## 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:  
1. \(u_{1}(t)\) 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. 2. \(u_{2}(t)\) 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. 3. \(u_{3}(t)\) 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. 4. \(u_{4}(t)\) represents the TB treatment effort that caters to people living with tuberculosis and diabetes. The cost of implementing the control \(u_{4}(t)\) 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 \(S(t)\) denotes the compartment of susceptible persons to both diabetes and TB, \(L(t)\) is the compartment for non-diabetic individuals exposed to tuberculosis, \(A(t)\) stands for the compartment of non-diabetic individuals infected with tuberculosis, \(D(t)\) is the compartment for diabetic individuals that do not develop complications, \(L_{d}(t)\) represents the compartment of individuals with diabetes without complications and exposed to tuberculosis, \(A_{d}(t)\) denote the compartment of actively infected individuals with tuberculosis and diabetic without complications, \(C(t)\) is the class of individuals with complications of NIDDM, \(L_{c}(t)\) denotes the compartment for individuals exposed to tuberculosis and have NIDDM with complications, and \(A_{c}(t)\) is the class of infected individuals with tuberculosis and have NIDDM with complications. For simplicity, we let \(S=:S(t),C=:C(t),A=:A(t),A_{d}=:A_{d}(t),D=:D(t),\)\(A_{c}=:A_{c}(t),L=:L(t),L_{d}=:L_{d}(t),\) and \(L_{c}:=L_{c}(t)\).