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.

Original article

Optimal control for a fractional tuberculosis infection model including

the impact of diabetes and resistant strains

N.H Sweilama,⇑, S.M AL-Mekhlafib, D Baleanuc,d

a Cairo University, Faculty of Science, Mathematics Department, 12613 Giza, Egypt

b

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

h i g h l i g h t s

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

g r a p h i c a l a b s t r a c t

Article history:

Received 3 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 interven-tions 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/)

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.

Peer review under responsibility of Cairo University.

⇑ Corresponding author.

E-mail address: nsweilam@sci.cu.edu.eg (N.H Sweilam).

Contents lists available atScienceDirect

Journal of Advanced Research

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j a r e

A new study suggests that millions of people with high blood

sugar may be more likely to develop tuberculosis (TB) than

previ-ously 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

peo-ple were infected with TB[2] Experts are concerned that a global

explosion in the number of diabetes cases will put millions of

peo-ple 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

proper-ties 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

Riemann-Liouville 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

litera-ture 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

Atangana-Beleanu 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

frac-tional 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

‘‘Funda-mental Relations” In ‘‘Fractional Model for TB Infection Including

the Impact of Diabetes and Resistant Strains”, the

fractional-order model with four control variables is introduced The

pro-posed 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 Simula-tions” In ‘‘Conclusions”, the conclusions are presented

Fundamental relations

In the following, the basic fractional-order derivative definitions used in this paper are given

Definition 1 The Liouville-Caputo FO derivative is defined as in

[53]:

C

aDatg tð Þ ¼ 1

Cð1aÞ

Z t 0

t q

ð Þ a_g qð Þdq; 0 <a 1: ð1Þ

Definition 2 The Atangana-Baleanu fractional derivative in the Liouville-Caputo sense is defined as in[14]:

ABC

a DatgðtÞ ¼ BðaÞ

ð1 aÞ

Z t 0

ðEaðaðt  qÞa ð1 aÞÞ _gðqÞdq ; ð2Þ

where BðaÞ ¼ 1 aþ a

C ð a Þis the normalization function

Definition 3 The corresponding fractional integral concerning the Atangana–Baleanu-Caputo derivative is defined as[14]

ABC

a IatgðtÞ ¼ð1 aÞ

BðaÞ gðtÞ þ

a

BðaÞCðaÞ

Z t 0

ðt  qÞa 1_gðqÞdq;

They found that whenais zero, they recovered the initial func-tion, and ifais 1, they obtained the ordinary integral In addition, they computed the Laplace transform of both derivatives and obtained the following:

fABC

0 Datg tð Þg ¼Bð ÞG pa ð Þpa pa1gð0Þ

1a

ð Þðpaþ a

1 a

ð ÞÞ

Theorem 1 For a function g2 C [a, b], the following result holds[9]:

jjABC

a Datg tð Þjj < Bð Þa

1a

ð Þjjg tð Þjj; where jjg tð Þjj ¼ maxatbjg tð Þj;

Further, the Atangana–Baleanu-Caputo derivatives fulfill the Lipschitz condition[9]:

jjABC

a Datg1ð Þ t ABC

a Datg2ð Þjj <t -jjg1ðtÞ  g2ðtÞjj

Fractional model for TB infection including the impact of diabetes and resistant strains

In this section, we study fractional optimal control for TB infec-tion 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 1shows 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, u3and u4; and four real positive model constants, xi; i ¼ 1; 2; 3; 4 and xi2 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 com-plete the treatments by taking TB and diabetes medications regu-larly This model consists of fourteen classes Let us consider the population to be divided into diabetic (index 1) and non-diabetic

(index 2) Then, we have susceptible individuals (S2and S1),

individ-uals exposed and sensitive to TB (E2sand E1s), individuals exposed

and resistant to TB (E2Rand E1R), individuals infected with and

sen-sitive to TB (I2sand I1s), individuals infected with and resistant to TB

(I2Rand 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 inTable 1,

depend on the FO because the use of the constant parametera

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

birth and death rates are equal and da1Ử da

2Ử 0 The resulting model with four controls is given as follows:

ABC

a DatS1Ử Aa đlaợaa

ABC

a DatS2Ửaa

ABC

a DatE1sỬ đ1  nỡđ1  P1ỡkTS1ợr31đ1  da

1R1sỡ  đ1  ra

1ỡđka

1

ợr1kTỡE1s đn ợaa

ABC

a DatE1RỬ nđ1  P1ỡkTS1ợ nE1sợr32đ1  da

1kTR1Rỡ  đ1  ra

1ỡ

 đka

1ợr1kTỡE1R đaa

ABC

a DatE2sỬ 1  nđ ỡ 1  Pđ 2ỡhkTS2ợr41 1 da

2

hkTR2sợaa

DE1s

 đ1  ra

2ỡđka

2ợr2hkTỡE2s đn ợlaỡE2s; đ7ỡ

ABC

a DatE2RỬ n 1  Pđ 2ỡhkTS2ợ nE2sợr42 1 da

2

hkTR2Rợaa

DE1R

 đ1  ra

2ỡđka

2ợr2hkTỡE2RlaE2R; đ8ỡ

ABC

a DatI1sỬ đ1  nỡP1kTS1ợ đ1  ra

1ỡđka

1ợr1kTỡE1sợ da

11R1s

 đs1aa

Dợg1nợca

11ợlaợ da

1ợx1u1ỡI1s; đ9ỡ

ABC

a DatI1RỬ nP1kTS1ợ đ1  ra

1ỡđka

1ợr1kTỡE1Rợg1nI1sợ da

12R1R

 đs1aa

Dợca

12ợlaợ da

ABC

a DatI2sỬ 1  nđ ỡP2hkTS2ợ 1  ra

2

ka21ợr2hkT

E2sợs1aa

DI1s

ợ da

21R2s đg2nợca

21ợlaợ da

2ợx3u3ỡI2s; đ11ỡ

ABC

a DatI2RỬ nP2hkTS2ợ đ1  ra

2ỡđka

2ợr2hkTỡE2Rợg2nI2sợs1aa

DI1R

ợ da

22R2R đca

22ợlaợ da

2ợx4u4ỡI2R; đ12ỡ

ABC

a DatR1sỬca

11I1sợx1u1I1sr31đ1  da

1ỡkTR1s

 đda

11ợ n ợaa

ABC

a DatR1RỬca

21I1Rợx2u2I1Rợ nR1sr32đ1  da

1kTR1Rỡ

 đda

12ợaa

ABC

a DatR2sỬca

21I2sợx3u3I2sợaa

DR1sr41hđ1  da

2ỡkTR2s

 đd21ợ n ợlaỡR2s; đ15ỡ

ABC

a DatR2RỬca

22I2Rợx4u4I2Rợ nR2sợaDR1R

r42h 1 da

2

kTR2R đda

where

kT Ử bI1sợeI1Rợe1I2sợe2I2R

N

Control problem formulation Let us consider the state system presented in Eqs.(3)Ờ(16), in

R14; with the set of admissible control functions

XỬ ufđ 1:đỡ; u2:đỡ; u3đ ỡ; u: 4đ ỡ: ỡjui is Lebsegue measurable on 0; 1ơ ;

0 u1đ:ỡ; u2đ:ỡ; u3đ:ỡ; u4đ:ỡ  1; 8t2 0; Tf

; i Ử 1; 2; 3; 4g;

where Tf is the final time and u1đ ỡ; u: 2đ ỡ; u: 3đ ỡ and u: 4đ ỡ: are controls functions:

Fig 1 Flowchart of the model [13]

The objective function is defined as follows:

Jðu1; u2; u3; u4Þ ¼

Z Tf

0

ðI1sþ I1Rþ I2sþ I2RþB1

2u

2ðtÞ þB2

2u

2ðtÞ

þB3

2u

2ðtÞ þB4

2u

where B1, B2, B3, and B4are 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; u3and u4that

mini-mize the cost function

Jðu1; u2; u3; u4Þ ¼

Z T f

0

gðS1; S2; E1s; E1R; E2s; E2R; I1s; I1R; I2s; I2R; R1s; R1R;

R2s; R2R; u1;u2;u3;u4;tÞdt; ð18Þ

subject to the constraint

ABC

a DatS1¼ n1; ABC

a DatS2¼ n2; ABC

a DatE1s¼ n3;

ABC

a DatE1R¼ n4; ABC

a DatE2s¼ n5; ABC

a DatE2R¼ n6;

ABC

a DatI1s¼ n7; ABC

a DatI1R¼ n8; ABC

a DatI2s¼ n9;

ABC

a DatI2R¼ n10; ABC

a DatR1s¼ n11; ABC

a DatR1R¼ n12;

ABC

a DatR2s¼ n13; ABC

a DatR2R¼ n14;

where

ni¼ nðS1;S2;E1s;E1R;E2s;E2R;I1s;I1R;I2s;I2R;R1s;R1R;R2s;R2R;u1;u2;u3;u4;tÞ;

i¼ 1; :::; 14; and the following initial conditions are satisfied:

S1ð0Þ ¼ S01; S2ð0Þ ¼ S02; E1sð0Þ ¼ E1s0; E1Rð0Þ ¼ E1R0; E2sð0Þ ¼ E2s0;

E2Rð0Þ ¼ E2R0; I1sð0Þ ¼ I1s0; I1Rð0Þ ¼ I1R0; I2sð0Þ ¼ I2s0; I2Rð0Þ ¼ I2R0;

R1sð0Þ ¼ R1s0; R1Rð0Þ ¼ R1R0; R2sð0Þ ¼ R2s0; R2Rð0Þ ¼ R2R0:

To define the FOCP, consider the following modified cost func-tion[31]:

J



¼

Z T f

0

½HaðS1;S2;E1s;E1R;E2s;E2R;I1s;I1R;I2s;I2R;R1s;R1R;R2s;R2R;uj;tÞ

X14 i¼1

ðkiniðS1;S2;E1s;E1R;E2s;E2R;I1s;I1R;I2s;I2R;R1s;R1R;R2s;R2R;uj;tÞdt;

ð19Þ where j¼ 1; 2; 3; 4; and i ¼ 1; :::; 14

The Hamiltonian is given as follows:

HaS1;S2;E1s;E1R;E2s;E2R;I1s;I1R;I2s;I2R;R1s;R1R;R2s;R2R;uj;ki;t

¼gS1;S2;E1s;E1R;E2s;E2R;I1s;I1R;I2s;I2R;R1s;R1R;R2s;R2R;uj;t

þX14 i¼1

kiniðS1;S2;E1s;E1R;E2s;E2R;I1s;I1R;I2s;I2R;R1s;R1R;R2s;R2R;uj;tÞ;

ð20Þ

where, j¼ 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:

ABC

t Datfk1¼ @Ha

@S1; ABC

t Datfk2¼ @Ha

@S2;

ABC

t Datfk3¼ @Ha

@E1s; ABC

t Datfk4¼ @Ha

@E1R;

ABC

t Dat

fk5¼ @Ha

@E2s; ABC

t Dat

fk6¼ @Ha

@E2R;

ABC

t Dat

fk7¼ @Ha

@I1s

; ABC

t Dat

fk8¼ @Ha

@I1R

ABC

t Datfk9¼ @Ha

@I2s; ABC

t Datfk10¼ @Ha

@I2R;

ABC

t Dat

fk11¼ @Ha

@R1s; ABC

t Dat

fk12¼ @Ha

@R1R;

ABC

t Dat

fk13¼ @Ha

@R2s; ABC

t Dat

fk14¼ @Ha

@R2R;

0¼ @H

Table 1

The parameters of systems (3)–(16) and their descriptions [13]

Parameter Descriptions Values

Aa Recruitment rate 667685.

aa

D Diabetes acquisition rate 9

1000 yr  a

ba Effective contact rate for TB infection f5; 8; 9g

ea Modification parameter 1:1

ea Modification parameter 1:1

ea Modification parameter 1:1

ha Modification parameter 2

la Rate of natural death 1

53:5 yra

n Rate of TB infection among diabetic individuals 0:04

P 1 Rate of TB infection among non-diabetic

individuals

0:03

P 2 Rate of TB infection among diabetic individuals 0:06

r a Non-diabetic individuals’ chemoprophylaxis rate 0yr  a

r a Diabetic individuals’ chemoprophylaxis rate 0yr  a

r1 Non-diabetic individuals’ degree of immunity 0:75P 1

r2 Diabetic individuals’ degree of immunity 0:7P 2

ka Non-diabetic individuals’ rate of endogenous

reactivation

0:00013yr  a

ka Diabetic individuals’ rate of endogenous

reactivation

2K 1 yr  a

ca

11 Non-diabetic individuals’ sensitive TB infection

recovery rate

0:7372yr  a

ca

12 Non-diabetic individuals’ resistant TB infection

recovery rate

0:7372yr  a

ca

21 Diabetic individuals’ sensitive TB infection

recovery rate

0:7372yr  a

ca

22 Diabetic individuals’ resistant TB infection

recovery rate

0:7372yr  a

da Rate of death due to TB 0yra

da Rate of death due to TB and diabetes 0yr  a

s1 Modification parameter 1:01

g1 Modification parameter 1:01

g2 Modification parameter 1:01

da Non-diabetic individuals of partial immunity 0:0986yr  a

da11 Non-diabetic individuals’ partial immunity for

sensitive recovered

0:0986yr  a

da12 Non-diabetic individuals’ partial immunity after

resistant recovery

0:0986yr  a

da Diabetic individuals’ of partial immunity 0:1yr  a

da21 Sensitive recovered diabetic individuals’ partial

immunity

0:1yr  a

ca

22 Resistant recovered diabetic individuals’ partial

immunity

0:1yr  a

ra

31 Sensitive recovered non-diabetic individuals’

degree of immunity

0:73P 1

ra

32 Resistant recovered non-diabetic individuals’

degree of immunity

0:73P 1

ra

41 Sensitive recovered diabetic individuals’ degree of

immunity

0:71P 2

ra

42 Recovered diabetic individuals’ degree of

immunity

0:71P 2

0 DatS1¼ @Ha

@k1

; ABC

0 DatS2¼ @Ha

@k2

;

ABC

0 DatE1s¼ @Ha

@k3; ABC

0 DatE1R¼ @Ha

@k4;

ABC

0 DatE2s¼ @Ha

@k5; ABC

0 DatE2R¼ @Ha

@k6;

ABC

0 DatI1s¼ @Ha

@k7; ABC

0 DatI1R¼ @Ha

@k8;

ABC

0 DatI2s¼ @Ha

@k9; ABC

0 DatI2R¼ @Ha

@k10;

ABC

0 DatR1s¼ @Ha

@k11; ABC

0 DatR1R¼ @Ha

@k12;

ABC

0 DatR2s¼ @Ha

@k13; ABC

0 DatR2R¼ @Ha

@k14;

Moreover,

kj Tf

 

¼ 0; ki; j ¼ 1; 2; 3; :::; 14; ð23Þ

are the Lagrange multipliers Eqs.(21) and (22)describe the

neces-sary conditions in terms of a Hamiltonian for the optimal control

problem defined above We arrive at the following theorem:

Theorem 2 Let S1, S2, E1R, E1s, E2R, E2s, I1R, I1s; I

2s, I2R; R 1R; R 1s; R 2R;

R2sbe the solutions of the state system and ui, i¼ 1; ; 4 be the given

optimal controls Then, there exists co-state variables kj; j ¼ 1; ; 14

satisfying the following:

(i) Co-state equations:

ABC

t Dat

fk1¼ ððlaþa

Dþ kTÞk

1þaa

Dk2þ ðð1  nÞð1  P1ÞkTÞk

3

þ ðnð1  P1ÞkTÞk

4þ ðð1  nÞP1kTÞk

7þ ðnP1kTÞk

8Þ; ð24Þ

ABC

t Dat

fk2¼ ððlaþ hkTÞk

2þ ðð1  nÞð1  P2ÞhkTÞk

5

þ ðnð1  P2ÞhkTÞk

6þ ð1  nÞP2hkTk9þ nP2hkTk10Þ; ð25Þ

ABC

t Datfk3¼ ðk

3ð1  ra

1Þðka

1þr1kTÞ þ nk

4þ ðaa

DE1sÞk 5

þ ð1  ra

1Þðka

1þrkTÞk

ABC

t Datfk4¼ ðk

4ð1  ra

1Þðka

1þr1kTÞ  ðaa

DþlaÞ þ k

6aa

D

þ k

8ð1  ra

1Þðka

ABC

t Datfk5¼ k

5 1 ra

2

ka2þr2hkT

þ nk

6þ k

9 1 ra

2

ka2þr2hkT

; ð28Þ

ABC

t Dat

fk6¼ ððk

6ð1  ra

2Þðka

2þr2hkTÞ þlaÞ þ k

10ð1  ra

2Þðka

2þr2hkTÞÞ;

ð29Þ

ABC

t Dat

fk7¼ ð1 b

NS



1k1b

NhS



2k2þ ð1  nÞð1  P1Þb

NS



1k3

þr31ð1  da

1Þb

NR

 1sk3

ð1  ra

1Þb

NE



1sk3þ nð1  P1Þb

NS



1k4þb

NR

 1Rk4r32ð1  da

1Þ

ð1  ra

1Þr1

b

NE

 1Rk4þ ð1  nÞð1  P2Þhb

NS



2k5þr41ð1  da

2Þb

NR

 2Rk5

ð1  ra

2Þr2hkTE2sk5þ nð1  P2Þhb

NS



2k6þb

Nk



6hr42

ð1  ra

2Þr2hkTE2Rk6þ ð1  nÞP1

b

NS



1k7þ ð1  ra

1Þr1hkTE1sk7

þðs1aa

Dþgnþc11þlaþ da

1þx1u1ðtÞÞk

7þ nP1

b

NS



1k8þ ð1  ra

1Þ

b

NE

 1Rk8r1

þg1nk7þ ð1  nÞP2

b

NS



2k9hþs1aa

Dk9þ ð1  ra

2Þr2hE2s

b

Nk

 9

þð1  ra

2Þr2hE2R

b

Nk



10þ nP2

b

NS



2k10hþc11k11r31ð1  da

1Þb

NR

 1sk11

þr32 1 da

1

NR

 1Rk12þr41 1 da

2

NR

 2shk13

þr42ð1  da

2Þb

NR



ABC

t Dat

fk8¼ ð1 be

NS



1k1be

NhS



2k2þ ð1  nÞð1  P1Þbe

NS



1k3

þr31 1 da

1

  be

NR

 1sk3 1  ra

1

  be

NE

 1sk3

þ n 1  Pð 1Þbe

NS



1k4þbe

NR

 1Rk4r32 1 da

1

 1  ra

1

r1

be

NE

 1Rk4þ 1  nð Þ 1  Pð 2Þhbe

NS



2k5

þr41 1 da

2

  be

NR

 2Rk5 1  ra

2

r2hkTE2sk5

þ n 1  Pð 2Þhbe

NS



2k6þbe

Nk



6hr42 1  ra

2

r2hkTE2Rk6

þ 1  nð ÞP1

be

NS



1k7þ 1  ra

1

r1hkTE1sk7

þ s1aa

Dþc12þlaþ daþx2u2ð Þt

k8þ nP1

be

NS



1k8

þ 1  ra

1

  be

NE

 1Rk8r1þ 1  nð ÞP2

be

NS



2k9hþs1aa

Dk9

þ 1  ra

2

r2hE2sbe

Nk

 9

þ 1  ra

2

r2hE2R

be

Nk



10þ nP2

be

NS



2k10hþ k

10s1aa

Dþc11k11

r31 1 da

1

  be

NR

 1sk11þr32 1 da

1

  be

NR

 1Rk12

þr41 1 da

2

  be

NR

 2shk13þ

r42ð1  da

2Þbe

NR



ABC

t Dat

fk9¼ ð1 be1

N S



1k1be1

N hS



2k2þ ð1  nÞð1  P1Þbe1

N S



1k3

þr31ð1  da

1Þbe1

N R

 1sk3 ð1  ra

1Þbe1

N E

 1sk3

þ nð1  P1Þbe1

N S



1k4þbe1

N R

 1Rk4r32ð1  da

1Þ

 ð1  ra

1Þr1

be1

N E

 1Rk4þ ð1  nÞð1  P2Þhbe1

N S



2k5

þr41ð1  da

2Þbe1

N R

 2Rk5 ð1  ra

2Þr2hkTE2sk5

þn 1  Pð 2Þhbe1

N S



2k6þbe1

N k



6hr42 1  ra

2

r2hkTE2Rk6þ

1 n

ð ÞP1

be1

N S



1k7þ 1  ra

1

r1hkTE1sk7þ nP1

be1

N S



1k8þ

1 ra

1

  be1

N E



1Rk8r1þ 1  nð ÞP2

be1

N S



2k9hþs1aa

Dk9

þ g2nþc21þlaþ da

2þx3u3ð Þt

k9þ 1  ra

2

r2hE2sbe1

N k

 9

þ 1  ra

2

r2hE2Rbe1

N k



10þ nP2

be1

N S



2k10hþ k

10s1aa

D

r31 1 da

1

  be1

N R

 1sk11þr32 1 da

1

  be1

N R

 1Rk12

þr41ð1  da

2Þbe1

N R

 2shk13þr42ð1  da

2Þbe1

N R

 2Rk14Þ; ð32Þ

ABC

t Dat

fk10¼ ð1 be2

N S



1k1be2

N hS



2k2þ ð1  nÞð1  P1Þbe2

N S



1k3

þr31 1 da

1

  be2

N R

 1sk3 1  ra

1

  be2

N E

 1sk3þ n 1  Pð 1Þbe2

N S



1k4

þbe2

N R



1Rk4r32 1 da

1

 1  ra

1

r1

be2

N E

 1Rk4

þ 1  nð Þ 1  Pð 2Þhbe2

N S



2k5þr41 1 da

2

  be2

N R

 2Rk5

 1  ra

2

r2hkTE2sk5þ n 1  Pð 2Þhbe1

N S



2k6þbe2

N k



6hr42

 1  ra

2

r2hkTE2Rk6þ 1  nð ÞP1

be2

N S



1k7

þ 1  ra

1

r1hkTE1sk7þ nP1

be1

N S



1k8þ 1  ra

1

  be1

N E

 1Rk8r1

þ 1  nð ÞP2

be2

N S



2þs1aa

Dk9þ 1  ra

2

r2hE2sbe2

N k

 9

þ 1  ra

2

r2hE2Rbe2

N k



10þ c22þlaþ da

2þx4u4ð Þt

k10þ nP2

be2

N S



2k10h

r31 1 da

1

  be2

N R

 1sk11þr32 1 da

1

  be2

N R

 1Rk12

þr41ð1  da

2Þbe2

N R

 2shk13þr42ð1  da

2Þbe2

N R

 2Rk14Þ; ð33Þ

ABC

t Dat

fk11¼ ðr31ð1  da

1ÞkTk3þr32ð1  da

1ÞkTk4þ da

11k7

þr31ð1  da

1ÞkTk11 k

11ðda

11þ n þaa

DþlaÞ

þx1u1k11 nk12þaa

ABC

t Datfk12¼ ðr32ð1  da

1ÞkTk4þ da

12k8 k

11ðda

12þx2u2k12

þaa

DþlaÞ þaa

ABC

t Datfk13¼ ðr41ð1  da

2ÞhkTk5þ k

9d21þx3u3k13 k

13ðda

12

þ n þlaÞ þ k

ABC

t Dat

fk14¼ ðr42ð1  da

2ÞhkTk6þ d22k10þr42ð1  da

2ÞhkTk14

 ðda

22þlaÞk

14þ u

(ii) Transversality conditions:

kjðTfÞ ¼ 0; j ¼ 1; 2; :::; 14: ð38Þ

(iii) Optimality conditions:

HaðS

1;S

2;E 1s;E 1R;E 2s;E 2R;I 1s;I 1R;I 2s;I 2R;R 2s;R 2R;u

1;u

2;u

3;u

4;kjÞ

¼ min

0u 

1 ;u 

2 ;u 

3 ;u 

4 1HððS

1;S

2;E 1s;E 1R;E 2s;E 2R;I 1s;I 1R;I 2s;I 2R;R 2s;R 2R;u

1;u

2;u

3;u

4;kjÞ; ð39Þ

u1¼ minf1; maxf0;ðx1I1sÞðk

11 k

7Þ

u2¼ minf1; maxf0;ðx2I1RÞðk

12 k

8Þ

u3¼ minf1; maxf0;ðx3I2sÞðk

13 k

9Þ

u4¼ minf1; maxf0;ðx4I2RÞðk

14 k

10Þ

Proof We find the co-state system Eqs.(24)–(37), from Eq.(21), where

Ha¼ I 1sþ I 1Rþ I 2sþ I 2RþB1

2u

2ðtÞ þB2

2u

2ðtÞ þB3

2u

2ðtÞ

þB4

2u

2ðtÞ þ k ABC

1 a DatS1þ k ABC

2 a DatS2þ k ABC

3 a DatE1s

þ k ABC

4 a DatE1Rþ k ABC

5 a DatE2sþ k ABC

6 a DatE2Rþ k c

7ABCDatI1s

þ k ABC

8 a DatI1Rþ k ABC

9 a DatI2sþ k ABC

10 aDatI2Rsþ k ABC

11 aRa1st

þ k ABC

12 aDatR1R þ k ABC

13 aDatR2sþ k ABC

14 aDatR2R; ð44Þ

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 ui, i = 1,2, .,4 in (3)-(16), we can obtain the fol-lowing state system:

ABc

a DatS1¼ Aa ðlaþaa

Dþ kTÞS

ABC

a DatS2¼aa

DS1 ðlaþ hkTÞS

ABC

a DatE1s¼ 1  nð Þ 1  Pð 1ÞkTS1þr31 1 da

1R1s

 1  ra

1

ðka

1þr1kTÞE

1s ðn þaa

DþlaÞE

ABC

a DatE1R¼ nð1  P1ÞkTS1þ nE

1sþr32ð1  da

1kTR1RÞ  ð1

 ra

1Þðka

1þr1kTÞE

1R ðaa

DþlaÞE

ABC

a DatE2s¼ 1  nð Þ 1  Pð 2ÞhkTS2þr41 1 da

2

hkTR2sþaa

DE1s

 1  ra

2

ðka

2þr2hkTÞE

2snþlaÞE

ABC

a DatE2R¼ n 1  Pð 2ÞhkTS2þ nE

2sþr42 1 da

2

hkTR2R

þaa

DE1R ð1  ra

2Þðka

2þr2hkTÞE

2RlaE2R; ð50Þ

ABC

a DatI1s¼ ð1  nÞP1kTS1þ ð1  ra

1Þðka

1þr1kTÞE

1sþ da

11R1s

 ðs1aa

Dþg1nþca

11þlaþ da

1þx1u1ÞI

ABC

a DatI1R¼ nP1kTS1þ ð1  ra

1Þðka

1þr1kTÞE

1Rþg1nI1sþ da

12R1R

 ðs aaþca þlaþ daþxu ÞI ð52Þ

a DatI2s¼ 1  nð ÞP2hkTS2þ 1  ra

2

ka21þr2hkT

E2sþs1aa

DI1s

þ da

21R2s ðg2nþca

21þlaþ da

2þx3u3ÞI

ABC

a DatI2R¼ nP2hkTS2þ 1  ra

2

ka2þr2hkT

E2Rþg2nI2s

þ s1aa

DI1Rþ da

22R2R ðca

22þlaþ da

2þx4u4ÞI

2R; ð54Þ

ABC

a DatR1s¼ca

11I1sþx1u1I1sr31ð1  da

1ÞkTR1s

 ðda

11þ n þaa

DþlaÞR

ABC

a DatR1R¼ca

21I1Rþx2u2I1Rþ nR1sr32 1 da

1kTR1R

 ðda

12þaa

DþlaÞR

ABC

a DatR2s¼ca

21I2sþx3u3I2sþaa

DR1sr41h 1 da

2

kTR2s

 dð 21þ n þ laÞR

ABC

a DatR2R¼ca

22I2Rþx4u4I2Rþ nR

2sþaDR1Rr42hð1

 da

2ÞkTR2R ðda

22þlaÞR

Numerical techniques for the fractional optimal control model

Let us consider the following general initial value problem:

ABC

a Day tð Þ ¼ g t; y tð ð ÞÞ; yð0Þ ¼ y0: ð59Þ

Applying the fundamental theorem of FC to Eq.(59), we obtain

yðtÞ  yð0Þ ¼1a

BðaÞgðt; yðtÞÞ þ a

CðaÞBðaÞ

Z t 0

gðh; yðhÞÞ t  hð Þa 1dh;

ð60Þ

where BðaÞ ¼ 1 aþ a

C ð a Þis a normalization function, and at tnþ1, we have

ynþ1 y0¼ CðaÞð1 aÞ

CðaÞð1 aÞ þagðtn; yðtnÞÞ þ a

CðaÞ það1 CðaÞ

Xn m¼0

Z tmþ1

t m

g tðnþ1 hÞa 1dh; ð61Þ

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]:

P¼gðtm; ymÞ

h ðh  tm1Þ gðtm1; ym1Þ

Eq.(62), is replaced in Eq (61), and by performing the same steps in[22], we obtain

Fig 2 Numerical simulations of ðS 1 þ S 2 þ I 1s þ I 1R þ I 2s þ I 2R þ E 1s þ E 1R þ E 2s þ

E 2R þ R 1s þ R 1R þ R 2s þ R 2R Þ=N anda¼ 1 with control cases using NS2LIM.

a

ynþ1 y0¼ Cð Þ 1 að aÞ

Cð Þ 1 að aÞ þagðtn; y tð Þ þn 1

aþ 1

ð Þ 1 ð aÞCð Þ þa a

Xn

m¼0

hagðtm; y tð ÞÞ n þ 1  mm ð Þa

ðn  m þ 2 þaÞ  n  mð Þaðn m þ 2 þ 2aÞ

 ha

g tðm1Þ; y tðm1ÞÞ n þ 1  mð Þa þ1

ðn  m þ 2 þaÞ  n  mð Þaðn  m þ 1 þaÞ; ð63Þ

To obtain high stability, we present a simple modification in Eq

(63) This modification is to replace the step size h with / hð Þ such

that

/ð Þ ¼ h þ O hh  2

: 0 < / hð Þ  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:

ynþ1 y0¼ CðaÞð1 aÞ

CðaÞð1 aÞ þagðtn; yðtnÞÞ þ 1

ðaþ 1Þð1 aÞCðaÞ þa

Xn

m¼0

/ð Þhagðtm; yðtmÞÞ

nþ 1  m

ð Þaðn  m þ 2 þaÞ  n  mð Þaðn m þ 2 þ 2aÞ

 / hð Þagðtm1; yðtm1ÞÞ

nþ 1  m

ð Þa þ1ðn  m þ 2 þaÞ  n  mð Þaðn m þ 1 þaÞ: 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

condi-tion in Eq (38) using the parameters given in Table 1 and

/ðhÞ ¼ Qð1  ehÞ, where Q is a positive number less than or equal

to 0.01 The initial conditions are S1ð0Þ ¼ 8741400, S2ð0Þ ¼ 200000,

E1sð0Þ ¼ 557800, E1Rð0Þ ¼ 7800, E2sð0Þ ¼ 4500, E2Rð0Þ ¼ 3000,

I1sð0Þ ¼ 20000; I1Rð0Þ ¼ 2000, I2sð0Þ ¼ 1800, I2Rð0Þ ¼ 800,

R1sð0Þ ¼ 8000, R1Rð0Þ ¼ 800, R2sð0Þ ¼ 200, andR2Rð0Þ ¼ 100 For

computational purposes, we use MATLAB on a computer with the

64-bit Windows 7 operating system and 4 GB of RAM We now

show some numerical aspects of the simulation of the proposed

model in Eqs (3)–(16).Fig 2 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 3shows the numerical solutions

of I1s, I1R, I2sand I2Rusing the scheme in Eq.(64)when Tf ¼ 200 in

the controlled case We note that the solutions for different values

ofavary 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, I2sand I2Rin the case

a¼ 0:95; 1, and we note that there are fewer infected individuals Fig 4 Numerical simulations of Icontrol cases using NS2LIM. 1s, I1R, I2sand I2Rwitha¼ 0:95 and b ¼ 9 without

in the control case These results agree with the results given in

Table 2.Fig 7illustrates the behaviour of relevant variables from

the proposed model in Eqs.(3)–(16) for differentavalues using

the scheme in Eq.(64) We note that the relevant variables change

under different values ofa following the same behaviour.Fig 8

shows the behaviours of the relevant variables from the proposed model in Eqs.(3)–(16)fora¼ 0:8 using the scheme in Eq.(63) We note that the relevant variables exhibit the same behaviour.Fig 9 Fig 5 Numerical simulations of I 1s , I 1R , I 2s and I 2R when B 1 ¼ B 2 ¼ B 3 ¼ B 4 ¼ 100 anda¼ 0:95, b ¼ 9 with control cases using NS2LIM.

Fig 6 Numerical simulations of I 1s , I 1R , I 2s and I 2R when B 1 ¼ B 3 ¼ 5000, B 2 ¼ B 4 ¼ 100, b ¼ 8, anda¼ 1 with and without control cases using NS2LIM.

shows the behaviour of the control variables u2and u3at different

values ofa and Tf ¼ 200 using NS2LM We note that the control

variables exhibit the same behaviour in the integer and fractional

cases.Fig 10shows that the proposed scheme in Eq.(64)is more

stable than the scheme in Eq.(63).Table 2shows a comparison of

the value of the objective function system using Eq.(64)with and

without control cases when Tf¼ 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

func-tion system without the controls for all values of 0:6 <a 1

Table 3shows a comparison of the two proposed schemes in Eqs

(64) and (63)under different values ofa 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

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 local and non-singular kernel Four optimal control variables, u1, u2, u3and 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 real-world 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

Table 2

Comparison of the values of the objective function system using NS2LIM and T f ¼ 50

with and without control cases.

a Jðu 

1 ; u 

2 ; u 

3 ; u 

4 Þ with control Jðu 

1 ; u 

2 ; u 

3 ; u 

4 Þ without controls

1 8:7371  10 5 1:0721  10 6

0.98 8:6240  10 5 1:0581  10 6

0.95 8:4617  10 5 1:0383  10 6

0.90 8:2138  10 5 1:0082  10 6

0.80 7:8340  10 5 9:6373  10 5

0.75 7:7330  10 5 9:5414  10 5

0.60 8:2733  10 5 1:0502  10 6

¼ B ¼ 500, B ¼ B ¼ 100 and b ¼ 5 with different values ofa