## Epidemiological Modeling

Overview and background

The COVID-19 pandemic has infected individuals all around the world. The virus, known as SARS-CoV-2, caused many deaths, and even survivors have reported long-term health effects. As it threatened the lives of many, it also delivered a significant blow to the livelihoods of many more.

From day one, scientists from many disciplines have been deeply involved in efforts to combat the disease. As the members of Mehta Research Group and our collaborators at Coordinated Science Laboratory, we were no exception.

Our early efforts focused on using individual-level data to inform the construction of compartmental models. With the help of our colleagues from the Department of Pathobiology in the College of Veterinary Medicine at UIUC, we acquired ten de-identified open-source datasets online. We proposed a principled approach for constructing compartmental models using these datasets and some well-known properties of Markov chains. We presented this work at the American Control Conference (ACC) 2021.

Later, we published two conference papers on modeling the spread of disease in a heterogeneous population with the aim to improve classical compartmental models by bringing in the human decision-making aspect. We model the individuals in the population as rational agents, who must find a balance between their own health risks, the health of the population, and their own economic rewards. We make the optimistic assumption that if agents know that they are infected, they isolate to protect others. We show that even with this assumption, the existence of asymptomatic agents may still cause an epidemic.

Data-informed approach for compartmental models

Since their introduction in 1927 by Kermack and McKendrick, compartmental models have been used as the dominant approach in describing the progression of diseases. These models divide the population into compartments with labels such as "Susceptible", "Infected", "Recovered" and "Deceased". Once the compartments are defined, one can write ordinary differential equations (ODEs) to describe movements between compartments. In the COVID-19 pandemic, many compartmental models have been proposed: SIR, SIRD, SEIR, SAIRD... We acknowledge the significance of many contributions by fellow researchers who used these models to derive insights. However, one should note that compartmental models are mean-field approximations of the underlying Markov chain. It is well-known that jump times are exponentially distributed in a Markov chain. However, when we look at individual-level data that we acquired online, we see that transitions from "Infected" to "Recovered" or "Deceased" are indeed not exponentially distributed.

Infected to Recovered Transition Time Distribution

Infected to Deceased Transition Time Distribution

In our ACC 2021 paper, we document 10 de-identified online open-source datasets. Then using some of these datasets and well-known properties of Markov chains, we propose a data-informed approach for the analysis, validation, and identification of COVID-19 models. Our initial results show that this approach helps us develop compartmental models that are compliant with the underlying Markov chain.

Partially Observed Mean Field Games

The analysis given in the ACC 2021 paper aims to explain in-host dynamics, i.e. the progression of the disease once an individual gets infected. In order to model the spread of the disease among individuals, we propose a mean field game framework. In the most simple terms, we model the decisions of individuals that make up the population during an epidemic and the effect of such decisions on disease progression. In our work, these individuals are considered to be rational agents, who choose contact rate with other agents as a control input. The dynamics of a single agent then become a controlled Markov process, as shown below.

Here, the transition rate from the "Susceptible" to "Asymptomatic" state essentially represents the rate at which disease spreads in the population. It depends on three things: the intrinsic infectivity of the virus, the contact rate with other agents (control input), and the fraction of infected agents in the population, weighted by their activity level. This last term is called the mean-field process. It combines the effect of states and the actions of all individuals in the population. Several other researchers have proposed mean-field game models for epidemics. Our novelty lies in our information structure. Specifically, in our formulation, agents have the same observations for the states "Symptomatic", "Asymptomatic" and "Recovered". Therefore the problem is partially observed until they start showing symptoms. In the fully observed formulation of our problem, all infected agents, regardless of showing symptoms or not, are isolated. This is because they know that they are infected and they want to protect others. Therefore an epidemic does not occur. However, even under this somewhat excessively optimistic setting, we may have an epidemic if the observations are partial. You may watch our presentation video below for further details.

Presentations

Slides (Summary) from 2021 American Control Conference - New Orleans, LA, USA May 25-28, 2021

Slides from 2021 American Control Conference - New Orleans, LA, USA May 25-28, 2021

Slides from the 61st IEEE Conference on Decision and Control - Cancun, Mexico Dec 6-9, 2022

Publications

S. Y. Olmez, J. Mori, E. Miehling, T. Başar, R. L. Smith, M. West and P. G. Mehta, "A Data-Informed Approach for Analysis, Validation, and Identification of COVID-19 Models," 2021 American Control Conference (ACC), 2021.

S. Y. Olmez, S. Aggarwal, J. W. Kim, E. Miehling, T. Başar, M. West and P. G. Mehta, "Modeling Presymptomatic Spread in Epidemics via Mean-Field Games," 2022 American Control Conference (ACC), 2022.

S. Y. Olmez, S. Aggarwal, J. W. Kim, E. Miehling, T. Başar, M. West and P. G. Mehta, "How does a Rational Agent Act in an Epidemic?," 2022 IEEE 61st Conference on Decision and Control (CDC), 2022.

Acknowledgement

Research supported in part by the C3.ai Digital Transformation Institute sponsored by C3.ai Inc. and the Microsoft Corporation, in part by the Jump ARCHES endowment through the Health Care Engineering Systems Center of the University of Illinois at Urbana-Champaign, and in part by the National Science Foundation grant NSF-ECCS 20-32321.