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

Abstract

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 (E0) and endemic equilibrium (E1) was studied. Therefore, the local and global stability analysis are carried out. It is proved that E0 is locally asymptotically stable once R is less than. However, if R > 1 then E0 is unstable. We proved also that E1 is locally asymptotically stable once R > 1. The global stability of both equilibrium E0 and E1 is discussed where we proved that E0 is globally asymptotically stable once R ≤ 1, and E1 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.