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

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

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A generalized "SVEIR" epidemic gul rune d2 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 quadruple topical ointment for dogs $, 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.