Optimal control strategy for the effectiveness of TB treatment taking into account the influence of HIV/AIDS and diabetes

: The aim of this paper is to present an optimal control problem to reduce the MDR-TB (multidrug-resistant tuberculosis) and XDR-TB (extensively drug-resistant TB) cases, using controls in these compartments and controlling reinfection/reactivation of the bacteria. The model used studies the efficacy of the tuberculosis treatment taking into account the influence of HIV/AIDS and diabetes, and we prove the global stability of the disease-free equilibrium point based on the behavior of the basic reproduction number. Various control strategies are proposed with the combinations of controls. We show the existence of optimal control using Pontryagin’s maximum principle. We solve the optimality system numerically with an algorithm based on forward/backward Runge-Kutta method of the fourth-order. The numerical results indicate that the implementation of the strategy that activates all controls and of type I (starting with the highest controls) is the most cost-effective of the strategies studied. This strategy reduces significantly the number of MDR-TB and XDR-TB cases in all sub-populations, which is an important factor in combating tuberculosis and its resistant strains.


Introduction
T uberculosis is a chronic bacterial infectious disease caused by "Mycobacterium Tuberculosis". The main medical problems faced is the effectiveness of anti-TB treatments and extensively drug-resistant TB (XDR-TB) cause alarm of future non-effective TB drugs [1].
The number of cases of tuberculosis in the world has been increasing annually. For example in 2015, 10.4 million new cases were estimated in the world, 1.4 million TB-induced deaths, and 0.4 million deaths resulting from TB-HIV/AIDS co-infection [2].
The rifampcin, isoniazid, pyrazinamide, ethambtol are some of the first-line drugs for TB treatment and amikacin, capriomycim, cycloserine, azithromycin, clavithromycin, moxifloxacin, levofloxacin are the second line of treatment.
The treatment applied for active forms of tuberculosis is using first-line drugs (rifampcin, isoniazid, pyrazinamide, ethambtol) which belong to the first line of treatment for 4 months and followed by a daily intake of rifampcin and isoniazid for a period of four months.
Multidrug-resistant tuberculosis (MDR-TB) is caused by Mycobacterium tuberculosis strains with resistance to at least isoniazid and rifampin. Extensively drug-resistant tuberculosis (XDR-TB) is defined as a strain resistant to any type of fluoroquinolone and mainly to one of three injectable drugs: amikacin, capreomycin, or kanamycin in addition to isoniazid and rifampfin [3]. According to Gandhi et al., [4], about 3.2% of all new tuberculosis cases are multidrug-resistant (MDR-TB).
Diabetes is a risk factor for lower respiratory infections including TB and is a significant factor for TB infections at the population level [5]. Stevenson et al., reported that diabetes increases TB risk 1.5 to 7.8 times [6] and Jeon and Murray found that the relative risk for TB among diabetes patients was 3.11 [7]. Diabetes can This paper is organized as follows: §2 briefly describes a mathematical model for the study of resistance to treatment of tuberculosis in the presence of HIV/AIDS and diabetes presented in [25]. In §3 is incorporated four controls in order to reduce the impact of TB treatment resistance and we study the optimal control problem. §4 includes numerical experimentation and §5 is about some conclusions.

Model formulation
In this section, we present the model with ordinary differential equations introduced in [25] and its basic properties. This model is designed for the study of MDR-TB and XDR-TB, taking into account the influence of HIV/AIDS and diabetes.
The model compartments are: uninfected of TB, (S T , S H , S D ), latently infected, (E T , E H , E D ), individuals infected and drug-sensitive to TB (I T 1 , I H 1 , I D 1 ), MDR-TB infected (I T 2 , I H 2 , I D 2 ), XDR-TB infected (R T 1 , R H 1 , R D 1 ) and recovered of TB (R T , R H , R D ) individuals where T represent TB-Only individuals, H are HIV/AIDS cases and D diabetics.
The M 1 , M 2 and M 3 are recruitment rates only-infected-TB, HIV/AIDS and diabetes respectively. The rate of acquiring diabetes by use of antiretroviral treatment is α 4 and the rate of an individual acquiring HIV and of developing diabetes are α 2 and α 1 respectively.
The TB infection rate is defined as where α * is the effective contact rate and N is the total population. The parameters ϵ j , j = 1, 2 are modifications parameters, modeling the increased infectiousness in HIV/AIDS and diabetics. The natural death rate µ is the same from any compartment and µ 11 and µ 12 are death rate associated with HIV/AIDS and diabetes. The η is defined as the natural rate of progression of tuberculosis. The β * represents the proportion of active TB cases that are resistant. The ω 1 , ω 2 , are the parameters associated with the transmission of tuberculosis in the HIV/AIDS and diabetes compartments where ω 1 , ω 2 > 1.
We assume that TB-recovered R i , i = T, H, D acquire partial immunity so that from the recovered compartment (the cases that recover) enter the latent compartment with a parameter associated with TB reinfection and reactivation β ′ 1 with β ′ 1 ≤ 1. We define ϵ * j , j = 1, 2 as the parameters modification associated with resistance to tuberculosis treatment in HIV/AIDS and diabetics.
The t 1 and t 2 are modifications parameters associated with diabetes or HIV infection from the compartments of active TB infection. We define death from TB with a rate d 1 , deaths from the combination TB and HIV/AIDS with a rate d 2 and deaths from the combination TB and diabetes with a rate d 3 and we assumed that d 3 ≥ d 1 and d 2 ≥ d 1 as diabetes and HIV/AIDS experience greater disease induced deaths than their corresponding only TB and we assume death from TB after the use of treatment. The rates l 1 , l 2 and l 3 represent the cases that will be MDR-TB (first resistance). The p 1 , p 2 and p 3 represent the rates related to developing XDR-TB. The t 3 parameter is associated with the combination of antirretroviral therapy and the treatment of tuberculosis and the possibility of developing diabetes. The η 11 , η 12 and η 13 is the recovery rate after being sensitive TB and m 1 , m 2 and m 3 is the recovery rate after being MDR-TB. Let's assume that η 1l > m l for l = 1, 2, 3. The t ′ l , l = 1, 2, 3 are modification parameters associated with TB deaths in MDR-TB cases. The η * 11 , η * 12 and η * 13 are the recovery rate after being XDR-TB. Let's assume that η 1l > η * 1l and m l > η * 1l for l = 1, 2, 3. The t * 1 , t * 2 and t * 3 are modification parameters associated with death by TB, death by combination TB-HIV/AIDS and by combination TB-Diabetes after being XDR-TB.
The variables and parameters of the model are represented in the Table 1. Rates related to developing XDR-TB The effectiveness of the TB treatment with the presence of HIV/AIDS and diabetes is modeled with the following system of differential equations: with initial conditions, The following results present the existence and positivity of the solution of the system (1) and the region where all variables are always non-negative that is defined as a biologically feasible region. These results and their proof are presented in [25].
is positively-invariant and attracts all positive solutions of the model (1).
The model (1) is mathematically and epidemiologically well posed in Ω which is the biologically feasible region.
The basic reproduction number is found for the different sub-populations (TB-HIV/AIDS, TB-Diabetes and TB-Only) in order to study the transmission of tuberculosis in these sub-populations, using the next generation matrix theory [26]. The basic reproduction numbers and the results relating the stability at the disease-free equilibrium points with the basic reproduction number are given in [25].
The basic reproduction number for the sub-model where we study cases without the presence of HIV/AIDS and diabetes (TB-Only where We have that: The disease-free equilibrium point ϵ T 0 is locally asymptotically stable when ℜ T 0 < 1 and unstable when ℜ T 0 > 1.

Global stability
Now, we list two conditions that if, also guarantee the global asymptotic stability of the disease-free equilibrium point. Following [27], we can rewrite the model (1) as where S ∈ R 6 + is the vector whose components are the number of uninfected and recovered individuals (S T , S H , S D , R T , R H , R D ) and I ∈ R 12 + denotes the number of infected individuals including the latent and the infectious (the other variables of the model (1)).
The disease-free equilibrium is now denoted by The conditions H 1 and H 2 below must be satisfied to guarantee the global asymptotic stability of E G 0 .
where A = D I G((S * 0 , 0)) ( D I G((S * 0 , 0)) is the jacobian of G at (S * 0 , 0)), A is a M-matrix (the off-diagonal elements of A are nonnegative) and D is the biologically feasible region.
If model (1) satisfies the conditions H 1 and H 2 , then the following results holds; Theorem 3. The fixed point E G 0 is a globally asymptotically stable equilibrium of model (1) provided that ℜ 0 < 1 and that the conditions H 1 and H 2 are satisfied.

Proof. Let
As F(S, 0) is a linear equation, we have that S * 0 is globally stable, hence H 1 is satisfied. Then, The E G 0 is a globally asymptotically stable.

Definition of controls and its policy
Our aim with the help of the control optimal theory is to decrease the total number of patients with MDR-TB and XDR-TB during a period of time [t 0 , t f ]. The control strategy is decomposed in four controls u 0 , u 11 , u 12 and u 13 defined as follows: • u 0 (t) (control over reinfection and reactivation)-this refers to preparing patients recovering from TB to avoid possible reinfection or reactivation of the bacteria, scheduling medical consultations, and lab tests periodically. Control of entry of new genotypes of TB into the population. Also, to inform patients how to maintain an active immune system, particularly immunocompromised patients (HIV/AIDS) with adherence to antiretroviral treatment, stimulation of a good (healthy) diet, physical exercise, among others. • u 11 (t) (control for TB-Only)-this includes personal respiratory protection, educational programs for public health, activities that ensure treatment completion to reduce relapse following treatment. Patients receiving treatment for MDR-TB should be monitored to ensure the completion of the treatment. As part of this control, it is needed to check blood glucose levels and make HIV tests to determine if the person is diabetic and/or HIV positive. • u 12 (t) (control for HIV/AIDS cases)-the control will be based on clinical follow-up (we assume all cases are diagnosed), and we consider all cases are using antiretroviral therapy and have a follow-up on their CD4 count and viral load. In particular, from the beginning of treatment for TB, the return of the patient should occur in up to 15 days. Monthly consultations until the end of the TB treatment. Consultations by other members of the multi-professional team, with the objective of promoting adherence to treatment, identifying interoccurrences that may interfere with the correct use of TB drugs and antiretrovirals. Another important element is to check blood glucose levels and determine if the person is diabetic. • u 13 (t) (control for diabetics cases)-the control is focused on monitoring glycemic parameters throughout TB treatment, and promoting adherence to treatment, identifying interoccurrences that may interfere with the efficacy of TB treatment. Another important factor is to make HIV tests to control the exits of this subpopulation.
In particular, u 11 (t) is the control in the entrance to compartment I T 2 , R T 1 , u 12 (t) is the control in the entrance to compartment I H 2 , R H 1 and u 13 (t) is the control in the entrance to compartment I D 2 , R D 1 . The u 0 (t) controls entry into the E T , E H and E D compartments by reinfection or reactivation. Eqs (7) shows the incorporation of the controls in the compartments of model (1) and the other equations remain the same as the uncontrolled model (1). The (1 − u 0 ), (1 − u 11 ), (1 − u 12 ) and (1 − u 13 ) representing effort that prevents failure of the treatment. The Figure 1 shows the control dynamics. Figure 1. Schematic representation of model with controls, the arrows (discontinued) and boxes red represents the inputs and compartments to be controlled.

Optimal control problem and its analysis
Our objective functional to be minimized is The structure of our functional is consistent with recent works as [12][13][14][15][16][17][18][19]. The coefficients B m , m = 0, 1, ..., 6 represent the constant weight associated with the relative costs of implementing the respective control strategies on a finite time horizon [t 0 , t f ] (were initial time t 0 = 0, final time t f = 10 year period) and consists in the cost induced by the efforts of the three different types of control. The B 1 , B 2 and B 3 are associated with the implementation of control on the MDR-TB and B 4 , B 5 and B 6 to the XDR-TB. Given the characteristics of resistance to tuberculosis and its treatment, which in some cases may include hospitalization, high drug costs, the use of other drugs to stimulate the immune system, among others, let's assume that B 1 < B 4 , B 2 < B 5 and B 3 < B 6 and these constants cannot be neither zeros nor very large (realistic values). Given the characteristics of resistance to tuberculosis and its treatment, which in some cases may include hospitalization, high drug costs, the use of other drugs to stimulate the immune system, among others, let's assume that B 1 < B 4 , B 2 < B 5 and B 3 < B 6 and these constants cannot be neither zeros nor very large (realistic values).

The necessary and sufficient conditions of optimal control
We will study the sufficient conditions for the existence of an optimal control for our controls system using the conditions in Theorem 4.1 and its corresponding Corollary in [28]. After that, we will characterize the optimal control functions by using Pontryagin's Maximum Principle and then we derive necessary conditions for our control problem. We have that solutions of the controls system are bounded and non-negative for finite time interval in the biologically feasible region. These results are important to establish the existence of an optimal control. We denote the vector of states ⃗ x = [S T , S H , S D , E T , E H , E D , I T 1 , I T 2 , I H 1 , I H 2 , I D 1 , I D 2 , R T 1 , R H 1 , R D 1 , R T , R H , R D ] T and the controls vector ⃗ u = [u 0 , u 11 , u 12 , u 13 ] T .
Proof. We use the requirements of Theorem 4.1 and Corollary 4.2 in [28] to prove the Theorem 4. Let l(t, ⃗ x, ⃗ u) as the right-hand of (1) with controls. We will show that the following requirements are satisfied: I. l is of class C 1 and there exist a constant C such that |l(t, 0, 0)| ≤ C, |l x (t,⃗ x, ⃗ u)| ≤ C(1 + |⃗ u|), |l u (t,⃗ x, ⃗ u)| ≤ C. II. The admissible set F of all solution to system (1) with controls in U ad is non empty; III. l(t, ⃗ x, ⃗ u) = a 1 (t,⃗ x) + a 2 (t,⃗ x)⃗ u; IV. The control set U = [0, 1] × [0, 1] × [0, 1] is closed, convex and compact; V. The integrand of the objective functional is convex in U. We can write system (1) with controls as where N(t) is the total population. Then, there exists a constant C such that Thus condition I. is satisfied. By the construction of the model and the condition I., system (1) with controls has a unique solution for constant controls, this implies that condition II. is satisfied.
We can write the system (1) with controls as Then, l(t, ⃗ x, ⃗ u) = a 1 (t,⃗ x) + a 2 (t,⃗ x)⃗ u. This means that condition III. holds. By construction the sets U is closed, convex and compact and condition IV. is satisfied. Now, we are going to prove the convexity of the integrand in the objective functional this implies proving that where ⃗ u, ⃗ v are two control vectors with q ∈ [0, 1]. It follows that Then, we have With this, we prove the requirement V. and the proof of the theorem is complete.

Numerical Results
The aim of this section is to simulate the application of the controls in the population. First, we solve the optimality system numerically using an iterative method based on the fourth-order Runge-Kutta. We use the forward-backward sweep method for finding the solution of the optimality system which has the state Eq. (1), adjoint Eq. (11), control characterizations (10), and initial/final condition (initial and transversality conditions). The method starts with initial values for the optimal control and we solve the state system forward in time using the Runge-Kutta method of the fourth-order. Following, we solve the adjoint equation backward in time with Runge-Kutta of the fourth-order using the state variables, initial control guesses, and transversality conditions. The controls u 0 (t), u 11 (t), u 12 (t) and u 13 (t) are updated and used to solve the state and adjoint system respectively. This iterative process continues until that current state, adjoint, and control values converge [19,30].
The initial conditions for the TB-Only and TB-Diabetes sub-populations are taken from [31], and the values for the sub-population of HIV/AIDS are assumed and do not represent a specific demographic area, but fall within the range of actual achievable data, see Table 2. The values of the parameters and initial conditions that are assumed are discussed and validated with the specialists. We assume that B 0 = 200, B 1 = 50, B 2 = 150, B 3 = 75, B 4 = 100, B 5 = 250 and B 3 = 150.
We assume that always apply control in the HIV/AIDS sub-population because tuberculosis is classified as an opportunistic disease and HIV/AIDS cases are monitored by the use of antiretroviral therapy. Also, reinfection or reactivation of TB is always controlled in all strategies because of its impact on treatment resistance. Then, our control strategies are defined as the combination of efforts and are defined as: Strategy I. We activate all controls (u 0 (t) > 0, u 11 (t) > 0, u 12 (t) > 0, u 13 (t) > 0). Strategy II. Combination of u 0 (t), u 11 (t), u 12 (t) while setting u 13 (t) = 0 (u 0 (t) > 0, u 11 (t) > 0, u 12 (t) > 0 and u 13 (t) = 0).
We are going to study how we start the control process, with high efficient control (type I) and with minimum value (type II). Figure 2 shows the profiles of the resistance controls (u 11 (t), u 12 (t), u 13 (t)) over time for the different strategies and control types.
Strategy I. In this strategy all controls are active (u 0 (t) > 0, u 11 (t) > 0, u 12 (t) > 0 and u 13 (t) > 0). In other words, reinfection or reactivation and resistance are controlled. We show the behavior of the controls over time, see Figure 2a for control type I and see Figure 2b for control type II. It is observed that when this control strategy is implemented, there is a significant decrease in the number of TB resistant compared with the model without control, see Figure 3. In the case of MDR-TB in the different sub-populations before the study year, a decrease in the number of reported cases was observed, see Figures 3a-3c. In the case of XR-TB, the reduction in the number of cases will occur over a longer period of time, but this reduction is significant, mainly in XDR-TB diabetics, which has a strong incidence in the dynamics, see Figure 3d-3f. In the case of MDR-TB and XDR-TB in the TB-HIV/AIDS sub-population, the control manages to avoid the growth of the number of cases because in the dynamics these compartments tend to decrease initially and then grow. This strategy takes advantage of the decrease in the number of cases by preventing future growth. The type I control showed better results so it is recommended to start with higher efficacy and this evolves showing better results.
Strategy II. Here we activate the controls u 0 (t) > 0, u 11 (t) > 0 and u 12 (t) > 0 and u 13 (t) = 0, this means that we control resistance in the TB-HIV/AIDS and TB-Only sub-populations and reinfection or reactivation TB in the full model. The behavior of the controls is shown in Figure 2c-2d. This strategy succeeds in reducing the number of resistant cases, but this reduction is lower than that of strategy I. It is important to keep in mind that the largest number of resistant cases are XDR-TB diabetics and this strategy reduces this compartment but not sufficiently. It is recommended to maintain control over diabetic resistant compartments, due to their impact on resistance dynamics, mainly of XR-TB. In XDR-TB, TB-HIV/AIDS and TB-Diabetes sub-populations the controls decreased the number of cases but did not take advantage of the decrease in the number of cases and with the controls, the asymptotic behavior was maintained. The type I control was more effective.
Strategy III. This strategy does not control resistance in the TB-Only sub-population. As in the previous strategies, the objective of reducing resistance in the dynamics is met. Controls in XDR-TB HIV/AIDS and diabetes reduced the number of cases and asymptotic behavior was maintained, but the results were better than strategy II for the different types of controls. This strategy also failed to take advantage of the decrease in the number of cases, reducing the number of cases but not avoiding future asymptotic growth, see Figures 5a, 5e and 5f. Here too, type I controls achieved better results.
In general, all strategies and types of controls met the objective of reducing the number of cases of MDR-TB and XDR-TB. The most efficient strategy was the strategy I with type I control. In addition to significantly reducing the number of cases in all resistance compartments and also takes advantage of the decrease in dynamics and prevents the future growth of cases. In all strategies, type I control showed better results, so it is recommended to start with low control. In [25] the numerical results show the need to reduce XDR-TB in diabetics due to the growth that occurs in this sub-population, so strategy II does not meet significantly this objective. Strategy II is not recommended because it fails to significantly reduce resistance in diabetics and diabetes is a risk factor for adherence to TB treatment.

Conclusions
In this work, we have studied the problem of control in patients to achieve greater adherence to treatment taking into account the influence of HIV/AIDS and diabetes and thus avoid MDR-TB and XDR-TB. We present the stability at the disease-free equilibrium point related to the basic reproduction number for the sub-models and proved the global stability of this point for the full model (1). The controls are defined as u 0 , u 11 , u 12 and u 13 and are based on avoiding reinfection or reactivation of the bacteria and on differentiated care and follow-up in cases who are neither HIV+ nor diabetics, HIV/AIDS, and diabetics. The optimal control theory for the model is derived analytically by applying Pontryagin's maximum principle and we demonstrate the existence of optimal control. For the computational simulations, we used a fourth-order Runge-Kutta forward/backward scheme. We experimented in different scenarios built with different combinations of the controls. We present the results of the resistance compartments (I T 2 , I H 2 , I D 2 , R T 1 , R H 1 and R D 1 ). We concluded that all variants of strategies with the different types of controls met the objective of reducing the number of resistant cases. The strategy that obtained the best results was the strategy I (activating all controls) with type I controls (starting with the highest controls). Recommend keeping all sub-populations under control and starting with maximum control. However, if we have to use only three control, we recommended using strategy III, because all the resistance compartments and mainly the diabetic XDR-TB are reduced compared with strategy II. We do not recommend the use of strategy II, since one of the main factors of resistance to TB treatment is diabetes and this strategy did not manage to reduce significantly the number of resistant cases to TB treatment in diabetics. The results of this work help those responsible for health systems to make decisions for the efficacy of tuberculosis treatment, taking into account the influence of HIV/AIDS and diabetes. In future works, we propose to experiment in a real scenario and use differential inclusions for the control system.    Cases (in thousand persons)   Cases (in thousand persons)  Author Contributions: All authors contributed equally in this paper. All authors read and approved final version of this paper.

Conflicts of Interest:
The author declares no conflict of interest.
Data Availability: All data required for this research is included within this paper.
Funding Information: No funding is available for this research.