Optimal control of an age-structured SVIRS model with treatment and vaccination
We develop and analyze an age-structured SVIRS epidemic model withВ ages of vaccination  and recovery which allows for variations inВ the exit rate of the removed as a function of age (time since recovery)В and variations in the vaccine loss rate as a function of age (timeВ since vaccination). We show that the age-structured model has oneВ disease-free steady state and an endemic steady state. Local andВ global stability analysis show that the disease-free steady stateВ is locally and globally stable if the reproduction number is belowВ one. This means that if the reproduction number is reduced belowВ one, say through vaccination or treatment, the disease will beВ eliminated. In addition, to provide a better understanding of theВ interaction between treatment and vaccination, we formulate anВ optimal control problem  for the age-structured model and then derive the existence of solutions to the optimal control problemВ from optimal control theory. The optimal treatment and vaccinationВ strategies are obtained by solving the corresponding optimalityВ system numerically. It is demonstrated by numerical simulationsВ that the effectivenessВ of the optimal therapy and vaccination protocolsВ would be strongly affected by the recovery age of the removed.
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