A discrete-time vaccination SEIR model

In the present we consider a discrete-time version of a vaccination SEIR epidemic model with a cubic incidence rate. We study the dynamics of the model based on the theory of cubic stochastic operators. We find all the fixed points of the cubic operator and show that allmost the fixed points are non-hyperbolic. Furthermore, by analysing the orbits, we prove that the cubic operator has a regularity property. Numerical simulations are carried out to illustrate the theoretical results. Next, we solve an optimal control problem to minimise the infectious burden while balancing the costs. Using the discrete Pontryagin maximum principle we find the optimal strategies via adjoint dynamics. We demonstrated that the proposed discrete-time model also represents realistic scenarios of the disease propagation process, and that such analytical models can be used in the decision of some epidemiological problems.