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.

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 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], [5], [6], [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], [10], [11], [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], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [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], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [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], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [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 Atangana-Beleanu derivative has been applied in the modeling of various real-world problems in different fields, as previously discussed [15], [16], [17], [18], [19], [20], [21], [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], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [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 fractional-order 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.where is the normalization function.

The Liouville-Caputo FO derivative is defined as in [53]:

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

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

They found that when is zero, they recovered the initial function, and if is 1, they obtained the ordinary integral. In addition, they computed the Laplace transform of both derivatives and obtained the following:

For a function  ∈ C [a, b], the following result holds [9]:

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

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. 1 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, , , and and four real positive model constants, . These controls are given to prevent the failure of treatment in , , and , 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 (index 2). Then, we have susceptible individuals ( and ), individuals exposed and sensitive to TB ( and ), individuals exposed and resistant to TB ( and ), individuals infected with and sensitive to TB ( and ), individuals infected with and resistant to TB ( and ), individuals recovering from and sensitivite to TB ( and ), and individuals recovering from and resistant to TB ( and ). All the parameters for the modified model in Table 1, depend on the FO because the use of the constant parameter 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 is a constant in time, i.e., the birth and death rates are equal and . The resulting model with four controls is given as follows:where

Fig. 1

Flowchart of the model [13].

Table 1

The parameters of systems (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16) and their descriptions [13].

Parameter	Descriptions	Values
A^α	Recruitment rate	667685.
αDα	Diabetes acquisition rate	91000yr^-α
β^α	Effective contact rate for TB infection	{5,8,9}
ε^α	Modification parameter	1.1
ε1α	Modification parameter	1.1
ε2α	Modification parameter	1.1
θ^α	Modification parameter	2
μ^α	Rate of natural death	153.5yr^-α
ξ	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
r1α	Non-diabetic individuals’ chemoprophylaxis rate	0yr^-α
r2α	Diabetic individuals’ chemoprophylaxis rate.	0yr^-α
σ_1	Non-diabetic individuals' degree of immunity	0.75P_1
σ_2	Diabetic individuals’ degree of immunity	0.7P_2
k1α	Non-diabetic individuals’ rate of endogenous reactivation	0.00013yr^-α
k2α	Diabetic individuals’ rate of endogenous reactivation	2K_1yr^-α
γ11α	Non-diabetic individuals’ sensitive TB infection recovery rate	0.7372yr^-α
γ12α	Non-diabetic individuals’ resistant TB infection recovery rate	0.7372yr^-α
γ21α	Diabetic individuals’ sensitive TB infection recovery rate	0.7372yr^-α
γ22α	Diabetic individuals’ resistant TB infection recovery rate	0.7372yr^-α
d1α	Rate of death due to TB	0yr^-α
d2α	Rate of death due to TB and diabetes	0yr^-α
τ_1	Modification parameter	1.01
η_1	Modification parameter	1.01
η_2	Modification parameter	1.01
δ1α	Non-diabetic individuals of partial immunity	0.0986yr^-α
δ11α	Non-diabetic individuals’ partial immunity for sensitive recovered	0.0986yr^-α
δ12α	Non-diabetic individuals’ partial immunity after resistant recovery	0.0986yr^-α
δ2α	Diabetic individuals’ of partial immunity	0.1yr^-α
δ21α	Sensitive recovered diabetic individuals’ partial immunity	0.1yr^-α
γ22α	Resistant recovered diabetic individuals’ partial immunity	0.1yr^-α
σ31α	Sensitive recovered non-diabetic individuals’ degree of immunity	0.73P_1
σ32α	Resistant recovered non-diabetic individuals’ degree of immunity	0.73P_1
σ41α	Sensitive recovered diabetic individuals’ degree of immunity	0.71P_2
σ42α	Recovered diabetic individuals’ degree of immunity	0.71P_2

Control problem formulation

Let us consider the state system presented in Eqs. (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), in with the set of admissible control functionswhere is the final time and

The objective function is defined as follows: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 and that minimize the cost functionsubject to the constraintwhere

and the following initial conditions are satisfied:

To define the FOCP, consider the following modified cost function [31]:where and .

The Hamiltonian is given as follows:where, and .

From Eqs. (19), (20), the necessary and sufficient conditions for the FOCP [34], [35], [36], [37] are as follows:

Moreover,are the Lagrange multipliers. Eqs. (21), (22) describe the necessary conditions in terms of a Hamiltonian for the optimal control problem defined above. We arrive at the following theorem:is the Hamiltonian. Moreover, the condition in Eq. (23) holds, and the optimal control characterization in Eqs. (40), (41), (42), (43) can be derived from Eq. (22). #

Let , , , , , , , , be the solutions of the state system and , be the given optimal controls. Then, there exists co-state variables satisfying the following:

(i) Co-state equations:

(ii) Transversality conditions:

(iii) Optimality conditions:

We find the co-state system Eqs. (24), (25), (26), (27), (28), (29), (30), (31), (32), (33), (34), (35), (36), (37), from Eq. (21), where

Substituting , i = 1,2,…,4 in (3)-(16), we can obtain the following state system:

Numerical techniques for the fractional optimal control model

Let us consider the following general initial value problem:

Applying the fundamental theorem of FC to Eq. (59), we obtainwhere is a normalization function, and at , we have

Now, 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]:

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

To obtain high stability, we present a simple modification in Eq. (63). This modification is to replace the step size such that

For more details, see [54]. Then, the new scheme is called the nonstandard two-step Lagrange interpolation method (NS2LIM) and is given as follows:

Then, we use the new scheme in Eq. (64) to numerically solve the state system in Eqs. (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), (58), and we use the implicit finite difference method to solve the co-state system Eqs. (24), (25), (26), (27), (28), (29), (30), (31), (32), (33), (34), (35), (36), (37) with the transversality conditions in Eq. (38).

Numerical simulations

In this section, we present two new schemes in Eqs. (63), (64) to numerically simulate the fractional- order optimal system in Eqs. (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), (58) and Eqs. (24), (25), (26), (27), (28), (29), (30), (31), (32), (33), (34), (35), (36), (37) with the transversality condition in Eq. (38) using the parameters given in Table 1 and , where is a positive number less than or equal to 0.01. The initial conditions are , , , , , , , , , , , , . 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), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16). Fig. 2 shows that the summation of all the unknown of variables in the proposed model in Eqs. (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (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. 3 shows the numerical solutions of , , and using the scheme in Eq. (64) when in the controlled case. We note that the solutions for different values of 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 Fig. 4, Fig. 5, Fig. 6, we examined the numerical results of , , and in the case , and we note that there are fewer infected individuals in the control case. These results agree with the results given in Table 2. Fig. 7 illustrates the behaviour of relevant variables from the proposed model in Eqs. (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16) for different using the scheme in Eq. (64). We note that the relevant variables change under different values of following the same behaviour. Fig. 8 shows the behaviours of the relevant variables from the proposed model in Eqs. (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16) for using the scheme in Eq. (63). We note that the relevant variables exhibit the same behaviour. Fig. 9 shows the behaviour of the control variables and at different values of . 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 2 shows a comparison of the value of the objective function system using Eq. (64) with and without control cases when and under different values of . 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 . Table 3 shows a comparison of the two proposed schemes in Eqs. (64), (63) under different values of 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).

Fig. 2

Numerical simulations of and with control cases using .

Fig. 3

Numerical simulations of , , and under different values of with control cases using .

Fig. 4

Numerical simulations of , , and with and without control cases using .

Fig. 5

Numerical simulations of , , and when and , with control cases using .

Fig. 6

Numerical simulations of , , and when , , , and with and without control cases using .

Table 2

Comparison of the values of the objective function system using NS2LIM and with and without control cases.

α	J(u1∗,u2∗,u3∗,u4∗) with control	J(u1∗,u2∗,u3∗,u4∗) 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

Fig. 7

Numerical simulations of the relevant variables with control cases when , and with different values of using .

Fig. 8

Dynamics of relevant variables of the system in Eqs. (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), (58) when and with control cases using .

Fig. 9

Numerical simulations of the control variables using .

Fig. 10

Numerical simulations of when and , with control case using and

Table 3

Comparison of 2LIM and NS2LIM in the controlled case with ,

Variables	2LIM	NS2LIM	α
I1R	6.0500 × 103	1.9694 × 103	0.8
I2s	1.7822 × 103	1.5554 × 103
I1R	4.0922 × 103	1.9382 × 103	0.7
I2s	3.1513 × 103	1.6662 × 103
I1R	2.9203 × 103	1.9168 × 103	0.6
I2s	6.2551 × 103	2.3815 × 103

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 non-local and non-singular kernel. Four optimal control variables, , , and , 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.

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.