# Mathematical Study for Chikungunya Virus with Nonlinear General Incidence Rate
## 2 Mathematical Model and Results
### Local Stability  
In epidemiology, the basic reproduction number \(\mathcal{R}_{0}\) of an infection is the average number of secondary cases caused by an individual with a communicable disease in a fully susceptible population. More precisely, \(\mathcal{R}_{0}\) is the average number of people a contagious person can infect. This rate is calculated from a population that is fully susceptible to infection and which has not yet been vaccinated or immunized against an infectious agent. Theorem 1: _If \(\mathcal{R}_{0}<1\), then the disease-free steady-state \(E_{0}\) is locally asymptotically stable, and if \(\mathcal{R}_{0}>1\), it is unstable._  
Proof: The Jacobian matrix at point \((s,y,p,x)\) is given by:  
\[J=\left( \begin{array}{cccc}-\delta-\eta_{1}pf^{\prime}(s)-\eta_{2}yf^{ \prime}(s)&-\eta_{2}f(s)&-\eta_{1}f(s)&0\\ \eta_{1}pf^{\prime}(s)+\eta_{2}yf^{\prime}(s)&\eta_{2}f(s)-\epsilon&\eta_{1}f (s)&0\\ 0&\pi&-(c+rx)&-rp\\ 0&0&\rho x&\rho p-m\end{array} \right).\]  
Its value at \(E_{0}\) is given by:  
\[J_{0}=\left( \begin{array}{cccc}-\delta&-\eta_{2}f(s_{0})&-\eta_{1}f(s_{0})& 0\\ 0&\eta_{2}f(s_{0})-\epsilon&\eta_{1}f(s_{0})&0\\ 0&\pi&-(c+rx_{0})&0\\ 0&0&\rho x_{0}&-m\end{array} \right).\]  
\(J_{0}\) admits four eigenvalues; \(\lambda_{1}=-\delta<0\) and \(\lambda_{2}=-m<0\). \(\lambda_{3}\) and \(\lambda_{4}\) are eigenvalues of the sub-matrix  
\[S_{j_{0}}:=\left( \begin{array}{cccc}\eta_{2}f(s_{0})-\epsilon&\eta_{1}f(s_{ 0})\\ \pi&-(c+rx_{0})\end{array} \right).\]  
The trace of \(S_{j_{0}}\) is given by  
\[\mathrm{Tr}(S_{j_{0}}) = \eta_{2}f(s_{0})-\epsilon-(c+rx_{0})\] \[= -(c+rx_{0})-\epsilon\Big{(}1-\frac{\eta_{2}f(s_{0})}{\epsilon} \Big{)}\] \[\leq -(c+rx_{0})-\epsilon\Big{(}1-\frac{\eta_{2}f(s_{0})}{\epsilon}- \frac{\pi\eta_{1}f(s_{0})}{\epsilon(c+rx_{0})}\Big{)}\] \[\leq -(c+rx_{0})-\epsilon\Big{(}1-\mathcal{R}_{0}\Big{)}\]  
and the determinant of \(S_{j_{0}}\) is given by  
\[\mathrm{Det}(S_{j_{0}}) = -(c+rx_{0})\Big{(}\eta_{2}f(s_{0})-\epsilon\Big{)}-\pi\eta_{1}f(s _{0})\] \[= -\epsilon(c+rx_{0})\Big{(}\frac{\eta_{2}f(s_{0})}{\epsilon}-1+ \frac{\pi\eta_{1}f(s_{0})}{\epsilon(c+rx_{0})}\Big{)}\] \[= -\epsilon(c+rx_{0})\Big{(}\mathcal{R}_{0}-1\Big{)}\] \[= \epsilon(c+rx_{0})\Big{(}1-\mathcal{R}_{0}\Big{)}.\]  
Then, the steady-state \(E_{0}\) is locally asymptotically stable if \(\mathcal{R}_{0}<1\), and it is unstable if \(\mathcal{R}_{0}>1\). Theorem 2: _If \(\mathcal{R}_{0}>1\), then the infected steady-state \(E_{1}\) is locally asymptotically stable._  
Proof: The Jacobian matrix at a point \(E_{1}=(s_{1},y_{1},p_{1},x_{1})\) is given by:  
\[I_{1} = \left( \begin{array}{cccc}-\delta-\eta_{1}p_{1}f^{\prime}(s_{1} )-\eta_{2}y_{1}f^{\prime}(s_{1})&-\eta_{2}f(s_{1})&-\eta_{1}f(s_{1})&0\\ \eta_{1}p_{1}f^{\prime}(s_{1})+\eta_{2}y_{1}f^{\prime}(s_{1})&\eta_{2}f(s_{1} )-\epsilon&\eta_{1}f(s_{1})&0\\ 0&\pi&-(c+rx_{1})&-rp_{1}\\ 0&0&\rho x_{1}&\rho p_{1}-m\end{array} \right).\]The characteristic polynomial is then given by:  
\[P(X)=\left| \begin{array}{cccc}-X-\delta-\eta_{1}p_{1}f^{\prime}(s_{1})-\eta_{ 2}y_{1}f^{\prime}(s_{1})&-\eta_{2}f(s_{1})&-\eta_{1}f(s_{1})&0\\ \eta_{1}p_{1}f^{\prime}(s_{1})+\eta_{2}y_{1}f^{\prime}(s_{1})&-X+\eta_{2}f(s_{ 1})-\epsilon&\eta_{1}f(s_{1})&0\\ 0&\pi&-X-(c+rx_{1})&-rp_{1}\\ 0&0&\rho x_{1}&-X+\rho p_{1}-m\\ \end{array} \right|\]  
\[=\left| \begin{array}{cccc}-(X+\delta)&-(X+\epsilon)&0&0\\ \eta_{1}p_{1}f^{\prime}(s_{1})+\eta_{2}y_{1}f^{\prime}(s_{1})&-X+\eta_{2}f(s_{ 1})-\epsilon&\eta_{1}f(s_{1})&0\\ 0&\pi&-X-(c+rx_{1})&-rp_{1}\\ 0&0&\rho x_{1}&-X+\rho p_{1}-m\\ \end{array} \right|\]  
\[=-(X+\delta)\left| \begin{array}{cccc}-X+\eta_{2}f(s_{1})-\epsilon&\eta_{1}f (s_{1})&0\\ \pi&-X-(c+rx_{1})&-rp_{1}\\ 0&\rho x_{1}&-X+\rho p_{1}-m\\ \end{array} \right|\]  
\[+(X+\epsilon)\left| \begin{array}{cccc}\eta_{1}p_{1}f^{\prime}(s_{1})+\eta_{ 2}y_{1}f^{\prime}(s_{1})&\eta_{1}f(s_{1})&0\\ 0&-X-(c+rx_{1})&-rp_{1}\\ 0&\rho x_{1}&-X+\rho p_{1}-m\\ \end{array}