The standard approach for epidemic modeling, at the community scale and larger, is compartmental models in which individuals are in one of a small number of states (for example, susceptible, infected, recovered, exposed, latent), with individuals moving between states. The COVID-19 epidemic can be modeled in this way, with resistance as part of the dynamics. The simplest examples of such models for large populations are coupled ordinary differential equations that describe the fraction of a population in each of the states. To model the stochasticity of infection and latency, models with self-exciting point processes can be fit to real-world data. This project compared the dynamical systems and stochastic models of relevance to COVID-19 transmission. The models also incorporate network structure for the transmission pathways. The project extends prior research on contagions on multilayer networks by incorporating multiple transmission methods and coupling between the spread of the contagion itself and human behavior patterns. 

Deterministic compartmental models for infectious diseases give the mean behavior of stochastic agent-based models. These models work well for counterfactual studies in which a fully mixed large-scale population is relevant. However, with finite size populations, chance variations may lead to significant departures from the mean. In real-life applications, finite size effects arise from the variance of individual realizations of an epidemic course about its fluid limit. We considered the classical stochastic Susceptible-Infected-Recovered (SIR) model, and derived a martingale formulation consisting of a deterministic and a stochastic component. The deterministic part coincides with the classical deterministic SIR model and we provided an upper bound for the stochastic part. Through analysis of the stochastic component depending on varying population size, we provide a theoretical explanation of finite size effects. Our theory is supported by quantitative and direct numerical simulations of theoretical infinitesimal variance. Case studies of coronavirus disease 2019 (COVID-19) transmission in smaller populations illustrate that the theory provides an envelope of possible outcomes that includes the field data. 

We develop a method for analyzing spatial and spatiotemporal anomalies in geospatial data using topological data analysis (TDA). To do this, we use persistent homology (PH), which allows one to algorithmically detect geometric voids in a data set and quantify the persistence of such voids. We constructed an efficient filtered simplicial complex (FSC) such that the voids in our FSC are in one-to-one correspondence with the anomalies. Our approach went beyond simply identifying anomalies; it also encoded information about the relationships between anomalies. We used vineyards, which one can interpret as time-varying persistence diagrams (which are an approach for visualizing PH), to track how the locations of the anomalies change with time. We conducted two case studies using spatially heterogeneous COVID-19 data. First, we examined vaccination rates in New York City by zip code at a single point in time. Second, we studied a year-long data set of COVID-19 case rates in neighborhoods of the city of Los Angeles.

During the COVID-19 pandemic, conflicting opinions on physical distancing swept across social media, affecting both human behavior and the spread of COVID-19. Inspired by such phenomena, we constructed a two-layer multiplex network for the coupled spread of a disease and conflicting opinions. We modeled each process as a contagion. On one layer, we considered the concurrent evolution of two opinions -- pro-physical-distancing and anti-physical-distancing -- that compete with each other and have mutual immunity to each other. The disease evolved on the other layer, and individuals were less likely (respectively, more likely) to become infected when they adopted the pro-physical-distancing (respectively, anti-physical-distancing) opinion. We developed approximations of mean-field type by generalizing monolayer pair approximations to multilayer networks; these approximations agreed well with Monte Carlo simulations for a broad range of parameters and several network structures. Through numerical simulations, we illustrated the influence of opinion dynamics on the spread of the disease from complex interactions both between the two conflicting opinions and between the opinions and the disease. We found that lengthening the duration that individuals hold an opinion may help suppress disease transmission, and we demonstrated that increasing the cross-layer correlations or intra-layer correlations of node degrees may lead to fewer individuals becoming infected with the disease.

Last Modified: 07/18/2022
Modified by: Andrea L Bertozzi