# Mathematical Study for Chikungunya Virus with Nonlinear General Incidence Rate
## 2 Mathematical Model and Results  
Consider the following mathematical model describing the CHIKV dynamics with two routes of infection which is a generalization of the mathematical model given in [18].  
\[\left\{\begin{array}{ll}\delta&=\Gamma-\delta s-\eta_{1}pf(s)-\eta_{2}yf(s),\\ \dot{y}&=\eta_{1}pf(s)+\eta_{2}yf(s)-\epsilon y,\\ \dot{p}&=\pi y-cp-rxp,\\ \dot{x}&=\lambda+\rho xp-mx.\end{array}\right. \tag{1}\]  
The variables \(s,y,p\) and \(x\) describe the number of uninfected and infected cells, the CHIKV virus, and antibodies, respectively. The model parameters are nonnegative and are described as follows.  
\begin{tabular}{c l} \hline
**Parameter** & **Description** \\ \hline \(\Gamma\) & uninfected cells recruitment rate \\ \hline \(\delta\) & Uninfected cells mortality rate \\ \hline \(\epsilon\) & Infected cells mortality rate \\ \hline \(c\) & CHIKV mortality rate \\ \hline \(m\) & Antibodies loss rate \\ \hline \(\pi\) & Generation rate of the virus by infected cells \\ \hline \(r\) & Rate at which Antibodies attack the virus \\ \hline \(\lambda\) & Antibodies expansion rate \\ \hline \(\rho\) & Proliferative rate of antibodies, once antigen is encountered \\ \hline \(\eta_{1},\eta_{2}\) & Incidence rate constants \\ \hline \end{tabular}  
Note that \((\eta_{1}pf(s)+\eta_{2}yf(s))\) represents the number of cells disappearing from recruitment in the susceptible compartment and entering into an infected compartment, where \(f\) is a nonlinear increasing function.  
Using nonlinear incidence rates of the form \(\eta_{1}f(s)p\) and \(\eta_{2}f(s)y\) into this model is important because the number of effective contacts between infective cells and susceptible cells may increase at high infective levels due to crowding of susceptible cells. If the function \(f(s)\) is increasing for large values of \(s\), used for interpreting the \"psychological\" effects: for high number of susceptible cells, the infection risk increase as the number of susceptible cells increases, since the total number of cells may tend to increase the number of contacts.  
Let \(s_{0}=\dfrac{\Gamma}{\delta}\) and \(x_{0}=\dfrac{\lambda}{m}\). We make the following assumption that will be used in the rest of the paper.  
**Assumption 1**.: \(f\) _is an increasing, nonnegative \(C^{1}(\mathbb{R}_{+})\) function such that \(f(0)=0\) and \(f(s_{0})<\dfrac{\epsilon}{\eta_{2}}\)._  
Assumption 1 states that the CHIKV-cell and the infected-cell incidence rates increase with the number of susceptible cells. Assumption 1 also considers that no CHIKV-cell and infected-cell infections can take place in the absence of susceptible cells.  
The classical Monod functions can be used to express the transmission rate of infections from infected to susceptible cells as well as from CHIKV to susceptible cells:  
\[f(s)=\dfrac{\bar{\mu}s}{k+s},\]  
where \(\bar{\mu}\) represents the transmission rates of the disease and \(k\) is the Monod constant which is proportional both to the number of CHIKV pathogens and infected cells when the saturated incidence rate is \(\bar{\mu}/2\).  
The last condition of Assumption 1 is of a mathematical artifice that we used to prove the existence and uniqueness of the infected equilibrium point.