## 3 Analysis of the Mathematical Model with Constant Control Parameters
### Equilibrium Points  
The autonomous model (when we assume constant control parameters) will have the following tuberculosis-free equilibrium point \[\mathcal{T}^{0}=\bigg{(}\frac{\Lambda}{\mu+(1-u_{2})\sigma_{1}},\frac{\sigma_{1}(1-u _{2})\Lambda}{(\mu+(1-u_{2})\sigma_{1})(\mu+(1-u_{2})\sigma_{2})},\frac{\sigma_{ 1}(1-u_{2})^{2}\sigma_{2}\Lambda}{\mu(\mu+(1-u_{2})\sigma_{1})(\mu+(1-u_{2}) \sigma_{2})},0,0,0,0,0\bigg{)}. \tag{3}\]  
Hence, using the Next Generation Matrix approach [29; 31; 44; 45], the basic reproduction number of the autonomous model is given below as  
\[\mathcal{R}_{0}=\mathcal{R}_{c}+\mathcal{R}_{d}+\mathcal{R}_{s_{1}}+\mathcal{ R}_{s_{2}} \tag{4}\]  
where  
\[\mathcal{R}_{c} =\frac{\sigma_{1}(1-u_{2})^{2}\sigma_{2}\Lambda r_{3}\beta_{1} \varphi\omega}{(\mu+(1-u_{2})\sigma_{1})(\mu+d_{c})(\mu+(1-u_{2})\sigma_{2})B_ {5}B_{6}},\] \[\mathcal{R}_{d} =\frac{\phi\sigma_{1}(1-u_{2})\Lambda\omega}{(\mu+(1-u_{2}) \sigma_{1})(\mu+(1-u_{2})\sigma_{2})}\bigg{(}\frac{\beta_{1}r_{3}(1-u_{2}) \tau_{2}}{B_{3}B_{5}B_{6}}+\frac{r_{2}\beta_{2}}{B_{3}B_{4}}+\frac{\beta_{1}r_ {2}\gamma_{2}(1-u_{2})\sigma_{2}}{B_{3}B_{4}B_{6}}\bigg{)},\] \[\mathcal{R}_{s_{1}} =\frac{\Lambda\omega}{(\mu+(1-u_{2})\sigma_{1})}\bigg{(}\frac{r_{ 1}}{B_{1}B_{2}}+\frac{r_{1}\beta_{2}\gamma_{1}(1-u_{2})\sigma_{1}}{B_{1}B_{2} B_{4}}+\frac{r_{1}\beta_{1}\gamma_{1}\gamma_{2}(1-u_{2})^{2}\sigma_{1} \sigma_{2}}{B_{1}B_{2}B_{4}B_{6}}\bigg{)},\] \[\mathcal{R}_{s_{2}} =\frac{\Lambda\omega}{(\mu+(1-u_{2})\sigma_{1})}\bigg{(}\frac{ \beta_{1}r_{3}(1-u_{2})^{2}\tau_{1}\tau_{2}}{B_{1}B_{3}B_{5}B_{6}}+\frac{ \beta_{2}r_{2}(1-u_{2})\tau_{1}}{B_{1}B_{3}B_{4}}+\frac{\beta_{1}\gamma_{2} \sigma_{2}(1-u_{2})^{2}r_{2}\tau_{1}}{B_{1}B_{3}B_{4}B_{6}}\bigg{)},\]  
and \(B_{1}=r_{1}+\mu+u_{1}+(1-u_{2})\tau_{1}\), \(B_{2}=\mu+\delta_{1}+u_{3}+\gamma_{1}(1-u_{2})\sigma_{1}\), \(B_{3}=r_{2}+(1-u_{2})\tau_{2}+\mu+u_{1}\), \(B_{4}=\mu+\delta_{2}+u_{4}+\gamma_{2}(1-u_{2})\sigma_{2}\), \(B_{5}=\mu+r_{3}+d_{1}+u_{1}\), 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.