Numerical and Theoretical Treatments of the Optimal Control Model for the Interaction Between Diabetes and Tuberculosis

Abstract

We primarily focus on the formulation, theoretical, and numerical analyses of a non-autonomous model for tuberculosis (TB) prevention and control programs in a population where individuals suffering from the double trouble of tuberculosis and diabetes are present. The model incorporates four time-dependent control functions, saturated treatment of non-infectious individuals harboring tuberculosis, and saturated incidence rate. Furthermore, the basic reproduction number of the autonomous form of the proposed optimal control mathematical model is calculated. Sensitivity indexes regarding the constant control parameters reveal that the proposed control and preventive measures will reduce the tuberculosis burden in the population. This study establishes that the combination of campaigns that teach people how the development of tuberculosis and diabetes can be prevented, a treatment strategy that provides saturated treatment to non-infectious individuals exposed to tuberculosis infections, and prompt effective treatment of individuals infected with tuberculosis disease is the optimal strategy to achieve zero TB by 2035.

1. Introduction

Tuberculosis (TB) disease outbreak is a pandemic that is still present and its eradication has been a global health challenge for a long time [1,2,3]. It is an agelong respiratory illness. As noted by World Health Organization (WHO) in 2024, several years of progress achieved in the fight to end TB have been reversed because of the pandemic and countless other formidable challenges [4]; making TB the world’s deadliest infectious disease today, only second to the novel coronavirus during the pandemic in 2020. Subsequently, another report circulated by the Centers for Disease Control and Prevention (CDC) [5], in 2024, emphasized that there are still increasing TB incidences in the US.

Mycobacterium tuberculosis is a germ that causes tuberculosis infections. It is a pathogen that is ejected into the air when humans are sick with active tuberculosis disease and cough, talk, sing, or sneeze. When this airborne germ is inhaled by other humans, they contract tuberculosis in latent or active form depending on their immune response [6,7]. People with latent tuberculosis infections (LTBIs) are non-infectious and by the WHO’s estimate, around 1.7 billion people have LTBIs globally. If those exposed to the latent form of tuberculosis are not treated using the treatment regimen recommended by the WHO, preventing and controlling will be a difficult task since around 5 to 10 percent of the population exposed to TB would develop active TB in their lifetime [3].

Treatments are available for both inactive and active TB infections. Individuals hosting the sleeping bacteria in their bodies, i.e., latently infected persons, are administered some antibiotics by their healthcare providers, and this treatment regimen could last for three to four months [4,8]. Active TB disease can also be treated by several medications like isoniazid, rifabutin (Mycobutin), rifampin (Rimactane), pyrazinamide, ethambutol (Myambutol) and so on. Another way to ensure TB patients adhere to and comply with the treatment regimen is through Directly Observed Therapy, Short-course (DOTS) for TB infections [9]. This TB control strategy (DOTS for TB) was recommended by WHO towards the end of the 20th century and it is touted as the most proven, cost-effective tuberculosis treatment strategy [8]. Several countries have hence adopted DOTS for TB as a control measure towards TB prevention and control [4,9].

While tuberculosis is associated with poverty and its incidence is higher in poorer nations, noninsulin-dependent diabetes mellitus (NIDDM), alternatively named Type 2 diabetes, is a noncommunicable disease mostly common in rich nations in the past, but it is now prevalent in many TB-endemic, low- and middle-income countries (LMIC) [10,11,12]. NIDDM is also a global health threat and it happens due to insulin failure or the inability of the body to properly regulate the glucose in the blood plasma. Risk factors for NIDDM include obesity, poor diets, family history of diabetes, sedentary lifestyles, ethnicity, triglycerides (high cholesterol), non-alcoholic fatty liver disease, tuberculosis disease, etc. It has also been confirmed that there is a relationship between hyperglycemia (too much glucose present in the blood plasma) and the chronic complications, such as nephropathy (diabetic kidney disease), retinopathy, neuropathy, among people living with NIDDM [13,14].

There have also been reported cases of comorbidity of tuberculosis and other non-infectious diseases, particularly cardiovascular disease, cancer, noninsulin-dependent diabetes mellitus, etc. The bidirectional link between tuberculosis and noninsulin-dependent diabetes has been established through several research findings [3,12,14,15,16,17]. NIDDM is an increasing risk factor for tuberculosis disease as a result of immunodeficiency. People who are suffering from NIDDM with untreated tuberculosis infections in the latent form are at a greater risk of acquiring active tuberculosis disease due to weaker immune systems [3,14]. It is also a known fact that NIDDM is responsible for a high risk of treatment failure, mortality, and relapse among patients with active tuberculosis infections. Tuberculosis can also lead to glucose intolerance and severely affect glycemic control [18]. As of today, there is no known cure for NIDDM, but it can be well managed with patient education, healthy diets, lifestyle changes, expertise availability, medication, screening for complications associated with diabetes, proper glycemic control, and sustained observational follow-up [19,20].

The convergence, comorbidity, and concomitant conditions of tuberculosis and noninsulin-dependent diabetes mellitus have been presented in several epidemiological studies [7,12,14,17,21,22,23]. Moualeu et al. [12] developed and analyzed a compartmental TB model that assesses the impact of NIDDM in the spread of the tuberculosis disease in a community. Pan et al. [24] presented a mathematical modeling study on the effect of NIDDM on TB disease control in thirteen countries with a high burden of TB disease. Carvalho et al. [25] studied and assessed the role of diabetes in the progression of TB using a non-integer order model. Jajarmi et al. [26] used a fractional modeling approach to investigate the clinical implications of the comorbidity of tuberculosis and NIDDM. Moya et al. [27] examined the effectiveness of TB disease treatment therapy while taking into consideration the impact of associated diseases (NIDDM and HIV/AIDS). Awad et al. [28] used a mathematical modeling approach that utilized a recently developed age-structure NIDDM and TB co-infection dynamics to predict, estimate, and investigate the impact of noninsulin-dependent diabetes mellitus on mycobacterium tuberculosis epidemiology in Indonesia between 2020 and 2050. Agwu et al. [7] extended the study carried out in [12] by presenting the optimal control and some cost-effectiveness strategies for the mathematical model of TB and NIDDM co-infection dynamics. In [29], a deterministic mathematical model of the SEIS type that accounts for the saturated treatment of individuals exposed to mycobacterium TB in the latent form, along with crowding and psychological effects, was introduced to further study the impact of NIDDM in the transmission mechanism of TB in a community. Recently, Afolabi et al. [30] employ supervised machine learning algorithms to develop models that effectively predict diabetes from patients electronic health records (EHRs). For more detailed work on integer and non-integer compartmental models, see [31,32,33,34,35,36,37].

Over the years, several mathematical epidemiologists have developed and incorporated control strategies into disease models. Oke et al. [38] incorporated both pharmaceutical and non-pharmaceutical intervention strategies into a deterministic COVID-19 model used to garner insight into the transmission mechanism of the pandemic in the USA, South Africa, Italy, and Nigeria. Omede et al. [39] proposed and analyzed the autonomous and non-autonomous mathematical model of the third wave of the novel coronavirus in Nigeria. Olaniyi et al. [40] analyzed a malaria transmission model with protected travelers and partial immunity, where four preventive and control measures are implemented.

Following the lead of the aforementioned studies above, we formulate and analyze a non-autonomous deterministic model that evaluates the effect of the optimal treatment and prevention of the active and latent forms of tuberculosis infections among diabetic patients and nondiabetic individuals towards attaining the WHO’s global strategy to end the present TB epidemic by 2035. The remainder of this manuscript proceeds as follows: the autonomous form of the proposed non-autonomous model formulated in Section 2 is detailed in Section 3 and analyzed; Section 4 is dedicated to the theoretical analysis of non-autonomous mathematical model using the Pontryagin’s Maximum Principle; we illustrate the numerical simulation outputs of the several intervention strategies considered in this work in Section 5 and the last section summarizes our findings.

2. Non-Autonomous Mathematical Model System Formulation

This section is about formulating the optimal control model problem for the co-morbid association of noninsulin-dependent diabetes mellitus and tuberculosis. The model that we present here is a slight modification of the model for tuberculosis transmission proposed in [29], and it is its generalization. The optimal control’s primary goal is to identify the most effective approach to employ in the effort to eradicate TB in the population, that is, we want to derive optimal prevention and treatment strategies that come with minimal costs of implementation. Below are the definitions of the time-dependent controls that will be incorporated into the model that characterizes the dynamical transmission of mycobacterium TB disease in a human population compose of non-diabetic and diabetic individuals:

(a)

$u_1\left(t\right)$ represents a TB prevention effort that screens for LTBIs among key groups of people in the population (i.e., people living with HIV or NIDDM, TB patients, family members, coworkers, and roommates) and promptly provides saturated treatment to those who have TB infections in the latent form. The cost of implementing this control includes identifying and screening key individuals who are at risk of being latently infected with tuberculosis, providing medications or directly observed treatment (DOT) to patients, and also non-death productivity losses by the patients.

(b)

$u_2\left(t\right)$ is the tuberculosis and diabetes intervention to prevent people from developing either or both diseases. This is carried out through health and wellness promotions in the community, i.e., community awareness campaigns that encourage those diagnosed with pre-diabetes or gestational diabetes to go for routine screening with a diagnostic test for diabetes mellitus. Healthy diets and lifestyles advocacy. Prevention efforts towards developing complications are aimed at diabetic patients who are free of complications. Mass sensitization in the community on tuberculosis prevention. The cost of implementing this intervention strategy includes providing health education at home, hospitals, schools, and through mass and social media.

(c)

$u_3\left(t\right)$ denotes TB control strategy that provides treatment to non-diabetic individuals suffering from active tuberculosis disease. The cost of implementing this control involves the amount spent on receiving treatment for tuberculosis by the patients, health care, allocation of resources for the treatment and management of people suffering from active tuberculosis infections and so on.

(d)

$u_4\left(t\right)$ represents the TB treatment effort that caters to people living with tuberculosis and diabetes. The cost of implementing the control $u_4\left(t\right)$ includes the expenses when receiving simultaneous treatment for tuberculosis and noninsulin-dependent diabetes mellitus, management of diabetic complications, health care workers, funding TB-NIDDM clinics, and so on.

The population is divided into nine categories where $S\left(t\right)$ denotes the compartment of susceptible persons to both diabetes and TB, $L\left(t\right)$ is the compartment for non-diabetic individuals exposed to tuberculosis, $A\left(t\right)$ stands for the compartment of non-diabetic individuals infected with tuberculosis, $D\left(t\right)$ is the compartment for diabetic individuals that do not develop complications, $L_d\left(t\right)$ represents the compartment of individuals with diabetes without complications and exposed to tuberculosis, $A_d\left(t\right)$ denote the compartment of actively infected individuals with tuberculosis and diabetic without complications, $C\left(t\right)$ is the class of individuals with complications of NIDDM, $L_c\left(t\right)$ denotes the compartment for individuals exposed to tuberculosis and have NIDDM with complications, and $A_c\left(t\right)$ is the class of infected individuals with tuberculosis and have NIDDM with complications. For simplicity, we let $S=:S\left(t\right),C=:C\left(t\right),A=:A\left(t\right),A_d=:A_d\left(t\right),D=:D\left(t\right)$, $A_c=:A_c\left(t\right),L=:L\left(t\right),L_d=:L_d\left(t\right),\mathrm{and}{L}_c:=L_c\left(t\right).$

The following model assumptions are considered:

• The homogeneous mixing of individuals in the population and people in different classes interact.
• It is assumed that no individual in the population has permanent immunity to tuberculosis.
• The tuberculosis infection progression is from latency to the active disease stage, i.e., people first harbor tuberculosis infections in latent form and later become infectious either through exogenous reinfection or endogenous reactivation.
• We also assume the bidirectional link between tuberculosis and diabetes, i.e., diabetic patients can develop tuberculosis and tuberculosis patients can also become diabetic.

Hence, the non-autonomous model is as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dS}{dt}}& =\Lambda -((1-u_2\left(t\right)){\lambda }_T+\mu +(1-u_2\left(t\right)){\sigma }_1)S+u_3\left(t\right)A+{\displaystyle \frac{u_1\left(t\right)L}{1+q_{a}L}}\hfill \\ \hfill {\displaystyle \frac{dD}{dt}}& ={\sigma }_1(1-u_2\left(t\right))S-(\varphi (1-u_2\left(t\right)){\lambda }_T+\mu +(1-u_2\left(t\right)){\sigma }_2)D+u_4{A}_d\left(t\right)+{\displaystyle \frac{u_1\left(t\right)L_d}{1+q_a{L}_d}}\hfill \\ \hfill {\displaystyle \frac{dC}{dt}}& ={\sigma }_2(1-u_2\left(t\right))D-(\mu +\phi (1-u_2\left(t\right)){\lambda }_T+d_c)C+u_4\left(t\right)A_c+{\displaystyle \frac{u_1\left(t\right)L_c}{1+q_a{L}_c}}\hfill \\ \hfill {\displaystyle \frac{dL}{dt}}& =(1-u_2\left(t\right)){\lambda }_{T}S-({\alpha }_1(1-u_2\left(t\right)){\lambda }_T+(1-u_2\left(t\right)){\mathsf{\tau }}_1+r_1+\mu )L-{\displaystyle \frac{u_1\left(t\right)L}{1+q_{a}L}}\hfill \\ \hfill {\displaystyle \frac{d{L}_d}{dt}}& =\varphi (1-u_2\left(t\right)){\lambda }_{T}D+(1-u_2\left(t\right)){\mathsf{\tau }}_{1}L-({\alpha }_2(1-u_2\left(t\right)){\lambda }_T+r_2+(1-u_2\left(t\right)){\mathsf{\tau }}_2+\mu )L_d-{\displaystyle \frac{u_1\left(t\right)L_d}{1+q_a{L}_d}}\hfill \\ \hfill {\displaystyle \frac{d{L}_c}{dt}}& =\phi (1-u_2\left(t\right)){\lambda }_{T}C+(1-u_2\left(t\right)){\mathsf{\tau }}_2{L}_d-(\mu +{\alpha }_3(1-u_2\left(t\right)){\lambda }_T+r_3+d_1)L_c-{\displaystyle \frac{u_1\left(t\right)L_c}{1+q_a{L}_c}}\hfill \\ \hfill {\displaystyle \frac{dA}{dt}}& =({\alpha }_1(1-u_2\left(t\right)){\lambda }_T+r_1)L-(\mu +{\delta }_1+u_3\left(t\right)+{\gamma }_1(1-u_2\left(t\right)){\sigma }_1)A\hfill \\ \hfill {\displaystyle \frac{d{A}_d}{dt}}& =({\alpha }_2(1-u_2\left(t\right)){\lambda }_T+r_2)L_d+{\gamma }_1(1-u_2\left(t\right)){\sigma }_{1}A-(\mu +{\delta }_2+u_4\left(t\right)+{\gamma }_2(1-u_2\left(t\right)){\sigma }_2)A_d\hfill \\ \hfill {\displaystyle \frac{d{A}_c}{dt}}& =({\alpha }_3(1-u_2\left(t\right)){\lambda }_T+r_3)L_c+{\gamma }_2(1-u_2\left(t\right)){\sigma }_2{A}_d-(\mu +u_4\left(t\right)+{\delta }_3+d_2)A_c\hfill \end{array}$$

(1)

subject to the following initial conditions $0<S\left(0\right)=S^0,0<D\left(0\right)=D^0,0<C\left(0\right)=C^0$, $0\le L\left(0\right)=L^0,0<L_d\left(0\right)=L_d^0,0<L_c\left(0\right)=L_c^0,0\le A\left(0\right)=A^0,0<A_d\left(0\right)=A_d^0$, $0<A_c\left(0\right)=A_c^0$, with ${\lambda }_T={\lambda }_1\left(A\right)+{\lambda }_2\left(A_d\right)+{\lambda }_3\left(A_c\right)$ where

$${\lambda }_1\left(A\right)={\displaystyle \frac{\omega A}{1+bA}},{\lambda }_2\left(A_d\right)={\displaystyle \frac{\omega {\beta }_2{A}_d}{1+b{A}_d}}\mathrm{and}{\lambda }_3\left(A_c\right)={\displaystyle \frac{\omega {\beta }_1{A}_c}{1+b{A}_c}}$$

are saturated incidence rates first introduced by Capasso and Serio [41]. We assume that ${\beta }_1>{\beta }_2>1$ to take into account the increased infectiousness of people suffering from the comorbidity of complications of NIDDM and active tuberculosis disease [10,12]. We emphasize that without the controls (i.e., $u_1\left(t\right),u_2\left(t\right),u_3\left(t\right),u_4\left(t\right)=0$), the model Equation (1) reduces to the autonomous compartmental model introduced in [29] with no saturated treatment of LTBIs and per-capita treatment of active TB. Refer to

Table 1. Model parameters, their definitions and numerical values.

Model Parameter	Description	Baseline Values Per Year	Range	Reference
$\Lambda$	Constant per-capita recruitment rate	667,685	[600,000, 700,000]	[12]
${\sigma }_1$	Rate of acquiring diabetes mellitus without developing complications	0.009	[0.00466, 0.0133]	[12]
${\sigma }_2$	Rate of acquiring complicated form of diabetes mellitus	0.01	[0.00413, 0.0159]	[7]
$d_1$	Death rate as a result of severe complications associated with diabetes in the $L_c$ compartment	0.005	[0.00376, 0.00624]	[7]
$d_2$	Death rate as a result of severe complications associated with diabetes in the $A_c\left(t\right)$ compartment	0.1 $\times d_1$	[0.000398, 0.000602]	[7]
${\alpha }_1$	Exogenous reinfection rate	0.05	[0.0221, 0.0779]	[6]
${\alpha }_2$	Exogenous reinfection rate	$1.01\times {\alpha }_1$	[0.00897, 0.0920]	[7]
${\alpha }_3$	Exogenous reinfection rate	$1.01\times {\alpha }_2$	[0.0349, 0.0671]	[7]
$d_c$	Death rate as a result of complications linked directly to diabetes in the $C\left(t\right)$ compartment	0.1 $\times d_1$	[0.000398, 0.000602]	[7]
${\delta }_1$	Mortality rate owing to active tuberculosis in the A compartment	0.0025	[0.00219, 0.00281]	[6]
${\delta }_2$	Mortality rate owing to active tuberculosis in the $A_d$ compartment	$1.25\times {\delta }_1$	[0.00158, 0.00467]	[7]
${\delta }_3$	Mortality rate owing to active tuberculosis in the $A_c$ compartment	$1.25\times {\delta }_2$	[0.000807, 0.00701]	[7]
$\varphi$	Modification parameter that adjusts the accelerated rate of contracting the Mycobaterium TB in the latent form among diabetic individuals that experienced no complication	1.01	[0.459, 1.543]	[7]
$\phi$	Modification parameter that adjusts the accelerated rate of developing LTBIs among those having diabetes with complications	2.851	[2.247, 3.455]	Assumed
${\gamma }_1$	Modification parameter that adjusts the accelerated rate at which individuals in the $A\left(t\right)$ class acquire noncomplicated form of diabetes	1.01	[1, 1.5]	[7]
${\gamma }_2$	Modification parameter that adjusts the accelerated rate at which those in the $A_d\left(t\right)$ class acquire complicated form of diabetes	1.05	[1, 1.5]	[7]
${\mathsf{\tau }}_1$	Rate of acquiring diabetes mellitus without developing complications in the L compartment	1.01	[1, 2]	[7]
${\mathsf{\tau }}_2$	Rate of acquiring diabetes mellitus without developing complications in the $L_d$ compartment	1.01	[1, 2]	[7]
$\omega$	Transmission probability per contact with infectious diabetic and non-diabetic individuals	$1.65\times {10}^{-7}$	$[1.43\times {10}^{-7},1.87\times {10}^{-7}]$	[42]
$q_a$	Saturating factor	$6.7\times {10}^{-10}$	$[4.53\times {10}^{-10},8.87\times {10}^{-10}]$	[6]
${\beta }_1$	Modification parameter	5.597	[5.168, 6.026]	Assumed
${\beta }_2$	Modification parameter	5.14	[4.723, 5.557]	Assumed
$r_1$	Per-capita rate at which latent TB infections progress to infectious TB disease among individuals without diabetes	0.023	[0.00565, 0.0404]	[42]
$r_2$	Per-capita rate at which latent TB infections progress to infectious TB disease among individuals with noncomplicated form of diabetes	$2\times r_1$	[0.0169, 0.0751]	[7]
$r_3$	Per-capita rate at which latent TB infections progress to infectious TB disease among individuals with complicated form of diabetes	$2\times r_2$	[0.0827, 0.101]	[7]
b	Saturation	0.7	[0, 1]	Assumed
$\mu$	Natural mortality rate per-capita	0.02041	[0.0202, 0.02189]	[43]

for the biological definitions and values of the parameters used in model (1).

We also let $N:=S\left(t\right)+D\left(t\right)+C\left(t\right)+L\left(t\right)+L_d\left(t\right)+L_c\left(t\right)+A\left(t\right)+A_d\left(t\right)+A_c\left(t\right)$ be the total human population. Suppose that $\eta =\min \{d_c,d_1,d_2,{\delta }_1,{\delta }_2,{\delta }_3\}.$ From (1), it can be shown that

$$\begin{array}{cc}\hfill N^{\prime }\left(t\right)& =\Lambda -\mu N\left(t\right)-(d_{c}C+d_1{L}_c+d_2{A}_c+{\delta }_{1}A+{\delta }_2{A}_d+{\delta }_3{A}_c)\hfill \\ \hfill & \le \Lambda -\mu N\left(t\right)-\eta (C+L_c+A_c+A+A_d+A_c)\hfill \\ \hfill & \le \Lambda -\mu N\left(t\right).\hfill \end{array}$$

Assume that N is constant and $\Lambda$ (the recruitment rate) is positive, so, $N\le {\displaystyle \frac{\Lambda }{\mu }}$ when the initial value is $N\left(0\right)\le {\displaystyle \frac{\Lambda }{\mu }}$. Define the domain where the model is meaningful biologically below

$$\mathcal{D}=\left\{(S\left(t\right),D\left(t\right),C\left(t\right),L\left(t\right),L_d\left(t\right),L_c\left(t\right),A\left(t\right),A_d\left(t\right),A_c\left(t\right))\in {\mathbb{R}}_{+}^9\mathrm{such}\mathrm{that}N\left(t\right)\le {\displaystyle \frac{\Lambda }{\mu }}\right\}.$$

Hence, in this work, the state variables $S,D,C,L,L_d,L_c,A,A_d,$ and $A_c$ are restricted to the set $\mathcal{D}$. Let $u_j=u_j\left(t\right)$ for $j=1,\cdots ,4$. $\mathcal{D}$ is a set (domain) such that it is positively invariant with respect to (1), that is, any solution that begins in $\mathcal{D}$ stays there for all $t>0$. Hence, the proposed optimal control mathematical model (1) is mathematically and epidemiologically (biologically insightful) well posed in the region $\mathcal{D}$.

Assume that for $j=1,\cdots ,4$, $u_j\left(t\right)$ are Lebesgue integrable and bounded control functions. Define a Lebesgue measurable control set by

$$U=\left\{u_j\left(t\right)|0\le t\le t_f,u_j\left(t\right)\in [0,1],\mathrm{for}j=1,\cdots ,4\right\}.$$

Minimizing the total count of infectious and exposed individuals in the population and also the implementation costs of those intervention strategies ($u_1,u_2,u_3,$ and $u_4$) that are put in place is the objective of the optimal control technique. We propose the following objective functional J, defined as

$$J(u_1,u_2,u_3,u_4)={\int }_0^{t_f}\left(P_{1}L+P_2{L}_d+P_3{L}_c+P_{4}A+P_5{A}_d+P_6{A}_c+{\displaystyle \frac{1}{2}}\sum _{j=1}^4{Q}_i{u}_i^2\left(t\right)\right)dx,$$

(2)

$t_f$ is the elapsed time for the implementation of those control strategies, $P_j$ for $j=1,\cdots ,6$ represent the positive weight constants of both individuals exposed to and infected with TB, and $Q_1,\cdots ,Q_4$ denote the positive weight constants of $u_j,$ for $j=1,\cdots ,4$, the optimal controls. Note that quadratic treatment costs are assumed, that is, they are not linear in nature. Thus, we want to have $u_1^{*},u_2^{*},u_3^{*},$ and $u_4^{*}$, the optimal control that satisfies

$$J(u_1^{*},u_2^{*},u_3^{*},u_4^{*})=min\left\{(J\left(u_j\right):u_j\in U,j=1,\cdots ,4)\right\},$$

where $U\ne \varnothing$ is the Lebesgue measurable control set.

3. Analysis of the Mathematical Model with Constant Control Parameters

The four time-dependent control parameters, in this section, are considered as constants. We then find the equilibrium point (tuberculosis-free equilibrium), basic reproduction number, sensitivity analysis, and stability criteria for the model (1) where $u_1=:u_1\left(t\right),u_2=:u_2\left(t\right),u_3=:u_3\left(t\right)$, and $u_4=:u_4\left(t\right)$.

3.1. Equilibrium Points

The autonomous model (when we assume constant control parameters) will have the following tuberculosis-free equilibrium point

$${\mathcal{T}}^0=\left({\displaystyle \frac{\Lambda }{\mu +(1-u_2){\sigma }_1}},{\displaystyle \frac{{\sigma }_1(1-u_2)\Lambda }{(\mu +(1-u_2){\sigma }_1)(\mu +(1-u_2){\sigma }_2)}},{\displaystyle \frac{{\sigma }_1{(1-u_2)}^2{\sigma }_2\Lambda }{\mu (\mu +(1-u_2){\sigma }_1)(\mu +(1-u_2){\sigma }_2)}},0,0,0,0,0,0\right).$$

(3)

Hence, using the Next Generation Matrix approach [29,31,44,45], the basic reproduction number of the autonomous model is given below as

$${\mathcal{R}}_0={\mathcal{R}}_c+{\mathcal{R}}_d+{\mathcal{R}}_{s_1}+{\mathcal{R}}_{s_2}$$

(4)

where

$$\begin{array}{cc}\hfill {\mathcal{R}}_c& ={\displaystyle \frac{{\sigma }_1{(1-u_2)}^2{\sigma }_2\Lambda r_3{\beta }_1\phi \omega }{(\mu +(1-u_2){\sigma }_1)(\mu +d_c)(\mu +(1-u_2){\sigma }_2)B_5{B}_6}},\hfill \\ \hfill {\mathcal{R}}_d& ={\displaystyle \frac{\varphi {\sigma }_1(1-u_2)\Lambda \omega }{(\mu +(1-u_2){\sigma }_1)(\mu +(1-u_2){\sigma }_2)}}\left({\displaystyle \frac{{\beta }_1{r}_3(1-u_2){\mathsf{\tau }}_2}{B_3{B}_5{B}_6}}+{\displaystyle \frac{r_2{\beta }_2}{B_3{B}_4}}+{\displaystyle \frac{{\beta }_1{r}_2{\gamma }_2(1-u_2){\sigma }_2}{B_3{B}_4{B}_6}}\right),\hfill \\ \hfill {\mathcal{R}}_{s_1}& ={\displaystyle \frac{\Lambda \omega }{(\mu +(1-u_2){\sigma }_1)}}\left({\displaystyle \frac{r_1}{B_1{B}_2}}+{\displaystyle \frac{r_1{\beta }_2{\gamma }_1(1-u_2){\sigma }_1}{B_1{B}_2{B}_4}}+{\displaystyle \frac{r_1{\beta }_1{\gamma }_1{\gamma }_2{(1-u_2)}^2{\sigma }_1{\sigma }_2}{B_1{B}_2{B}_4{B}_6}}\right),\hfill \\ \hfill {\mathcal{R}}_{s_2}& ={\displaystyle \frac{\Lambda \omega }{(\mu +(1-u_2){\sigma }_1)}}\left({\displaystyle \frac{{\beta }_1{r}_3{(1-u_2)}^2{\mathsf{\tau }}_1{\mathsf{\tau }}_2}{B_1{B}_3{B}_5{B}_6}}+{\displaystyle \frac{{\beta }_2{r}_2(1-u_2){\mathsf{\tau }}_1}{B_1{B}_3{B}_4}}+{\displaystyle \frac{{\beta }_1{\gamma }_2{\sigma }_2{(1-u_2)}^2{r}_2{\mathsf{\tau }}_1}{B_1{B}_3{B}_4{B}_6}}\right),\hfill \end{array}$$

and $B_1=r_1+\mu +u_1+(1-u_2){\mathsf{\tau }}_1$, $B_2=\mu +{\delta }_1+u_3+{\gamma }_1(1-u_2){\sigma }_1$, $B_3=r_2+(1-u_2){\mathsf{\tau }}_2+\mu +u_1$, $B_4=\mu +{\delta }_2+u_4+{\gamma }_2(1-u_2){\sigma }_2$, $B_5=\mu +r_3+d_1+u_1$, and $B_6=\mu +u_4+{\delta }_3+d_2$. The following result on the local stability of the equilibrium point (${\mathcal{T}}^0$) was discussed in [29].

Theorem 1

([29]). The tuberculosis-free equilibrium point denoted by ${\mathcal{T}}^0$ will be locally asymptotically stable if ${\mathcal{R}}_0<1$ and unstable if ${\mathcal{R}}_0>1$.

It should be noted that Theorem 1 is a direct consequence of a theorem in [44] and its epidemiological significance is that tuberculosis disease will be under control in a case when the important threshold ${\mathcal{R}}_0$ is below unity since a small number of people infected with tuberculosis in the population will not lead to a spike in new TB cases whenever ${\mathcal{R}}_0<1.$

Theorem 2.

The partial derivatives of ${\mathcal{R}}_0$ with respect to the constant control parameters $u_1,u_3,$ and $u_4$ are negative.

Proof. 

See Appendix A. □

By Theorem 2, we can conclude that increasing the constant controls $u_1,u_3$ and $u_4$ will result in a reduction in the value of the threshold parameter, ${\mathcal{R}}_0$, thereby helping in lowering the cases of tuberculosis in the population.

3.2. Sensitivity Analysis

Since the model system (1) with constant control parameters has a lot of parameters, we consider it imperative to perform the sensitivity analysis to know those parameters that contribute significantly to the transmission mechanism of TB in a population where people are either diabetes free or living with diabetes. We perform the local sensitivity analysis on the variable ${\mathcal{R}}_0$ using the elastic index or normalized forward sensitivity index technique and the model parameter values in Table 1. We describe below the elastic index on ${\mathcal{R}}_0$ with respect to a parameter $u_1$:

$${\mathcal{E}}_{u_1}^{{\mathcal{R}}_0}={\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial u_1}}\times {\displaystyle \frac{u_1}{{\mathcal{R}}_0}}.$$

Those parameters in

Table 2. Sensitivity index of parameters of model system (1) with constant control parameters.

Parameter	Index	Parameter	Index
$r_1$	0.0247	$r_2$	0.2561
$r_3$	0.5733	${\beta }_1$	0.6768
${\beta }_2$	0.2891	${\mathsf{\tau }}_1$	0.2228
${\sigma }_1$	0.3219	${\sigma }_2$	0.0591
$\omega$	1	$\Lambda$	1
$\varphi$	0.3438	$\phi$	0.1575
${\gamma }_1$	$0.0012$	$u_2$	−0.3958
$u_1$	−1.1599	${\gamma }_2$	$0.00024755$
$d_c$	−0.0038	$u_4$	−0.9211
${\delta }_1$	−0.00016905	${\delta }_2$	−0.0017
${\delta }_3$	−0.0050	$\mu$	−1.4621
$d_1$	−0.0055	$d_2$	−0.00064477
${\mathsf{\tau }}_2$	0.1358	$u_3$	−0.0338

whose sensitive indices are negative (positive) will decrease (increase) the value of ${\mathcal{R}}_0$. It should be emphasized that the parameters $u_1,u_2,u_3,$ and $u_4$ have a negative sensitivity index, and as a consequence, they will lower the value of ${\mathcal{R}}_0$, hence, they can be used as control parameters. If the values of the parameters, such as ${\beta }_1,{\beta }_2,{\sigma }_1,\varphi ,$ and $r_3$ with positive sensitivity, are decreased, the value of the reproduction number will also be decreased. For example, reducing each of the following parameters ($\varphi ,{\sigma }_1,$ and $r_3$) values by 75%, the value of ${\mathcal{R}}_0$ is reduced by 25.79%, 27.93%, and 48.40%, respectively. Also, if ${\beta }_1$ and ${\beta }_2$ are simultaneously decreased by $75\%$ (since ${\beta }_1>{\beta }_2)$, the basic reproduction number (${\mathcal{R}}_0)$ is decreased by 72.44%. This suggests that reducing the number of individuals with uncomplicated diabetes exposed to latent Mycobacterium tuberculosis (TB), preventing the development of uncomplicated diabetes, and controlling infectious TB among people with diabetes and its complications can significantly lower the community burden of TB. Furthermore, reducing the rate of TB transmission from infectious individuals living with the comorbidity of diabetes can help control the TB epidemic in a community. Finally, decreasing all the values of the five parameters $\varphi ,{\beta }_1,{\beta }_2,{\sigma }_1,$ and $r_3$ by 75% at the same time will give a corresponding reduction in the value of the basic reproduction number, such that ${\mathcal{R}}_0=0.6399<1.$ It should be emphasized that the parameters $u_1,u_2,u_3,$ and $u_4$ have negative sensitivity index, and as a consequence, will lower the value of ${\mathcal{R}}_0$, hence, they can be used as control parameters.

4. Theoretical Study of the Optimal Control Model Problem

Dynamic optimization is also a term for the theory of optimal control. This is a crucial tool in Mathematics, and is particularly used in decision-making that involves complex biological systems [46]. This section provides a detailed analysis of the non-autonomous, time-dependent TB-DM model (1). The approach we will take has been implemented extensively in several mathematical models that incorporate control strategies [7,37,38,39,40].

4.1. Existence of an Optimal Control

Theorem 3.

Suppose that J is the objective functional as given in (2) and defined on the non-empty Lebesgue measurable control set U, and subject to (1) such that the initial conditions, at $t=0$, are non-negative. Therefore, there exists $(u_1^{*},u_2^{*},u_3^{*},u_4^{*})$, an optimal control quadruple, denoted by $u^{*}$ for which

$$J(u_1^{*},u_2^{*},u_3^{*},u_4^{*})=min\{\left(J\left(u_j\right)\right):u_j\in U,j=1,2,3,4)\}.$$

Suppose that $x=(S,D,C,L,L_d,L_c,A,A_d,A_c)$, $U={[0,1]}^4$ is the Lebesgue measurable admissible control set, $\nu =(u_1,u_2,u_3,u_4)$, and $g(t,x,\nu )$ denotes the right-hand side of (1) given by

$$\begin{array}{cc}\hfill g(t,x,\nu )& =\left(\begin{array}{cc}& \Lambda -((1-u_2){\lambda }_T+\mu +(1-u_2){\sigma }_1)S+u_{3}A+{\displaystyle \frac{u_{1}L}{1+q_{a}L}}\\ & {\sigma }_1(1-u_2)S-(\varphi (1-u_2){\lambda }_T+\mu +(1-u_2){\sigma }_2)D+u_4{A}_d+{\displaystyle \frac{u_1{L}_d}{1+q_a{L}_d}}\\ & {\sigma }_2(1-u_2)D-(\mu +\phi (1-u_2){\lambda }_T+d_c)C+u_4{A}_c+{\displaystyle \frac{u_1{L}_c}{1+q_a{L}_c}}\\ & (1-u_2){\lambda }_{T}S-({\alpha }_1(1-u_2){\lambda }_T+r_1+(1-u_2\left(t\right)){\mathsf{\tau }}_1+\mu )L-{\displaystyle \frac{u_{1}L}{1+q_{a}L}}\\ & \varphi (1-u_2){\lambda }_{T}D+(1-u_2\left(t\right)){\mathsf{\tau }}_{1}L-({\alpha }_2(1-u_2){\lambda }_T+r_2+(1-u_2){\mathsf{\tau }}_2+\mu )L_d-{\displaystyle \frac{u_1{L}_d}{1+q_a{L}_d}}\\ & \phi (1-u_2){\lambda }_{T}C+(1-u_2){\mathsf{\tau }}_2{L}_d-(\mu +{\alpha }_3(1-u_2){\lambda }_T+r_3+d_1)L_c-{\displaystyle \frac{u_1{L}_c}{1+q_a{L}_c}}\\ & ({\alpha }_1(1-u_2){\lambda }_T+r_1)L-(\mu +{\delta }_1+u_3+{\gamma }_1(1-u_2){\sigma }_1)A\\ & ({\alpha }_2(1-u_2){\lambda }_T+r_2)L_d+{\gamma }_1(1-u_2){\sigma }_{1}A-(\mu +{\delta }_2+u_4+{\gamma }_2(1-u_2){\sigma }_2)A_d\\ & ({\alpha }_3(1-u_2){\lambda }_T+r_3)L_c+{\gamma }_2(1-u_2){\sigma }_2{A}_d-(\mu +u_4+{\delta }_3+d_2)A_c\end{array}\right).\hfill \end{array}$$

(5)

Let $f(t,x,u_j)$ for $j=1,2,3,4$ be the integrand of J, the objective functional, mentioned in (2). By Theorem 2.2 in [46], it is sufficient to prove the following properties to ensure that an optimal control exists:

(I) The permissible control set, U, is closed and convex.

(II) The boundedness of the state system by a linear function in both the control and state variables.

(III) For each $j=1,2,3,4$, $f(t,x,u_j)$ is convex in $u_j$.

(IV) There exist positive constants ${\Psi }_1,{\Psi }_2$ and ${\Psi }_3>1$ such that the Lagrangian is bounded below by

$${\Psi }_1\left(\right|u_j{\left|\right)}^{{\displaystyle \frac{{\Psi }_3}{2}}}-{\Psi }_2,j=1,2,3,4.$$

We establish properties (I)–(IV) as follows:

Proof. 

See Appendix B. □

4.2. Characterization of the Optimal Control

Pontryagin’s maximum principle [47] is used in this section to establish the necessary conditions that an optimal control pair must meet. Now, if we use the principle, we see that (1) and (2) will be transformed into a problem whereby the Hamiltonian function, denoted by H, is minimized pointwise with respect to the following control variables $u_j$, for $j=1,2,3,4.$ Define the Hamiltonian H below:

$$\begin{array}{cc}\hfill H=& P_{1}L+P_2{L}_d+P_3{L}_c+P_{4}A+P_5{A}_d+P_6{A}_c\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left(Q_1{u}_1^2+Q_2{u}_2^2+Q_3{u}_3^2+Q_4{u}_4^2\right)\hfill \\ \hfill & +{\lambda }_S\left[\Lambda -((1-u_2){\lambda }_T+\mu +(1-u_2){\sigma }_1)S+u_{3}A+{\displaystyle \frac{u_{1}L}{1+q_{a}L}}\right]\hfill \\ \hfill & +{\lambda }_D\left[{\sigma }_1(1-u_2)S-(\varphi (1-u_2){\lambda }_T+\mu +(1-u_2){\sigma }_2)D+u_4{A}_d\left(t\right)+{\displaystyle \frac{u_1{L}_d}{1+q_a{L}_d}}\right]\hfill \\ \hfill & +{\lambda }_C\left[{\sigma }_2(1-u_2)D-(\mu +\phi (1-u_2){\lambda }_T+d_c)C+u_4{A}_c+{\displaystyle \frac{u_1{L}_c}{1+q_a{L}_c}}\right]\hfill \\ \hfill & +{\lambda }_L\left[(1-u_2){\lambda }_{T}S-({\alpha }_1(1-u_2){\lambda }_T+(1-u_2){\mathsf{\tau }}_1+r_1+\mu )L-{\displaystyle \frac{u_{1}L}{1+q_{a}L}}\right]\hfill \\ \hfill & +{\lambda }_{L_d}\left[\varphi (1-u_2){\lambda }_{T}D+(1-u_2){\mathsf{\tau }}_{1}L-({\alpha }_2(1-u_2){\lambda }_T+r_2+(1-u_2){\mathsf{\tau }}_2+\mu )L_d-{\displaystyle \frac{u_1{L}_d}{1+q_a{L}_d}}\right]\hfill \\ \hfill & +{\lambda }_{L_c}\left[\phi (1-u_2){\lambda }_{T}C+(1-u_2){\mathsf{\tau }}_2{L}_d-(\mu +{\alpha }_3(1-u_2){\lambda }_T+r_3+d_1)L_c-{\displaystyle \frac{u_1{L}_c}{1+q_a{L}_c}}\right]\hfill \\ \hfill & +{\lambda }_A\left[({\alpha }_1(1-u_2){\lambda }_T+r_1)L-(\mu +{\delta }_1+u_3+{\gamma }_1(1-u_2){\sigma }_1)A\right]\hfill \\ \hfill & +{\lambda }_{A_d}\left[({\alpha }_2(1-u_2){\lambda }_T+r_2)L_d+{\gamma }_1(1-u_2){\sigma }_{1}A-(\mu +{\delta }_2+u_4+{\gamma }_2(1-u_2){\sigma }_2)A_d\right]\hfill \\ \hfill & +{\lambda }_{A_c}\left[({\alpha }_3(1-u_2){\lambda }_T+r_3)L_c+{\gamma }_2(1-u_2){\sigma }_2{A}_d-(\mu +u_4+{\delta }_3+d_2)A_c\right],\hfill \end{array}$$

(6)

where ${\lambda }_S,{\lambda }_L,{\lambda }_A,{\lambda }_D,{\lambda }_{L_d},{\lambda }_{A_d},{\lambda }_C,{\lambda }_{L_c}$, and ${\lambda }_{A_c}$ are the adjoint variables. The next result is the necessary conditions for the optimal control.

Theorem 4.

If there is an optimal control quadruple given by $(u_1^{*},u_2^{*},u_3^{*},u_4^{*})$ such that the objective functional, J, in (2), is minimized over the admissible control set U, and subject to the state system (1), then there are the adjoint variables ${\lambda }_S,{\lambda }_D,{\lambda }_C,{\lambda }_L,{\lambda }_{L_d},{\lambda }_{L_c},{\lambda }_A,{\lambda }_{A_d},{\lambda }_{A_c}$ satisfying the following adjoint system:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_S}{dt}}& =(1-u_2)\omega \left({\displaystyle \frac{A}{1+bA}}+{\displaystyle \frac{{\beta }_2{A}_d}{1+b{A}_d}}+{\displaystyle \frac{{\beta }_1{A}_c}{1+b{A}_c}}\right)({\lambda }_S-{\lambda }_L)+\mu {\lambda }_S+{\sigma }_1(1-u_2)({\lambda }_S-{\lambda }_D)\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_D}{dt}}& =\varphi (1-u_2)({\lambda }_D-{\lambda }_{L_d})\omega \left({\displaystyle \frac{A}{1+bA}}+{\displaystyle \frac{{\beta }_2{A}_d}{1+b{A}_d}}+{\displaystyle \frac{{\beta }_1{A}_c}{1+b{A}_c}}\right)+\mu {\lambda }_D+{\sigma }_2(1-u_2)({\lambda }_D-{\lambda }_C)\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_C}{dt}}& =\phi (1-u_2)\omega \left({\displaystyle \frac{A}{1+bA}}+{\displaystyle \frac{{\beta }_2{A}_d}{1+b{A}_d}}+{\displaystyle \frac{{\beta }_1{A}_c}{1+b{A}_c}}\right)({\lambda }_C-{\lambda }_{L_c})+(\mu +d_c){\lambda }_C\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_L}{dt}}& ={\displaystyle \frac{u_1}{{(1+q_{a}L)}^2}}({\lambda }_L-{\lambda }_S)-P_1+\left({\alpha }_1(1-u_2)\omega \left({\displaystyle \frac{A}{1+bA}}+{\displaystyle \frac{{\beta }_2{A}_d}{1+b{A}_d}}+{\displaystyle \frac{{\beta }_1{A}_c}{1+b{A}_c}}\right)+r_1\right)({\lambda }_L-{\lambda }_A)+\mu {\lambda }_L+(1-u_2){\mathsf{\tau }}_1({\lambda }_L-{\lambda }_{L_d})\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{L_d}}{dt}}& ={\displaystyle \frac{u_1}{{(1+q_a{L}_d)}^2}}({\lambda }_{L_d}-{\lambda }_D)-P_2+\left({\alpha }_2(1-u_2)\omega \left({\displaystyle \frac{A}{1+bA}}+{\displaystyle \frac{{\beta }_2{A}_d}{1+b{A}_d}}+{\displaystyle \frac{{\beta }_1{A}_c}{1+b{A}_c}}\right)+r_2\right)({\lambda }_{L_d}-{\lambda }_{A_d})\hfill \\ \hfill & +{\mathsf{\tau }}_2(1-u_2)({\lambda }_{L_d}-{\lambda }_{L_c})+\mu {\lambda }_{L_d}\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{L_c}}{dt}}& =\left({\alpha }_3(1-u_2)\omega \left({\displaystyle \frac{A}{1+bA}}+{\displaystyle \frac{{\beta }_2{A}_d}{1+b{A}_d}}+{\displaystyle \frac{{\beta }_1{A}_c}{1+b{A}_c}}\right)+r_3\right)({\lambda }_{L_c}-{\lambda }_{A_c})-P_3\hfill \\ \hfill & +(\mu +d_1){\lambda }_{L_c}+{\displaystyle \frac{u_1}{{(1+q_a{L}_c)}^2}}({\lambda }_{L_c}-{\lambda }_C)\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_A}{dt}}& =({\lambda }_A-{\lambda }_{A_d}){\gamma }_1(1-u_2){\sigma }_1+(\mu +{\delta }_1){\lambda }_A-P_4+({\lambda }_A-{\lambda }_S)u_3+{\displaystyle \frac{\widetilde{{\rm Y}}}{{(1+bA)}^2}}\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{A_d}}{dt}}& =({\lambda }_{A_d}-{\lambda }_D)u_4-P_5+{\lambda }_{A_d}(\mu +{\delta }_2)+{\gamma }_2{\sigma }_2(1-u_2)({\lambda }_{A_d}-{\lambda }_{A_c})+{\displaystyle \frac{{\beta }_2\widetilde{{\rm Y}}}{{(1+b{A}_d)}^2}}\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_{A_c}}{dt}}& =({\lambda }_{A_c}-{\lambda }_C)u_4-P_6+(\mu +{\delta }_3+d_2){\lambda }_{A_c}+{\displaystyle \frac{{\beta }_1\widetilde{{\rm Y}}}{{(1+b{A}_c)}^2}},\hfill \end{array}$$

(7)

with transversality conditions

$${\lambda }_S\left(t_f\right)={\lambda }_L\left(t_f\right)={\lambda }_A\left(t_f\right)={\lambda }_D\left(t_f\right)={\lambda }_{L_d}\left(t_f\right)={\lambda }_{A_d}\left(t_f\right)={\lambda }_C\left(t_f\right)={\lambda }_{L_c}\left(t_f\right)={\lambda }_{A_c}\left(t_f\right)=0$$

(8)

and

$$\begin{array}{cc}\hfill u_1^{*}& =min\left\{max\left\{0,{\displaystyle \frac{({\lambda }_L-{\lambda }_S)\frac{L}{1+q_{a}L}+({\lambda }_{L_d}-{\lambda }_D)\frac{L_d}{1+q_a{L}_d}+({\lambda }_{L_c}-{\lambda }_C)\frac{L_c}{1+q_a{L}_c}}{Q_1}}\right\},1\right\}\hfill \\ \hfill u_2^{*}& =min\left\{max\left\{0,{\displaystyle \frac{{\mathcal{T}}_1+{\mathcal{T}}_2}{Q_2}}\right\},1\right\}\hfill \\ \hfill u_3^{*}& =min\left\{max\left\{0,{\displaystyle \frac{({\lambda }_A-{\lambda }_S)A}{Q_3}}\right\},1\right\}\hfill \\ \hfill u_4^{*}& =min\left\{max\left\{0,{\displaystyle \frac{({\lambda }_{A_d}-{\lambda }_D)A_d+({\lambda }_{A_c}-{\lambda }_C)A_c}{Q_4}}\right\},1\right\}\hfill \end{array}$$

(9)

where

$$\begin{array}{cc}\hfill {\mathcal{T}}_1& =\omega \left({\displaystyle \frac{A}{1+bA}}+{\displaystyle \frac{{\beta }_2{A}_d}{1+b{A}_d}}+{\displaystyle \frac{{\beta }_1{A}_c}{1+b{A}_c}}\right)(({\lambda }_L-{\lambda }_S)S+{\alpha }_{1}L({\lambda }_A-{\lambda }_L)\hfill \\ \hfill & +\varphi D({\lambda }_{L_d}-{\lambda }_D)+{\alpha }_2{L}_d({\lambda }_{A_d}-{\lambda }_{L_d})+\phi C({\lambda }_{L_c}-{\lambda }_C)+{\alpha }_3{L}_c({\lambda }_{A_c}-{\lambda }_{L_c})),\hfill \\ \hfill {\mathcal{T}}_2& ={\sigma }_{1}S({\lambda }_D-{\lambda }_S)+{\gamma }_1{\sigma }_{1}A({\lambda }_{A_d}-{\lambda }_A)+({\lambda }_C-{\lambda }_D){\sigma }_{2}D\hfill \\ \hfill & +({\lambda }_{L_c}-{\lambda }_{L_d}){\mathsf{\tau }}_2{L}_d+{\gamma }_2{\sigma }_2{A}_d({\lambda }_{A_c}-{\lambda }_{A_d})+({\lambda }_{L_d}-{\lambda }_L){\mathsf{\tau }}_{1}L,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \widetilde{{\rm Y}}& =\omega (1-u_2)[S({\lambda }_S-{\lambda }_L)+\varphi D({\lambda }_D-{\lambda }_{L_d})+\phi C({\lambda }_C-{\lambda }_{L_c})]\hfill \\ \hfill & +\omega (1-u_2)[{\alpha }_{1}L({\lambda }_L-{\lambda }_A)+{\alpha }_2{L}_d({\lambda }_{L_d}-{\lambda }_{A_d})+{\alpha }_3{L}_c({\lambda }_{L_c}-{\lambda }_{A_c})].\hfill \end{array}\end{array}$$

Proof. 

We take the partial derivatives of the piecewise continuous Hamiltonian function, H, given by (6) with respect to each corresponding state variable to obtain the adjoint system stated in (7). By the Pontryagin maximum principle [47], there exist adjoint variables (${\lambda }_S,{\lambda }_D,{\lambda }_C,{\lambda }_L,{\lambda }_{L_d},{\lambda }_{L_c},{\lambda }_A,{\lambda }_{A_d},{\lambda }_{A_c}$ ) such that

$$\begin{array}{cccccc}\hfill {\displaystyle \frac{d{\lambda }_S}{dt}}& =-{\displaystyle \frac{\partial H}{\partial S}},\hfill & \hfill {\displaystyle \frac{d{\lambda }_L}{dt}}& =-{\displaystyle \frac{\partial H}{\partial L}},\hfill & \hfill {\displaystyle \frac{d{\lambda }_A}{dt}}& =-{\displaystyle \frac{\partial H}{\partial A}},\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_D}{dt}}& =-{\displaystyle \frac{\partial H}{\partial D}},\hfill & \hfill {\displaystyle \frac{d{\lambda }_{L_d}}{dt}}& =-{\displaystyle \frac{\partial H}{\partial L_d}},\hfill & \hfill {\displaystyle \frac{d{\lambda }_{A_d}}{dt}}& =-{\displaystyle \frac{\partial H}{\partial A_d}},\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_C}{dt}}& =-{\displaystyle \frac{\partial H}{\partial C}},\hfill & \hfill {\displaystyle \frac{d{\lambda }_{L_c}}{dt}}& =-{\displaystyle \frac{\partial H}{\partial L_c}},\hfill & \hfill {\displaystyle \frac{d{\lambda }_{A_c}}{dt}}& =-{\displaystyle \frac{\partial H}{\partial A_c}},\hfill \end{array}$$

with transversality conditions stated in (8).

Also, for us to obtain the optimal controls (9), we solve the following equations:

$${\displaystyle \frac{\partial H}{\partial u_j}}=0$$

for $u_j^{*}$, $j=1,2,3,4.$

Lastly, using the standard control arguments involving the bounds on the control parameters, we then conclude that for $j=1,2,3,4$

$$\begin{array}{c}\hfill u_j^{*}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}{\mathcal{J}}_j^{*}\le 0\hfill \\ {\mathcal{J}}_j^{*},\hfill & \mathrm{if}0\le {\mathcal{J}}_j^{*}\le 1\hfill \\ 1,\hfill & \mathrm{if}{\mathcal{J}}_j^{*}\ge 1\hfill \end{array}\right.\end{array}$$

and for which

$$\begin{array}{cc}\hfill {\mathcal{J}}_1^{*}& ={\displaystyle \frac{({\lambda }_L-{\lambda }_S)\frac{L}{1+q_{a}L}+({\lambda }_{L_d}-{\lambda }_D)\frac{L_d}{1+L_d}+({\lambda }_{L_c}-{\lambda }_C)\frac{L_c}{1+q_a{L}_c}}{Q_1}}\hfill \\ \hfill {\mathcal{J}}_2^{*}& ={\displaystyle \frac{{\mathcal{T}}_1+{\mathcal{T}}_2}{Q_2}}\hfill \\ \hfill {\mathcal{J}}_3^{*}& ={\displaystyle \frac{({\lambda }_A-{\lambda }_S)A}{Q_3}}\hfill \\ \hfill {\mathcal{J}}_4^{*}& ={\displaystyle \frac{({\lambda }_{A_d}-{\lambda }_D)A_d+({\lambda }_{A_c}-{\lambda }_C)A_c}{Q_4}}\hfill \end{array}$$

□

5. Numerical Simulations

The goal next is to execute the numerical simulations of the optimality system, which is comprised of the model system (1), and adjoint equations (co-state system) (7) with the following initial conditions $S^0=3,850,000,D^0=250,000;C^0=10,050;$$L^0=78,000$; $L_d^0=95,000;L_c^0=2500;A^0=9200;A_d^0=15,000;A_c^0=590,$ along with the characterization of the optimal control (9), and transversality conditions (8) using the forward-backward sweep procedure in [46]. We choose the following weight constants $P_1=8000,P_2=2$, $P_3=41,P_4=1,P_5=1,P_6=45,Q_1=6,Q_2=1,Q_3=1,Q_4=8000$ for the aim of demonstrating the optimal level of preventive and treatment strategies needed to minimize the number of people actively and latently infected with Mycobacterium TB.

The forward-backward sweep methods begin with initial guesses of $u_j,$ for $j=1,\cdots ,4$ (control variables) and solve (1) and the non-autonomous state system in forward time by employing a Runge-Kutta iterative solver of order four accuracy. The adjoint (co-state) system (7) is then solved in reverse time with both iterative solutions of the state systems and the transversality constraints (8). We keep the admissible controls current by utilizing a convex combination of the controls that have been used previously and the characterization value.

Next, we will illustrate the impact of the control strategies that focus on the prevention of active TB/LTBI and Diabetes; the treatment of active cases of TB among diabetic and non-diabetic individuals and the prevention of the incidence of diabetes/TB and the prompt treatment of TB infections, either in the active or latent form. The baseline values of the model parameters in Table 1 were used for the simulations. The final time for implementing these control strategies is 10 years.

5.1. Intervention Strategy with Prevention of TB and Diabetes

Under this intervention strategy, the focus is mainly on preventing the populace from developing diabetes and tuberculosis infections in either the latent or active form. This preventive effort involves a treatment effort that provides a saturated treatment of TB infections in the latent form and mass sensitization of the community on how to prevent developing tuberculosis and diabetes. We use the control parameters $u_1\left(t\right)$ and $u_2\left(t\right)$ to minimize J, the objective functional (2), while $u_3,u_4=0$. In Figure 1, we observe that TB disease persists in the infected classes ($A,A_d,\mathrm{and}{A}_c$) with or without the control strategy being executed (Figure 1a–c). At the end of the 10-year implementation period of this control strategy, only 1476 and 1082 TB infections were averted in the A and $A_d$ classes, respectively. We observe in Figure 1d,e that the number of people exposed to Mycobaterium TB in latent form in each L and $L_d$ compartments approach zero at the final time $t_f$. The number of individuals latently infected with TB and diabetic (with its complications) decreased significantly with this control strategy and persists without control (Figure 1f).

5.2. Intervention Strategy with Only Treatment of Active Cases of TB

Here, the control strategy involves the combination of treatment efforts of tuberculosis among diabetic and non-diabetic individuals, that is, the control variables $u_3$ and $u_4$ are used to minimize the objective functional, J. The simulation results in Figure 2a,b establish that this intervention strategy will significantly reduce the population of infected humans in A and $A_d$ compartments to two and eight people, respectively, at the end of executing this intervention strategy. The infected individuals in the $A_c$ class (Figure 2c) increased rapidly, and the mycobacterium TB disease is not under control, in fact, more than ten times its original population size, even with the implementation of this intervention strategy in place. The TB disease persists in all the infected classes $A,A_d,$ and $A_c$ (Figure 2a–c) when the control variables are not used and also in the latently infected population $L_c$ with or without implementing the control strategy. We do not notice any clear difference in the populations of the exposed individuals ($L,L_d,\mathrm{and}{L}_c$) when the intervention strategy is used or not. It is observed that the individuals in the $L_d$ and L compartments decrease sharply at the end of 10 years (see Figure 2d–f).

5.3. Intervention Strategy with Prevention and Treatment

The control strategy combines all the control functions ($u_1\left(t\right),u_2\left(t\right),u_3\left(t\right)$ and $u_4\left(t\right)$) and we assume they are nonzero, i.e., providing saturated treatment of latent tuberculosis infections (LTBIs), the treatment of infectious TB cases among diabetic and nondiabetic patients, and preventing the incidence of both diabetes and tuberculosis diseases. Figure 3 demonstrates how the implementation of all the control measures helps to control the spread of TB disease in the population. In Figure 3a–c, we see that the infected population A, and the latently infected populations $L,L_d$ approach zero at the end of the implementation period. The populations in the $A_d,A_c$ and $L_c$ compartments reduced considerably by 99.94%, 95.06% and 99.9%, respectively (Figure 3e,f). In the uncontrolled case, tuberculosis disease persists in all the following compartments ($A,A_d,A_c,L_c$), and the population of latently infected individuals (L and $L_d$ ) declines sharply in 10 years.

In Figure 4, the optimal control profile graphs of different intervention strategies implemented are shown. It is observed that in the intervention strategy with the prevention of active TB and diabetes (Figure 4a), the control parameter $u_1\left(t\right)$ should be sustained at the maximum bound throughout the control period and $u_2\left(t\right)$ may be kept at the minimum level for the first 4 years and ought to be carried on at the highest bound throughout the rest of the implementation of this control strategy. For the intervention strategy with treatment effort towards active tuberculosis disease (Figure 4b), the control variables $u_3\left(t\right)$ and $u_4$ should be maintained at the peak level for the implementation of this intervention strategy. Lastly, for the control strategy that combines all the control parameters, it is significant to mention that the control variable $u_1\left(t\right)$ is to be upheld at the maximum bound throughout. $u_3\left(t\right)$ and $u_4\left(t\right)$ should sustain maximum efforts for 9 and 6 years, respectively. The control variable $u_2\left(t\right)$ may be used at the lowest bound for the first 4.5 years and then should be maintained at its peak bound for the remaining years during the implementation period (see Figure 4c).

6. Conclusions

The analysis of a non-autonomous (optimal control) mathematical model that captures the transmission mechanism of tuberculosis in a population where individuals are either diabetic or not was presented in this study. By taking the time-dependent control variables as constants, we compute ${\mathcal{R}}_0$ (basic reproduction number), an important threshold in epidemiology, of the autonomous form of (1). Subsequently, the tuberculosis-free equilibrium point, ${\mathcal{T}}^0$, was established to be locally asymptotically stable whenever ${\mathcal{R}}_0$ is below unity and unstable whenever ${\mathcal{R}}_0$ exceeds unity. Through sensitivity indexes of the autonomous model parameters relative to ${\mathcal{R}}_0$, we justified the development of (1). Using the normalized forward sensitivity/elasticity index approach, those parameters that would have direct and indirect relationships with the future course of tuberculosis transmission in a population where some individuals are diabetic were identified. We further established that by reducing the rate at which people develop noncomplicated form of diabetes, rate (per-capita) at which noninfectious TB advances to active TB disease among those living with diabetes and its associated complications, and the modification parameter for increased rate of contracting active tuberculosis disease from those suffering from the double burden of tuberculosis and diabetes with/without complications, will result in a considerable decrease in the value of ${\mathcal{R}}_0$. Also, by increasing the time-independent control parameters $u_1,u_2,u_3$ and $u_4$, the tuberculosis disease burden in the population will be diminished.

The analysis of the non-autonomous model (1) is built on the theory of optimal control. To find optimal intervention strategies for tuberculosis prevention and control, we considered the following four time-dependent functions which, in short, are the treatment efforts that screen for and diagnose LTBIs among key groups and immediately provide saturated treatment, preventive measures towards the development of either or both tuberculosis and diabetes diseases through community sensitization programs on radio, social media, television, etc., and the treatment efforts of infectious TB disease for people living with or without diabetes. We presented the theoretical analysis of the non-autonomous model problemand several simulation results of the four intervention (control) strategies considered in this work. It was noted that without the control strategies utilized, achieving control over the tuberculosis epidemic in the population will be a considerable challenge. The intervention strategy that has all four controls implemented is the optimal control strategy for preventing and controlling tuberculosis in the population, thereby, resulting in very few latent and active tuberculosis cases at the end of the implementation period of 10 years.