## 2 Non-Autonomous Mathematical Model System Formulation
\begin{table}
\begin{tabular}{c l c c c} \hline \hline
**Model Parameter** & **Description** & **Baseline Values Per Year** & **Range** & **Reference** \\ \hline \(\delta_{1}\) & Mortality rate owing to active & \multirow{2}{*}{0.0025} & \multirow{2}{*}{[0.00219, 0.00281]} & [6] \\  & tuberculosis in the \(A\) compartment & & & \\ \hline \(\delta_{2}\) & Mortality rate owing to active & \multirow{2}{*}{1.25 \(\times\)\(\delta_{1}\)} & \multirow{2}{*}{[0.00158, 0.00467]} & [7] \\  & tuberculosis in the \(A_{d}\) compartment & & & \\ \hline \(\delta_{3}\) & Mortality rate owing to active & \multirow{2}{*}{1.25 \(\times\)\(\delta_{2}\)} & \multirow{2}{*}{[0.000807, 0.00701]} & [7] \\  & tuberculosis in the \(A_{c}\) compartment & & & \\ \hline \multirow{4}{*}{\(\phi\)} & Modification parameter that adjusts & \multirow{2}{*}{1.01} & \multirow{2}{*}{[0.459, 1.543]} & [7] \\  & the Mycobacterium TB in the latent & & & \\  & form among diabetic individuals & & & \\  & that experienced no complication & & & \\ \hline \multirow{4}{*}{\(\varphi\)} & Modification parameter that adjusts & \multirow{4}{*}{2.851} & \multirow{4}{*}{[2.247, 3.455]} & \multirow{4}{*}{Assumed} \\  & the accelerated rate of developing & & & \\  & LTBIs among those having diabetes & & & \\  & with complications & & & \\ \hline \multirow{4}{*}{\(\gamma_{1}\)} & Modification parameter that adjusts & \multirow{4}{*}{1.01} & \multirow{4}{*}{[1, 1.5]} & [7] \\  & the accelerated rate at which & & & \\  & individuals in the \(A(t)\) class acquire & & & \\  & noncomplicated form of diabetes & & & \\ \hline \multirow{4}{*}{\(\gamma_{2}\)} & Modification parameter that adjusts & \multirow{4}{*}{1.05} & \multirow{4}{*}{[1, 1.5]} & [7] \\  & the accelerated rate at which those in & & & \\  & the \(A_{d}(t)\) class acquire complicated & & & \\  & form of diabetes & & & \\ \hline \multirow{4}{*}{\(\gamma_{1}\)} & Rate of acquiring diabetes mellitus & \multirow{4}{*}{1.01} & \multirow{4}{*}{[1, 2]} & [7] \\  & without developing complications in & & & \\  & the \(L\) compartment & & & \\ \hline \multirow{4}{*}{\(\gamma_{2}\)} & Rate of acquiring diabetes mellitus & \multirow{4}{*}{1.01} & \multirow{4}{*}{[1, 2]} & [7] \\  & without developing complications in & & & \\  & the \(L_{d}\) compartment & & & \\ \hline \multirow{4}{*}{\(\omega\)} & Transmission probability per contact & \multirow{4}{*}{1.65 \(\times\)\(10^{-7}\)} & \multirow{4}{*}{[1.43 \(\times\)\(10^{-7}\), \(1.87\times\)\(10^{-7}\)]} & [42] \\  & with infectious diabetic and & & & \\  & non-diabetic individuals & & & \\ \hline \(q_{a}\) & Saturating factor & \(6.7\times\)\(10^{-10}\) & \([4.53\times\)\(10^{-10}\), \(8.87\times\)\(10^{-10}]\) & [6] \\ \hline \(\beta_{1}\) & Modification parameter & \multirow{2}{*}{5.597} & \multirow{2}{*}{[5.168, 6.026]} & Assumed \\ \hline \(\beta_{2}\) & Modification parameter & \multirow{2}{*}{5.14} & \multirow{2}{*}{[4.723, 5.557]} & Assumed \\ \hline \multirow{4}{*}{\(r_{1}\)} & Per-capita rate at which latent TB & \multirow{4}{*}{0.023} & \multirow{4}{*}{[0.00565, 0.0404]} & [42] \\  & infections progress to infectious TB & \multirow{4}{*}{0.023} & \multirow{4}{*}{[0.00565, 0.0404]} & [42] \\  & disease among individuals without & & & \\  & diabetes & & & \\ \hline \multirow{4}{*}{\(r_{2}\)} & Per-capita rate at which latent TB & \multirow{4}{*}{2 \(\times\)\(r_{1}\)} & \multirow{4}{*}{[0.0169, 0.0751]} & [7] \\  & infections progress to infectious TB & \multirow{4}{*}{Incomplicated form of diabetes} & & \\  & diseases among individuals with & & & \\  & complicated form of diabetes & & & \\ \hline \multirow{4}{*}{\(r_{3}\)} & Per-capita rate at which latent TB & \multirow{4}{*}{2 \(\times\)\(r_{2}\)} & \multirow{4}{*}{[0.0827, 0.101]} & [7] \\  & infections progress to infectious TB & \multirow{4}{*}{2 \(\times\)\(r_{2}\)} & \multirow{4}{*}{[0.0827, 0.101]} & [7] \\  & disease among individuals with & & & \\  & complicated form of diabetes & & & \\ \hline \(b\) & Saturation & \multirow{2}{*}{0.7} & \multirow{2}{*}{[0, 1]} & Assumed \\ \hline \(\mu\) & Natural mortality rate per-capita & \multirow{2}{*}{0.02041} & \multirow{2}{*}{[0.0202, 0.02189]} & [43] \\ \hline \hline \end{tabular}
\end{table}