Nonlinear bias toward complex contagion in uncertain transmission settings (preprint)

Current epidemics in the biological and social domains are challenging the standard assumptions of mathematical contagion models. Chief among them are the complex patterns of transmission caused by heterogeneous group sizes and infection risk varying by orders of magnitude in different settings, like indoor versus outdoor gatherings in the COVID-19 pandemic or different moderation practices in social media communities. However, quantifying these heterogeneous levels of risk is difficult and most models typically ignore them. Here, we include these novel features in an epidemic model on weighted hypergraphs to capture group-specific transmission rates. We study analytically the consequences of ignoring the heterogeneous transmissibility and find an induced superlinear infection rate during the emergence of a new outbreak, even though the underlying mechanism is a simple, linear contagion. The dynamics produced at the individual and group levels are therefore more similar to complex, nonlinear contagions, thus blurring the line between simple and complex contagions in realistic settings. We support this claim by introducing a Bayesian inference framework to quantify the nonlinearity of contagion processes. We show that simple contagions on real weighted hypergraphs are systematically biased toward the superlinear regime if the heterogeneity of the weights is ignored, greatly increasing the risk of erroneous classification as complex contagions. Our results provide an important cautionary tale for the challenging task of inferring transmission mechanisms from incidence data. Yet, it also paves the way for effective models that capture complex features of epidemics through nonlinear infection rates.

The unintended consequences of inconsistent pandemic control policies (preprint)

Controlling the spread of COVID-19 -- even after a licensed vaccine is available -- requires the effective use of non-pharmaceutical interventions, e.g., physical distancing, limits on group sizes, mask wearing, etc.. To date, such interventions have neither been uniformly nor systematically implemented in most countries. For example, even when under strict stay-at-home orders, numerous jurisdictions granted exceptions and/or were in close proximity to locations with entirely different regulations in place. Here, we investigate the impact of such geographic inconsistencies in epidemic control policies by coupling search and mobility data to a simple mathematical model of SARS-COV2 transmission. Our results show that while stay-at-home orders decrease contacts in most areas of the United States of America (US), some specific activities and venues often see an increase in attendance. Indeed, over the month of March 2020, between 10 and 30% of churches in the US saw increases in attendance; even as the total number of visits to churches declined nationally. This heterogeneity, where certain venues see substantial increases in attendance while others close, suggests that closure can cause individuals to find an open venue, even if that requires longer-distance travel. And, indeed, the average distance travelled to churches in the US rose by 13% over the same period. Strikingly, our mathematical model reveals that, across a broad range of model parameters, partial measures can often be worse than no measures at all. In the most severe cases, individuals not complying with policies by traveling to neighboring jurisdictions can create epidemics when the outbreak would otherwise have been controlled. Taken together, our data analysis and modelling results highlight the potential unintended consequences of inconsistent epidemic control policies and stress the importance of balancing the societal needs of a population with the risk of an outbreak growing into a large epidemic.

Deep learning of contagion dynamics on complex networks (preprint)

Forecasting the evolution of contagion dynamics is still an open problem to which mechanistic models only offer a partial answer. To remain mathematically or computationally tractable, these models must rely on simplifying assumptions, thereby limiting the quantitative accuracy of their predictions and the complexity of the dynamics they can model. Here, we propose a complementary approach based on deep learning where the effective local mechanisms governing a dynamic on a network are learned from time series data. Our graph neural network architecture makes very few assumptions about the dynamics, and we demonstrate its accuracy using different contagion dynamics of increasing complexity. By allowing simulations on arbitrary network structures, our approach makes it possible to explore the properties of the learned dynamics beyond the training data. Finally, we illustrate the applicability of our approach using real data of the COVID-19 outbreak in Spain. Our results demonstrate how deep learning offers a new and complementary perspective to build effective models of contagion dynamics on networks.

Stochasticity and heterogeneity in the transmission dynamics of SARS-CoV-2 (preprint)

SARS-CoV-2 causing COVID-19 disease has moved rapidly around the globe, infecting millions and killing hundreds of thousands. The basic reproduction number, which has been widely used and misused to characterize the transmissibility of the virus, hides the fact that transmission is stochastic, is dominated by a small number of individuals, and is driven by super-spreading events (SSEs). The distinct transmission features, such as high stochasticity under low prevalence, and the central role played by SSEs on transmission dynamics, should not be overlooked. Many explosive SSEs have occurred in indoor settings stoking the pandemic and shaping its spread, such as long-term care facilities, prisons, meat-packing plants, fish factories, cruise ships, family gatherings, parties and night clubs. These SSEs demonstrate the urgent need to understand routes of transmission, while posing an opportunity that outbreak can be effectively contained with targeted interventions to eliminate SSEs. Here, we describe the potential types of SSEs, how they influence transmission, and give recommendations for control of SARS-CoV-2.

The role of directionality, heterogeneity and correlations in epidemic risk and spread (preprint)

Most models of epidemic spread, including many designed specifically for COVID-19, implicitly assume that social networks are undirected, i.e., that the infection is equally likely to spread in either direction whenever a contact occurs. In particular, this assumption implies that the individuals most likely to spread the disease are also the most likely to receive it from others. Here, we review results from the theory of random directed graphs which show that many important quantities, including the reproductive number and the epidemic size, depend sensitively on the joint distribution of in- and out-degrees ("risk" and "spread"), including their heterogeneity and the correlation between them. By considering joint distributions of various kinds we elucidate why some types of heterogeneity cause a deviation from the standard Kermack-McKendrick analysis of SIR models, i.e., so called mass-action models where contacts are homogeneous and random, and some do not. We also show that some structured SIR models informed by complex contact patterns among types of individuals (age or activity) are simply mixtures of Poisson processes and tend not to deviate significantly from the simplest mass-action model. Finally, we point out some possible policy implications of this directed structure, both for contact tracing strategy and for interventions designed to prevent superspreading events. In particular, directed networks have a forward and backward version of the classic "friendship paradox" -- forward links tend to lead to individuals with high risk, while backward links lead to individuals with high spread -- such that a combination of both forward and backward contact tracing is necessary to find superspreading events and prevent future cascades of infection.

Social confinement and mesoscopic localization of epidemics on networks (preprint)

Reports concerning epidemics, and COVID-19 is no exception, have mainly focused on individual behavior, while event cancellations and gathering size restrictions are decided with little information on their impact. This is quite the norm since most models lack the higher-order structure necessary to describe large gatherings. Through a higher-order description of contagions on networks, we find that epidemics can suddenly collapse when interventions operate over groups of individuals rather than at the level of individuals themselves. We relate this phenomenon to the onset of mesoscopic localization, where contagions concentrate around dominant groups.

Beyond R0: the importance of contact tracing when predicting epidemics (preprint)

The basic reproductive number --- R0 --- is one of the most common and most commonly misapplied numbers in public health. Although often used to compare outbreaks and forecast pandemic risk, this single number belies the complexity that two different pathogens can exhibit, even when they have the same R0. Here, we show how to predict outbreak size using estimates of the distribution of secondary infections, leveraging both its average R0 and the underlying heterogeneity. To do so, we reformulate and extend a classic result from random network theory~\cite{Newman2001} that relies on contact tracing data to simultaneously determine the first moment (R0) and the higher moments (representing the heterogeneity) in the distribution of secondary infections. Further, we show the different ways in which this framework can be implemented in the data-scarce reality of emerging pathogens. Lastly, we demonstrate that without data on the heterogeneity in secondary infections for emerging infectious diseases like COVID-19, the uncertainty in outbreak size ranges dramatically. %, in the case of COVID-19 from 10-70\% of susceptible individuals. Taken together, our work highlights the critical need for contact tracing during emerging infectious disease outbreaks and the need to look beyond R0 when predicting epidemic size.

Beyond $R_0$: Heterogeneity in secondary infections and probabilistic epidemic forecasting (preprint)

The basic reproductive number -- $R_0$ -- is one of the most common and most commonly misapplied numbers in public health. Although often used to compare outbreaks and forecast pandemic risk, this single number belies the complexity that two different pathogens can exhibit, even when they have the same $R_0$. Here, we show how to predict outbreak size using estimates of the distribution of secondary infections, leveraging both its average $R_0$ and the underlying heterogeneity. To do so, we reformulate and extend a classic result from random network theory that relies on contact tracing data to simultaneously determine the first moment ($R_0$) and the higher moments (representing the heterogeneity) in the distribution of secondary infections. Further, we show the different ways in which this framework can be implemented in the data-scarce reality of emerging pathogens. Lastly, we demonstrate that without data on the heterogeneity in secondary infections for emerging infectious diseases like COVID-19, the uncertainty in outbreak size ranges dramatically. Taken together, our work highlights the critical need for contact tracing during emerging infectious disease outbreaks and the need to look beyond $R_0$ when predicting epidemic size.