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Abstract The effectiveness of non-pharmaceutical interventions (NPIs) during a pandemic is challenging to assess due to the multifaceted interactions between interventions and population dynamics. Significant difficulty arises from the overlapping effects of various NPIs applied to different subgroups within a population. To address this, we propose a new mathematical model that incorporates various intervention strategies, including total and partial lockdowns, school closures, and reduced interactions among specific subgroups, such as the elderly. Our model extends previous work by explicitly accounting for the quadratic nature of control costs and the interplay between overlapping controls targeting the same population segments. Using optimal control theory, we identify intervention policies that effectively mitigate disease transmission while balancing economic and societal costs. To demonstrate the utility of our approach, we apply the model to real-world data from the COVID-19 pandemic in the State of New Jersey. Our results provide insights into the trade-offs and synergies of different NPIs and the importance of accurately capturing the relationship between a policy and the population affected. 

The COVID-19 pandemic highlighted the need to quickly respond, via public policy, to the onset of an infectious disease breakout. Deciding the type and level of interventions a population must consider to mitigate risk and keep the disease under control could mean saving thousands of lives. Many models were quickly introduced highlighting lockdowns, testing, contact tracing, travel policies, later on vaccination, and other intervention strategies along with costs of implementation. Here, we provided a framework for capturing population heterogeneity whose consideration may be crucial when developing a mitigation strategy based on non-pharmaceutical interventions. Precisely, we used age-stratified data to segment our population into groups with unique interactions that policy can affect such as school children or the oldest of the population, and formulated a corresponding optimal control problem considering the economic cost of lockdowns and deaths. We applied our model and numerical methods to census data for the state of New Jersey and determined the most important factors contributing to the cost and the optimal strategies to contained the pandemic impact. 

The COVID-19 pandemic lit a fire under researchers who have subsequently raced to build models which capture various physical aspects of both the biology of the virus and its mobility throughout the human population. These models could include characteristics such as different genders, ages, frequency of interactions, mutation of virus, etc. Here, we propose two mathematical formulations to include virus mutation dynamics. The first uses a compartmental epidemiological model coupled with a discrete-time finite-state Markov chain. If one includes a nonlinear dependence of the transition matrix on current infected, the model is able to reproduce pandemic waves due to different variants. The second approach expands such an idea to a continuous state-space leveraging a combination of ordinary differential equations with an evolution equation for measure. This approach allows to include reinfections with partial immunity with respect to variants genetically similar to that of first infection. 

Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs) and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible \begin{document}$ S $$\end{document} and removed \begin{document}$$ R $$\end{document} populations by ODEs and the infected \begin{document}$$ I $$\end{document} population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for \begin{document}$$ S $$\end{document} and \begin{document}$$ R $$\end{document} contains terms that are related to the measure \begin{document}$$ I $$\end{document}$. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in case of constant or time-dependent parameters as special cases.