# Mathematical Study for Chikungunya Virus with Nonlinear General Incidence Rate
## 2 Mathematical Model and Results
### Basic Reproduction Number and Equilibrium Points  
The basic reproduction rate is a dimensionless quantity which allows the measurement of the ability of an infectious cell to spread infection through a given cell's population immediately after its introduction. From a mathematical point of view, it allows, under certain conditions, the stability of the points of equilibrium of a dynamic system to be established. [PERSON] et al. (2017, 2018) designed a way to calculate the basic reproduction number \(\mathcal{R}_{0}\), named the next-generation matrix method, and this was adapted by [PERSON] and [PERSON] (2019). \[\text{Here, }F=\left( \begin{array}{ccc}\eta_{2}f(s_{0})&\eta_{1}f(s_{0})&0\\ 0&0&0\\ 0&0&0\\ \end{array} \right)\text{ and }V=\left( \begin{array}{ccc}\epsilon&0&0\\ -\pi&rx_{0}+c&0\\ 0&-\rho x_{0}&m\\ \end{array} \right).\text{ The determinant of }V\text{ is given by }\det(V)=me(rx_{0}+c)>0;\text{ thus,}\]\[V^{-1}=\frac{1}{me(rx_{0}+c)}\left( \begin{array}{ccc}m(rx_{0}+c)&0&0\\ m\pi&me&0\\ \pi\rho x_{0}&\epsilon px_{0}&\epsilon(rx_{0}+c)\end{array} \right)\text{ and the next-generation matrix is given by}\] \[FV^{-1}=\frac{1}{me(rx_{0}+c)}\left( \begin{array}{ccc}m(rx_{0}+c) \eta_{2}f(s_{0})+m\pi\eta_{1}f(s_{0})&me\eta_{1}f(s_{0})&0\\ 0&0&0\\ 0&0&0\end{array} \right).\]  
Then, the basic reproduction number of system (1) is calculated as the spectral radius of the matrix \(FV^{-1}\):  
\[\mathcal{R}_{0}=\frac{m(rx_{0}+c)\eta_{2}f(s_{0})+m\pi\eta_{1}f(s_{0})}{m \epsilon(rx_{0}+c)}=\frac{\eta_{2}f(s_{0})}{\epsilon}+\frac{\pi\eta_{1}f(s_{0} )}{\epsilon(rx_{0}+c)}.\]  
**Lemma 3**. :
* _If_ \(\mathcal{R}_{0}\leq 1\)_, then (_1_) admits only_ \(E_{0}\in\Omega\) _as an equilibrium point._
* _If_ \(\mathcal{R}_{0}>1\)_, then (_1_) admits two equilibrium points,_ \(E_{0}\in\Omega\) _and_ \(E_{1}\in\overset{\circ}{\Omega}\)_, and here,_ \(\overset{\circ}{\Omega}\) _represents the interior of the set_ \(\Omega\)_._  
Proof. : Let \(E(s,y,p,x)\) be an equilibrium point satisfying  
\[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\] \[0 =\eta_{1}pf(s)+\eta_{2}yf(s)-\epsilon y,\] (3) \[0 =\pi y-cp-rxp,\] (4) \[0 =\lambda+\rho xp-mx. \tag{5}\]  
The resolution of Equations (2)-(5) gives us the CHIKV-free equilibrium point \(E_{0}=(s_{0},0,0,x_{0})\). Moreover, we have  
\[x=\frac{\lambda}{m-\rho p},\] \[y=\frac{cp+rxp}{\pi}=\frac{cp}{\pi}+\frac{r\lambda p}{r(m-\rho p )},\] \[s=\frac{\Gamma-\epsilon y}{\delta}=s_{0}-\frac{cep}{\pi\delta}- \frac{r\lambda ep}{\pi\delta(m-\rho p)},\] \[\eta_{1}pf(s)=(\epsilon-\eta_{2}f(s))y.\]  
We define the function  
\[g(p) = \eta_{1}f(s)+(\eta_{2}f(s)-\epsilon)\frac{y}{p}\] \[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\]0 \[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\]1  
Then, we obtain  
\[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\]2 \[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\]3 \[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\]4Now, we have \(\lim_{p\to(m/\rho)^{-}}\Big{(}s_{0}-\frac{cep}{\pi\delta}-\frac{r\lambda ep}{\pi \delta(m-\rho p)}\Big{)}=-\infty\), and then, \(\lim_{p\to(m/\rho)^{-}}\eta_{1}f\Big{(}s_{0}-\frac{cep}{\pi\delta}-\frac{r \lambda ep}{\pi\delta(m-\rho p)}\Big{)}<0\) and \(\lim_{p\to(m/\rho)^{-}}\eta_{2}f\Big{(}s_{0}-\frac{cep}{\pi\delta}-\frac{r \lambda ep}{\pi\delta(m-\rho p)}\Big{)}<0\). One deduces, therefore, that  
\[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\]5  
The derivative of the function \(g\) is given by  
\[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\]6 \[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\]7 \[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\]8 \[0 =\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s), \tag{2}\]9 \[0 =\eta_{1}pf(s)+\eta_{2}yf(s)-\epsilon y,\]0  
and thus, by Assumption 1,  
\[0 =\eta_{1}pf(s)+\eta_{2}yf(s)-\epsilon y,\]1  
Therefore, the equation \(g(p)=0\) admits a unique solution \(p_{1}\in(0,\frac{m}{\rho})\).