# Mathematical Study for Chikungunya Virus with Nonlinear General Incidence Rate
## 4 Optimal Susceptible Recruitment Rate
### Existence and Uniqueness of the Solution  
Define the variable \(\varphi=(s,y,p,x)^{l}\); then, the model (1) takes the simple form  
\[\dot{\varphi}=G(\varphi)=A\varphi+F(\varphi), \tag{10}\]  
\[\text{where }A=\left( \begin{array}{cccc}-\delta&0&0&0\\ 0&-\varepsilon&0&0\\ 0&\pi&-c&0\\ 0&0&0&-m\end{array} \right)\text{and }F(\varphi)=\left( \begin{array}{c} \Gamma-pf(s)-yf(s)\\ pf(s)+yf(s)\\ -rxp\\ \lambda+\rho xp\end{array} \right).\]  
**Proposition 1**. : _The function \(G\) is continuous and uniformly Lipschitz._Proof. : The function \(F\) is continuous and uniformly Lipschitz, since  
\[ \begin{array}{rcl}\left\|F(\varphi_{1})-F(\varphi_{2})\right\|_{1}&=&\Big{|}-p_ {1}f(s_{1})-y_{1}f(s_{1})+p_{2}f(s_{2})+y_{2}f(s_{2})\Big{|}\\ &&+\Big{|}p_{1}f(s_{1})+y_{1}f(s_{1})-p_{2}f(s_{2})-y_{2}f(s_{2})\Big{|}+r\Big{|} -x_{1}p_{1}+x_{2}p_{2}\Big{|}+\rho\Big{|}x_{1}p_{1}-x_{2}p_{2}\Big{|}\\ &\leq&2\Big{|}p_{1}f(s_{1})+y_{1}f(s_{1})-p_{2}f(s_{2})-y_{2}f(s_{2})\Big{|}+2 \max(r,\rho)\Big{|}x_{1}p_{1}-x_{2}p_{2}\Big{|}\\ &\leq&2\Big{|}(p_{1}+y_{1})f(s_{1})-(p_{2}+y_{2})f(s_{2})\Big{|}+2\max(r,\rho )\Big{|}x_{1}(p_{1}-p_{2})+p_{2}(x_{1}-x_{2})\Big{|}\\ &\leq&2\Big{|}(p_{1}+y_{1})f(s_{1})-(p_{1}+y_{1})f(s_{2})+(p_{1}+y_{1})f(s_{2} )-(p_{2}+y_{2})f(s_{2})\Big{|}\\ &&+2\max(r,\rho)M_{3}\big{|}p_{1}-p_{2}\Big{|}+2\max(r,\rho)M_{2}\big{|}x_{1}-x_ {2}\Big{|}\\ &\leq&2\big{|}p_{1}+y_{1}\left|f(s_{1})-f(s_{2})\right|+2\left|f(s_{2})\right| \left|\left(p_{1}+y_{1}\right)-(p_{2}+y_{2})\right|\\ &&+2\max(r,\rho)M_{3}\big{|}p_{1}-p_{2}\Big{|}+2\max(r,\rho)M_{2}\big{|}x_{1}- x_{2}\Big{|}\\ &\leq&2(M_{1}+M_{2})\left|L|s_{1}-s_{2}\right|+2\left|f\left(p_{1}-p_{2}+|y_{1}-y_{2} \right|\right)\\ &&+2\max(r,\rho)M_{3}\big{|}p_{1}-p_{2}\Big{|}+2\max(r,\rho)M_{2}\big{|}x_{1}- x_{2}\Big{|}\\ &\leq&M\big{\|}\varphi_{1}-\varphi_{2}\big{\|}_{1},\end{array} \]  
where \(M=2\max((M_{1}+M_{2})\ L,f,f+M_{3}\max(r,\rho),M_{2}\max(r,\rho))\). Since  
\[\left\|A\varphi_{1}-A\varphi_{2}\right\|_{1}\leq\left\|A\right\|_{1}\big{\|} \varphi_{1}-\varphi_{2}\big{\|}_{1}, \tag{11}\]  
where \(\left\|A\right\|_{1}:=\sup_{X\
eq 0}\frac{\left\|AX\right\|_{1}}{\left\|X \right\|_{1}}\) is the matrix norm of \(A\) subordinate to the vector norm \(\left\|\cdot\right\|_{1}\). Therefore,  
\[\left\|G(\varphi_{1})-G(\varphi_{2})\right\|_{1}\leq K\big{\|}\varphi_{1}- \varphi_{2}\big{\|}_{1}, \tag{12}\]  
where \(K=\max(M,\left\|A\right\|)\), and then the function \(G\) is uniformly Lipschitz and continuous. Since the function \(G\) is uniformly Lipschitz and continuous, then the solution of system (10) exists and is unique. Let us apply the maximum principle of [PERSON] [33; 34; 35] to derive the necessary conditions for the considered optimal control and the corresponding states. The Hamiltonian is  
\[ \begin{array}{rcl}H&=&-\alpha_{1}s+\alpha_{2}y+\frac{\alpha_{3}}{2}\Gamma^{ 2}+\lambda_{1}s+\lambda_{2}\dot{y}+\lambda_{3}\dot{p}+\lambda_{4}\dot{x}\\ &=&-\alpha_{1}s+\alpha_{2}y+\frac{\alpha_{3}}{2}\Gamma^{2}+\lambda_{1}(\Gamma- \delta s-pf(s)-yf(s))+\lambda_{2}(pf(s)+yf(s)-\epsilon y)\\ &&\quad+\lambda_{3}(\pi y-cp-rxp)+\lambda_{4}(\lambda+\rho xp-mx).\end{array}  \tag{13}\]  
For a given optimal control, \(\Gamma^{*}\), there exist adjoint functions \(\lambda_{1},\lambda_{2},\lambda_{3}\) and \(\lambda_{4}\) associated to the states \(s,y,p\) and \(x\) such that:  
\[\left\{ \begin{array}{rcl}\dot{\lambda}_{1}&=&-\frac{\partial H}{\partial s}= \alpha_{1}+\lambda_{1}(\delta+\eta_{1}pf^{\prime}(s)+\eta_{2}yf^{\prime}(s))- \lambda_{2}(\eta_{1}pf^{\prime}(s)+\eta_{2}yf^{\prime}(s)),\\ \dot{\lambda}_{2}&=&-\frac{\partial H}{\partial y}=-\alpha_{2}+\lambda_{1}\eta _{2}f(s)+\lambda_{2}(\epsilon-\eta_{2}f(s))-\lambda_{3}\pi,\\ \dot{\lambda}_{3}&=&-\frac{\partial H}{\partial p}=\lambda_{1}\eta_{1}f(s)- \lambda_{2}\eta_{1}f(s)+\lambda_{3}(c+rx)-\lambda_{4}\rho x,\\ \dot{\lambda}_{4}&=&-\frac{\partial H}{\partial x}=\lambda_{3}rp+\lambda_{4}(m -\rho p)\end{array} \right. \tag{14}\]  
where \(\lambda_{i}(T)=0\), for \(i=1,2,3,4\), are the transversality conditions.