Analytical Modelling of the Spread of Disease in Confined and Crowded Spaces

Abstract

Since 1927 and until recently, most models describing the spread of disease have been of compartmental type, based on the assumption that populations are homogeneous and well-mixed. Recent models have utilised agent-based models and complex networks to explicitly study heterogeneous interaction patterns, but this leads to an increasing computational complexity. Compartmental models are appealing because of their simplicity, but their parameters, especially the transmission rate, are complex and depend on a number of factors, which makes it hard to predict how a change of a single environmental, demographic, or epidemiological factor will affect the population. Therefore, in this contribution we propose a middle ground, utilising crowd-behaviour research to improve compartmental models in crowded situations. We show how both the rate of infection as well as the walking speed depend on the local crowd density around an infected individual. The combined effect is that the rate of infection at a population scale has an analytically tractable non-linear dependency on crowd density. We model the spread of a hypothetical disease in a corridor and compare our new model with a typical compartmental model, which highlights the regime in which current models may not produce credible results.

The Subways Seeded the Massive Coronavirus Epidemic in New York City

“…Quite apart from the present study and the above-cited work by Goscé and colleagues (Gosce, Barton, andJohansson 2014, Gosce andJohansson 2018), a few other researchers have attempted to test whether public transport has served as a critical vehicle for the propagation of contagious respiratory diseases (Sun et al 2013, Troko et al 2011, Cooley et al 2011. One distinguishing factor between the present study and prior work is that seasonal influenza has generally had a reproductive number R in the range of 1.2-1.4, while pandemic influenza has had an R in the range of 1.4-1.8, with the high end representing the 1918 pandemic (Biggerstaff et al 2014).…”

“…The model was borrowed from the basic law of mass action in chemistry, where S and I molecules bombard against each other, bounding around in a gas or a liquid. In an innovative series of papers, Goscé and colleagues generalized this model to consider contagion when the S's and I's move along a corridor (Gosce, Barton, andJohansson 2014, Gosce andJohansson 2018). They applied their framework to the study of the spread of influenza-like illness in the London Underground, a vast network opened just nine years after Dr. John Snow got public officials to disable a pump at Broad (now Broadwick) and Lexington Streets, now about a five-minute walk from the Oxford Circus station.…”

The Relationship between In-Person Voting and COVID-19: Evidence from the Wisconsin Primary

“…As a result, our work relates to Harris (2020), Almagro and Orane-Hutchinson (2020) and Kuchler, Russel, and Stroebel (2020) among others looking at how geographical differences in behavior (e.g., public transit availability or occupation characteristics) affects the spread of COVID-19. As we measure the impact of polling locations, we also inform models such as Goscé, Barton, and Johansson (2014) who analyze the impact of the proximity of persons on the spread of a disease. Relatedly, an emerging literature examines the determinants and effects of social distancing orders on the spread of COVID-19 cases (Andersen, 2020;Courtemanche et al, 2020a,b;Friedson, McNichols, Sabia, and Dave, 2020).…”

Reducing community infection by institutional interventions: a minimal SEIRJ model applied to COVID-19

“…While this may be difficult to fully implement, government mandate coupled with increased media mileage brings awareness to the community. Models have shown that increased consciousness or explicitly minimizing contacts reduce disease transmission [9][10][11][12][13][14] . While social distancing is normally implemented in models via closures of establishments 12,13 and thus akin to a lockdown approach, strategies based on a network model have shown how selective social distancing delays and flattens the infection curve by limiting interaction and disease transmission into within small groups 14 .…”