## 4 Theoretical Study of the Optimal Control Model Problem
### Characterization of the Optimal Control  
[PERSON]'s maximum principle [47] is used in this section to establish the necessary conditions that an optimal control pair must meet. Now, if we use the principle, we see that (1) and (2) will be transformed into a problem whereby the Hamiltonian function, denoted by \(H\), is minimized pointwise with respect to the following control variables \(u_{j}\), for \(j=1,2,3,4\). Define the Hamiltonian \(H\) below:  
_and_  
\[u_{1}^{*} = min\bigg{\{}max\bigg{\{}0,\frac{(\lambda_{L}-\lambda_{S})\frac{L}{1+ q_{s}L}+(\lambda_{L_{d}}-\lambda_{D})\frac{L_{d}}{1+q_{s}L_{d}}+(\lambda_{L_{c}}- \lambda_{C})\frac{L_{c}}{1+q_{s}L_{c}}}\bigg{\}},1\bigg{\}}\] \[u_{2}^{*} = min\bigg{\{}max\bigg{\{}0,\frac{\mathcal{T}_{1}+\mathcal{T}_{2}} {Q_{2}}\bigg{\}},1\bigg{\}} \tag{9}\] \[u_{3}^{*} = min\bigg{\{}max\bigg{\{}0,\frac{(\lambda_{A}-\lambda_{S})A}{Q_{ 3}}\bigg{\}},1\bigg{\}}\] \[u_{4}^{*} = min\bigg{\{}max\bigg{\{}0,\frac{(\lambda_{A_{d}}-\lambda_{D})A_ {d}+(\lambda_{A_{c}}-\lambda_{C})A_{c}}{Q_{4}}\bigg{\}},1\bigg{\}}\]  
_where_  
\[\mathcal{T}_{1} = \omega\bigg{(}\frac{A}{1+bA}+\frac{\beta_{2}A_{d}}{1+bA_{d}}+ \frac{\beta_{1}A_{c}}{1+bA_{c}}\bigg{)}((\lambda_{L}-\lambda_{S})S+\alpha_{1} L(\lambda_{A}-\lambda_{L})\] \[+\phi D(\lambda_{L_{d}}-\lambda_{D})+\alpha_{2}L_{d}(\lambda_{A_ {d}}-\lambda_{L_{d}})+\varphi\mathcal{C}(\lambda_{L_{c}}-\lambda_{C})+\alpha_ {3}L_{c}(\lambda_{A_{c}}-\lambda_{L_{c}})),\] \[\mathcal{T}_{2} = \sigma_{1}S(\lambda_{D}-\lambda_{S})+\gamma_{1}\sigma_{1}A( \lambda_{A_{d}}-\lambda_{A})+(\lambda_{C}-\lambda_{D})\sigma_{2}D\] \[+(\lambda_{L_{c}}-\lambda_{L_{d}})\tau_{2}L_{d}+\gamma_{2}\sigma _{2}A_{d}(\lambda_{A_{c}}-\lambda_{A_{d}})+(\lambda_{L_{d}}-\lambda_{L})\tau_ {1}L,\]  
_and_  
\[\tilde{\Upsilon} = \omega(1-u_{2})[S(\lambda_{S}-\lambda_{L})+\phi D(\lambda_{D}- \lambda_{L_{d}})+\varphi\mathcal{C}(\lambda_{C}-\lambda_{L_{c}})]\] \[+ \omega(1-u_{2})[\alpha_{1}L(\lambda_{L}-\lambda_{A})+\alpha_{2}L_ {d}(\lambda_{L_{d}}-\lambda_{A_{d}})+\alpha_{3}L_{c}(\lambda_{L_{c}}-\lambda_ {A_{c}})].\]  
Proof. : We take the partial derivatives of the piecewise continuous Hamiltonian function, \(H\), given by (6) with respect to each corresponding state variable to obtain the adjoint system stated in (7). By the Pontryagin maximum principle [47], there exist adjoint variables \((\lambda_{S},\lambda_{D},\lambda_{C},\lambda_{L},\lambda_{L_{d}},\lambda_{L_{ c}},\lambda_{A},\lambda_{A_{d}},\lambda_{A_{c}})\) such that  
\[u_{2}^{*} = min\bigg{\{}max\bigg{\{}0,\frac{\mathcal{T}_{1}+\mathcal{T}_{2}} {Q_{2}}\bigg{\}},1\bigg{\}} \tag{9}\]0 \[u_{2}^{*} = min\bigg{\{}max\bigg{\{}0,\frac{\mathcal{T}_{1}+\mathcal{T}_{2}} {Q_{2}}\bigg{\}},1\bigg{\}} \tag{9}\]1 \[u_{2}^{*} = min\bigg{\{}max\bigg{\{}0,\frac{\mathcal{T}_{1}+\mathcal{T}_{2}} {Q_{2}}\bigg{\}},1\bigg{\}} \tag{9}\]2  
with transversality conditions stated in (8). Also, for us to obtain the optimal controls (9), we solve the following equations:  
\[u_{2}^{*} = min\bigg{\{}max\bigg{\{}0,\frac{\mathcal{T}_{1}+\mathcal{T}_{2}} {Q_{2}}\bigg{\}},1\bigg{\}} \tag{9}\]3  
for \(u_{j}^{*}\), \(j=1,2,3,4\). Lastly, using the standard control arguments involving the bounds on the control parameters, we then conclude that for \(j=1,2,3,4\)  
\[u_{j}^{*}= \begin{cases}0,&\text{if}\quad\mathcal{J}_{j}^{*}\leq 0\\ \mathcal{J}_{j}^{*},&\text{if}\quad 0\leq\mathcal{J}_{j}^{*}\leq 1\\ 1,&\text{if}\quad\mathcal{J}_{j}^{*}\geq 1\end{cases} \]and for which  
\[\mathcal{J}_{1}^{*} =\frac{(\lambda_{L}-\lambda_{S})\frac{L}{1+q_{L}L}+(\lambda_{L_{d}}- \lambda_{D})\frac{L}{1+L_{d}}+(\lambda_{L_{c}}-\lambda_{C})\frac{L}{1+q_{L}L_{c }}}{\mathbb{Q}_{1}}\] \[\mathcal{J}_{2}^{*} =\frac{\mathcal{T}_{1}+\mathcal{T}_{2}}{\mathbb{Q}_{2}}\] \[\mathcal{J}_{3}^{*} =\frac{(\lambda_{A}-\lambda_{S})A}{\mathbb{Q}_{3}}\] \[\mathcal{J}_{4}^{*} =\frac{(\lambda_{A_{d}}-\lambda_{D})A_{d}+(\lambda_{A_{c}}- \lambda_{C})A_{c}}{\mathbb{Q}_{4}}\]