Backward bifurcation and optimal control of an SVI age-structured epidemic model
Résumé
In this work, we study a Susceptible-Vaccinated-Infectious (SVI) epidemiological model with an infection age structure, an imperfect preventive vaccine administered to susceptible individuals and a therapeutic treatment of infectious ones. First, using integrated semigroup theory, we handle the well-posedness property of the model proposed and derive the explicit expression of the basic reproduction number R 0 that determines whether the disease dies out or persists. Using the Lyapunov-Schmidt type technique, we derive a necessary and sufficient condition for the occurrence of a backward bifurcation. Second, we extend the model by considering the vaccination and therapeutic treatment as time-dependent control variables. An optimal control problem is formulated with the aim of minimizing the number of infected individuals and the cost associated with vaccination and treatment. The existence of an optimal control pair is proved and the first-order necessary condition for optimality are established. Finally, the optimal control problem is solved numerically and simulations are provided. They show that the combination of these two control strategies considerably reduce the spread of the disease by keeping the population of infected individuals relatively low.
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