# Mathematical Study for Chikungunya Virus with Nonlinear General Incidence Rate
## 2 Mathematical Model and Results
### Local Stability
\right|\]  
\[=(X+\delta)\left[(X+\epsilon-\eta_{2}f(s_{1}))\Big{(}(X+c+rx_{1})(X+m-\rho p_ {1})+r\rho p_{1}x_{1}\Big{)}\right.\]  
\[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]  
\[\Big{(}(X+c+rx_{1})(X+m-\rho p_{1})+r\rho p_{1}x_{1}\Big{)}.\]  
The characteristic polynomial \(P(X)=0\) if, and only if  
\[\left[(X+\delta)(X+\epsilon-\eta_{2}f(s_{1}))+(\eta_{1}p_{1}f^{ \prime}(s_{1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\right]\] \[\Big{(}(X+c+rx_{1})(X+m-\rho p_{1})+r\rho p_{1}x_{1}\Big{)}=\pi \eta_{1}f(s_{1})(X+\delta)(X+m-\rho p_{1})\]  
or also,  
\[\left[(X+\delta)(X+\epsilon-\eta_{2}f(s_{1}))+(\eta_{1}p_{1}f^{ \prime}(s_{1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\right]=\] \[\frac{\pi\eta_{1}f(s_{1})(X+\delta)(X+m-\rho p_{1})}{\Big{(}(X+c+ rx_{1})(X+m-\rho p_{1})+r\rho p_{1}x_{1}\Big{)}}.\]  
Suppose that \(X\) is an eigenvalue with \(Re(X)\geq 0\); then, since \((\epsilon-\eta_{2}f(s_{1}))=\frac{\eta_{1}p_{1}f(s_{1})}{y_{1}}\) and \(\frac{p_{1}}{y_{1}}=\frac{\pi}{(c+rx_{1})}\), the left-hand side satisfies  
\[\left|(X+\delta)(X+\epsilon-\eta_{2}f(s_{1}))+(\eta_{1}p_{1}f^{ \prime}(s_{1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\right| > (\epsilon-\eta_{2}f(s_{1}))|X+\delta|\] \[= \frac{\eta_{1}p_{1}f(s_{1})}{y_{1}}|X+\delta|\] \[= \frac{\pi\eta_{1}f(s_{1})}{c+rx_{1}}|X+\delta|,\]  
and the right-hand side satisfies\[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]0 \[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]1 \[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]2  
Hence, the contradiction occurs; thus, \(Re(X)<0\) for all eigenvalues \(X\), and therefore, \(E_{1}\) is locally asymptotically stable. **3. Global Stability**  
Consider the function \(G(z)=z-1-\ln z\) that will be used in this section. Theorem 3.: _If \(\mathcal{R}_{0}\leq 1\), then the CHIKV-free equilibrium point \(E_{0}\) is globally asymptotically stable._  
Proof. : Assume that \(\mathcal{R}_{0}\leq 1\) and define the following Lyapunov function \(U_{0}(s,y,p,x)\):  
\[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]3  
Clearly, \(U_{0}(s,y,p,x)>0\) for all \(s,y,p,x>0\) and \(U_{0}(s_{0},0,0,x_{0})=0\). The derivative of \(U_{0}\) with respect to time along the system (1) is given by:  
\[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]4 \[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]5 \[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]6 \[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]7  
\[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]8 \[\left.-\pi\eta_{1}f(s_{1})(X+m-\rho p_{1})\right]+(\eta_{1}p_{1}f^{\prime}(s_{ 1})+\eta_{2}y_{1}f^{\prime}(s_{1}))(X+\epsilon)\]9 \[\Big{(}(X+c+rx_{1})(X+m-\rho p_{1})+r\rho p_{1}x_{1}\Big{)}.\]0 \[\Big{(}(X+c+rx_{1})(X+m-\rho p_{1})+r\rho p_{1}x_{1}\Big{)}.\]1 \[\Big{(}(X+c+rx_{1})(X+m-\rho p_{1})+r\rho p_{1}x_{1}\Big{)}.\]2 \[\Big{(}(X+c+rx_{1})(X+m-\rho p_{1})+r\rho p_{1}x_{1}\Big{)}.\]3  
If \(\mathcal{R}_{0}\leq 1\), then \(\dot{U}_{0}\leq 0\) for all \(s,y,p,x>0\). Let \(W_{0}=\{(s,y,p,x):\dot{U}_{0}=0\}\). It can be easily shown that \(W_{0}=\{E_{0}\}\). Applying [PERSON]'s invariance principle [30] (see [20; 21; 31; 32] for other applications), we deduce that \(E_{0}\) is GAS when \(\mathcal{R}_{0}\leq 1\).