For the last decades, mathematical epidemiological models have been used to understand the dynamics of infectious diseases and guide public health policy. In particular, several continuous models have been considered to study influenza outbreaks and their controls policies. However, most epidemiological data is discrete; therefore, a discrete formulation is more convenient to compare collected data with the output of the model. We introduce a discrete time model in order to study optimal control strategies for influenza transmission.

In our model, we divide the population into four classes: susceptible, infectious, treated, and recovered individuals. In particular, we evaluate the potential effect of control measures, such as social distancing and antiviral treatment, on the dynamics of a single influenza outbreak. The objective is to reduce the total number of infected individuals at the end of the epidemic in a most economical way. We solve the problem by using two different techniques. The first one is a discrete version of Pontryagin's maximum principle, which uses the forward-backward algorithm. In the second approach, we propose to solve the problem by using the primal-dual interior-point method that enforces physical conditions explicitly. We conclude that the primal-dual interior-point algorithm solves the problem more efficiently than the forward-backward algorithm, in terms of number of iterations and with a competitive value at the solution.

Finally, we include age structure in the model and analyze disease dynamics in different age classes. Our goal is to determine how treatment doses should be distributed and how social distancing should be implemented in each age group in order to reduce the final epidemic size.