Constrained optimal control for a multi-group Discrete Time Influenza Model

During 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 single influenza outbreaks and the impact of different control policies. In this dissertation, a discrete time model is introduced in order to study optimal control strategies for influenza transmission; since epidemiological data is collected on discrete units of time, a discrete formulation is more efficient. From a mathematical point of view, continuous time model are easier to analyze, however, the numerical solution of discrete-time models is simpler and therefore can be easily implemented. We formulated a discrete susceptible-infected-treated-recovered (SITR) model. We evaluated the potential effect of control measures, such as social distancing and antiviral treatment in the context of a single influenza outbreak. The objective was to minimize the total number of infected individuals at the end of the epidemic in a most economical way. The potential effect of antiviral treatment was evaluated by considering both unlimited and limited supply. We found that the use of single and dual strategies (social distancing and antiviral treatment) resulted in reductions in the cumulative number of infected individuals. In the case of limited resources, our results showed that in order to control the epidemic, most of the resources must be utilized at the beginning of the epidemic until all the resources exhausted. The optimal control problem was solved by using two different techniques: 1) By using a discrete version of Pontryagin\u27s maximum principle, with the commonly used forward-backward algorithm; and 2) by using the primal-dual interior-point method. The later approach allowed the inclusion of constraints more efficiently and solved the problem in fewer iterations. The main advantage of interior-point methods was that constraints can be included in a simple way; in particular, the isoperimetric constraint in the case of limited resources. However, since this technique is based on the Newton Method, the solution was very sensitive to initial conditions. In addition, the role of heterogeneity in the population was considered. The total population was divided into subgroups according to activity or susceptibility levels. The goal was to determine how treatment doses should be distributed and how social distancing should be implemented in each group in order to reduce the final epidemic size. We presented numerical results for two and three groups, both for the case of seasonal and pandemic influenza. The results were sensitive to different population sizes; that is, our results showed that in order to control an epidemic, more resources need to be channeled towards the group with the largest population and most active. Finally, we considered another optimization problem derived for the same epidemiological model in which, instead of minimizing the number of infected individuals, we estimated parameter values from historical epidemic data in order to identify features, such as contact rate and recovery rate. Although the parameter values used in the model were estimated from epidemiological data, there is still some uncertainty in their values. Thus, we introduced some basic ideas and examples of parameter estimation applied to discrete epidemiological models. Our goal was to present all the tools necessary to better estimate parameter values and implementation of optimal control measures. We found that a discrete formulation allows us to solve the parameter estimation problem in a simpler way