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

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 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 (80) and endemic equilibrium (81) was studied. Therefore, the local and global stability analysis are carried out. It is proved that 80 is locally asymptotically stable once R is less than. However, if R > 1 then 80 is unstable. We proved also that 81 is locally asymptotically stable once R > 1. The global stability of both equilibrium 80 and 81 is discussed where we proved that 80 is globally asymptotically stable once R <= 1, and 81 is globally asymptotically stable once R > 1. The sensitivity analysis of the basic reproduction number 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.