In this paper, we presented a novel multi-strain TB model of variable-order fractional derivatives, which incorporates three strains: drug-sensitive, emerging multi-drug resistant (MDR) and extensively drug-resistant (XDR), as an extension for multi-strain TB model of nonlinear ordinary differential equations which developed in 2014 by Arino and Soliman [1]. Numerical simulations for this variable-order fractional model are the main aim of this work, where the variable-order fractional derivative is defined in the sense of Gru¨nwald–Letnikov definition. Two numerical methods are presented for this model, the standard finite difference method (SFDM) and nonstandard finite difference method (NSFDM). Numerical comparison between SFDM and NSFDM is presented. It is concluded that, NSFDM preserves the positivity of the solutions and numerically stable in large regions than SFDM.
Journal of Advanced Research (2016) 7, 271–283 Cairo University Journal of Advanced Research ORIGINAL ARTICLE Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives Nasser H Sweilam a b a,* , Seham M AL-Mekhlafi b Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt Department of Mathematics, Faculty of Education, Sana’a University, Sana’a, Yemen A R T I C L E I N F O Article history: Received February 2015 Received in revised form 30 May 2015 Accepted 15 June 2015 Available online 27 June 2015 Keywords: Nonstandard finite difference Epidemic model Tuberculosis M/XDR-TB Variable-order fractional GruănwaldLetnikov denition A B S T R A C T In this paper, we presented a novel multi-strain TB model of variable-order fractional derivatives, which incorporates three strains: drug-sensitive, emerging multi-drug resistant (MDR) and extensively drug-resistant (XDR), as an extension for multi-strain TB model of nonlinear ordinary differential equations which developed in 2014 by Arino and Soliman [1] Numerical simulations for this variable-order fractional model are the main aim of this work, where the variable-order fractional derivative is defined in the sense of GruănwaldLetnikov denition Two numerical methods are presented for this model, the standard finite difference method (SFDM) and nonstandard finite difference method (NSFDM) Numerical comparison between SFDM and NSFDM is presented It is concluded that, NSFDM preserves the positivity of the solutions and numerically stable in large regions than SFDM ª 2015 Production and hosting by Elsevier B.V on behalf of Cairo University Introduction Variable-order fractional calculus (i.e., the fractional differentiation and integration of variable order) is the generalization of classical calculus and fractional calculus, which were invented by Newton and Leibnitz hundreds of years ago Now the study on it becomes a hot pot in recent ten years [2–7] It has turned out that many problems in physics, * Corresponding author Tel.: +20 1003543201; fax: +20 3572 884 E-mail address: nsweilam@sci.cu.edu.eg (N.H Sweilam) Peer review under responsibility of Cairo University Production and hosting by Elsevier biology, engineering, and finance can be described excellently by models using mathematical tools from variable-order fractional calculus, such as mechanical applications [2], diffusion process [5], multifractional Gaussian noise [8], and FIR filters [9] For more details, see [7,10] and references therein Understanding the transmission characteristics of infectious diseases in communities, regions and countries can lead to better approaches to decrease the transmission of these diseases [11] Variable-order fractional derivative is good at depicting the memory property which changes with time or spatial location [3,5] TB is growing more resistant to treatment worldwide according to study released in August 2012 in the journal Lancet, a finding that suggests the potentially fatal disease is becoming more difficult and costly to treat [12] In this article we focused our attention in Egypt http://dx.doi.org/10.1016/j.jare.2015.06.004 2090-1232 ª 2015 Production and hosting by Elsevier B.V on behalf of Cairo University 272 N.H Sweilam and S.M AL-Mekhlafi We consider in this work a model developed by Arino and Soliman for TB [1] The model incorporates three strains, drug-sensitive, MDR and XDR Several papers considered modeling TB such as [13,14], but the model we consider here includes several factors of spreading TB such as the fast infection, the exogenous reinfection and secondary infection along with the resistance factor The main aim of this paper was to study numerically the multi-strain TB model of variable-order fractional derivatives which incorporates three strains: drugsensitive, MDR and XDR We develop a special class of numerical method, known as NSFDM for solving this model This technique, developed by Mickens (1980) [15–23] has brought a creation of new numerical schemes preserving the physical properties, especially the stability properties of equilibria, of the approximated system Numerical comparison between NSFDM and SFDM is presented When the secondary infection generated by an infected individual exceeds the unity, there are no analytical results proved for the model, such as the existence and stability of the endemic equilibrium ðEEÞ In this case we use the developed NSFD numerical scheme to approximate the endemic solution numerically and investigate its stability Furthermore, with the help of the NSFDM, we answer the following question: Given the data provided by the World Health Organization (2012) on the current parameters corresponding to the propagation of the TB in Egypt, what would be the required rate of treatment to achieve in order to control the disease? The proposed method showed its superiority in preserving the positivity (compared to the numerical standard method considered in this work) of the state variables of the systems under study This is an essential requirement when simulating systems especially those arising in biology This paper is organized as follows: In Section ‘Mathematical model’, Mathematical model is presented Preliminaries and notations on variable-order fractional differential equations are given, in Section ‘Preliminaries and notations’ Equilibrium points and their asymptotic stability are presented in Section ‘Variable-order fractional derivatives for multi-strain TB model’ Variable-order fractional derivatives for the multi-strain TB model are presented; moreover, the construction of the proposed nonstandard numerical scheme is carried out in Section ‘Equilibrium points and their asymptotic stability’ In Section ‘Numerical results and simulations’, Numerical results and simulation are discussed Finally, in Section ‘Conclusions’ we presented the conclusions Mathematical model The multistrain TB-model given in [1] can be formulated as follows: S0 ¼ b À dS À bs SIs SIm SIx À bm À bx ; N N N SIs RIs L s Is L s Im ¼ ks bs ỵ rs ks bs ỵ cs Is ass bs À asm bm N N N N Ls Ix asx bx d ỵ es ỵ t1s ịLs ; N SIm RIm Ls Im L0m ¼ km bm þ rm km bm þ cm Im þ asm bm km N N N Lm Im ỵ P1 ịt1s Ls ỵ P2 ịt2s Is amm bm N Lm Ix amx bx d ỵ em ÞLm ; N SIx RIx L s Ix Lm Ix ỵ rx kx bx ỵ cx Ix ỵ asx bx kx ỵ amx bx kx N N N N L x Ix d ỵ ex ịLx ; 4ị þ ð1 À P3 Þt2m Im À axx bx N L0x ẳ kx bx I0s ẳ ass bs 5ị I0m ¼ amm bm L m Im SIm RIm Ls Im þ ð1 À km Þbm þ rm þ asm N N N N 6ị L x Ix I0x ẳ axx bx ỵ kx ịbx N SIx RIx Ls Ix Lm Ix ỵ ex L x ỵ rx ỵ asx ỵ amx N N N N d ỵ dx ỵ t2x ỵ cx ịIx ; R0 ẳ P1 t1s Ls ỵ P2 t2s Is þ P3 t2m Im þ t2x Ix À rs bs À rm bm RIm RIx À rx bx À dR: N N ð7Þ RIs N ð8Þ All variables in above system and their definition are in Table Also, all parameters and their interpretation are in Table The basic reproduction number R0 The basic reproduction number R0 for system (1)–(8) is given by [1] R0 ¼ maxðR0s ; R0m ; R0x ị; 9ị where R0s ẳ bs es ỵ ks ịd ỵ t1s ịị ; es ỵ d ỵ t1s ịt2s ỵ ds ỵ dị ỵ cs t1s ỵ dị Table All variables of the system (1)–(8) and their interpretation Variable Definition SðtÞ The susceptible population individuals who have never encountered TB The individuals infected with the drug-sensitive TB strain but who are in a latent stage, i.e., who are neither showing symptoms nor infecting others Individuals latently infected with MDR-TB Individuals latently infected with XDR-TB Individuals infected with the drug-sensitive TB strain who are infectious to others (and most likely, showing symptoms as well) Those individuals who are infectious with the MDR-TB strain Individuals who infectious with the XDR-TB strain Those individuals for whom treatment was successful The total population N ẳ S ỵ Ls þ Lm þ Lx þ Is þ Im þ Ix þ R Ls ðtÞ Lm ðtÞ Lx ðtÞ Is ðtÞ 2ị Im tị Ix tị Rtị Ntị 3ị ỵ em Lm d ỵ dm ỵ t2m ỵ cm ÞIm ; ð1Þ L0s L s Is SIs RIs ỵ ks ịbs ỵ rs ỵ es Ls d ỵ ds ỵ t2s ỵ cs ịIs ; N N N General multi-strain tuberculosis (TB) model Table 273 All parameters of the system (1)–(8) and their interpretation Parameter Interpretation b d Birth/recruitment rate Per capita natural death rate Disease dynamics Transmission coefficient for strain r Proportion of newly infected individuals developing LTBI with strain r Proportion of newly infected individuals progressing to active TB with strain r due to fast infection Per capita rate of endogenous reactivation of Lr Proportion of exogenous reinfection of Lr1 due to contact with Ir2 Per capita rate of natural recovery to the latent stage Lr Per capita rate of death due to TB of strain r Treatment related Per capita rate of treatment for Ls Per capita rate of treatment for Ir Note that t2x is the rate of successful treatment of Ix ; r fx; m; sg Efficiency of treatment in preventing infection with strain r Probability of treatment success for Ls Proportion of treated Ls moved to Lm due to incomplete treatment or lack of strict compliance in the use of drugs Probability of treatment success for Is Proportion of treated Is moved to Lm due to incomplete treatment or lack of strict compliance in the use of drugs Probability of treatment success for Im Proportion of treated Im moved to Lx due to incomplete treatment or lack of strict compliance in the use of drugs br kr À kr er ar1 ; ar2 cr dr t1s t2r À rr P1 À P1 P2 À P2 P3 À P3 R0m ¼ bm em ỵ km ịdị ; em ỵ dịt2m ỵ dm ỵ dị ỵ dcm R0x ẳ bx ex ỵ kx ịdị : ex ỵ dịt2x þ dx þ dÞ þ dcx To apply Miken’s scheme, we have chosen this Gruănwald Letnikov approximation variable-order fractional derivative as follows [15]: ẵht X atị atị atị 16ị yt jhị; Dt ytị ẳ limh 1ị j h!0 j j¼0 Theorem [1] assumes that ass ð1 À ks Þ; ð10Þ amm ð1 À km Þ; ð11Þ axx ð1 À kx Þ: ð12Þ Then the disease free equilibrium is globally asymptotically stable when R0 < and endemic equilibria are locally asymptotically stable when R0 > Preliminaries and notations In this section, some basic definitions and properties in the theory of the variable-order fractional calculus are presented GruănwaldLetnikov approximation We will begin with the signal variable-order fractional differential Dtatị ytị ẳ ft; ytịị; T P t P 0; and; yt0 ị ẳ 0; 13ị where atị > 0, and DtaðtÞ denotes the variable fractional order derivative, defined by Dtatị ytị ẳ Jnatị Dtatị ytị; 14ị where n À < aðtÞ n, n N and Jn is the nth-order Riemann–Liouville integral operator defined as Z Jn ytị ẳ t sịn1 ysịds; with t > 0; ð15Þ CðtÞ t where CðÁÞ is the gamma function where ½t denotes the integer part of t and h is the step size; therefore, Eq (16) is discretized as t ẵh X at ị xj n ytnj Þ ¼ fðtn ; yðtn ÞÞ n ¼ 1; 2; 3; : 17ị jẳ0 at ị where tn ẳ nh, and xj n , are the GruănwaldLetnikov coefcients dened as ỵ atn ị atn Þ aðt Þ aðt Þ xj n ¼ À xj1 and x0 n ẳ hatn ị ; j ẳ 1; 2;3; : j Variable-order fractional derivatives for multi-strain TB model In the following, we introduce the multi-strain TB model of variable-order fractional derivatives which is the integer order given in system (1)–(8), and the new system is described by variable-order fractional differential equations as follows: Dtatị S ẳ b À dS À bs SIs SIm SIx À bm À bx ; N N N SIs RIs L s Is ỵ rs ks bs ỵ cs Is ass bs À asm bm N N N Ls Im Ls Ix asx bx d ỵ es ỵ t1s ịLs ; N N 18ị Dtatị Ls ẳ ks bs SIm RIm ỵ rm km bm ỵ cm Im þ asm bm km N N L s Im  þ ð1 À P1 Þt1s Ls þ ð1 À P2 Þt2s Is N Lm Im Lm Ix À amm bm amx bx d ỵ em ịLm ; N N 19ị Dtatị Lm ẳ km bm 20ị 274 N.H Sweilam and S.M AL-Mekhla SIx RIx L s Ix ỵ rx kx bx ỵ cx Ix ỵ asx bx kx N N N Lm Ix L x Ix ỵ amx bx kx ỵ P3 ịt2m Im axx bx N N d ỵ ex ịLx ; To evaluate the asymptotic stability let Dtatị Lx ẳ kx bx Dtatị Is ẳ ass bs 21ị Stị ẳ Seq ỵ e1 tị; Ls tị ẳ Leq s tị ỵ e2 tị; Lm tị ẳ Leq m tị ỵ e3 tị; 22ị Lx tị ẳ Leq x tị ỵ e4 tị; Is tị ẳ Ieq tị ỵ e5 tị; s eq Im tị ẳ Im tị ỵ e6 tị; Ix tị ẳ Ieq x tị ỵ e7 tị; Ls Is SIs RIs ỵ es L s ỵ ks ịbs ỵ rs N N N d ỵ ds ỵ t2s ỵ cs ịIs ; Rtị ẳ Req þ e8 ðtÞ: eq eq eq eq eq eq So the equilibrium point ðSeq ; Leq s ; Lm ; Lx ; Is ; Im ; Ix ; R Þ is locally asymptotically stable if all eigenvalues of Jacobian evaluated at the equilibrium point satisfy [16] Lm Im ỵ N SIm RIm Ls Im ỵ em Lm þ rm þ asm À km Þbm N N N Dtatị Im ẳ amm bm d ỵ dm ỵ t2m ỵ cm ịIm ; Datị t Ix ẳ axx bx ð23Þ Lx Ix SIx RIx Ls Ix Lm Ix ỵ kx ịbx ỵ rx ỵ asx þ amx N N N N N þ ex Lx d ỵ dx ỵ t2x ỵ cx ịIx ; Dtatị R 24ị RIs ẳ P1 t1s Ls ỵ P2 t2s Is ỵ P3 t2m Im ỵ t2x Ix À rs bs N RIm RIx À rm bm À rx bx À dR; N N ð25Þ Let aðtÞ 0; and consider the system (18)(25) Dtatị Stị ẳ f1 ðS; Ls ; Lm ; Lx ; Is ; Im ; Ix ; Rị; Dtatị Ls tị ẳ f2 ðS; Ls ; Lm ; Lx ; Is ; Im ; Ix ; Rị; Dtatị Lm tị ẳ f3 S; Ls ; Lm ; Lx ; Is ; Im ; Ix ; Rị; Dtatị Lx tị ẳ f4 S; Ls ; Lm ; Lx ; Is ; Im ; Ix ; Rị; ẳ f5 S; Ls ; Lm ; Lx ; Is ; Im ; Ix ; Rị; ẳ f6 ðS; Ls ; Lm ; Lx ; Is ; Im ; Ix ; RÞ; To evaluate the equilibrium points, let atị atị atị atị atị Datị t S ẳ Dt Ls ¼ Dt Lm ¼ Dt Lx ¼ Dt Is ¼ Dt Im aðtÞ ¼ DaðtÞ t Ix ¼ D t R ¼ i Now, if Is ¼ Im ¼ Ix ¼ ) Ls ¼ Lm ¼ Lx ¼ 0, R ¼ and S ¼ bd Then the diseaseÁÉ free equilibrium (DFE) is ÈÀ E0 ¼ bd ; 0; 0; 0; 0; 0; 0; We calculate the Jacobian matrix of the system (18)–(25) at the disease free equilibrium point as follows: a 0 b c d B0 e 0 f 0 0C B C B C B0 g h p q 0C B C B0 0 r s t 0C B C JðE0 Þ ¼ B C; B0 u 0 v 0 0C B C B0 w 0 x 0C B C B C @0 0 y 0 z 0A m 0 n j k a where a ¼ Àd; b ¼ Àbs ; c ¼ bm ; d ẳ bx ; e ẳ d ỵ es ỵ t1s ị; f ẳ cs ỵ ks bs , v ẳ d ỵ ds ỵ t2s ỵ cs Þ; g ¼ ð1 À P1 Þt1s ; h ¼ d ỵ em ị; p ẳ P2 ịt2s ; q ẳ cm ỵ km bm ; r ẳ d ỵ ex ị, s ẳ P3 ịt2m ; t ẳ cx ỵ kx bx ; u ẳ es ; x ẳ d ỵ dm ỵ t2m ỵ cm ị; w ẳ em Dtatị Rtị ẳ f8 ðS; Ls ; Lm ; Lx ; Is ; Im ; Ix ; RÞ: With the initial values ðSð0Þ; Ls ð0Þ; Lm ð0Þ; Lx ð0Þ; Is ð0Þ; Im ð0Þ; Ix ð0Þ; Rð0ÞÞ To evaluate the equilibrium points let aðtÞ aðtÞ atị atị Dtatị S ẳ Datị t Ls ẳ Dt Lm ¼ Dt Lx ¼ Dt Is ¼ Dt Im atị ẳ Datị t Ix ẳ D t R ẳ The characteristic equation associated with above matrix is jJðE0 Þ À kIj ¼ ) ða À kÞ2 ðk2 r ỵ zịk yt ỵ zrị k2 ỵ h ỵ xịk xh ỵ wqị k2 ỵ e ỵ vịk ỵ uf veị ẳ Then the eigenvalues of Jacobian matrix are k1;2 ¼ Àd, k3;4 ¼ rỵzặ i ẳ 1; 2; 3; ; 8: equilibrium y ẳ ex ; z ẳ d ỵ dx ỵ t2x ỵ cx ị; m ẳ P1 t1s ; n ¼ P2 t2s ; j ¼ P3 t2m ; k ¼ t2x : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eq eq eq eq eq eq ) fi ðSeq ; Leq s ; Lm ; Lx ; Is ; Im ; Ix ; R Þ ¼ 0; the ð26Þ DtaðtÞ Ix ðtÞ ¼ f7 ðS; Ls ; Lm ; Lx ; Is ; Im ; Ix ; RÞ; From which we can get eq eq eq eq eq eq ðSeq ; Leq s ; Lm ; Lx ; Is ; Im ; Ix ; R ị i ẳ 1; 2; ; 8: ¼ 1; 2; 3; ; 8: Equilibrium points and their asymptotic stability DtaðtÞ Im ðtÞ aðtÞp ; aðtÞ ð0; 1; t P where eq eq eq eq eq eq ) fi ðSeq ; Leq s ; Lm ; Lx ; Is ; Im ; Ix ; R ị ẳ 0; where Dtatị is the Caputo variable fractional order derivative Because model (18)–(25) monitors the dynamics of human populations, all the parameters are assumed to be nonnegative DtaðtÞ Is ðtÞ jarg ki j > points r2 ỵ2zrỵz2 ỵ4ytị ; p xỵhặ x2 2xhỵh ỵ4wqị ;k7;8 p vỵeặ v2 ỵ2veỵe2 ỵ4ufị , by using Theorem (Routh Hurwitz criteria) [17], these roots are negative or have negative real parts and DFE is locally asymptotically stable if all eigenvalues of the Jacobian matrix satisfies k5;6 ¼ jarg ki j ¼ j À pj > atịp , ẳ atị 0; 1, t P For General multi-strain tuberculosis (TB) model 275 Table All parameters in the system (18)–(25) and the reference of the parameters Parameter Value Reference b d bs ¼ bm ¼ bx ks ¼ km ¼ kx es ¼ em ¼ ex ar1;r2 cs ¼ cm ¼ cx t1s t2r : r ðs; m; xÞ rr Pr dr 3190 0.38 14 0.5 0.5 0.05 0.3 0.88 t2s ¼ 0:88; t2m ¼ t2x ¼ 0:034 0.25 0.88 0.045 Assumed [26] [26] Assumed Assumed Assumed Assumed [26] [26] [26] [26] [26] simplicity, we will determine the stability of the DFE numerically by using Table and put bs ¼ bm ¼ bx ¼ 0:1 Then eigenvalues are k1 ¼ À0:3800, k2 ¼ À0:3800, k3 ¼ À0:3675, k4 ¼ À0:3675, k5 ¼ À1:2215, k6 ¼ À1:2215, k7 ¼ À2:0882, k8 ¼ À1:2268 So, if R0 < 1, the DFE is locally asymptotically , aðtÞ ð0; 1, t P stable since jarg ki j ¼ j À pj > aðtÞp If at least one of the infected variables is non-zero, then the solutions for model (18)–(25) are the endemic equilibrium [1] This system is highly nonlinear in Is , Im and Ix , and hence explicit solutions are not obtainable So we solved the system (18)–(25) numerically to obtain endemic fixed point using NSFDM following two main steps: 1- the derivative at the left-hand side of (27) is replaced by a discrete representation in the form dy ynỵ1 yn ẳ ; uk; hị dt where ynỵ1 is an approximation of yðtn Þ, 2-the nonlinear term in (27) is replaced by a nonlocal discrete representation Ft; ynỵ1 ; yn ; ; kÞ depending on some of the previous approximations For example, if there are nonlinear terms such as yðtÞxðtÞ in NðtÞ the differential equation, these are replaced by ytỵhịxtị Ntị Let us denote by Sn , Lns , Lnm , Lnx , Ins , Inm , Inx and Rn the values of the approximations of SðnhÞ, Ls ðnhÞ, Lm ðnhÞ, Lx ðnhÞ, Is ðnhÞ, Im nhị, Ix nhị and Rnhị respectively, for n ẳ 0; 1; 2; and h is the timestep of the scheme The sequences Sn , Lns , Lnm , Lnx , Ins , Inm , Inx and Rn should be nonnegative in order to be consistent with the biological nature of the model [21] NSFDM has many advantages than SFDM, for more details see [20–24] Generally speaking, we can say that NSFDM is more efficient and accurate than SFDM [15,25] NSFD for variable-order fractional derivatives system The system (18)(25) can be discretized as follows: nỵ1 X Snỵ1 Ins Snỵ1 Inm Snỵ1 Inx at ị xj n Snỵ1j ẳ b dSnỵ1 bs bm bx ; n n N N Nn jẳ0 28ị SFD discretization SFD methods are simple numerical methods for approximating the solutions of differential equations using finite differences to approximate the derivatives The forward Euler method is one of these methods, in this is replaced by ytỵhịytị , where h method the derivative term dy dt h is the step size, for more details see [18] NSFD discretization The nonstandard finite difference schemes were introduced by Mickens in the 1980s as a powerful numerical method that preserves significant properties of exact solutions of the involved differential equation [19] The concept of the nonstandard nite difference method is discussed in [20] nỵ1 X atn ị xj Lnỵ1j ẳ ks bs s jẳ0 Nonlocal approximation is used The discretization of derivative is not traditional and uses a nonnegative function [19,20] ð27Þ where k is a possibly vector, parameter Given a mesh-grid tn ¼ t0 ỵ hn that just for simplicity we assume to be equispaced with step-size h > 0, NSFD schemes are constructed by the n n Lnỵ1 Lnỵ1 n s Ix s Im ; n d ỵ es ỵ t1s ịLs asm bm n N N 29ị nỵ1 X Snỵ1 Inm Rnỵ1 Inm Lnỵ1 In at ị xj n Lnỵ1j ẳ km bm ỵ km asm bm s n m n ỵ rm km bm n m N N N jẳ0 ỵ cm Inm ỵ t1s Lnỵ1 P1 t1s Lnỵ1 ỵ t2s Ins P2 t2s Ins s s Lnỵ1 In Lnỵ1 In amm bm m n m amx bx m n x d ỵ em ịLnỵ1 m ; N N 30ị nỵ1 X Snỵ1 Inx Rnỵ1 Inx Lnỵ1 In at ị xj n Lnỵ1j ẳ kx bx ỵ rx kx bx ỵ kx asx bx s n x n n x N N N jẳ0 ỵ cx Inx ỵ kx amx bx n Lnỵ1 m Ix ỵ t2m Inm P3 t2m Inm n N n Lnỵ1 x Ix d ỵ ex ịLnỵ1 31ị n x ; N Lnỵ1 In Snỵ1 Ins Rnỵ1 Ins ẳ ass bs s n s ỵ ks ịbs ỵ r s N Nn Nn axx bx nỵ1 X at ị xj n Inỵ1j s j¼0 To describe the main aspects of NSFD schemes, we consider an ODE in the form n Snỵ1 Ins Rnỵ1 Ins Lnỵ1 n s Is n ỵ rs ks bs n ỵ cs Is ass bs n N N N À asx bx Definition A numerical scheme is called NSFD discretization if at least one of the following conditions is satised [18]: dy ẳ ft; y; kị; dt or xtỵhịytị Ntị ỵ es Lnỵ1 d þ ds ÞInþ1 À ðcs þ t2s ÞIns ; s s nỵ1 X Lnỵ1 In at ị xj n Inỵ1j ẳ amm bm m n m ỵ km ịbm m N jẳ0 nỵ1 n n S Im Rnỵ1 Inm Lnỵ1 s Im ỵ r ỵ a m sm Nn Nn Nn nỵ1 nỵ1 ỵ em Lm d ỵ dm ịIm cm ỵ t2m ịInm ; 32ị 33ị 276 nỵ1 X N.H Sweilam and S.M AL-Mekhla atn ị nỵ1j Ix xj ẳ axx bx jẳ0 n Lnỵ1 x Ix ỵ kx ịbm n N at ị nỵ1 n n S Ix Rnỵ1 Inx Lnỵ1 x Im ỵ r ỵ a x mx Nn Nn Nn ỵ ex Lnỵ1 x And x0 n ẳ ui hịịatn ị , i ẳ 1; 2; ; where the nonlocal approximations are used for the nonlinear terms and the following denominator functions are used: where the discretization for NðtÞ is given as edh edỵes ỵt1s ịh ; u2 hị ẳ ; d d ỵ es ỵ t1s ị edỵem ịh edỵex ịh ; u4 hị ẳ ; u3 hị ẳ d ỵ em ị d ỵ ex ị edỵds ịh edỵdm ịh ; u6 hị ẳ ; u5 hị ẳ cs ỵ t2s ị cm ỵ t2m ị edỵdx ịh edh ; u8 hị ẳ : u7 hị ẳ cx ỵ t2x ị d Nn ẳ Sn ỵ Lns ỵ Lnm ỵ Lnx ỵ Ins ỵ Inm ỵ Inx ỵ Rn : We obtain, d þ dx ÞInþ1 x À ðcx þ t2x ÞInx ; 34ị nỵ1 X at ị xj n Rnỵ1j ẳ P1 t1s Lnỵ1 ỵ P2 t2s Ins ỵ P3 t2m Inm þ t2x Inx À dRnþ1 s j¼0 À rs bs Rnỵ1 Ins Rnỵ1 Inm Rnỵ1 Inx rx bx : ð35Þ n À rm bm n N N Nn u1 hị ẳ Fig Proles obtained by using NSFDM for solving variable-order fraction model with different atị; h ẳ 0:5, bs ¼ bm ¼ bx ¼ 0:1, an R0 < General multi-strain tuberculosis (TB) model 277 Fig Profiles obtained by using NSFDM for solving variable-order fraction model with different atị; h ẳ 0:2, bs ẳ bm ẳ bx ẳ 14, and R0 > Snỵ1 ẳ b u1 hịị Pnỵ1 jẳ1 atn ị atn ị xj Snỵ1j n n ỵ d ỵ bs Is ỵbmNInm ỵbx 36ị ; In x Lnỵ1 ẳ x P atn ị nỵ1j bs Ins k Snỵ1 ỵ rs Rnỵ1 ị ỵ cs Ins nỵ1 Ls jẳ1 xj Nn s nỵ1 Ls ẳ ; atn ị u2 hịị ỵ d ỵ t1s ỵ es ị ỵ N1n ass bs Ins ỵ asm bm Inm ỵ asx bx Ix ị bx kx Inx n Snỵ1 ỵ rx Rnỵ1 ỵ asx Lnỵ1 ỵ amx Lnỵ1 s m ị ỵ t2s Im P3 ị Nn u4 hịịatn ị ỵ d ỵ ex ị ỵ N1n axx bx Inx ị ỵ cx Inx Pnỵ1 jẳ1 atn ị xj Lnỵ1j x u4 hịịatn ị ỵ d ỵ ex ị ỵ N1n axx bx Inx ị ; 39ị 37ị Lnỵ1 m ẳ bm km Inm n nỵ1 Snỵ1 ỵ rm Rnỵ1 ỵ asm Lnỵ1 s ị þ cm Im þ t1s Ls ð1 À P1 Þ Nn u3 hịịatn ị ỵ d ỵ em ị ỵ N1n amm bm Inm ỵ amx bx Inx ị ỵ t2s Ins P2 ị Pnỵ1 jẳ1 atn ị xj Lnỵ1j m u3 hịịatn ị ỵ d ỵ em ị ỵ N1n amm bm Inm ỵ amx bx Inx ị n Inỵ1 ẳ s u5 hịbs NIsn ass Lnỵ1 ỵ ks ịSnỵ1 ỵ rs Rnỵ1 ịị s ỵ ; 38ị u5 hịịatn ị ỵ d ỵ ds ị P atn ị nỵ1j cs t2s ịịIns þ es Lnþ1 À nþ1 Is s j¼1 xj ðu5 hịịatn ị ỵ d ỵ ds ị ; 40ị 278 N.H Sweilam and S.M AL-Mekhlafi Fig Profiles obtained by using different methods with atị ẳ 1, h ẳ 0:02, bs ¼ bm ¼ bx ¼ 14, and R0 > Table Result obtained by SFDM and NSFDM for R0 < 1; atị ẳ 0:98 0:01=100t, Bs ẳ Bm ẳ Bx ẳ 0:1, t ẵ0; 100 and initial conditions as ð5000; 50; 50; 50; 30; 30; 30; 60Þ with different time step size Table Result obtained by SFDM and NSFDM for R0 > 1; atị ẳ 0:98 À 0:01=100t, Bs ¼ Bm ¼ Bx ¼ 14, t ½0; 100 and initial conditions as ð5000; 50; 50; 50; 30; 30; 30; 60Þ with different time step size h SFDM NSFDM h SFDM NSFDM 0.01 0.1 20 100 Convergent Convergent Convergent Divergent Divergent Convergent Convergent Convergent Convergent Convergent 0.01 0.1 20 100 Convergent Convergent Divergent Divergent Divergent Convergent Convergent Convergent Convergent Convergent General multi-strain tuberculosis (TB) model 279 Fig Illustrate propagation of multi-strain TB along the time atị ẳ 0:98 0:03=100t, h ẳ 3, bs ¼ bm ¼ bx ¼ 14, and R0 > 1, by using NSFDM Fig Profiles obtained by using NSFDM and SFDM with atị ẳ 0:98 0:01=100t, h ¼ 0:2, bs ¼ bm ¼ bx ¼ 14, and R0 > 280 N.H Sweilam and S.M AL-Mekhlafi Inỵ1 ẳ m bm Inm Nn nỵ1 amm Lnỵ1 ỵ rm Rnỵ1 ỵ asm Lnỵ1 m ỵ km ÞðS s ÞÞ Numerical results and simulations Àaðtn Þ ðu6 hịị ỵ cm t2m ịịInm ỵ u6 hịị em Lnỵ1 m atn ị ỵ d ỵ dm ị P atn ị nỵ1j nỵ1 Im jẳ1 xj ỵ d ỵ dm ị ; 41ị n Inỵ1 ẳ x bx NIxn axx Lnỵ1 ỵ kx ịSnỵ1 ỵ rx Rnỵ1 ỵ asx Lnỵ1 ỵ amx Lnỵ1 x s m ịị ỵ u7 hịịatn ị ỵ d ỵ dx ị P atn ị nỵ1j cx t2x ịịInx ỵ ex Lnỵ1 nỵ1 Ix x jẳ1 xj u7 hịịatn ị ỵ d ỵ dx ị ; 42ị Rnỵ1 ẳ t1s P1 Lnỵ1 ỵ P2 t2s Ins ỵ t2m P3 Inm ỵ t2x Inx s Pnỵ1 jẳ1 atn ị xj Rnỵ1j u8 hịịatn ị ỵ d ỵ N1n rs bs Ins ỵ rm bm Inm ỵ rx bx Inx ị : ð43Þ Fig Since most of the variable-order fractional differential equations not have exact analytic solutions, so approximation and numerical techniques must be used Several analytical and numerical methods have been proposed to solve variable-order fractional differential equations For numerical solutions of the system (18)–(25) one can use NSFDM, the approximate solution SðtÞ, Ls ðtÞ, Lm ðtÞ, Lx ðtÞ, Is ðtÞ, Im ðtÞ, Ix ðtÞ, RðtÞ is displayed in Fig 1, when R0 < and in Fig 2, when R0 > 1, in each figures, and three different values of atị ẳ 1, atị ẳ 0:95 0:01=100t, atị ẳ 0:85 0:01=100t are considered The approximate solutions are displayed in Fig that, the equilibrium point ðS; 0; 0; Lx ; 0; 0; Ix ; RÞ of NSFDM is locally asymptotically stable when atị ẳ 0:95 À 0:01=100t, t ½0; 20, where the eigenvalues are given as k1 ¼ À9:8100, k2 ¼ À0:4098; k3 ¼ À0:3688, k4 ¼ À2:7660, k5 ¼ À2:4591, k6 ¼ À1:2392, k7 ¼ À1:6005, Profiles obtained by using NSFDM and SFDM with atị ẳ 0:98 0:01=100t, h ẳ 1, bs ¼ bm ¼ bx ¼ 14, and R0 > General multi-strain tuberculosis (TB) model k8 ¼ À1:4465 By applying the relationship (26) we obtained , aðtÞ ð0; 1 When atị ẳ 1, systhat, jarg ki j ẳ j À pj > aðtÞp tem (18)–(25) is the classical integer-order system Moreover, we observed that, the integer order derivative can be used to characterize the short memory of systems, and the variableorder fractional derivative can be employed to depict the variable memory of systems In Fig 3, we presented the result obtained by NSFDM and SFDM and ode45 schemes with step size h ẳ 0:02 and atị ẳ 1, and we observed that, all numerical methods converge almost to the equilibrium point when R0 > In Table 4, we reported the convergence behavior of numerical methods to the disease free equilibrium, and in Table 5, we reported the convergence behavior of numerical methods to the equilibrium point ðS; 0; 0; Lx ; 0; 0; Ix ; RÞ From Table 4, we can conclude that NSFDM unconditionally converges to the correct disease free equilibria for large h, while the SFDM converges only when h is small From Table 5, we can conclude that NSFD scheme unconditionally converges to the equilibrium point ðS; 0; 0; Lx ; 0; 0; Ix ; RÞ for large h, while the SFD scheme converges only when Fig 281 h is small Moreover, the system (28)–(35) is unconditionally locally asymptotically stable Previous Fig 4(a)–(d), illustrates propagation of TB along the time when atị ẳ 0:98 À 0:03=100t as follows: In Fig 4(a), the relationship between RðtÞ and Is ðtÞ illustrates that, there are individuals succeeded treatment with them and may exposed to infection again by contagious members Is ðtÞ of the first strain At the beginning of the period of the time the number of Is ðtÞ members increases and the number of RðtÞ members decreases, then after time steps the curves intersect again, Is ðtÞ will be responsible to treatment and their numbers will be decreased In Fig 4(b), the relationship between SðtÞ and Ix ðtÞ, describes the spread of infection from the members of the third strain to healthy people, then the number of infectious people increases and the number of healthy people decreases with proper time In Fig 4(c), the relationship between SðtÞ and Im ðtÞ, describes the spread of contagious from the members Im ðtÞ of the second strain to healthy people, then the number of Profiles obtained by using NSFDM for h ẳ 1, atị ẳ 0:85 0:02=100t, bs ¼ bm ¼ bx ¼ 14, and t2s ¼ t2m ¼ t2x ¼ 17, R0 > 282 N.H Sweilam and S.M AL-Mekhlafi infectious people increases and the number of healthy people decreases with proper time In Fig 4(d), the relationship between Ls ðtÞ and Is ðtÞ, describes the spread of contagious from the members Is ðtÞ of the first strain to individuals who carry the disease latent of the first strain Ls ðtÞ, after time steps the curves intersect again then Is ðtÞ will be responsible to treatment and the number of them decreases In Fig 5, we presented the result obtained by NSFDM and SFDM schemes with step size h ẳ 0:1 and atị ẳ 0:98 0:01=100t, t ½0; 100 We can clearly see, all schemes converge to correct equilibrium point when R0 > In Fig 6, we presented the results obtained by NSFD and SFD schemes with step size h ¼ and atị ẳ 0:98 0:01=100t As we can clearly see, the SFD scheme is unstable and the solutions are divergent, so we cannot use this scheme to solve the system when step size is large From these numerical results obtained in this work we can control the disease and turn the endemic point to the disease free point as follows: Let us consider: R0s < ) t22s ỵ 5:3950t2s þ 8:6060 < 0; where t1s ¼ t2s : t22s þ 1:6050t2s þ 1:050 ð44Þ R0m < ) 9:1720 0:8800t2m < 0; 0:8800t2m ỵ 0:4880 45ị R0x < ) 9:1720 0:8800t2x < 0: 0:8800t2x ỵ 0:4880 46ị t1s ẳ t2s P 6:6828; t2m P 10:4227; t2x P 10:4227: 47ị Then, T ẳ maxf t2s ; t2m ; t2x g ) T ¼ t2m ¼ t2x P 10:4227: ð48Þ So, we derive the rate of treatment required for achieving control of the disease For example, if we choose the following elements which belong to such as t2s ¼ t2m ¼ t2x ¼ 17, Bs ¼ Bm ¼ Bx ẳ 14, h ẳ and atị ẳ 0:85 À 0:02=100t, we obtained the disease free point (see Fig 7) Conclusions In this article, a novel multi-strain TB model of variable-order fractional derivatives which incorporates three strains: drugsensitive, MDR and XDR, is studied It can be concluded from the numerical results presented in this paper, that the variableorder fractional TB model given here is a general model than the integer and fractional order models Furthermore, the integer order model can be used to characterize the short memory of systems, and the variable-order fractional model can be employed to depict the variable memory of systems Moreover, we can conclude that NSFDM is more efficient for solving variable-order fractional mathematical model for multi-strain TB, than the SFDM, because it preserves the positivity of the solution and the stability regions using it are bigger than the SFDM stability regions All results in this paper are obtained using MATLAB (R2013a), on a computer machine with intel (R) core i3-3110M @ 2.40 GHz and GB RAM Conflict of Interest The authors have declared no conflict of interest Compliance with Ethics Requirements This article does not contain any studies with human or animal subjects References [1] Arino J, Soliman IA A model for the spread of tuberculosis with drug-sensitive and emerging multidrug-resistant and extensively drug resistant strains In: Mathematical and computational modelling Wiley; 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Section Variable-order fractional derivatives for multi-strain TB model Variable-order fractional derivatives for the multi-strain TB model are presented; moreover, the construction of the proposed... Þ ; j ¼ 1; 2;3; : j Variable-order fractional derivatives for multi-strain TB model In the following, we introduce the multi-strain TB model of variable-order fractional derivatives which is... TB model of variable-order fractional derivatives which incorporates three strains: drugsensitive, MDR and XDR We develop a special class of numerical method, known as NSFDM for solving this model