A mathematical investigation of an "SVEIR" epidemic model for the measles transmission

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<abstract><p>A generalized "SVEIR" epidemic model with general nonlinear incidence rate has been proposed as a candidate model for measles virus dynamics. The basic reproduction number $\mathcal{R}$, an important epidemiologic index, was calculated using the next generation matrix method. The existence and uniqueness of the steady states, namely, disease-free equilibrium ($\mathcal{E}_0$) and endemic equilibrium ($\mathcal{E}_1$) was studied. Therefore, the local and global stability analysis are carried out. It is proved that $\mathcal{E}_0$ is locally asymptotically stable once $\mathcal{R}$ is less than. However, if $\mathcal{R} > 1$ then $\mathcal{E}_0$ is unstable. We proved also that $\mathcal{E}_1$ is locally asymptotically stable once $\mathcal{R} > 1$. The global stability of both equilibrium $\mathcal{E}_0$ and $\mathcal{E}_1$ is discussed where we proved that $\mathcal{E}_0$ is globally asymptotically stable once $\mathcal{R}\leq 1$, and $\mathcal{E}_1$ is globally asymptotically stable once $\mathcal{R} > 1$. The sensitivity analysis of the basic reproduction number $\mathcal{R}$ with respect to the model parameters is carried out. In a second step, a vaccination strategy related to this model will be considered to optimise the infected and exposed individuals. We formulated a nonlinear optimal control problem and the existence, uniqueness and the characterisation of the optimal solution was discussed. An algorithm inspired from the Gauss-Seidel method was used to resolve the optimal control problem. Some numerical tests was given confirming the obtained theoretical results.</p></abstract>