ON THE TIME-OPTIMAL VACCINATION CONTROL FOR AN SEIR EPIDEMIC MODEL WITH EVENTUAL MODELLING ERRORS

This paper presents a time-optimal vaccination control for an SEIR (susceptible-exposed-infectious-recovered by immunity, or immune, subpopulations) epidemic model under a bang-bang vaccination control. The model can eventually include generic uncertainties with parameterization errors and unmodeled dynamics. The designed bang-bang control operates with two design "a priori" vaccination control levels and chooses the switching time instants between both of them. Both values are chosen being compatible with the positivity and global stability of the epidemic model. The two constant vaccination controls define two possible disease-free equilibrium points in the absence of switching actions which are stable if the disease transmission rate lies below a certain critical value. It is assumed that the disease transmission rate is below such a critical value so that the resulting disease-free equilibrium point under any constant vaccination control, or, in general, if the vaccination is time-varying but it converges to a constant value, is asymptotically stable. The time-optimal vaccination control is generated from a design chosen constant value plus an incremental value which is generated by the minimization of the Hamiltonian associated with the minimal-time loss function. The targeted state final value is defined as a certain closed ball around some point being a reasonable approximate measure of both existing disease-free equilibrium points associated with the two vaccination levels used for the time-optimal control. Numerical examples are discussed to evaluate the proposed optimization method.