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. 2024 Jul 7;4(1):134.
doi: 10.1038/s43856-024-00561-4.

Epidemic modelling suggests that in specific circumstances masks may become more effective when fewer contacts wear them

Peter Klimek  1   2   3   4 , Katharina Ledebur  5   6 , Stefan Thurner  5   6   7
Affiliations

Epidemic modelling suggests that in specific circumstances masks may become more effective when fewer contacts wear them

Peter Klimek et al. Commun Med (Lond). .

Abstract

Background: The effectiveness of non-pharmaceutical interventions to control the spread of SARS-CoV-2 depends on many contextual factors, including adherence. Conventional wisdom holds that the effectiveness of protective behaviours, such as wearing masks, increases with the number of people who adopt them. Here we show in a simulation study that this is not always true.

Methods: We use a parsimonious network model based on the well-established empirical facts that adherence to such interventions wanes over time and that individuals tend to align their adoption strategies with their close social ties (homophily).

Results: When these assumptions are combined, a broad dynamic regime emerges in which the individual-level reduction in infection risk for those adopting protective behaviour increases as adherence to protective behaviour decreases. For instance, at 10 % coverage, we find that adopters face nearly a 30 % lower infection risk than at 60 % coverage. Based on surgical mask effectiveness estimates, the relative risk reduction for masked individuals ranges from 5 % to 15 %, or a factor of three. This small coverage effect occurs when the outbreak is over before the pathogen is able to invade small but closely knit groups of individuals who protect themselves.

Conclusions: Our results confirm that lower coverage reduces protection at the population level while contradicting the common belief that masking becomes ineffective at the individual level as more people drop their masks.

Plain language summary

Face masks have been used as one tool to protect people against COVID-19 infection. Here, we perform mathematical simulations to investigate how well mask-wearing works in different scenarios. Counterintuitively, our simulations showed that as fewer people follow these measures, the risk of infection decreases for those who still do. For instance, when only 10% of people follow them, their infection risk gets reduced by almost 30% compared to situations where 60% follow. Our findings challenge the idea that masks become ineffective when fewer people wear them. The overall public health benefit still increases as more people wear masks, but their protective effect at the individual level can still be substantial if only a few people wear them.

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Conflict of interest statement

The authors declare no competing interests.

Figures

Fig. 1
Fig. 1. Illustration of a small-world network with assortative adoption of protective behaviour (homophily).
We show N = 100 individuals (nodes) out of which 20 adopt (red) and 80 do not adopt the behaviour (blue). Nodes are positioned on a ring and linked to their k = 6 nearest neighbours; individuals with the same behaviour cluster in the network. To reflect imperfect assortative mixing, with some probability (ϵ) a link is chosen and rewired toward a randomly chosen different node and with another probability (1-η) the behaviour state (colour) is swapped between two randomly chosen nodes. The resulting networks are illustrated for (ϵ,η) being (a) (0.05, 1), (b) (0.05, 0.8), (c) (0.2, 1) and (d) (0.2, 0.8).
Fig. 2
Fig. 2. Demonstration of the small coverage effect of protective behaviour.
The number of new cases over time for those adopting protective behaviour (C(t, M = 1), red) and those who do not (C(t, M = 0), blue) are shown for coverage from (a) m = 0.1 over (b) m = 0.3 to (c) m = 0.5; shaded areas denote the 68% CI. Insets illustrate the networks for these settings, respectively. Results for the (d) individual infection risk, IIR(M = 0, 1), as a function of coverage m are shown as solid lines and compared to a simulation for which mask-wearing adherence does not wane (dotted lines, using q = 1 and qr(0) = qs(0) = 0.4); error bars denote a standard deviation. If adherence wanes, the infection risk increases with m for m < 0.6; for higher values of m the distribution of outbreak sizes becomes bimodal (marker size is proportional to the fraction of simulation runs in each mode). The infection risk is always a monotonously decreasing function of m if adherence does not wane. On the (e) population level we observe a plateau with similar risks, PIR, at intermediate coverage next to the bifurcation for high coverage.
Fig. 3
Fig. 3. Homophily amplifies the small coverage effect.
We show the reduction of the individual infection risk [%] due to the small coverage effect from a coverage of of 60% to 10%. We find (a) an infection risk reduction of about 20–25% for small world parameters representing strong homophily (large η). The reduction decreases with decreasing homophily and plateaus at values about 10%, for η ≤ 0.5. Increasing the small world parameter from ϵ = 0.1 (blue) to 0.5 (orange) reduces the small coverage effect even for high levels of homophily. Error bars show a standard deviation. b A parameter sweep over the homophily and small world parameters, η and ϵ, respectively, shows that the small coverage effect is most pronounced (close to 30%) for small ϵ and large η and plateaus between 10 and 15% for other parameter settings.
Fig. 4
Fig. 4. Robustness of the small coverage effect.
For various values of the (a, b) transmission rate α, (c, d) network degree k and (e, f) waning parameter, q, we show the individual-level infection risk for adopters, IIR(M = 1), and the population-level infection risk, PIR, respectively. For low values of α or k, outbreaks are suppressed and there is no small coverage effect. Above this suppression regime, we find a regime in which the effect can be observed with different magnitudes.
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