Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains

The objective of this paper is to study the optimal control problem for the fractional tuberculosis (TB) infection model including the impact of diabetes and resistant strains. The governed model consists of 14 fractional-order (FO) equations. Four control variables are presented to minimize the cost of interventions. The fractional derivative is defined in the Atangana-Baleanu-Caputo (ABC) sense. New numerical schemes for simulating a FO optimal system with Mittag-Leffler kernels are presented. These schemes are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation. We introduce a simple modification of the step size in the two-step Lagrange polynomial interpolation to obtain stability in a larger region. Moreover, necessary and sufficient conditions for the control problem are considered. Some numerical simulations are given to validate the theoretical results.

Journal of Advanced Research 17 (2019) 125–137 Contents lists available at ScienceDirect Journal of Advanced Research journal homepage: www.elsevier.com/locate/jare Original article Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains N.H Sweilam a,⇑, S.M AL-Mekhlafi b, D Baleanu c,d a Cairo University, Faculty of Science, Mathematics Department, 12613 Giza, Egypt Sana’a University, Faculty of Education, Mathematics Department, Sana’a, Yemen c Cankaya University, Department of Mathematics, 06530, Ankara, Turkey d Institute of Space Sciences, P.O Box MG 23, Magurele, 077125 Bucharest, Romania b h i g h l i g h t s g r a p h i c a l a b s t r a c t  Optimal control problem for the fractional TB infection model is presented  The nonstandard two-step Lagrange interpolation method is presented for numerically solving the optimality system  Necessary and sufficient conditions that guarantee the existence and the uniqueness of the solution of the control problem are given  Four controls variables are proposed to minimize the cost of interventions  New numerical schemes for simulating fractional order optimality system with Mittag-Leffler kernel are given a r t i c l e i n f o Article history: Received November 2018 Revised 22 December 2018 Accepted 13 January 2019 Available online 19 January 2019 Keywords: Tuberculosis model Diabetes and resistant strains Atangana-Baleanu fractional derivative Lagrange polynomial interpolation Nonstandard two-step Lagrange interpolation method a b s t r a c t The objective of this paper is to study the optimal control problem for the fractional tuberculosis (TB) infection model including the impact of diabetes and resistant strains The governed model consists of 14 fractional-order (FO) equations Four control variables are presented to minimize the cost of interventions The fractional derivative is defined in the Atangana-Baleanu-Caputo (ABC) sense New numerical schemes for simulating a FO optimal system with Mittag-Leffler kernels are presented These schemes are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation We introduce a simple modification of the step size in the two-step Lagrange polynomial interpolation to obtain stability in a larger region Moreover, necessary and sufficient conditions for the control problem are considered Some numerical simulations are given to validate the theoretical results Ó 2019 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer review under responsibility of Cairo University ⇑ Corresponding author E-mail address: nsweilam@sci.cu.edu.eg (N.H Sweilam) https://doi.org/10.1016/j.jare.2019.01.007 2090-1232/Ó 2019 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 126 N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125–137 Introduction A new study suggests that millions of people with high blood sugar may be more likely to develop tuberculosis (TB) than previously expected TB is a severe infection that is caused by bacteria in the lungs and kills many people each year, in addition to HIV/AIDS and malaria, according to the Daily Mail website [1] In 2017, according to the World Health Organization nearly 10 million people were infected with TB [2] Experts are concerned that a global explosion in the number of diabetes cases will put millions of people at risk [3] Many mathematical models have been proposed to elucidate the patterns of TB [4–7], Recently, Khan et al., [8], presented a new fractional model for tuberculosis In addition, several papers considered modeling TB with diabetes; see, for example, [9–12] Recently, Carvalho and Pinto presented non-integer-order analysis of the impact of diabetes and resistant strains in a model of TB infection [13] Fractional-order (FO) models provide more accurate and deeper information about the complex behaviors of various diseases than can classical integer-order models FO systems are superior to integer-order systems due to their hereditary properties and description of memory [14–28] Fractional optimal control problems (FOCPs) are optimal control problems associated with fractional dynamic systems Fractional optimal control theory is a very new topic in mathematics FOCPs may be defined in terms of different types of fractional derivatives However, the most important types of fractional derivatives are the RiemannLiouville and Caputo fractional derivatives [29–40] In addition, the theory of FOCPs has been under development Recently, some interesting real-life models of optimal control problems (OCPs) were presented elsewhere [41–52] A new concept of differentiation was introduced in the literature whereby the kernel was converted from non-local singular to non-local and non-singular One of the great advantages of this new kernel is its ability to portray fading memory as well as the well-defined memory of the system under investigation A new FO derivative, based on the generalized Mittag-Leffler function as a non-local and non-singular kernel, was presented by Atangana and Baleanu [14] in 2016 The newly introduced AtanganaBeleanu derivative has been applied in the modeling of various real-world problems in different fields, as previously discussed [15–22] This derivative, based on the Mittag-Leffler function, is more suitable for describing real-world complex problems Numerical and analytical methods are very useful because they can play very necessary roles in characterizing the behavior of the solution of the fractional differential equations, as shown in [15–27] To the best of our knowledge, the optimal control for a FO tuberculosis infection model that includes diabetes and resistant strains has never been explored The main contribution of this work is to propose a class of FOCPs and develop a numerical scheme to provide an approximate solution for those FOCPs We consider the mathematical model in Khan et al [8], and the fractional derivative is defined here in the Atangana-Baleanu-Caputo (ABC) sense A new generalized numerical scheme for simulating a FO optimal system with Mittag-Leffler kernels is established These schemes are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation This paper was organized as follows Fundamental relations are given in ‘‘Fundamental Relations” In ‘‘Fractional Model for TB Infection Including the Impact of Diabetes and Resistant Strains”, the fractionalorder model with four control variables is introduced The proposed control problem with the optimality conditions is given in ‘‘Formulation of the Fractional Optimal Control Problem” In ‘‘Numerical Techniques for the Fractional Optimal Control Model”, numerical schemes with exponential and Mittag-Leffler laws are presented Numerical experiments are given in ‘‘Numerical Simulations” In ‘‘Conclusions”, the conclusions are presented Fundamental relations In the following, the basic fractional-order derivative definitions used in this paper are given Definition The Liouville-Caputo FO derivative is defined as in [53]: C a a Dt g t ị ẳ Cð1 À aÞ Z t ðt À qÞÀa g_ ðqÞdq; < a ð1Þ 1: Definition The Atangana-Baleanu fractional derivative in the Liouville-Caputo sense is defined as in [14]: ABC a a Dt gtị ẳ Baị aÞ Z t ðEa ðÀa ðt À qÞa _ ÞgðqÞdq ; aị 2ị where Baị ẳ a ỵ Caaị is the normalization function Definition The corresponding fractional integral concerning the Atangana–Baleanu-Caputo derivative is defined as [14] ABC a a It gtị ẳ aị a gtị ỵ Baị BaịCaị Z t _ t À qÞaÀ1 gðqÞdq; They found that when a is zero, they recovered the initial function, and if a is 1, they obtained the ordinary integral In addition, they computed the Laplace transform of both derivatives and obtained the following: a fABC Dt g t ịg ẳ BaịGpịpa pa1 g0ị aịpa ỵ 1a aịị Theorem For a function g C [a, b], the following result holds [9]: a jjABC a Dt g ðt Þjj < Baị jjg t ịjj; where jjg t ịjj ẳ maxa ð1 À aÞ t b jg ðt Þj; Further, the Atangana–Baleanu-Caputo derivatives fulfill the Lipschitz condition [9]: a ABC a jjABC a Dt g ðt Þ À a Dt g ðt Þjj < -jjg ðtÞ À g ðtÞjj Fractional model for TB infection including the impact of diabetes and resistant strains In this section, we study fractional optimal control for TB infection including the impact of diabetes and resistant strains, as given in Carvalho and Pinto [13] So that the reader can make sense of the model, Fig shows the flowchart of the model as given in Carvalho and Pinto [13] The fractional derivative here is defined in the ABC sense We add four control functions, u1 , u2 , u3 and u4 ; and four real positive model constants, xi ; i ¼ 1; 2; 3; and xi ð0; 1Þ These controls are given to prevent the failure of treatment in I1s , I1R , I2s and I2R , e.g., patients’ health care providers encourage them to complete the treatments by taking TB and diabetes medications regularly This model consists of fourteen classes Let us consider the population to be divided into diabetic (index 1) and non-diabetic 127 N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125–137 Fig Flowchart of the model [13] (index 2) Then, we have susceptible individuals (S2 and S1 ), individuals exposed and sensitive to TB (E2s and E1s ), individuals exposed and resistant to TB (E2R and E1R ), individuals infected with and sensitive to TB (I2s and I1s ), individuals infected with and resistant to TB (I2R and I1R ), individuals recovering from and sensitivite to TB (R2s and R1s ), and individuals recovering from and resistant to TB (R2R and R1R ) All the parameters for the modified model in Table 1, depend on the FO because the use of the constant parameter a instead of an integer parameter can lead to better results, as one has an extra degree of freedom [40] The main assumption of this model is that the total population N is a constant in time, i.e., the a a birth and death rates are equal and d1 ¼ d2 ¼ The resulting model with four controls is given as follows: ABC a a Dt S1 ẳ Aa la ỵ aaD ỵ kT ịS1 ; 3ị ABC a a Dt S2 ẳ aaD S1 la ỵ hkT ịS2 ; 4ị ABC a a Dt E1s ẳ n1 P1 ịkT S1 ỵ nE1s þ r32 ð1 À da1 kT R1R Þ À ð1 r a1 ị 6ị ẳ nị1 P2 ịhkT S2 ỵ r41 da2 hkT R2s ỵ aaD E1s a ra2 ịk2 ỵ r2 hkT ịE2s n ỵ la ịE2s ; 8ị a a s1 aaD ỵ g1 n þ ca11 þ la þ d1 þ x1 u1 ÞI1s ; a da11 ỵ n ỵ aaD ỵ la ÞR1s ABC a a Dt R1R ð13Þ ¼ ca21 I1R þ x2 u2 I1R þ nR1s À r32 ð1 À da1 kT R1R ị da12 ỵ aaD ỵ la ÞR1R ; ABC a a Dt R2s ABC a a Dt R2R 12ị 14ị ẳ ca21 I2s ỵ x3 u3 I2s ỵ aaD R1s r41 h1 da2 ịkT R2s 15ị ẳ ca22 I2R ỵ x4 u4 I2R ỵ nR2s ỵ aD R1R r42 h da2 kT R2R da22 ỵ la ịR2R 16ị where I1s ỵ eI1R ỵ e1 I2s ỵ e2 I2R N Let us consider the state system presented in Eqs (3)–(16), in R14 ; with the set of admissible control functions X ẳ fu1 :ị; u2 :ị; u3 :ị; u4 :ịịjui is Lebsegue measurable on ẵ0; 1; ẳ nP1 kT S1 ỵ r1 ịk1 ỵ r1 kT ịE1R ỵ g1 nI1s ỵ d12 R1R a a ẳ ca11 I1s ỵ x1 u1 I1s r31 da1 ịkT R1s 9ị a s1 aaD ỵ ca12 þ la þ d1 þ x2 u2 ÞI1R a Control problem formulation ẳ nịP1 kT S1 ỵ r a1 ịk1 ỵ r1 kT ịE1s ỵ da11 R1s a a ỵ d22 R2R c22 ỵ l ỵ d2 ỵ x4 u4 ịI2R ; kT ẳ b ẳ n1 P2 ịhkT S2 ỵ nE2s þ r42 À da2 hkT R2R þ aaD E1R a ABC a a Dt I1R a ¼ nP hkT S2 ỵ r a2 ịk2 ỵ r2 hkT ịE2R ỵ g2 nI2s ỵ s1 aaD I1R a ABC a a Dt R1s ð11Þ ð7Þ À ð1 À ra2 ịk2 ỵ r2 hkT ịE2R la E2R ; ABC a a Dt I1s ABC a a Dt I 2R d21 ỵ n ỵ la ịR2s ; 5ị a ABC a a Dt E2R a ỵ da21 R2s g2 n ỵ ca21 ỵ la ỵ d2 ỵ x3 u3 ịI2s ; a k1 ỵ r1 kT ịE1R aaD ỵ la ịE1R ; ABC a a Dt E2s a ẳ nịP2 hkT S2 ỵ r a2 k21 ỵ r2 hkT E2s ỵ s1 aaD I1s ẳ nị1 P1 ịkT S1 ỵ r31 da1 R1s ị r a1 ịk1 ỵ r1 kT ịE1s n ỵ aaD ỵ la ịE1s ; ABC a a Dt E1R ABC a a Dt I 2s ð10Þ u1 ð:Þ; u2 ð:Þ; u3 ð:Þ; u4 ð:Þ where T f is the final are controls functions:  à 1; 8t 0; T f ; i ¼ 1; 2; 3; 4g; time and u1 ð:Þ; u2 ð:Þ; u3 ð:Þ and u4 ð:Þ 128 N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125–137 subject to the constraint Table The parameters of systems (3)–(16) and their descriptions [13] Parameter Descriptions Values Aa Recruitment rate Diabetes acquisition rate 667685 aaD ba ABC a a Dt S1 ABC a a Dt E1R Àa 1000 yr Effective contact rate for TB infection Modification parameter Modification parameter Modification parameter Modification parameter Rate of natural death f5; 8; 9g 1:1 1:1 1:1 0:04 0:03 a d1 Rate of TB infection among diabetic individuals Rate of TB infection among non-diabetic individuals Rate of TB infection among diabetic individuals Non-diabetic individuals’ chemoprophylaxis rate Diabetic individuals’ chemoprophylaxis rate Non-diabetic individuals’ degree of immunity Diabetic individuals’ degree of immunity Non-diabetic individuals’ rate of endogenous reactivation Diabetic individuals’ rate of endogenous reactivation Non-diabetic individuals’ sensitive TB infection recovery rate Non-diabetic individuals’ resistant TB infection recovery rate Diabetic individuals’ sensitive TB infection recovery rate Diabetic individuals’ resistant TB infection recovery rate Rate of death due to TB d2 a Rate of death due to TB and diabetes 0yrÀa s1 g1 g2 Modification parameter Modification parameter Modification parameter Non-diabetic individuals of partial immunity Non-diabetic individuals’ partial immunity for sensitive recovered Non-diabetic individuals’ partial immunity after resistant recovery Diabetic individuals’ of partial immunity Sensitive recovered diabetic individuals’ partial immunity Resistant recovered diabetic individuals’ partial immunity Sensitive recovered non-diabetic individuals’ degree of immunity Resistant recovered non-diabetic individuals’ degree of immunity Sensitive recovered diabetic individuals’ degree of immunity Recovered diabetic individuals’ degree of immunity 1:01 1:01 1:01 0:0986yr Àa 0:0986yr Àa ea ea1 ea2 la n P1 P2 r a1 r a2 r1 r2 a k1 a k2 ca11 ca12 ca21 ca22 da1 da11 da12 da2 da21 ca22 ra31 ra32 ra41 ra42 Àa 53:5 yr Z Tf I1s ỵ I1R ỵ I2s ỵ I2R ỵ B3 B4 u tị ỵ u24 tịịdt; Z Tf ¼ n7 ; ABC a a Dt I 2R ¼ n10 ; ABC a a Dt R1s ABC a a Dt R2s ¼ n13 ; ABC a a Dt R2R ABC a a Dt I 2s ¼ n8 ; ¼ n11 ; ¼ n6 ; ¼ n9 ; ABC a a Dt R1R ¼ n12 ; ¼ n14 ; 2K yr a R1s 0ị ẳ R1s0 ; R1R 0ị ẳ R1R0 ; R2s 0ị ẳ R2s0 ; R2R 0ị ẳ R2R0 : 0:7372yra To define the FOCP, consider the following modified cost function [31]: Z Tf $ J¼ ½Ha ðS1 ;S2 ;E1s ;E1R ;E2s ; E2R ;I1s ;I1R ;I2s ; I2R ;R1s ;R1R ;R2s ; R2R ;uj ;tÞ 0:7372yrÀa 0:7372yrÀa i ¼ 1; :::; 14; and the following initial conditions are satisfied: S1 0ị ẳ S01 ; S2 0ị ẳ S02 ; E1s 0ị ẳ E1s0 ; E1R 0ị ẳ E1R0 ; E2s 0ị ẳ E2s0 ; E2R 0ị ẳ E2R0 ; I1s 0ị ẳ I1s0 ; I1R 0ị ẳ I1R0 ; I2s 0ị ẳ I2s0 ; I2R 0ị ẳ I2R0 ; 0:7372yra 0yr a 14 X ðki ni ðS1 ;S2 ; E1s ;E1R ;E2s ;E2R ; I1s ;I1R ;I2s ; I2R ;R1s ; R1R ;R2s ; R2R ;uj ; tịdt; iẳ1 0:0986yr a 0:1yra 0:1yra 19ị where j ẳ 1; 2; 3; 4; and i ¼ 1; :::; 14 The Hamiltonian is given as follows: À Á Ha S1 ; S2 ; E1s ; E1R ;E2s ; E2R ;I1s ;I1R ; I2s ; I2R ;R1s ; R1R ;R2s ; R2R ;uj ; ki ;t À Á ¼ g S1 ; S2 ; E1s ;E1R ; E2s ; E2R ;I1s ; I1R ; I2s ;I2R ;R1s ; R1R ;R2s ; R2R ;uj ; t ỵ 14 X ki ni ðS1 ;S2 ;E1s ; E1R ;E2s ; E2R ; I1s ;I1R ;I2s ; I2R ; R1s ;R1R ; R2s ; R2R ;uj ;tị; iẳ1 20ị 0:1yra where, j ¼ 1; 2; 3; 4; and i ¼ 1; :::; 14 From Eqs (19) and (20), the necessary and sufficient conditions for the FOCP [34–37] are as follows: 0:73P 0:73P 0:71P 0:71P B1 B2 u tị ỵ u22 tị 2 17ị gS1 ; S2 ; E1s ; E1R ; E2s ; E2R ; I1s ; I1R ; I2s ; I2R ; R1s ; R1R ; ABC a Dtf k1 t ¼ @Ha ; @S1 ABC a Dtf k2 t ¼ @Ha ; @S2 ABC a Dtf k3 t ¼ @Ha ; @E1s ABC a Dtf k4 t ¼ @Ha ; @E1R ABC a Dtf k5 t ¼ @Ha ; @E2s ABC a Dtf k6 t ¼ @Ha ; @E2R ABC a Dtf k7 t ¼ @Ha ; @I1s ABC a Dtf k8 t ¼ @Ha ; @I1R ABC a Dtf k9 t ¼ @Ha ; @I2s ABC a Dtf k10 t ð18Þ ¼ ¼ @Ha ; @R1s ABC a Dtf k12 t ¼ @Ha ; @R1R ABC a Dtf k13 t ¼ @Ha ; @R2s ABC a Dtf k14 t ¼ @Ha ; @R2R 0ẳ @H ; @uk 21ị @Ha ; @I2R ABC a Dtf k11 t R2s ; R2R ; u1; u2; u3; u4; tÞdt; ABC a a Dt I 1R ABC a a Dt E2R ¼ n5 ; ABC a a Dt I 1s ¼ n3 ; where where B1, B2, B3, and B4 are the measure of the relative cost of the interventions associated with the controls u1, u2, u3, and u4 Then, we find the optimal controls u1 ; u2 ; u3 and u4 that minimize the cost function Jðu1 ; u2 ; u3 ; u4 Þ ¼ ABC a a Dt E2s ¼ n4 ; ABC a a Dt E1s ¼ n2 ; ni ¼ nðS1 ; S2 ; E1s ; E1R ; E2s ; E2R ; I1s ; I1R ;I2s ; I2R ; R1s ; R1R ; R2s ; R2R ; u1 ; u2 ; u3 ;u4 ; tị; ỵ ABC a a D t S2 0:06 0yrÀa 0yrÀa 0:75P 0:7P 0:00013yr Àa The objective function is defined as follows: Jðu1 ; u2 ; u3 ; u4 ị ẳ ẳ n1 ; ð22Þ 129 N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125–137 ABC a Dt S1 @Ha ; @k1 ¼ ABC a D t S2 ¼ @Ha ; @k2 Àð1 À r a1 Þr1 b à b b E k ỵ nị1 P2 ịh S2 k5 ỵ r41 da2 ị RÃ2R kÃ5 N 1R N N ABC a Dt E1s ¼ @Ha ; @k3 ABC a Dt E1R ẳ @Ha ; @k4 r a2 ịr2 hkT E2s k5 ỵ n1 P2 ịh ABC a Dt E2s ¼ @Ha ; @k5 ABC a Dt E2R ¼ @Ha ; @k6 Àð1 À r a2 ịr2 hkT E2R k6 ỵ nịP1 ẳ @Ha ; @k7 ABC a Dt I1s @Ha ABC a ; Dt I2s ¼ @k9 ¼ ABC a Dt I 2R @Ha ¼ ; @k10 @Ha ABC a ; Dt R1s ¼ @k11 ABC a Dt R1R @Ha ; @k13 ABC a Dt R2R ABC a Dt R2s ¼ @Ha ; @k8 ABC a Dt I 1R a ỵ1 r a2 Þr2 hE2R @Ha ; @k14 ð23Þ are the Lagrange multipliers Eqs (21) and (22) describe the necessary conditions in terms of a Hamiltonian for the optimal control problem defined above We arrive at the following theorem: ỵr42 da2 Þ ABC a à Dtf k8 t Theorem Let SÃ1 , SÃ2 , EÃ1R , EÃ1s , EÃ2R , EÃ2s , IÃ1R , IÃ1s ; IÃ2s , IÃ2R ; RÃ1R ; RÃ1s ; RÃ2R ; RÃ2s be the solutions of the state system and uÃi , i ¼ 1; Á Á Á ; be the given optimal controls Then, there exists co-state variables kÃj ; j ¼ 1; Á Á Á ; 14 satisfying the following: (i) Co-state equations: ẳ la ỵaD ỵ kT ịk1 ỵ aaD k2 þ ðð1 À nÞð1 À P1 ÞkT ÞkÃ3 þ ðnð1 P1 ịkT ịk4 ỵ nịP kT ịk7 ỵ nP1 kT ịk8 ị; a ẳ l ỵ hkT ịk2 ỵ nị1 a a 26ị a ẳ k4 r a1 ịk1 ỵ r1 kT ị aaD ỵ la ị ỵ k6 aaD a ỵ k8 r a1 ịk1 ỵ r1 kT ịị; ABC a Dtf k5 t 25ị ẳ k3 r a1 ịk1 ỵ r1 kT ị þ nkÃ4 þ ðaaD Ễ1s ÞkÃ5 þ ð1 À ra1 ịk1 ỵ rkT ịk7 ị; ABC a Dtf k4 t 24ị P ịhkT ịk5 ỵ n1 P2 ịhkT ịk6 ỵ nịP2 hkT k9 ỵ nP hkT kÃ10 Þ; ABC a à Dtf k3 t À À à ÁÀ a a ¼ Àk5 À r k2 ỵ r2 hkT 27ị a ỵ nk6 ỵ k9 r a2 k2 ỵ r2 hkT ; 28ị ABC a Dtf k6 t a ẳ k6 ra2 ịk2 ỵ a r2 hkT ị ỵ l a ị ỵ k10 r a2 ịk2 ỵ b b b S k hS k ỵ À nÞð1 À P1 Þ SÃ1 kÃ3 N 1 N 2 N b ỵ r31 da1 Þ RÃ1s kÃ3 N ¼ ð1 À Àð1 À r a1 ị b b b k ỵ nP2 S2 k10 h ỵ c11 k11 r31 da1 Þ RÃ1s kÃ11 N 10 N N b à à b b E k ỵ n1 P1 ị S1 k4 ỵ R1R k4 r32 da1 ị N 1s N N b à à R k Þ; N 2R 14 ð30Þ be à à be à à be S k hS k ỵ nị1 À P1 Þ SÃ1 kÃ3 N 1 N 2 N À À Á a Á be à à a be ỵ r31 d1 R k À À r1 E k N 1s N 1s À Á be à à be à ỵ n1 P1 ị S1 k4 ỵ R1R k4 r32 À da1 N N À Á be be ra1 r1 E1R k4 ỵ À nÞð1 À P2 Þh SÃ2 kÃ5 N N À a be a ỵ r41 À d2 R k À À r r2 hkT E2s kÃ5 N 2R À Á be be ỵ n1 P2 ịh S2 k6 ỵ k6 hr42 À À r a2 r2 hkT EÃ2R kÃ6 N N be ỵ nịP1 S1 k7 ỵ r a1 r1 hkT E1s k7 N ẳ be a ỵ s1 aaD ỵ c12 ỵ la ỵ d ỵ x2 u2 t ị k8 ỵ nP1 S1 k8 N be a be ỵ r1 E k r1 ỵ nịP2 S2 k9 h ỵ s1 aaD k9 N 1R N be a ỵ r r2 hE2s k9 N be be ỵ r a2 r2 hE2R k10 ỵ nP2 S2 k10 h ỵ k10 s1 aaD ỵ c11 k11 N N À À Á be à à a Á be à r31 d1 R k ỵ r32 À da1 R k N 1s 11 N 1R 12 be ỵ r41 da2 R hk ỵ N 2s 13 r42 da2 Þ r2 hkT ÞÞ; ð29Þ ABC a à Dtf k7 t b b S k h ỵ s1 aaD k9 ỵ r a2 ịr2 hE2s kÃ9 N N À Áb à à À b R k ỵ r41 da2 R hk ỵr32 da1 N 1R 12 N 2s 13 À Á kj T f ¼ 0; ki ; j ¼ 1; 2; 3; :::; 14; ABC a Dtf k2 t b S k ỵ ð1 À r a1 Þ N b à E k r1 N 1R ỵg1 nk7 ỵ ð1 À nÞP2 Moreover, ABC a à Dtf k1 t b S k ỵ r a1 ịr1 hkT E1s k7 N ỵs1 aaD ỵ gn ỵ c11 ỵ la ỵ d1 ỵ x1 u1 tịịk7 ỵ nP1 @Ha ẳ ; @k12 ẳ b b S k ỵ k hr42 N N ABC a à Dtf k9 t be à à R k Þ; N 2R 14 be1 à à be1 à à be1 à à S k À hS2 k2 ỵ nị1 P1 ị S k N 1 N N b e b e ỵ r31 ð1 À da1 Þ R k À ð1 À r a1 Þ E k N 1s N 1s be1 be1 ỵ n1 P1 ị S k ỵ R k r32 da1 Þ N N 1R be1 à à be1 à à À ð1 À r a1 Þr1 E k ỵ nị1 P2 ịh S k N 1R N be1 ỵ r41 ð1 À da2 Þ R k À ð1 À r a2 ịr2 hkT E2s k5 N 2R ẳ ð1 À ð31Þ 130 N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125137 ỵn1 P2 ịh be1 be1 S k ỵ k hr42 À À r a2 r2 hkT E2R k6 ỵ N N (ii) Transversality conditions: kj T f ị ẳ 0; j ẳ 1; 2; :::; 14: Á be1 à à À be1 à à nịP1 S k ỵ r a1 r1 hkT E1s k7 ỵ nP1 S k ỵ N N À (iii) Optimality conditions: Á be1 be1 E k r1 ỵ nịP2 S k h ỵ s1 aaD k9 À r1 N 1R N À be1 a ỵ g2 n ỵ c21 ỵ la ỵ d2 ỵ x3 u3 tị k9 þ À r a2 r2 hEÃ2s k N be1 be1 ỵ r a2 r2 hE2R k ỵ nP2 S k h þ kÃ10 s1 aaD N 10 N 10 À Á be1 à à À Á be1 à à À r31 da1 R k ỵ r32 da1 R k N 1s 11 N 1R 12 a be1 be1 ỵr41 d2 ị R hk ỵ r42 da2 ị R k Þ; N 2s 13 N 2R 14 a ABC a Dtf k10 t ẳ ỵr31 À d1 Ha ðSÃ1 ;SÃ2 ; EÃ1s ; EÃ1R ; EÃ2s ;EÃ2R ; IÃ1s ; IÃ1R ; IÃ2s ; IÃ2R ; RÃ2s ; RÃ2R ;uÃ1 ; uÃ2 ;uÃ3 ; u4 ; kj ị ẳ N R1s k3 a À À r1 Á be2 N EÃ1s kÃ3 be2 ỵ n1 P1 ị S k N ẳ r31 d1 ịkT k3 a d1 ịkT k11 a ỵ r31 ỵx ABC a Dtf k12 t ỵ r32 à u1 k11 À d1 ÞkT kÃ4 kÃ11 ðda11 a ẳ r41 a k14 nị; ỵnỵl ịỵ ABC a Dtf k14 t ỵ ỵ d22 þ l a ÞkÃ14 a a ð34Þ ð35Þ Ã Ã u3 k13 k13 da12 36ị ỵ u4 k14 Þ uÃ4 ¼ minf1; maxf0; ðx4 IÃ2R ÞðkÃ14 À kÃ10 Þ gg: B4 ð43Þ Proof We find the co-state system Eqs (24)(37), from Eq (21), where Ha ẳ I1s ỵ I1R ỵ I2s ỵ I2R ỵ ỵ B1 B2 B3 u tị ỵ u22 tị ỵ u23 tị 2 ABC ABC ABC B4 u ðtÞ þ kÃ1a Dat SÃ1 þ kÃ2a Dat SÃ2 þ kÃ3a Dat E1s ỵ k4a Dat E1R ỵ k5a Dat E2s ỵ k6a Dat E2R ỵ k7ABC Dat I1s ABC ABC ABC c ỵ k8a Dat I1R ỵ k9a Dat I2s ỵ k10a Dat I2R s ỵ k11a Ra1st ABC ABC à ABC à ABC ABC ABC ð44Þ ABc a a Dt S1 ẳ Aa la ỵ aaD ỵ kT ịS1 ; 45ị ABC a a Dt S2 ẳ aaD S1 la ỵ hkT ịS2 ; 46ị ẳ nị1 P ịkT S1 ỵ r31 da1 R1s a r a1 k1 ỵ r1 kT ịE1s n ỵ aaD ỵ la ịE1s ; ð37Þ ABC a à a Dt E2s ABC a à a Dt E2R ẳ nị1 P ịhkT S2 ỵ r41 da2 hkT R2s ỵ aaD E1s a r a2 k2 ỵ r2 hkT ịE2s n ỵ la ÞỄ2s ; a ð49Þ ð50Þ ¼ ð1 À nÞP1 kT S1 ỵ r a1 ịk1 ỵ r1 kT ịE1s ỵ da11 R1s a s1 aD ỵ g1 n ỵ c11 ỵ l ỵ d1 ỵ x1 u1 ÞIÃ1s ; a ABC a à a Dt I 1R 48ị ẳ n1 P2 ịhkT S2 ỵ nE2s ỵ r42 da2 hkT R2R ỵ aaD E1R r a2 ịk2 ỵ r2 hkT ÞEÃ2R À la EÃ2R ; ABC a à a Dt I 1s 47ị ẳ n1 P1 ịkT S1 ỵ nE1s ỵ r32 da1 kT R1R ị a ẳ r42 da2 ịhkT k6 ỵ d22 k10 ỵ r42 da2 ịhkT k14 a 42ị r a1 ịk1 ỵ r1 kT ịE1R aaD ỵ la ịE1R ; ỵ n ỵ aD ỵ l ị ỵx x3 I2s ịk13 k9 ị gg; B3 ABC a à a Dt E1R d11 kÃ7 ¼ r32 da1 ịkT k4 ỵ da12 k8 k11 da12 ỵ x2 u2 k12 k9 d21 u3 ẳ minf1; maxf0; a nk12 ỵ aD k13 ị; d2 ÞhkT kÃ5 ð41Þ ð33Þ a a ðx2 IÃ1R ÞðkÃ12 À kÃ8 Þ gg: B2 ABC a à a Dt E1s þ aaD þ la Þ þ aaD kÃ14 Þ; ABC a à Dtf k13 t uÃ2 ¼ minf1; maxf0; B1 is the Hamiltonian Moreover, the condition in Eq (23)also holds, and the optimal control characterization in Eqs (40)–(43) can be derived from Eq (22) # Substituting uÃi , i = 1,2, .,4 in (3)-(16), we can obtain the following state system: Á be2 à à À Á be2 à da1 R k ỵ r32 da1 R k N 1s 11 N 1R 12 ABC a à Dtf k11 t ð40Þ ABC À Á À be2 a ỵ r a2 r2 hE2R k ỵ c22 ỵ la ỵ d2 ỵ x4 u4 tị k10 ỵ nP2 N 10 be2  S k h N 10 be2 à à be2 R hk ỵ r42 da2 Þ R k Þ; N 2s 13 N 2R 14 k7 ị ỵ k12a Dat R1R ỵ k13a Dat R2s ỵ k14a Dat R2R ; be2 be2 ỵ1 nịP S ỵ s1 aaD k9 ỵ r a2 r2 hE2s k N N ỵr41 da2 ị x à I 1s Þðk11 gg; ð32Þ À Á Á be1 be1 ỵ r a1 r1 hkT E1s k7 ỵ nP1 S k þ À ra1 E k r1 N N 1R Àr31 HððSÃ1 ; SÃ2 ; EÃ1s ;EÃ1R ; EÃ2s ; EÃ2R ;IÃ1s ; IÃ1R ; IÃ2s ; IÃ2R ; RÃ2s ;RÃ2R ; uÃ1 ; uÃ2 ; uÃ3 ;u4 ; kj ị; u1 ẳ minf1; maxf0; be2 be2 ỵ R k r32 À da1 À À r a1 r1 E k N 1R N 1R À be2 be2 S k ỵ r41 da2 R k ỵ nị1 À P2 Þh N N 2R À Á be1 à à be2 à À À r a2 r2 hkT E2s k5 ỵ n1 P2 ịh S k ỵ k hr42 N N À Á be2 à à À À r a2 r2 hkT E2R k6 ỵ nịP1 S k N À uÃ1 ;uÃ2 ;uÃ3 ;uÃ4 ð39Þ be2 à à be2 à à be2 à S k hS2 k2 ỵ nị1 À P1 Þ S k N 1 N N a Á be2 ð38Þ a a a ð51Þ ẳ nP kT S1 ỵ r a1 ịk1 ỵ r1 kT ịE1R ỵ g1 nI1s ỵ da12 R1R a s1 aaD ỵ ca12 ỵ la ỵ d1 ỵ x2 u2 ịI1R a 52ị 131 N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125–137 ABC a à a Dt I2s À ÁÀ a ¼ nịP2 hkT S2 ỵ r a k21 ỵ r2 hkT E2s ỵ s1 aaD I1s a a u3 ịI 2s ; ỵ d21 R2s g2 n ỵ ca21 ỵ la ỵ d2 ỵ x ABC a a Dt I2R ABC a à a Dt R1s ABC a à a Dt R1R s1 aaD I1R ỵ da22 R2R ca22 ỵ la ỵ da2 ỵ x4 u4 ịI2R ; ABC a a Dt R2s 56ị ẳ ca21 I2s þ x3 uÃ3 IÃ2s þ aaD RÃ1s À r41 h da2 kT R2s d21 ỵ n ỵ 54ị ABC a a Dt R2R ẳ ca11 I1s þ x1 uÃ1 IÃ1s À r31 ð1 À da1 ÞkT R1s da11 ỵ n ỵ aaD ỵ la ịR1s da12 ỵ aaD ỵ la ịR1R ; 53ị a ẳ nP2 hkT S2 ỵ r a2 k2 ỵ r2 hkT E2R ỵ g2 nI2s þ À Á ¼ ca21 IÃ1R þ x2 uÃ2 IÃ1R þ nR1s À r32 À da1 kT RÃ1R la ịR2s ; 57ị ẳ ca22 I2R ỵ x4 u4 I2R þ nRÃ2s þ aD RÃ1R À r42 hð1 À da2 ịkT R2R da22 ỵ la ịR2R : 55ị 58ị Numerical techniques for the fractional optimal control model Let us consider the following general initial value problem: ABC a a D yt ị ẳ g t; ytịị; y0ị ẳ y0 : ð59Þ Applying the fundamental theorem of FC to Eq (59), we obtain ytị y0ị ẳ 1a a gt; ytịị ỵ Baị CaịBaị Z t gh; yhịịt hịa1 dh; 60ị where Baị ẳ a ỵ Caaị is a normalization function, and at t nỵ1 , we have ynỵ1 y0 ẳ Caị1 aị a gt ; yt n ịị ỵ Caị1 aị ỵ a n Caị ỵ a1 Caị n Z X mẳ0 t mỵ1 g tnỵ1 hịa1 dh; ð61Þ tm Now, gðh; yðhÞÞ will be approximated in an interval [tk, tk+1] using a two-step Lagrange interpolation method The two-step Lagrange polynomial interpolation is given as follows [22]: P¼ Fig Numerical simulations of S1 ỵ S2 ỵ I1s þ I1R þ I2s þ I2R þ E1s þ E1R þ E2s þ E2R þ R1s þ R1R þ R2s ỵ R2R ị=N and a ẳ with control cases using NS2LIM gðt m ; ym Þ gðtmÀ1 ; ymÀ1 Þ ðh À t mÀ1 Þ À ðh À tm Þ: h h ð62Þ Eq (62), is replaced in Eq (61), and by performing the same steps in [22], we obtain Fig Numerical simulations of I1s , I1R , I2s and I2R under different values of a with control cases using NS2LIM 132 N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125137 ynỵ1 y0 ẳ Caị1 aị gt ; yt n ị ỵ Caị1 aị ỵ a n a ỵ 1ị1 aịCaị ỵ a n X a h gtm ; yt m ịịn ỵ mịa mẳ0 n m ỵ ỵ aị n mịa n m ỵ ỵ 2aị h g tm1 ị; yt m1 ịịn ỵ mịaỵ1 a n m ỵ ỵ aị n mịa n m ỵ ỵ aị; 63ị To obtain high stability, we present a simple modification in Eq (63) This modification is to replace the step size h with /hị such that   /hị ẳ h ỵ O h : < /ðhÞ 1: For more details, see [54] Then, the new scheme is called the nonstandard two-step Lagrange interpolation method (NS2LIM) and is given as follows: ynỵ1 y0 ẳ Caị1 aị gt ; yt n ịị ỵ a ỵ 1ị1 aịCaị ỵ a Caị1 aị ỵ a n n X a /hị gt m ; yt m ịị mẳ0 n þ À mÞa ðn À m þ þ aị n mịa n m ỵ þ 2aÞ a À /ðhÞ gðtmÀ1 ; yðtmÀ1 ÞÞ ðn þ À mÞaþ1 ðn À m þ þ aị n mịa n m ỵ þ aÞ: ð64Þ Then, we use the new scheme in Eq (64) to numerically solve the state system in Eqs (45)–(58), and we use the implicit finite difference method to solve the co-state system Eqs (24)–(37) with the transversality conditions in Eq (38) Numerical simulations In this section, we present two new schemes in Eqs (63) and (64) to numerically simulate the fractional- order optimal system in Eqs (45)–(58) and Eqs (24)–(37) with the transversality condition in Eq (38) using the parameters given in Table and /hị ẳ Q ð1 À eÀh Þ, where Q is a positive number less than or equal to 0.01 The initial conditions are S1 0ị ẳ 8741400, S2 0ị ẳ 200000, E1s 0ị ¼ 557800, E1R ð0Þ ¼ 7800, E2s ð0Þ ¼ 4500, E2R 0ị ẳ 3000, I1s 0ị ẳ 20000; I1R 0ị ¼ 2000, I2s ð0Þ ¼ 1800, I2R ð0Þ ¼ 800, R1s 0ị ẳ 8000, R1R 0ị ẳ 800, R2s 0ị ¼ 200, andR2R ð0Þ ¼ 100 For computational purposes, we use MATLAB on a computer with the 64-bit Windows operating system and GB of RAM We now show some numerical aspects of the simulation of the proposed model in Eqs (3)–(16) Fig shows that the summation of all the unknown of variables in the proposed model in Eqs (3)–(16) is strictly constant during the studied time in the controlled case when using the scheme in Eq (64) This result indicates that the proposed method is efficient Fig shows the numerical solutions of I1s , I1R , I2s and I2R using the scheme in Eq (64) when T f ¼ 200 in the controlled case We note that the solutions for different values of a vary close to the integer-order solution, i.e., the FO model is a generalization of the integer-order model and the FOCP systems and is more suitable for describing the real world In Figs 4–6, we examined the numerical results of I1s , I1R , I2s and I2R in the case a ¼ 0:95; 1, and we note that there are fewer infected individuals Fig Numerical simulations of I1s , I1R , I2s and I2R with a ¼ 0:95 and b ¼ without control cases using NS2LIM N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125–137 133 Fig Numerical simulations of I1s , I1R , I2s and I2R when B1 ¼ B2 ¼ B3 ¼ B4 ¼ 100 and a ¼ 0:95, b ¼ with control cases using NS2LIM Fig Numerical simulations of I1s , I1R , I2s and I2R when B1 ¼ B3 ¼ 5000, B2 ¼ B4 ¼ 100, b ¼ 8, and a ¼ with and without control cases using NS2LIM in the control case These results agree with the results given in Table Fig illustrates the behaviour of relevant variables from the proposed model in Eqs (3)–(16) for different avalues using the scheme in Eq (64) We note that the relevant variables change under different values of a following the same behaviour Fig shows the behaviours of the relevant variables from the proposed model in Eqs (3)–(16) for a ¼ 0:8 using the scheme in Eq (63) We note that the relevant variables exhibit the same behaviour Fig 134 N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125–137 Table Comparison of the values of the objective function system using NS2LIM and T f ¼ 50 with and without control cases JðuÃ1 ; uÃ2 ; uÃ3 ; uÃ4 Þ with control a 8:7371  10 JðuÃ1 ; uÃ2 ; uÃ3 ; uÃ4 Þ without controls 1:0721  106 0.98 8:6240  10 1:0581  106 0.95 8:4617  105 1:0383  106 0.90 8:2138  10 0.80 7:8340  105 9:6373  105 0.75 7:7330  10 9:5414  105 0.60 8:2733  105 1:0502  106 Table shows a comparison of the two proposed schemes in Eqs (64) and (63) under different values of a with the control case The solutions for the scheme in Eq (64) appear to be slightly more accurate than those for the scheme in Eq (63) Conclusions 1:0082  10 shows the behaviour of the control variables u2 and u3 at different values of a and T f ¼ 200 using NS2LM We note that the control variables exhibit the same behaviour in the integer and fractional cases Fig 10 shows that the proposed scheme in Eq (64) is more stable than the scheme in Eq (63) Table shows a comparison of the value of the objective function system using Eq (64) with and without control cases when T f ¼ 50 and under different values of a We note that the values of the objective function system with the control cases are lower than the values of the objective function system without the controls for all values of 0:6 < a In this article, an optimal control for a fractional TB infection model that includes the impact of diabetes and resistant strains is presented The fractional derivative is defined in the ABC sense The proposed mathematical model utilizes a non-local and nonsingular kernel Four optimal control variables, u1 , u2 , u3 and u4 , are introduced to reduce the number of individuals infected It is concluded that the proposed fraction-order model can potentially describe more complex dynamics than can the integer model and can easily include the memory effects present in many realworld phenomena Two numerical schemes are used: 2LIM and NS2LIM Some figures are given to demonstrate how the fractional-order model is a generalization of the integer-order model Moreover, we numerically compare the two methods It is found that NS2LIM is more accurate, more efficient, more direct and more stable than 2LIM Fig Numerical simulations of the relevant variables with control cases when B1 ¼ B3 ¼ 500, B2 ¼ B4 ¼ 100 and b ¼ with different values of a using NS2LIM N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125–137 Fig Dynamics of relevant variables of the system in Eqs (45)–(58) when B1 ¼ B2 ¼ B3 ¼ B4 ¼ 100 and b ¼ 5; with control cases using 2LIM Fig Numerical simulations of the control variables using NS2LIM 135 136 N.H Sweilam et al / Journal of Advanced Research 17 (2019) 125–137 Fig 10 Numerical simulations of R1 when B1 ¼ B2 ¼ B3 ¼ B4 ¼ 100 and a ¼ 0:9, h ¼ with control case using NS2LIM and 2LIM: Table Comparison of 2LIM h ¼ 0:1 and b ¼ 5: and NS2LIM in the controlled case with T f ¼ 10, Variables 2LIM NS2LIM a I1R I2s 6.0500  103 1.7822  103 1.9694  103 1.5554  103 0.8 I1R I2s 4.0922  103 3.1513  103 1.9382  103 1.6662  103 0.7 I1R I2s 2.9203  103 6.2551  103 1.9168  103 2.3815  103 0.6 Compliance with Ethics Requirements This article does not contain any studies with human or animal subjects Conflict of interest The authors have declared no conflict of interest References [1] Global Tuberculosis Report 2014, Geneva, World Health Organization, 2014, http://www.who.int/tb/publications/global report/en/ [2] World Health Organization The dual epidemic of TB and diabetes http:// www.who.int/tb [3] Geerlings SE, Hoepelman AI Immune dysfunction in patients with diabetes mellitus (DM) FEMS Immunol Med Microbiol 1999;26(3-4):259–65 [4] Sweilam NH, Soliman IA, Al-Mekhlafi SM Nonstandard finite difference method for solving the multi-strain TB model J Egyp Mathe Soc 2017;25 (2):129–38 [5] Yang Y, Wu J, Li J, Xu X Tuberculosis with relapse: a model Math Popul Stud 2017;24(1):3–20 [6] Wallis RS Mathematical models of tuberculosis reactivation and relapse Front Microbiol 2016;7:669 [7] Castilloc’avez C, Feng Z To treat or not to treat: the case of tuberculosis J Math Biol 1997;35(6):629–56 [8] Khan MA, Ullah S, Farooq M A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative Chaos Solitons Fractals 2018;116:227–38 [9] Coll C, Herrero A, Sanchez E, Thome N A dynamic model for a study of diabetes Math Comput Model 2009;50(5–6):713–6 [10] Appuhamy JA, Kebreab E, France J A mathematical model for determining agespecific diabetes incidence and prevalence using body mass index Ann Epidemio, l 2013;23(5):248–54 [11] Delavari1 H, Heydarinejad H, Baleanu D Adaptive fractional order blood Glucose regulator based on high order sliding mode observer IET Syst Biol 2018:1–13 [12] Moualeu DP, Bowong S, Tewa JJ, Emvudu Y Analysis of the impact of diabetes on the dynamical transmission of tuberculosis Math Model Nat Phenom 2012;7(3):117–46 [13] Carvalho ARM, Pinto CMA Non-integer order analysis of the impact of diabetes and resistant strains in a model for TB infection Commun Nonlinear Sci Numer Simulat 2018;61:104–26 [14] Atangana A, Baleanu D New fractional derivatives with non-local and nonsingular kernel: theory and application to heat transfer model Therm Sci 2016;20(2):763–9 [15] Kumar D, Singh J, Baleanu D A new analysis of Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler type kernel Eur J Phys Plus 2018;133(2):70 doi: https://doi.org/10.1140/epjp/i2018-11934-y [16] Singh J, Kumar D, Baleanu D On the analysis of chemical kinetics system pertaining to a fractional derivative with Mittag-Leffler type kernel Chaos 2017;27:103113 [17] Singh J, Kumar D, Baleanu D On the analysis of fractional diabetes model with exponential law Adv Diff Equat 2018 doi: https://doi.org/10.1186/s13662018-1680-1 [18] Kumar D, Singh J, Baleanu D, Rathore S Analysis of a fractional model of Ambartsumian equation Eur J Phys Plus 2018;133:259 [19] Singh J, Kumar D, Baleanu D, Rathore S An efficient numerical algorithm for the fractional Drinfeld-Sokolov-Wilson equation Appl Math Comput 2018;335:12–24 [20] Kumar D, Agarwal RP, Singh J A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation J Comput Appl Mathe 2018;339:405–13 [21] Singh J, Kumar D, Hammouch Z, Atangana A A fractional epidemiological model for computer viruses pertaining to a new fractional derivative Appl Math Comput 2018;316:504–15 [22] Solís-Pérez JE, Gómez-Aguilar JF, Atangana A Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws Chaos Solitons Fractals 2018;114:175–85 [23] Ullah S, Khan MK, Farooq M Modeling and analysis of the fractional HBV model with Atangana-Baleanu derivative Eur Phys J Plus 2018:133–313 [24] Oldham K, Spanier J The fractional calculus: theory and application of differentiation and integration to arbitrary order New York: Academic Press; 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