SARS-CoV-2 superspreading in cities vs the countryside

Abstract

The first wave of the COVID-19 pandemic was characterized by an initial rapid rise in new cases followed by a peak and a more erratic behaviour that varies between regions. This is not easy to reproduce with traditional SIR models, which predict a more symmetric epidemic. Here, we argue that superspreaders and population heterogeneity would predict such behaviour even in the absence of restrictions on social life. We present an agent-based lattice model of a disease spreading in a heterogeneous population. We predict that an epidemic driven by superspreaders will spread rapidly in cities, but not in the countryside where the sparse population limits the maximal number of secondary infections. This suggests that mitigation strategies should include restrictions on venues where people meet a large number of strangers. Furthermore, mitigating the epidemic in cities and in the countryside may require different levels of restrictions.

Keywords: COVID-19; epidemiology; model; population density; superspreading.

Fig. 1

Model: A superspreader in a city interacts a little with a lot of people and will infect some fraction of them. On the other hand, a superspreader outside the city will interact a lot with each of a smaller set of people. The superspreader then infects practically all of them, but there is a lower cap on the number of secondary infections.

Fig. 2

Epidemic trajectories: Infected fraction of the population over time changes with the countryside population density ρ. In the low‐ ρ regime, increasing the population density stretches the curve, as the epidemic spreads further from the city. When ρ is above ∼ 0.06, the epidemic again approaches the behaviour of a SIR model, as the epidemic now spreads unhindered across the whole system. Around ρ_crit , there is a large variation in the duration of the epidemic. The parameters used are γ = 0.1, r ₀ = 10, f_meet  = 10 and dispersion parameter k = 0.1.

Fig. 3

Comparison of models without and with superspreaders. Panel (A) shows the attack rate as a function of the number of neighbours within the radius of interaction ($\sim$ ρr₀ ) in a population where everyone infects with the same rate. (B) shows attack rate with heterogeneous infection rates, using a gamma distribution with dispersion factor k = 0.1. The two overlaid curves demonstrate that the parameter r ₀ does not affect the physics of the system, and what really determines the ability of the disease to percolate is the number of neighbours, proportional to ρr₀ . (C) Epidemic trajectory when superspreaders dominate (blue) and when infectiousness is evenly distributed (red) for equal countryside population density and radius of interaction (ρr₀  = 6). When superspreaders are the main drivers of the epidemic, it is strongly impeded once the city has reached herd immunity. When everyone infects equally, the epidemic simply spreads radially out from the city, leading to a ‘second wave’ in the countryside. Parameters are as in Fig. 2.

Fig. 4

Dependence of attack rate on density and k. Since the variable determining percolation is not the absolute density, but the number of neighbours, we plot ρr₀ on the x‐axis rather than ρ. It is seen that the disease percolates much more easily at a higher k, which implies a more homogeneous infectivity. The more overdispersed the disease (corresponding to lower k), the more neighbours are required for percolation.

Fig. 5

Epidemic trajectory in regions of varying density. When infectivity is homogeneous (top), the epidemic is a lot less sensitive to a lower population density than when the epidemic is driven by superspreaders (bottom). Here, the epidemic is nearly absent in the low‐density regions and appears to be driven by spillover. Inset illustrates the layout of the lattice.

Fig. 6

Simulation with multiple cities: Infected individuals are shown in red, susceptibles are green, empty sites are white and recovered are black. City size distribution mimics Zipf’s law [23], such that there is one city of with 40,000 inhabitants, 4 with 10,000, and so forth. The graphs show the fraction of the population currently infected as a function of time for three example runs of the simulation. Here, ρ = 0.03 and the other parameters are the same as the above figures.