized that the asymptotic stability of the constructed NSFD schemes is investigated by a novel approach based on extensions of the Lyapunov stability theorem. 7
10. 2.2.1. Complete global stability of the Amarasekare and Possingham’s metapopulation model Consider the metapopulation model proposed in Amarasekare and Possingham (2001): dI = βI SI − eI I + f L − gI, dt dS = eI I − βI SI + f R − gS, dt (2.2.1) dL = gI − f L − eL L + βL RI, dt dR = gS − f R + eL L − βL RI. dt Details for the model can be found in Amarasekare and Possingham (2001). From the biological meaning of the model, we only consider initial conditions I(0), S(0), L(0), R(0) belonging to the following set n o Ω := (I, S, L, R) ∈ R4+ : I + S + L + R = 1 . (2.2.2) It is easy to prove that Ω is a positively invariant set of the model (2.2.1). The model (2.2.1) has always a boundary equilibrium point E0∗ = (I0∗ , S0∗ , L0∗ , R∗0 ) given by f g I0∗ = 0, S0∗ = , L0∗ = 0, R∗0 = . (2.2.3) f +g g+ f We now define f a =βI βL , b = βI ( f + eL ) + βL (eI + g) − βI βL , f +g (2.2.4)  f   f  c =( f + eL ) eI − βI + g eL − β L . f +g f +g Then, if c < 0 then (2.2.1) has a unique positive equilibrium point E1∗ = (I1∗ , S1∗ , L1∗ , R∗1 ) given by f √ √ b + 2βI βL − b2 − 4ac 2 −b + b − 4ac f +g I1∗ := I ∗ = , S1∗ := S∗ = , 2a 2a g βI  β I g + eI  ∗ (2.2.5) R∗1 := R∗ = − I∗2 + − I , f +g f f +g f g βI  β I g + eI  ∗ L1∗ := L∗ = 1 − I ∗ − S∗ − R∗ = − R∗ = I ∗ 2 − − I . f +g f f +g f where I1∗ is the unique positive root of the equation aX 2 + bX + c = 0. We now establish the complete GAS for (2.2.1). The well-known results constructed in Thieme (1992) follow that the dynamics (2.2.1) is qualitatively equivalent to the dynamics of the limit system given by dI  f  = βI − I I − eI I + f L − gI, dt f +g (2.2.6) dL  g  = gI − f L − eL L + βL − L I, dt f +g where a feasible region of (2.2.6) can be described as   f g Σ := (I, L) ∈ R2 : 0 ≤ I ≤ , 0≤L≤ . (2.2.7) f +g f +g in [A3], we have proved the following result on the complete GAS of (2.2.1). Theorem 2.4. If c ≥ 0, then the equilibrium point E0∗ of the model (2.2.1) is GAS w.r.t. Ω. Theorem 2.5. If c < 0, then the equilibrium point E1∗ of the model (2.2.1) is GAS w.r.t. Ω − E0∗ . 8
11. 2.2.2. Semi-implicit NSFD schemes for metapopulation model (2.2.1) We propose semi-implicit NSFD schemes for (2.2.1) in the form Sk+1 − Sk = eI Ik − βI Sk+1 Ik + f Rk − gSk , ϕ Ik+1 − Ik = βI Sk+1 Ik − eI Ik + f LK − gIk , ϕ (2.2.8) Rk+1 − Rk = gSk − f Rk + eL Lk − βL Rk+1 Ik , ϕ Lk+1 − Lk = gIk − f Lk − eL Lk + βL Rk+1 Ik . ϕ Our task now is to determine the conditions for the function ϕ(h) so that the scheme (2.2.8) preserves the following essential properties of the model (2.2.1): (P1 ) The monotone convergence of the sums I(t) + S(t) and L(t) + R(t): For any initial condition belonging to the set Ω given by (2.2.2), the sums a(t) := I(t) + S(t) and b(t) := R(t) + L(t) monotonically converges to a∗ := f /( f + g) and b∗ := g/( f + g), respectively. (P2 ) Boundedness: The set Ω defined by (2.2.2) is a positively invariant set of (2.2.1). (P3 ) The LAS: E0∗ is LAS if c > 0 and E1∗ is GAS if c < 0. In [A2], by proposing a novel approach based on extensions of the Lyapunov stability theorem (see Iggidr and Bensoubaya 1998), we obtain the following theorem. Theorem 2.6. (i) In the case c > 0, the NSFD scheme (2.2.8) preserves Properties (P1 ) − (P3 ) of the model (2.2.1) if   ∗ 1 1 1 τ 2 ϕ(h) < ϕ := min , , , , , ∀h > 0, (2.2.9) eI + g f + eL f +g c τ where f τ := f + eL + eI + g − βI . (2.2.10) f +g (ii) In the case c < 0, consider the following polynomials λ1 (ϕ) := ϕ 3 [(βI βL )2 I ∗ 4 − α4 ] + ϕ 2 [2(βI + βL )βI βL I ∗ 3 − α3 ] + ϕ[(βI + βL )2 I ∗ 2 + 2βI βL I ∗ 2 − α2 ] + [2(βI + βL )I ∗ − α1 ], λ2 (ϕ) := [βI2 βL2 I ∗ 4 − γ4 + α4 ]ϕ 2 + [2(βI + βL )βI βL I ∗ 3 − γ3 + α3 ]ϕ + [(βI + βL )2 I ∗ 2 + 2βI βL I ∗ 2 − γ2 + α2 ], λ3 (ϕ) := [βI2 βL2 I ∗ 4 + γ4 + α4 ]ϕ 4 + [2(βI + βL )βI βL I ∗ 3 + γ3 + α3 ]ϕ 3 + [(βI + βL )2 I ∗ 2 + 2βI βL I ∗ 2 + γ2 + α2 ]ϕ 2 + [2(βI + βL )I ∗ + γ1 + α1 ]ϕ + 4, and define n o ϕi∗ := sup ϕ > 0 : λi (ϕ) > 0 , i = 1, 2, 3; ϕ0 := min {ϕi∗ }. i=1,2,3 9
12. Then, the NSFD scheme (2.2.8) preserves Properties (P1 ) − (P3 ) of the model (2.2.1) if  ∗ 1 1 1 o ϕ(h) < ϕ := min , , , ϕ0 , ∀h > 0. (2.2.11) eI + g f + eL f +g 2.2.3. Explicit NSFD schemes for metapopulation model (2.2.1) We consider the following nonstandard Euler difference scheme Ik+1 − Ik = βI Sk Ik − eI Ik + f Lk − gIk , ϕ(h) Sk+1 − Sk = eI Ik − βI Sk Ik + f Rk − gSk , ϕ(h) (2.2.12) Lk+1 − Lk = gIk − f Lk − eL Lk + βL Rk Ik , ϕ(h) Rk+1 − Rk = gSk − f Rk + eL Lk − βL Rk Ik , ϕ(h) where ϕ(h) = h + O(h2 ) as h → 0. In [A3], by using the approach proposed in Subsection 2.2.2, we obtain the following theorem. Theorem 2.7. (i) In the case c ≥ 0, the nonstandard Euler scheme (2.2.12) preserves the boundedness, monotone convergence, GAS of E0∗ and instability of E1∗ of the model (2.2.1) if   1 1 1 1 1 ϕ(h) < ϕ ∗ := min , , , , , ∀h > 0. (2.2.13) eI + g βI + g f + eL f + βL f +g (ii) In the case c < 0, the nonstandard Euler scheme (2.2.12) preserves the boundedness, monotone conver- gence, LAS of E1∗ and instability of E0∗ of the model (2.2.1) if   ∗ 1 1 1 1 1 τ1 2 ϕ(h) < ϕ := min , , , , , , , ∀h > 0. (2.2.14) eI + g βI + g f + eL f + βL f + g τ2 τ1 where f τ1 := f + eL + βL I ∗ + 2βI I ∗ + eI + g − βI > 0, f +g (2.2.15) ∗2 ∗ ∗ τ2 := βI βL I + βI ( f + eL )I + f βL L > 0. 2.2.4. A note on the stability analysis of NSFD schemes for metapopulation model (2.2.1) We now reconsider the difference scheme (2.1.4)-(2.1.5) formulated in Section 2.1. By using the approach proposed in Section 2.2.4, we obtain the following result. Theorem 2.8. The NSFD scheme (2.1.4)-(2.1.5) preserves Properties (P1 ) − (P5 ) of the model (2.1.1) if δ c2 ≥ max c6 , c∗ ,  c5 ≤ 0, c6 ≥ 0, c1 ≤ − . (2.2.16) λ +e In Section 2.1, we only proved the positivity of (2.1.4)-(2.1.5). The standard method failed to prove the remaining properties of the scheme. Therefore, Theorem 2.8 is an important improvement to the results constructed in Section 2.1. This confirms the advantages of the new approach. 10
13. 2.3. Numerical dynamics of NSFD schemes for a computer virus propagation model The aim of this section is to construct and analyze NSFD schemes preserving essential qualitative properties of a computer virus propagation model proposed in Yang et al. (2013). It is worth noting that the GAS of the constructed NSFD schemes is established by the Lyapunov stability theorem. 2.3.1. Mathematical model and its dynamics We now consider a computer virus model with graded cure rates proposed in Yang et al. (2013): S˙ = δ − β S(L + B) + γ1 L + γ2 B − δ S, L˙ = β S(L + B) − γ1 L − αL − δ L, (2.3.1) B˙ = αL − γ2 B − δ B, where S(t), L(t) and B(t) denote, at time t, the percentages of uninfected, latent and seizing computers in all internal computers, respectively. We refer to Yang et al. (2013) for more details of this model. Also, for simplicity, we refer to (2.3.1) as the full model. Since S(t) + L(t) + B(t) ≡ 1, it is sufficient to consider the following simplified two-dimensional system L˙ = β (1 − L − B)(L + B) − γ1 L − αL − δ L, (2.3.2) B˙ = αL − γ2 B − δ B. n o It is easy to verify that the set Ω = (L, B) : L ≥ 0, B ≥ 0, L + B ≤ 1 is a positively invariant set of (2.3.2). Mathematical analyses show that (2.3.2) has two equilibrium points E0 and E∗ defined by  1   1 ! (γ2 + δ ) 1 − α 1− R0 R0 E0 = (0, 0), E∗ = (L∗ , B∗ ) = , , (2.3.3) α + δ + γ2 α + δ + γ2 where β (α + γ2 + δ ) R0 = . (2.3.4) (α + γ1 + δ )(γ2 + δ ) Furthermore, E0 is GAS w.r.t. Ω if R0 ≤ 1, meanwhile E∗ is GAS w.r.t. Ω0 = Ω − E0 if 1 < R0 ≤ 4. 2.3.2. Nonstandard finite difference schemes for model (2.3.1) We now consider the nonstandard Euler difference scheme of the form Sk+1 − Sk = δ − β Sk (Lk + Bk ) + γ1 Lk + γ2 Bk − δ Sk , ϕ(h) Lk+1 − Lk = β Sk (Lk + Bk ) − γ1 Lk − αLk − δ Lk , (2.3.5) ϕ(h) Bk+1 − Bk = αLk − γ2 Bk − δ Bk , ϕ(h) where ϕ(h) = h + O(h2 ) as h → 0. The conditions on ϕ(h) will be determined such that the properties of (2.3.1) are preserved. In [A4], by using the Lyapunov stability theorem, we obtain the following results. 11
14. Theorem 2.9. (i) In the case R0 ≤ 1, the nonstandard Euler scheme (2.3.5) preserves the positivity, bound- edness and GAS of E0 and instability of E∗ of the full model (2.3.1) if   ∗ 1 1 1 ϕ(h) < ϕ := min , , , ∀h > 0, (2.3.6) β + δ γ1 + α + δ δ + γ2 (ii) In the case R0 > 1, the nonstandard Euler scheme (2.3.5) preserves the positivity, boundedness and LAS of E∗ and instability of E0 of the full model (2.3.1) if ( ) ∗ 1 1 1 2 τ1 ϕ(h) < ϕ := min , , , , , ∀h > 0, (2.3.7) β + δ γ1 + α + δ δ + γ2 τ1 τ2 where τ1 and τ2 are defined by τ1 =: 2β (L∗ + B∗ ) + α + 2δ + γ1 + γ2 − β , h i h i (2.3.8) τ2 =: 2β (L∗ + B∗ ) + α + γ1 + δ − β (γ + δ ) + 2β (L∗ + B∗ ) − β α. 2.4. NSFD schemes for a general predator-prey model In this section, we design NSFD schemes preserving essential qualitative properties including positivity and stability of a general predator-prey model. It is important to note that the GAS of the constructed NSFD schemes is established based on the Lyapunov stability theorem. We also show that the SFD schemes such as the Euler scheme, the RK2 scheme and the RK4 scheme cannot preserve the properties of the continuous model for large step sizes. 2.4.1. Continuous model and its properties We consider a mathematical model for a predator-prey system with general functional response and recruitment for both species formulated in Ladino et al. (2015)   ˙ = x(t) f (x(t), y(t)) = x(t) r(x(t)) − y(t)φ (x(t)) − m1 , x(t)   (2.4.1) ˙ = y(t)g(x(t), y(t)) = y(t) s(y(t)) + cx(t)φ (x(t)) − m2 , y(t) where x(t) and y(t) are prey population and predator population, respectively. Details for the model can be
15. found in Ladino et al. (2015). It is easy to verify that the set Ω = (x, y) ∈ R2
16. x ≥ 0, y ≥ 0 is a positively  invariant set of (2.4.1). Theorem 2.10. (Existence of equilibrium points (Ladino et al. 2015)) System (2.4.1) has four distinct kinds of possible equilibrium points in Ω: (i) A trivial equilibrium point P0∗ = (x0∗ , y∗0 ) = (0, 0), for all the values of the parameter. (ii) An equilibrium point of the form P1∗ = (x1∗ , y∗1 ) = (K, 0), with r(K) = m1 , if and only if m1 < r(0). (iii) An equilibrium point of the form P2∗ = (x2∗ , y∗2 ) = (0, M), with s(M) = m2 , if and only if m2 < s(0). (iv) An equilibrium point of the form P3∗ = (x3∗ , y∗3 ) = (x∗ , y∗ ), where x∗ satisfies the equation  r(x∗ ) − m  ∗ ∗ 1 cx φ (x ) + s − m2 = 0, φ (x∗ ) 12
17. and y∗ is given, as a function of x∗ , by r(x∗ ) − m1 y∗ = , φ (x∗ ) if and only if (m1 , m2 ) verifies m1 < r(0) − Mφ (0) and m2 < s(0) or m1 < r(0) and s(0) < m2 < s(0) + cKφ (K). The asymptotic stability of the model was completely established in Ladino et al. (2015). 2.4.2. Construction of NSFD scheme We propose a general NSFD scheme for the model (2.4.1) in the form xk+1 − xk = α1 xk r(xk ) + α2 xk+1 r(xk ) − α3 xk yk φ (xk ) − α4 xk+1 yk φ (xk ) − α5 m1 xk − α6 m1 xk+1 , ϕ(h) yk+1 − yk = β1 yk s(yk ) + β2 yk+1 s(yk ) + cβ3 xk yk φ (xk ) + cβ4 xk yk+1 φ (xk ) − β5 m2 yk − β6 m2 yk+1 , (2.4.2) ϕ(h) α j + α j+1 = β j + β j+1 = 1, j = 1, 3, 5; ϕ(h) = h + O(h2 ), h → 0.
18. Proposition 2.1. The region Ω = (x, y) ∈ R2
19. x ≥ 0, y ≥ 0 is a positively invariant set for the scheme (2.4.2)  if α1 ≥ 0, α2 ≤ 0, α3 ≤ 0, α4 ≥ 0, α5 ≤ 0, α6 ≥ 0, (2.4.3) β1 ≥ 0, β2 ≤ 0, β3 ≥ 0, β4 ≤ 0, β5 ≤ 0, β6 ≥ 0. Proposition 2.2. The scheme (2.4.2) preserves the set of equilibrium points of the system (2.4.1). 2.4.3. Stability analysis Through the rest of this subsection, we always assume that the condition (2.4.3) is satisfied. In [A5], by the Laypunov stability theory, we obtain the following results. Theorem 2.11. (LAS property of NSFD schemes) (i) The extinction equilibrium point P0∗ = (x0∗ , y∗0 ) = (0, 0) of the NSFD scheme (2.4.2) is LAS if m1 > r(0) and m2 > s(0), and unstable if m1 < r(0) or m2 < s(0). (ii) Consider the NSFD scheme (2.4.2) in the case m1 < r(0) and m2 > s(0) + cKφ (K) and assume that T1 := 2α6 m1 − 2α2 r(K) + Kr0 (K) > 0, (2.4.4) T2 := s(0) − m2 + cKφ (K) − 2β2 s(0) − 2β4 cKφ (K) + 2β6 m2 > 0. Then, the equilibrium point of the form P1∗ = (K, 0) is LAS. Moreover, P1∗ is and unstable if m1 ≥ r(0) or m2 < s(0) + cKφ (K). (iii) Consider the NSFD scheme (2.4.2) in the case m1 > r(0) − Mφ (0) and m2 < s(0) and assume that T3 := r(0) − Mφ (0) − m1 − 2α2 r(0) + 2α4 Mφ (0) + 2α6 m1 > 0, (2.4.5) T4 := Ms0 (M) − 2β2 s(M) + 2β6 m2 > 0. Then, the equilibrium point of the form P2∗ = (0, M) is LAS. Moreover, P2∗ is unstable if m1 < r(0) − Mφ (0) or m2 ≥ s(0). 13
20. (iv) Suppose that the equilibrium point of the form P3∗ = (x∗ , y∗ ) belongs to Ω. Consider the NSFD scheme (2.4.2) under the assumption T5 := − x∗ [r0 (x∗ ) − y∗ φ 0 (x∗ )][−β2 s(y∗ ) − β4 cx∗ φ (x∗ ) + β6 m2 ] − y∗ s0 (y∗ )[−α2 r(x∗ ) + α4 y∗ φ (x∗ ) + α6 m1 ] − x∗ y∗ s0 (y∗ )[r0 (x∗ ) − y∗ φ 0 (x∗ )] − cx∗ y∗ φ (x∗ )[φ (x∗ ) + x∗ φ 0 (x∗ )] > 0, (2.4.6) T6 := − α2 r(x∗ ) + α4 y∗ φ (x∗ ) + α6 m1 + x∗ [r0 (x∗ ) − y∗ φ 0 (x∗ )] > 0, T7 := − β2 s(y∗ ) − β4 cx∗ φ (x∗ ) + β6 m2 + y∗ s0 (y∗ ) > 0. Then, P3∗ is LAS. Theorem 2.12. Consider the NSFD scheme (2.4.2) in the case m1 ≥ r(0) and m2 ≥ s(0) and assume that α4 + β4 < 0. (2.4.7) Then, the extinction equilibrium point P0∗ = (0, 0) is GAS. 2.4.4. Dynamically consistent NSFD schemes Theorem 2.13. The NSFD scheme (2.4.2) is dynamically consistent with the model (2.4.1) if the parameters α j , β j ( j = 1, . . . , 6) satisfy the conditions listed in Table 2.2, where the columns list sufficient conditions for the scheme (2.4.2) preserve corresponding properties of the model (2.4.1) for different cases of the parameters. The symbol 00 ∗00 means that the set of equilibrium points of (2.4.1) is always preserved by the scheme (2.4.2). Table 2.2. The sufficient conditions for dynamic consistency (m1 , m2 ) Set of equilibria Positivity Stability m1 ≥ r(0) and m2 ≥ s(0) * (2.4.3) (2.4.7) m1 < r(0) and m2 > s(0) + cKφ (K) * (2.4.3) (2.4.4) m1 > r(0) − Mφ (0) and m2 < s(0) * (2.4.3) (2.4.5) m1 < r(0) − Mφ (0) and m2 < s(0) * (2.4.3) (2.4.6) m1 < r(0) and s(0) < m2 < s(0) + cKφ (K) * (2.4.3) (2.4.6) 2.5. A novel approach for studying global stability of NSFD schemes for a mixing prop- agation model of computer viruses In this section, NSFD schemes preserving essential qualitative properties including positivity and stability of a mixing propagation model of computer viruses are proposed and analyzed. We propose a new approach to prove theoretically that the GAS of the original model is preserved by the proposed NSFD schemes. This approach is based on the Lyapunov stability theorem and its extension in combination with a theorem on the GAS of discrete-time nonlinear cascade systems. The important result is that we obtain NSFD schemes 14