## 2 Non-Autonomous Mathematical Model System Formulation
The following model assumptions are considered:  
1. The homogeneous mixing of individuals in the population and people in different classes interact. 2. It is assumed that no individual in the population has permanent immunity to tuberculosis. 3. 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. 4. 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:\[ \begin{split}\frac{dS}{dt}&=\Lambda-((1-u_{2}(t))\lambda_{T} +\mu+(1-u_{2}(t))\sigma_{1})S+u_{3}(t)A+\frac{u_{1}(t)L}{1+q_{a}L}\\ \frac{dD}{dt}&=\sigma_{1}(1-u_{2}(t))S-(\phi(1-u_{2 }(t))\lambda_{T}+\mu+(1-u_{2}(t))\sigma_{2})D+u_{4}A_{d}(t)+\frac{u_{1}(t)L_{ d}}{1+q_{a}L_{d}}\\ \frac{dC}{dt}&=\sigma_{2}(1-u_{2}(t))D-(\mu+\varphi( 1-u_{2}(t))\lambda_{T}+d_{c})C+u_{4}(t)A_{c}+\frac{u_{1}(t)L_{c}}{1+q_{a}L_{c} }\\ \frac{dL}{dt}&=(1-u_{2}(t))\lambda_{T}S-(\alpha_{1 }(1-u_{2}(t))\lambda_{T}+(1-u_{2}(t))\tau_{1}+r_{1}+\mu)L-\frac{u_{1}(t)L}{1+ q_{a}L}\\ \frac{dL_{d}}{dt}&=\phi(1-u_{2}(t))\lambda_{T}D+(1- u_{2}(t))\tau_{1}L-(\alpha_{2}(1-u_{2}(t))\lambda_{T}+r_{2}+(1-u_{2}(t)) \tau_{2}+\mu)L_{d}-\frac{u_{1}(t)L_{d}}{1+q_{a}L_{d}}\\ \frac{dL_{c}}{dt}&=\varphi(1-u_{2}(t))\lambda_{T}C+ (1-u_{2}(t))\tau_{2}L_{d}-(\mu+\alpha_{3}(1-u_{2}(t))\lambda_{T}+r_{3}+d_{1}) L_{c}-\frac{u_{1}(t)L_{c}}{1+q_{a}L_{c}}\\ \frac{dA}{dt}&=(\alpha_{1}(1-u_{2}(t))\lambda_{T}+ \tau_{1})L-(\mu+\delta_{1}+u_{3}(t)+\gamma_{1}(1-u_{2}(t))\sigma_{1})A\\ \frac{dA_{c}}{dt}&=(\alpha_{2}(1-u_{2}(t))\lambda_{T }+\tau_{2})L_{d}+\gamma_{1}(1-u_{2}(t))\sigma_{1}A-(\mu+\delta_{2}+u_{4}(t)+ \gamma_{2}(1-u_{2}(t))\sigma_{2})A_{d}\\ \frac{dA_{c}}{dt}&=(\alpha_{3}(1-u_{2}(t))\lambda_{T }+\tau_{3})L_{c}+\gamma_{2}(1-u_{2}(t))\sigma_{2}A_{d}-(\mu+u_{4}(t)+\delta_{ 3}+d_{2})A_{c}\\ \end{split}  \tag{1}\]  
subject to the following initial conditions \(0<S(0)=S^{0},0<D(0)=D^{0},0<C(0)=C^{0}\), \(0\leq L(0)=L^{0},0<L_{d}(0)=L^{0}_{d},0<L_{c}(0)=L^{0}_{c},0\leq A(0)=A^{0},0< A_{d}(0)=A^{0}_{d}\), \(0<A_{c}(0)=A^{0}_{c}\), with \(\lambda_{T}=\lambda_{1}(A)+\lambda_{2}(A_{d})+\lambda_{3}(A_{c})\) where  
\[\lambda_{1}(A)=\frac{\omega A}{1+bA},\quad\lambda_{2}(A_{d})=\frac{\omega\beta _{2}A_{d}}{1+bA_{d}}\quad\text{and}\quad\lambda_{3}(A_{c})=\frac{\omega\beta_ {1}A_{c}}{1+bA_{c}}\]  
are saturated incidence rates first introduced by [PERSON] and [PERSON] [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., \(u_{1}(t),u_{2}(t),u_{3}(t),u_{4}(t)=0\)), 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 for the biological definitions and values of the parameters used in model (1).