# Small coverage effect in epidemic network models shows that masks can become
more effective with less people wearing them

Peter Klimek1,2,∗ Katharina Ledebur1,2 Stefan Thurner1,2,3

###### Abstract

The effectiveness of non-pharmaceutical interventions to curb the spread of
SARS-CoV-2 is determined by numerous contextual factors, including adherence.
Conventional wisdom holds that the effectiveness of protective behaviour such
as wearing masks always increases with the number of people adopting it. Here
we show in a simulation study that this is not true in general. We employ a
parsimonious network model based on the well-established empirical facts that
(i) adherence to such interventions wanes over time and (ii) individuals tend
to align their adoption strategies with their close social ties (homophily).
When combining these assumptions, a broad dynamical regime emerges where the
individual-level infection risk reduction for those adopting protective
behaviour increases as the adherence to protective behavior decreases. For
instance, for a protective coverage of 10% we find the infection risk for
adopting individuals can be reduced by close to 30% compared to situations
where the coverage is 60%. Using estimates for the effectiveness of surgical
masks, we find that reductions in relative risk of masking versus non-masking
individuals range between 5% and 15%, i.e., vary by a factor of three. This
small coverage effect originates from system-dynamical network properties that
conspire to increase the chance that an outbreak will be over before the
pathogen is able to invade small but tightly connected groups of individuals
that protect themselves. Our results contradict the popular belief that
masking becomes ineffectual as more people drop their masks and might have
far-reaching implications for the protection of vulnerable population groups
under resurgent infection waves.

Section for Science of Complex Systems, Medical University of Vienna,
Spitalgasse 23, A-1090 Vienna, Austria

Complexity Science Hub Vienna, Josefstädterstraße 39, A-1080 Vienna, Austria

Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA

∗Correspondence<EMAIL_ADDRESS>
##  1 Introduction

The SARS-CoV-2 pandemic led to the widespread and sustained implementation of
a number of non-pharmaceutical interventions (NPIs) such as social distancing,
wearing face masks, or test-trace-isolate strategies. [1, 2, 3] From a
scientific viewpoint, this opened the unique opportunity to accumulate real-
world evidence on the effectiveness of such NPIs.[4, 5, 6, 7] Meanwhile, there
is overwhelming empirical evidence that many of these interventions do indeed
curb the virus spread. [8, 9] The literature is less unanimous when it comes
to concise quantitative estimates of the effectiveness of NPIs. [5, 7, 8, 10]
The heterogeneity in these estimates can be explained by a plethora of
contextual factors impacting their adoption, ranging from geographic,
cultural, socio-economic and healthcare-related to behavioural factors. [11,
12, 13, 14]

It has been shown that the extent to which individuals adhere to an NPI
changes substantially over time even if no formal change in strengthening or
easing the NPIs has occurred.[15] Adherence to physical distancing measures
has been observed to wane over time. [15, 16, 17, 18] Adherence to low-cost
habituating measures like mask wearing increased during the early phases of
the pandemic, likely linked to extensive public health messaging. [19, 16, 20]
However, even for those low-cost measures adherence plateaued and started to
decline after vaccines were rolled out, the Omicron variant became dominant
and an increasing number of countries dropped mask mandates. [21, 22, 23, 24]

Further, adherence to NPIs is strongly related to homophily. [25] It is a
well-established fact that individual behaviour and attitudes cluster in
social networks, meaning that attitudes of a person often conform with the
attitudes of its close social contacts. [26, 27] Homophily is observed in
health behaviour in general [28, 29] and more recently was confirmed in the
context of the COVID-19 pandemic. [30, 31] For instance, the likelihood that
an individual wears a mask is closely related to the frequency of actual mask
wearing, both in the general population, and in close social contacts. [32,
33]

In this work by means of a simulation study we show that an additional factor
may modulate the effectiveness of a wide range of NPIs such as wearing face
masks. We consider the broad class of NPIs that may –or may not– be adopted by
individuals and that reduce the chance of a transmission whenever adopted by
either the transmitting or receiving individual. Obviously, face masks are of
this type but also other habituation measures such as keeping distance can
reduce the chance of transmission, independent from whether the adopting
individual is the transmitter or receiver.

In epidemiological models it is well-established that if the rate of adoption
(coverage rate) and the effectiveness of NPIs are high enough, epidemic
outbreaks can be successfully suppressed. Consequently, the effectiveness of
an NPI on the individual level (i.e., how much wearing a mask reduces one’s
personal risk of becoming infected) depends on system-dynamical properties
such as the fraction of the entire population adopting the NPI. As a higher
coverage rate moves the system closer to its epidemic threshold, higher
population-level adoption typically also means a stronger individual-level
infection risk reduction.

Here we aim to understand how the individual- and population-level
effectiveness of mask-wearing depends on the coverage rate. To put it more
colloquially, if my own benefit from wearing a mask increases with the number
of others wearing a mask, should I wear one, if no-one else is wearing one? We
study this question in a simple epidemiological model that acknowledges two
repeatedly confirmed properties regarding the adoption of NPIs that were
observed throughout the pandemic. First, we assume that NPI adherence wanes
over time; someone who initially wore a mask during almost all non-household
contacts will, over time, wear the mask at fewer and fewer occasions. Second,
we assume a certain degree of behavioural homophily, in the sense that the
adoption behavior of NPIs clusters in social networks: individuals with close
social ties tend to synchronize their individual decisions on protection
measures. In all other aspects, we opt for a parsimonious modelling approach
so that we can clearly isolate the effects originating from these assumptions
and how they play out on the relation between individual- and system-level
effectiveness of NPIs.

In the model, all individuals have a static state indicating whether they
adopt protective behaviour or not. The effectiveness by which the adoption of
the behaviour reduces the probability for a successful transmission decreases
over time to implement waning of NPI adherence. Each individual is either
susceptible, infected or recovered. [34] To include homophily, epidemic
spreading takes place on a social contact network that shows assortative
mixing with respect to the adoption of protective behaviour. Assortative
mixing is implemented with a small-world network model, see figure 1. [35] In
this network, $N$ nodes are positioned on a ring and are linked to their $k$
nearest neighbors, resulting in a strongly clustered network, i.e., two
neighbours of a given node are also likely to be connected. With a certain
probability, $\epsilon$, links are randomly rewired. The parameter $\epsilon$
reflects a random linking between (family, work, and leisure) groups within
the society. $\epsilon=0$ means a regular structure where everyone is linked
to exactly $k$ neighbors; $\epsilon=1$ leads to a random network. Finally, to
introduce homophily in the adoption of protective behaviour, we first group
those individuals (nodes) with the same behaviour on the ring and then
randomly swap the behavior state between any two nodes with probability
$1-\eta$, see figure 1. $\eta$ acts as a control parameter for the degree of
homophily; large values represent a behaviorally clustered society, small
values lead to a non-homophilic situation.

Figure 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 behaviou (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 ($\epsilon$) a link is chosen and
rewired toward a randomly chosen different node and with another probability
(parameter $\eta$) the behaviour state (colour) is swapped between two
randomly chosen nodes. The resulting networks are illustrated for
($\epsilon$,$\eta$) being (a) ($0.05$, $1$), (b) ($0.05$, $0.8$), (c) ($0.2$,
$1$), and (d) ($0.8$, $0.2$).

In the model, the paradigmatic SIR-type dynamics unfolds on networks of this
type. Initially, all nodes are in the susceptible state except for a small
fraction of nodes in the infected state. At each timestep a susceptible node
can get infected from an infected neighboring node with probability, $\alpha$,
if both nodes don’t adopt protective behaviour. Adoption of protective
behaviour reduces the probability of a successful transmission by a time-
dependent factor, $q_{r}(t)$, for the receiving individual and by $q_{s}(t)$
for the transmitter (source). We assume that $q_{r}(t)$ and $q_{s}(t)$
decrease at every timestep by a constant percentage to reflect waning of NPI
adherence. Nodes remain infected for $1/\beta$ timesteps, after which they
become recovered and remain in that state for the rest of the simulation.

We measure how likely adopting and non-adopting individuals get infected
during an outbreak. We demonstrate that individual- and population-level
infection risk depend on the coverage rate in a non-trivial way.

##  2 Methods

We consider a small world network of the Watts-Strogatz type with $N$ nodes
positioned on a ring where they are connected to their $k$ nearest neighbours.
With probability, $\epsilon$, a link is chosen, disconnected from one of its
randomly chosen end nodes and rewired to any randomly chosen other node of the
network (self-links are excluded). The resulting network can be described by
an $N\times N$ adjacency matrix, $A$, with entries, $A_{ij}=1$, if nodes $i$
and $j$ are connected (neighboring) and $A_{ij}=0$, otherwise. Every node $i$
has a static state, $M_{i}$, signaling whether it adopts protective behaviour
($M_{i}=1$) or not ($M_{i}=0$) at the beginning of the simulation. The
coverage rate, $m$, represents the probability that a node adopts protective
behaviour, $\sum_{i}M_{i}=mN$. The homophily parameter, $\eta$, gives the
fraction of nodes that do assortatively mix with each other. That is, a number
of $\eta mN$ nodes $i$, with $M_{i}=1$, are grouped next to each other on the
network, whereas a fraction of $(1-\eta)mN$ of nodes with $M_{i}=1$ is
randomly distributed across the ring, see figure 1.

Every node, $i$, is endowed with a dynamic epidemic state variable,
$X_{i}(t)$, that takes values in $X_{i}(t)\in\\{S,I,R\\}$ corresponding to the
states in the SIR model: susceptible, infected, or recovered. At every
timestep, a susceptible individual $i$ can be infected with probability
$\alpha$ from each of its neighbouring infected individuals $j$ if none of the
two adopts protective behaviour ($M_{i}=M_{j}=0$). If $M_{i}=1$, the infection
risk decreases by a factor of $1-q_{r}(t)$, if $M_{j}=1$, the risk decreases
by $1-q_{s}(t)$. Adherence to these behaviours decreases exponentially with a
rate $0<q\leq 1$, such that $q_{r}(t)=q_{r}(0)q^{t}$ and
$q_{s}(t)=q_{s}(0)q^{t}$, respectively. Infected individuals recover with
probability $\beta$.

At every timestep, $t$, the model loops over all nodes $i=1,\dots,N$ and
performs a state update according to the following protocol (parallel update).

  * •

If $X_{i}(t)=S$, identify the set of all close infectious contacts of $i$,
$Nb(i)$, as $Nb(i)=\\{j|A_{ij}=1\wedge X_{j}(t)=I\\}$. For each node $j\in
Nb(i)$, set $X_{i}(t+1)=I$ with probability,
$\alpha\cdot(1-q_{r}(t))^{M_{i}}\cdot(1-q_{s}(t))^{M_{j}}$.

  * •

If $X_{i}(t)=I$, set $X_{i}(t+1)=R$ with probability $\beta$.

  * •

Proceed with the next node. Once all nodes are updated at time $t$, proceed to
the next timestep, $t+1$.

We denote the number of new cases by $C(t,M=1)$ and $C(t,M=0)$, for those that
do and do not adopt protective behaviour, respectively.

We measure the risk of becoming infected during the outbreak on a individual
and on a population scale. While the individual infection risk gives the
probability for an individual becoming infected as a function of whether
protective behaviour is adopted or not, on the population-level infection risk
measures the total number of infections in the entire population. The
individual infection risks in the two groups are
$IIR(M=1)=\frac{\sum_{t}C(t,M=1)}{mN}$ and
$IIR(M=0)=\frac{\sum_{t}C(t,M=0)}{(1-m)N}$, respectively; the population
infection risk is given by $PIR=\frac{\sum_{t,M}C(t,M)}{N}$.

Unless stated otherwise, the following parameter settings were used.
Simulations were performed for $N=10^{5}$ nodes and averaged over 20
independent iterations. Parameters have been chosen such that $k=20$,
$\alpha=0.02$, $\beta=0.1$, $q=0.99$, $q_{r}(0)=q_{s}(0)=1$, $\eta=1$, and
$\epsilon=0.1$. For the initial condition we randomly choose ten nodes and set
their $X_{i}(t=0)=I$; for all the others we set $X_{i}(t=0)=S$. The model
halts once the outbreak has ended, i.e., $X_{i}(t)\neq I$ for all $i$.

##  3 Results

Example runs for different coverage are shown in figure 2(a)–(c). We show the
averages and the 68% confidence intervals (CI) of the number of new cases as a
function of time over multiple iterations for increasing coverage, $m$, i.e.,
increasing adoption of protective behaviour. For individuals $i$ that do not
adopt protective behaviour ($M_{i}=0$), the height of the epidemic peaks
decreases with increasing coverage as the source-controlling effects
indirectly protect them. The duration of the outbreak increases however,
leading to a situation where the total infection risk for non-adopting
individuals during the outbreak changes little for sufficiently low levels of
$m$, see the corresponding plateau for low $m$ in figure 2(d).

Intriguingly, for adopting individuals the individual infection risk, $IIR$
may exhibit a pronounced maximum at an intermediate coverage rate. Simulation
runs for different values of coverage ($m$) show comparable heights of the
epidemic peak for adopting individuals; see figure 2(a)–(c). However, the
duration of this peak increases substantially with increasing $m$. As a
result, there is a regime where the individual infection risk for adopting
individuals decreases with decreasing coverage, as shown in figure 2(d). For
the settings used, 61% of the adopting individuals get infected during the
outbreak if they make up only 10% of the population, but 78% get infected if
they account for 60% of the total population. We refer to this phenomenon as
the “small coverage effect”.

On a population level, this small coverage effect reveals itself in the form
of a plateau in the population infection risk, $PIR$ at intermediate coverage,
see figure 2(e). On this plateau the $PIR$-increasing small coverage effect
approximately cancels with the increase in the number of adopting individuals
with higher $IIR$. For reference, figure 2(e) also shows results for the $PIR$
without waning in which the monotonous decrease in $PIR$ under increasing
coverage is recovered.

Figure 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 network and node states, $M$, for these
settings, respectively. With increasing coverage, duration of the outbreak in
adopting individuals increases. 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 $q_{r}(0)=q_{s}(0)=0.4$); error bars denote the
68% CI. If adherence wanes, the infection risk has a clear maximum around
$m=0.6$, while 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 population infection risks, $PIR$, for a wide range of
intermediate coverage.

Different mechanisms are at work that reduce the individual infection risk for
adopting individuals when coverage becomes very high or low, respectively.
Starting from a coverage of $80\%$ or more, simulation runs start to occur in
which the epidemic dies out before adherence has waned sufficiently to trigger
a resurgence of cases. This can be seen in figure 2(d) in the increase of the
variance (confidence intervals) of the infection risk, $IR(M)$, for high
coverage. In this regime, the epidemic either quickly dies out (leading to
values of $IR(M)$ close to zero) or larger outbreaks occur in both population
groups if a small number of infections persists until adherence has
sufficiently decreased. One can think of this regime as an ”all or nothing”
scenario-driven by finite size effects in the network model.

In figure 2(d,e) we also show simulation results for a scenario where
adherence does not wane, i.e. $q=1$ (using $q_{r}(0)=q_{s}(0)=0.4$). In this
case, the frequently held belief of infection risk being a monotonously
decreasing function of coverage, $m$, is recovered for both population groups
and on an individual and population level. That is, the more people adopt
protective behaviour, the smaller the outbreak size.

Note that the small coverage effect is not a result of the normalization in
the definition of the individual infection risks, $IIR(M=1)$ and $IIR(M=0)$.
As the infection risk for adopting individuals increases with $m$ for small
coverage, figure 2(d) shows that the number of infections grows even faster
than the number of adopting individuals in that group. The situation is
slightly different on the population level, figure 2(e), where this effect is
compensated by an increasing number of individuals moving from the non-
adopting to the adopting state and thereby reducing their infection risks.

Figure 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
$\eta$). The reduction decreases with decreasing homophily and plateaus at
values about $10\%$, for $\eta\leq 0.5$. Increasing the small world parameter
from $\epsilon=0.1$ (blue) to $0.5$ (orange) reduces the small coverage effect
even for high levels of homophily. (b) A parameter sweep over the homophily
and small world parameters, $\eta$ and $\epsilon$, respectively, shows that
the small coverage effect is most pronounced (close to 30%) for small
$\epsilon$ and large $\eta$ and plateaus between $10\%-15\%$ for other
parameter settings.

As seen in figure 2(d), adopting behaviour at a coverage of 60% ($m=0.6$)
reduces the $IIR$ upon adopting protective behaviour from 95% for $M=0$ to 78%
for $M=1$. For a coverage of 10% ($m=0.1$), however, adoption reduces the risk
to 61%. To explore this effect more systematically for different parameter
settings, we consider the infection risk reduction (in percent) from a
coverage range of 60% down to 10%. For the example given above, the small
coverage effect thereby reduces the individual infection risk from 78% for
$m=0.6$ by 17% (absolute reduction) to 61% at $m=0.1$.

The extent to which homophily amplifies the small coverage effect is shown in
figure 3. Note that here we only show statistics from runs in which the
epidemic does not “die out” immediately at the beginning of the simulation, as
these runs would inflate the CIs. For strong homophily ($\eta$ close to one),
infection risk reduction falls within the range of 20% to 25%; see figure
3(a). With $\eta$ decreasing, the infection risk reduction decreases too. For
values of around $\eta\approx 0.5$ the infection risk reduction bottoms out at
10% where it assumes a plateau for weaker levels of homophily. Increasing the
small world parameter $\epsilon$ to 0.5 decreases assortative mixing and leads
to a smaller infection risk reduction for large $\eta$. These observations are
corroborated by a parameter sweep over a range of values for $\eta$ and
$\epsilon$; see figure 3(b). Under strong assortative mixing (high $\eta$,
small $\epsilon$), the small coverage effect reduces the infection risk by
close to 30%. As assortative mixing decreases, the small coverage effect
decreases and plateaus at around 10% to 15%.

So far we considered situations where the initial effectiveness of adopting
protective behaviour is high ($q_{r}(0)=q_{s}(0)=1$). Note that the small
coverage effect can also be observed for less effective measures. For
instance, for cloth masks the inward and outward protective effectiveness has
been estimated as 50% and 30%, respectively. [36] Using these values
($q_{r}(0)=0.5$, $q_{s}(0)=0.3$), we observe infection risk reductions (from a
coverage of 70% to 10%) of around $7\%$ (SD $1.6\%$). One can also compare the
relative risk ($RR$) of becoming infected between adopting and non-adopting
individuals. For a coverage of 70%, adopting individuals had $RR=0.95$
compared to non-adopting individuals, for small coverage ($m=10\%$) we found
$RR=0.88$. The magnitude of this reduction depends on the model parameters.
For instance, setting the connectivity of the network to $k=16$ gives a more
pronounced infection risk reduction (at 10% compared to a coverage 90%) of
$11\%$ (SD $2.1\%$), meaning that the relative risks of getting infected
reduce from $RR=0.95$ to $0.84$.

##  4 Discussion

Conventional wisdom holds that the more people wear a mask, the higher is the
protection it offers. In this study we show that this is not necessarily true
in general. Using the plausible assumptions that (i) adherence to protective
behaviour wanes over time and (ii) individuals tend to have a preference to
interact more with like-minded others (homophily), we demonstrate the
existence of a hitherto unknown small coverage effect: As the number of people
adopting the protective behaviour increases, the individual-level protection
offered by the behaviour decreases. This finding invalidates the frequently
held notion that the individual-level effectiveness of measures like mask-
wearing always increase with the number of people wearing a mask.[37, 38, 39,
40, 41, 42] We numerically observed this effect for a wide range of parameters
in a parsimonious network model. The small coverage effect can be observed
independently from how strong individuals cluster topologically in the network
(i.e., the friend of a friend is likely to be a friend of mine too) and in
terms of adopting the behaviour (homophily). However, both types of
clustering, when combined, do amplify the effect substantially.

The origin of the small coverage effect can be easily understood from system-
dynamical properties. If only a small fraction of the population adopts
protective behaviour, there is a good chance that the outbreak will already be
over in the non-adopting population group before measure adherence has
completely waned. The situation changes when there are more people that
initially adopt the behaviour. The overall susceptibility in the population
will grow faster as the outbreak unfolds, as measure adherence is waning in a
larger number of individuals. This fuels–and thereby prolongs–the outbreak.
Hence, the chance that the outbreak is over before adherence has vanished
completely decreases with coverage.

Homophily amplifies the small coverage effect. This is likely related to local
trapping of infection events in the network. [43, 44] Smaller groups of
adopting individuals that preferentially have ties with each other are less
likely to be affected in the initial phases of the outbreak, when adherence is
still high. As the outbreak continues, they are also less likely to be
affected by large infection numbers amongst non-adopting individuals since
they have a smaller number of ties with them. Therefore, it will take the
pathogen a longer to “invade” these protected groups, in contrast to a
situation with no homophily.

It has indeed been observed that adherence to protective measures like mask
wearing has strongly increased in the early phases of the pandemic [19, 16,
20] but has since then decreased. [23, 24] Many studies that tried to measure
the real-world effectiveness of mask wearing basically compared the infection
risk in adopting versus non-adopting individuals.[45] Systematic reviews found
that effectiveness can be quite heterogeneous across different settings and
studies.[10, 46] The reasons for this heterogeneity are not yet fully
understood. The present work demonstrates that such study designs can be
severely affected by the confounding influence of the population-wide mask
coverage, which is typically not considered as a covariate. Using conservative
estimates for the protective effectiveness of cloth masks, we find that
reductions in relative risk of masking versus non-masking individuals range
between $5\%$ and $15\%$. Further research is needed to understand whether
empirically measured real-world differences in the effectiveness of mask-
wearing could originate from the small coverage effect.

Academically, it would be challenging in a next step to demonstrate the
mechanisms behind the small coverage effects also in the framework of game
theory. One could think of a setting where adopters are cooperators and non-
adopters are defectors; the payoff function (infection risks for adopters and
non-adopters) would depend on the respective group sizes, and the waning
effect would have to be implemented by a continuous shift towards a larger
fraction of non-adopters over time. However, it is not clear from the outset
what would constitute a round or interaction in which this game would be
played. A conceivable option could be a model with waning immunity (i.e.,
transitions from the recovered to the susceptible state) and seasonal driving
of the transmission risk, $\alpha$. Individuals could observe their infection
risks during one outbreak and possibly adjust their behaviour (state $M$) in
the next seasonal outbreak depending on whether their past behaviour protected
them or not.

Our simulation study is limited due to the use of a parsimonious model that
makes idealized assumptions about several aspects, such as the infectious
disease dynamics or the contact network topology. This study design was a
deliberate choice in order to isolate the mechanism driving the small coverage
effect. It remains to be seen to what extent this effect is observed in more
sophisticated and calibrated epidemiological models and, above all, in real-
world settings.

If it were true that wearing a mask offers less protection as less individuals
mask themselves, it would be rational to give up on this behaviour as it won’t
offer protection anymore if everyone else drops his or her mask. An important
implication of our work is that quite the opposite might be true. Masking,
when adopted by a minority, might actually facilitate “sitting out” a wave
without becoming infected. This finding might have important consequences for
the protection of vulnerable population groups that are at increased risk for
severe disease.

References

## References

  * [1] C. Cheng, J. Barceló, A. S. Hartnett, R. Kubinec, and L. Messerschmidt, “Covid-19 government response event dataset (coronanet v. 1.0),” Nature human behaviour, vol. 4, no. 7, pp. 756–768, 2020.
  * [2] T. Hale, N. Angrist, R. Goldszmidt, B. Kira, A. Petherick, T. Phillips, S. Webster, E. Cameron-Blake, L. Hallas, S. Majumdar, et al., “A global panel database of pandemic policies (oxford covid-19 government response tracker),” Nature human behaviour, vol. 5, no. 4, pp. 529–538, 2021.
  * [3] A. Desvars-Larrive, E. Dervic, N. Haug, T. Niederkrotenthaler, J. Chen, A. Di Natale, J. Lasser, D. S. Gliga, A. Roux, J. Sorger, et al., “A structured open dataset of government interventions in response to covid-19,” Scientific data, vol. 7, no. 1, pp. 1–9, 2020.
  * [4] S. Flaxman, S. Mishra, A. Gandy, H. J. T. Unwin, T. A. Mellan, H. Coupland, C. Whittaker, H. Zhu, T. Berah, J. W. Eaton, et al., “Estimating the effects of non-pharmaceutical interventions on covid-19 in europe,” Nature, vol. 584, no. 7820, pp. 257–261, 2020.
  * [5] N. Haug, L. Geyrhofer, A. Londei, E. Dervic, A. Desvars-Larrive, V. Loreto, B. Pinior, S. Thurner, and P. Klimek, “Ranking the effectiveness of worldwide covid-19 government interventions,” Nature human behaviour, vol. 4, no. 12, pp. 1303–1312, 2020.
  * [6] M. Sharma, S. Mindermann, C. Rogers-Smith, G. Leech, B. Snodin, J. Ahuja, J. B. Sandbrink, J. T. Monrad, G. Altman, G. Dhaliwal, et al., “Understanding the effectiveness of government interventions against the resurgence of covid-19 in europe,” Nature communications, vol. 12, no. 1, pp. 1–13, 2021.
  * [7] Y. Liu, C. Morgenstern, J. Kelly, R. Lowe, and M. Jit, “The impact of non-pharmaceutical interventions on sars-cov-2 transmission across 130 countries and territories,” BMC medicine, vol. 19, no. 1, pp. 1–12, 2021\.
  * [8] A. Mendez-Brito, C. El Bcheraoui, and F. Pozo-Martin, “Systematic review of empirical studies comparing the effectiveness of non-pharmaceutical interventions against covid-19,” Journal of Infection, vol. 83, no. 3, pp. 281–293, 2021.
  * [9] D. K. Chu, E. A. Akl, S. Duda, K. Solo, S. Yaacoub, H. J. Schünemann, A. El-Harakeh, A. Bognanni, T. Lotfi, M. Loeb, et al., “Physical distancing, face masks, and eye protection to prevent person-to-person transmission of sars-cov-2 and covid-19: a systematic review and meta-analysis,” The lancet, vol. 395, no. 10242, pp. 1973–1987, 2020.
  * [10] S. Talic, S. Shah, H. Wild, D. Gasevic, A. Maharaj, Z. Ademi, X. Li, W. Xu, I. Mesa-Eguiagaray, J. Rostron, et al., “Effectiveness of public health measures in reducing the incidence of covid-19, sars-cov-2 transmission, and covid-19 mortality: systematic review and meta-analysis,” bmj, vol. 375, 2021.
  * [11] T. Pluemper and E. Neumayer, “The covid-19 pandemic predominantly hits poor neighborhoods, or does it? evidence from germany,” medRxiv, 2020.
  * [12] A. Buja, M. Paganini, S. Cocchio, M. Scioni, V. Rebba, and V. Baldo, “Demographic and socio-economic factors, and healthcare resource indicators associated with the rapid spread of covid-19 in northern italy: An ecological study,” PLoS One, vol. 15, no. 12, p. e0244535, 2020.
  * [13] N. Banholzer, S. Feuerriegel, and W. Vach, “Estimating and explaining cross-country variation in the effectiveness of non-pharmaceutical interventions during covid-19,” Scientific reports, vol. 12, no. 1, pp. 1–12, 2022.
  * [14] K. Ledebur, M. Kaleta, J. Chen, S. D. Lindner, C. Matzhold, F. Weidle, C. Wittmann, K. Habimana, L. Kerschbaumer, S. Stumpfl, et al., “Meteorological factors and non-pharmaceutical interventions explain local differences in the spread of sars-cov-2 in austria,” PLoS computational biology, vol. 18, no. 4, p. e1009973, 2022.
  * [15] A. Petherick, R. Goldszmidt, E. B. Andrade, R. Furst, T. Hale, A. Pott, and A. Wood, “A worldwide assessment of changes in adherence to covid-19 protective behaviours and hypothesized pandemic fatigue,” Nature Human Behaviour, vol. 5, no. 9, pp. 1145–1160, 2021.
  * [16] M. A. Crane, K. M. Shermock, S. B. Omer, and J. A. Romley, “Change in reported adherence to nonpharmaceutical interventions during the covid-19 pandemic, april-november 2020,” Jama, vol. 325, no. 9, pp. 883–885, 2021.
  * [17] P. Goldstein, E. Levy Yeyati, and L. Sartorio, “Lockdown fatigue: The diminishing effects of quarantines on the spread of covid-19,” CID Working Paper Series, 2021.
  * [18] T. Reisch, G. Heiler, J. Hurt, P. Klimek, A. Hanbury, and S. Thurner, “Behavioral gender differences are reinforced during the covid-19 crisis,” Scientific reports, vol. 11, no. 1, pp. 1–12, 2021.
  * [19] L. E. Smith, H. W. Potts, R. Amlot, N. T. Fear, S. Michie, and G. J. Rubin, “Engagement with protective behaviours in the uk during the covid-19 pandemic: a series of cross-sectional surveys (the covid-19 rapid survey of adherence to interventions and responses [corsair] study),” BMC public health, vol. 22, no. 1, pp. 1–11, 2022.
  * [20] M. Clinton, J. Sankar, V. Ramesh, and M. Madhusudan, “Changes in pattern of adherence to npis during the covid-19 pandemic,” Indian Journal of Pediatrics, vol. 88, no. 8, pp. 837–837, 2021.
  * [21] A. H. Tjaden, M. Gibbs, M. Runyon, W. S. Weintraub, Y. J. Taylor, S. L. Edelstein, C.-. C. R. P. S. Group, et al., “Association between self-reported masking behavior and sars-cov-2 infection wanes from pre-delta to omicron-predominant periods—north carolina covid-19 community research partnership,” medRxiv, 2022.
  * [22] V. R. Peixoto, A. J. Mason-Jones, A. McCallum, et al., “Is it really time to ditch the mask?,” 2022.
  * [23] “Covid personal precautions on the decline.” https://apnorc.org/projects/worries-about-being-infected-with-the-coronavirus-remain-low-and-americans-are-taking-fewer-personal-precautions/.  Accessed: 2022-08-08.
  * [24] “Public opinions and social trends, great britain: 20 to 31 july 2022.” https://www.gov.uk/government/statistics/public-opinions-and-social-trends-great-britain-20-to-31-july-2022.  Accessed: 2022-08-08.
  * [25] M. McPherson, L. Smith-Lovin, and J. M. Cook, “Birds of a feather: Homophily in social networks,” Annual review of sociology, pp. 415–444, 2001.
  * [26] M. E. Newman, “Assortative mixing in networks,” Physical review letters, vol. 89, no. 20, p. 208701, 2002.
  * [27] R. M. Bond, C. J. Fariss, J. J. Jones, A. D. Kramer, C. Marlow, J. E. Settle, and J. H. Fowler, “A 61-million-person experiment in social influence and political mobilization,” Nature, vol. 489, no. 7415, pp. 295–298, 2012\.
  * [28] K. P. Smith and N. A. Christakis, “Social networks and health,” Annual review of sociology, vol. 34, no. 1, pp. 405–429, 2008.
  * [29] D. Centola, “An experimental study of homophily in the adoption of health behavior,” Science, vol. 334, no. 6060, pp. 1269–1272, 2011.
  * [30] J. J. V. Bavel, K. Baicker, P. S. Boggio, V. Capraro, A. Cichocka, M. Cikara, M. J. Crockett, A. J. Crum, K. M. Douglas, J. N. Druckman, et al., “Using social and behavioural science to support covid-19 pandemic response,” Nature human behaviour, vol. 4, no. 5, pp. 460–471, 2020.
  * [31] M. Agranov, M. Elliott, and P. Ortoleva, “The importance of social norms against strategic effects: The case of covid-19 vaccine uptake,” Economics Letters, vol. 206, p. 109979, 2021.
  * [32] A. Woodcock and P. W. Schultz, “The role of conformity in mask-wearing during covid-19,” Plos one, vol. 16, no. 12, p. e0261321, 2021.
  * [33] J. Shin, A. Yang, W. Liu, H. Min Kim, A. Zhou, and J. Sun, “Mask-wearing as a partisan issue: Social identity and communication of party norms on social media among political elites,” Social Media+ Society, vol. 8, no. 1, p. 20563051221086233, 2022.
  * [34] R. M. Anderson and R. M. May, Infectious diseases of humans: dynamics and control.  Oxford university press, 1992.
  * [35] D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’networks,” nature, vol. 393, no. 6684, pp. 440–442, 1998\.
  * [36] J. Pan, C. Harb, W. Leng, and L. C. Marr, “Inward and outward effectiveness of cloth masks, a surgical mask, and a face shield,” Aerosol Science and Technology, vol. 55, no. 6, pp. 718–733, 2021.
  * [37] R. O. Stutt, R. Retkute, M. Bradley, C. A. Gilligan, and J. Colvin, “A modelling framework to assess the likely effectiveness of facemasks in combination with ‘lock-down’ in managing the covid-19 pandemic,” Proceedings of the Royal Society A, vol. 476, no. 20200376, 2020.
  * [38] D. Kai, G.-P. Goldstein, A. Morgunov, V. Nangalia, and A. Rotkirch, “Universal masking is urgent in the covid-19 pandemic: Seir and agent based models, empirical validation, policy recommendations,” 2020.
  * [39] A. Catching, S. Capponi, M. Te Yeh, S. Bianco, and R. Andino, “Examining the interplay between face mask usage, asymptomatic transmission, and social distancing on the spread of covid-19,” Scientific Reports, vol. 11, no. 15998, 2021.
  * [40] C. N. Ngonghala, E. Iboi, S. Eikenberry, M. Scotch, C. R. MacIntyre, M. H. Bonds, and A. B. Gumel, “Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel coronavirus,” Mathematical Biosciences, vol. 325, no. 108364, 2020.
  * [41] L. Tian, X. Li, Q.-Y. Tang, V. Tang, J. Liu, Z. Li, X. Cheng, X. Li, Y. Shi, H. Liu, and L.-H. Hang, “Harnessing peak transmission around symptom onset for non-pharmaceutical intervention and containment of the covid-19 pandemic,” Nature Communications, vol. 12, no. 1147, 2021.
  * [42] J. Howard, A. Huang, Z. Li, Z. Tufekci, V. Zdimal, H.-M. van der Westhuizen, A. von Delft, A. Price, L. Fridman, L.-H. Tang, V. Tang, G. L. Watson, C. E. Bax, R. Shaikh, F. Questier, D. Hernandez, L. F. Chu, C. M. Ramirez, and A. W. Rimoin, “An evidence review of face masks against covid-19,” Proceedings of the National Academy of Sciences, vol. 118, no. 4, p. e2014564118, 2021.
  * [43] A. Allard, C. Moore, S. V. Scarpino, B. M. Althouse, and L. Hébert-Dufresne, “The role of directionality, heterogeneity and correlations in epidemic risk and spread,” 2020.
  * [44] S. Thurner, P. Klimek, and R. Hanel, “A network-based explanation of why most covid-19 infection curves are linear,” Proceedings of the National Academy of Sciences, vol. 117, no. 37, pp. 22684–22689, 2020.
  * [45] B. Alihsan, A. Mohammed, Y. Bisen, J. Lester, C. Nouryan, and J. Cervia, “The efficacy of facemasks in the prevention of covid-19: A systematic review,” medRxiv, 2022.
  * [46] R. Chou, T. Dana, R. Jungbauer, C. Weeks, and M. S. McDonagh, “Masks for prevention of respiratory virus infections, including sars-cov-2, in health care and community settings,” Annals of Internal Medicine, vol. 173, no. 7, pp. 542–555, 2020.