Optimal vaccination strategy of a constrained time-varying SEIR epidemic model

The optimal control strategy for time-varying epidemic models remains a wide open research area. In this paper, the optimal vaccination strategy for a constrained time-varying SEIR (Susceptible, Exposed, Infected and Recovered) epidemic model is solved under the frame of constrained optimal control problems. Three time-varying factors, i.e., seasonally varying incidence coefficient, monotonic decreasing successfully immune rate and monotonic increasing vaccine yield are considered. Constraints on vaccination rate, susceptible population and vaccine supply at each time instant, which are pure-control constraint, pure-state constraint and mixed state-control constraint, respectively, are all taken into consideration. The characterization for the optimal control is derived with the help of the Pontryagin's maximum principle. And optimal control problems are successfully solved by a symplectic pseudospectral method numerically. Numerical results are consistent with the analytical ones. Finally, comparisons of different cases demonstrate that time-varying factors could alter the optimal vaccination strategy, which implies that omitting the time-varying factors may result in less optimal even unreasonable control strategy.