Bifurcation analysis and optimal control of an infection age-structured epidemic model with vaccination and treatment
Résumé
This paper deals with the study and control of the transmission of an infectious disease described
by a Susceptible–Vaccinated–Infected (SVI) epidemiological model. This model takes into account the
age structure of the infection, imperfect vaccination and therapeutic treatment of infected people who
become susceptible again. We show that the rate of therapeutic treatment can lead to bistability of
disease–free and an endemic equilibria. This phenomenon is known as backward bifurcation and can
allow the disease to persist in a population even if the basic reproduction number is less than one. The
existence of this bifurcation means that controlling this infectious disease remains extremely difficult.
This is why we are developing an optimal control strategy combining vaccination and treatment in
order to minimize the disease spread in the population. We then consider vaccination and treatment
rates as time–dependent functions. This allows us to formulate an optimal control problem whose
objective is to minimize the number of infected individuals as well as the cost of applying controls. We
establish first–order necessary conditions for optimality and characterize optimal controls. Numerical
simulations illustrate the fact that a combination of intervention measures can significantly reduce
the transmission dynamics of infectious diseases by keeping the population of infected individuals
relatively low.
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