## 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  
Figure 4: Simulation results showing the profiles of the control functions used for each intervention strategy implemented.  
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.  
Conceptualization, S.R.; Methodology, S.R., O.S.I., and S.I.O.; Software, S.R., O.S.I. and S.I.O.; Validation, S.I.O., O.S.I. and B.A.W.; Formal analysis, S.R.; Investigation, S.R., O.S.I. and B.A.W.; Data curation, S.R.; Writing--original draft, S.R.; Writing--review & editing, S.R., O.S.I., S.I.O. and B.A.W.; Visualization, S.R. and S.I.O.; Supervision, O.S.I. and B.A.W. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The Matlab codes employed to run the numerical experiments are available upon request to the authors.  
The authors declare that they have no conflicts of interest.  
We are grateful to the three anonymous reviewers and the handling editor for careful reading, constructive comments, and helpful suggestions, which have led to an improvement of the paper.