# The advantage of self-protecting interventions in mitigating epidemic
circulation at the community level

Romualdo Pastor-Satorras  Departament de Física, Universitat Politècnica de
Catalunya, Campus Nord B4, 08034 Barcelona, Spain  Claudio Castellano
Istituto dei Sistemi Complessi (ISC-CNR), Via dei Taurini 19, I-00185 Rome,
Italy  Centro Ricerche Enrico Fermi, Piazza del Viminale, 1, I-00184 Rome,
Italy

###### Abstract

Protecting interventions of many types (both pharmaceutical and non-
pharmaceutical) can be deployed against the spreading of a communicable
disease, as the worldwide COVID-19 pandemic has dramatically shown. Here we
investigate in detail the effects at the population level of interventions
that provide an asymmetric protection between the people involved in a single
interaction. Masks of different filtration types, either protecting mainly the
wearer or the contacts of the wearer, are a prominent example of these
interventions. By means of analytical calculations and extensive simulations
of simple epidemic models on networks, we show that interventions protecting
more efficiently the adopter (e.g the mask wearer) are more effective than
interventions protecting primarily the contacts of the adopter in reducing the
prevalence of the disease and the number of concurrently infected individuals
(“flattening the curve”). This observation is backed up by the study of a more
realistic epidemic model on an empirical network representing the patterns of
contacts in the city of Portland. Our results point out that promoting wearer-
protecting face masks and other self-protecting interventions, though deemed
selfish and inefficient, can actually be a better strategy to efficiently
curtail pandemic spreading.

##  I Introduction

The recent COVID-19 pandemic has shown how fragile world societies are when
confronted to the runaway spreading of a new virus strain Christakis (2020).
In stark contrast with other diseases, however, the combination of recent
scientific advances, liberal funding by governments and the availability of
large scale manufacturing, has allowed to create and deploy world-wide
vaccines for the SARS-CoV-2 virus at unprecedented speed Ball (2021), a fact
that has without any doubts greatly diminished the death toll expected in this
pandemic. Despite this huge effort, however, more than nine months lapsed
between the declaration of the COVID-19 pandemic by WHO in March 2020 and the
approval of the first vaccine in December 2020\. The large-scale deployment of
vaccines worldwide required additional months and is still an unsolved problem
in many areas of the planet. Moreover, the protection guaranteed by vaccines
against recently emerged new variants is far from perfect. In this context,
societies have had and still have to fight the propagation of the virus by
adopting also more traditional non-pharmaceutical interventions Perra (2021).
These interventions, aiming at limiting the spread of a disease Christakis
(2020), include collective actions, such as school closures, prohibition of
large meetings, closing of public transportation, curfews or lockdown orders
European Centre for Disease Prevention and Control (2020). At an individual
level, non-pharmaceutical interventions include, among others, physical
distancing, hand and respiratory hygiene, and face masks. Understanding the
impact of each of these interventions at the level of the single transmission
event among individuals and how this translates at the community level is a
crucial goal, which has attracted a huge scientific interest Lin  _et al._
(2020); Reguly  _et al._ (2022); Gozzi  _et al._ (2021); Jarvis  _et al._
(2020); Kraemer  _et al._ (2020); Brauner  _et al._ (2021); Davies  _et al._
(2020); Stutt  _et al._ (2020); Li  _et al._ (2020); Althouse  _et al._
(2020); Dehning  _et al._ (2020); Hsiang  _et al._ (2020); Bo  _et al._
(2021).

In this paper we present an analysis of how the reduction of the chance that
the disease is transmitted in a single contact, induced by a generic type of
protecting intervention, results in an overall mitigation of the epidemic
circulation in the population. In particular, we focus on asymmetric
interventions having different efficacies at the individual level, i.e., in
reducing the infectivity of an infected individual with respect to decreasing
the susceptibility of a susceptible one.

As one of the most widely adopted means to fight the current pandemic, mask
wearing is the paradigmatic example of these types of interventions. Given the
evidence for the airborne transmission of SARS-CoV-2 by means of droplets and
aerosols Greenhalgh  _et al._ (2021), recent research has shown that face
masks are a very effective way to reduce the spread of COVID-19 Mitze  _et
al._ (2020); Behring  _et al._ (2021); Liang  _et al._ (2020); Eikenberry  _et
al._ (2020); Ngonghala  _et al._ (2020); Howard  _et al._ (2021); Watanabe and
Hasegawa (2021); Robinson  _et al._ (2022). Different types of masks exist,
characterized by very different protecting performance, both qualitatively and
quantitatively Koh  _et al._ (2022); Leith  _et al._ (2021). Some protect
mainly the mask wearer, such as valved masks endowed with an exhalation valve.
Others mainly protect the contacts of the mask wearer, such as surgical or
cloth masks. Such asymmetric efficacy is however more general. For example,
vaccines reduce the susceptibility of individuals to being infected upon
contact with an infectious person, but they need not be equally effective in
reducing the capability of vaccinated infected individuals to transmit the
pathogen further if they ever catch the disease Braeye  _et al._ (2021). Also,
symptomatic treatment reducing cough and sneezing hinders the possibility to
transmit the disease onward but it does not reduce the chance to be infected.

Here we present a comparative analysis of the global efficiency of generic
protective interventions (PI) in the context of epidemic models on networks
Pastor-Satorras  _et al._ (2015), representing the patterns of social contacts
among people that mediate the propagation of an infective pathogen Jackson
(2010). We use a formulation over networks with a static topology,
parametrizing the effect of PIs in terms of the fraction of adopters and of
the efficacy of adoption in reducing the chance to receiving/transmitting the
disease.

We analyze the effect of different types of PIs at the community level by
considering two paradigmatic models of disease propagation in networks Keeling
and Rohani (2007); Kiss  _et al._ (2017a): the Susceptible-Infected-Removed
(SIR) model for non-recurrent diseases, that confer immunity and develop in
outbreaks (similarly to COVID-19, at least on short time scales), and the
Susceptible-Infected-Susceptible (SIS) model for recurrent diseases that may
lead to a steady endemic state. Using a combination of numerical simulations
and analytical calculations we show that the adoption of asymmetric PIs leads
to surprisingly different scenarios. In particular interventions protecting
the adopter turn out to have a stronger effect at the population level. If PIs
of the same efficacy in reducing infectivity/susceptibility are available, it
is better to use PIs protecting the adopter, as they suppress more the global
circulation of the disease. We investigate the origin of this unexpected
finding and show that the conclusion holds also when considering more
realistic interaction patterns and disease dynamics.

##  II Results

###  II.1 Modeling the effect of protecting interventions

The effect of protecting interventions on the propagation of a disease can be
modeled by considering the modification of the probability that the disease is
transmitted between an infected and a susceptible individual Miller (2007);
Behring  _et al._ (2021). Consider a population of $N$ individuals, whose
pattern of contacts is determined by a complex network Newman (2010), in which
nodes represent individuals and edges the presence of social contacts between
pairs of individuals. The network is fully described by its adjacency matrix
$a_{ij}$, taking value $1$ if nodes $i$ an $j$ are connected, and zero
otherwise. From a statistical perspective, the network can be characterized by
its degree distribution $P(k)$, representing the probability that a randomly
chosen node has degree $k$ (i.e., it is connected to $k$ other individuals).

Disease transmission is mediated by contacts between infected and susceptible
individuals. The properties of this transmission are mathematically encoded
Keeling and Rohani (2007); Kiss  _et al._ (2017a) in terms of an infection
rate $\beta$, representing the probability per unit time that the infection is
transmitted from an infected to a susceptible along a single contact.
Protecting interventions induce a reduction of this infection rate in two
possible manners. An intervention involving the infected individual diminishes
the infection rate along any edge emanating from him/her by a factor
$\alpha_{\mathrm{o}}$. An intervention affecting the susceptible individual
decreases, by a factor $\alpha_{\mathrm{i}}$, the infection rate of any edge
pointing to him/her. For example, in the case of airborne transmission, a mask
may reduce the number of viral particles exhaled by an infected individual and
also reduce the number of viral particles that a susceptible can inhale from
the environment Behring  _et al._ (2021). As the example of masks shows, the
factors $\alpha_{\mathrm{o}}$ and $\alpha_{\mathrm{i}}$ are not necessarily
equal. For example valved masks strongly protect the wearer
$\alpha_{\mathrm{i}}\ll 1$ while they do not impede viral transmission towards
the others $\alpha_{\mathrm{o}}\approx 1$. For cloth masks the opposite is
true. We compare scenarios where a single type of PI (with given
$\alpha_{\mathrm{o}}$ and $\alpha_{\mathrm{i}}$) is adopted by a fraction $f$
of the population. For a given edge pointing from node $i$ to node $j$, the
infection rate associated to the disease transmission from $i$ to $j$ takes
the form

$\beta_{ij}=\beta
a_{ij}\left[1-m_{i}(1-\alpha_{\mathrm{o}})\right]\left[1-m_{j}(1-\alpha_{\mathrm{i}})\right],$
(1)

where $m_{i}$ are quenched Bernoulli stochastic variables taking value $1$
with probability $f$ and zero with probability $1-f$. From this definition it
is obvious that, even if the contact network is undirected (symmetric
$a_{ij}$) Newman (2010), the transmission network given by (1) is effectively
directed, with $\beta_{ij}\neq\beta_{ji}$ unless $f=0$ (no PI adoption in the
population) or $f=1$ (whole population adopting PI), which lead to
$\beta_{ij}=\beta a_{ij}$ and
$\beta_{ij}=\beta\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}a_{ij}$, respectively.
In these cases, the infection process is symmetric if the underlying contact
network is undirected. In our study we are concerned with the differences
between PIs that mostly protect the adopter, characterized by
$\alpha_{\mathrm{i}}\ll\alpha_{\mathrm{o}}$, and PIs that mostly protect
his/her contacts, given by $\alpha_{\mathrm{o}}\ll\alpha_{\mathrm{i}}$. For
the sake of simplicity, we focus on what we call SELF interventions (with
$\alpha_{\mathrm{i}}<1$, $\alpha_{\mathrm{o}}=1$) that offer protection to the
adopter and no protection for his/her contacts, and OTHER interventions
($\alpha_{\mathrm{i}}=1$, $\alpha_{\mathrm{o}}<1$), that offer protection to
the contacts, and no protection to the adopter.

In the following, we investigate the effects of PIs in fundamental models of
disease propagation, either non-recurrent or recurrent.

###  II.2 Non-recurrent diseases

As an example of a communicable non-recurrent disease (that is, a disease that
confers permanent immunity), we consider the simple Susceptible-Infected-
Removed (SIR) model Keeling and Rohani (2007). The SIR model is defined in
terms of three compartments. Susceptible individuals are healthy and can
contract the disease. Infected individuals carry the disease and can infect
susceptible ones upon contact with a rate (probability per unit time) given by
the factor $\beta_{ij}$ in Eq. (1). In their turn, infected individuals can
recover and become removed with a constant rate $\mu$. By rescaling time, we
can absorb the rate $\mu$ and consider a single parameter, the spreading rate
$\lambda=\beta/\mu$.

Focusing for simplicity on homogeneous networks with a uniform degree
$k_{i}=K$, a mean-field theoretical analysis (see Methods IV.1) shows that the
SIR model exhibits a transition between a phase with non-extensive outbreaks
only and a phase with macroscopic ones, at a critical value of the spreading
rate $\lambda$ (see Methods IV.1.1)

$\lambda_{c}=\frac{1}{K-1}\frac{1}{f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+1-f}.$
(2)

From this expression, we see that the threshold is symmetric in the parameters
$\alpha_{\mathrm{i}}$ and $\alpha_{\mathrm{o}}$, in agreement with the results
in Ref. Behring  _et al._ (2021), and that the maximum system-wide protection
is obtained, for a given value of adoption probability $f$, when
$\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}=0$, which can be attained either in
the perfect SELF scenario ($\alpha_{\mathrm{i}}=0$) or in the perfect OTHER
case ($\alpha_{\mathrm{o}}=0$). In the same theoretical framework, it is also
possible to calculate the total prevalence $R_{\infty}$, i.e., the overall
fraction of individuals infected by the disease throughout the outbreak,
obtaining, slightly above the threshold (see Methods IV.1.2)

$R_{\infty}\simeq\frac{2\Delta}{\lambda_{c}^{2}(K-1)}\frac{f\alpha_{\mathrm{i}}+1-f}{f\alpha_{\mathrm{i}}^{2}\alpha_{\mathrm{o}}+1-f},$
(3)

where $\Delta=\lambda-\lambda_{c}$ quantifies the distance from the critical
threshold. This expression reveals that, at variance with the position of the
threshold, the size of an outbreak is not symmetric with respect to the
efficacy of PIs: At fixed values of $f$, $\lambda$ and of the threshold
$\lambda_{c}$ (i.e. for the product $\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}$
fixed to a constant $A$), if we substitute
$\alpha_{\mathrm{o}}=A/\alpha_{\mathrm{i}}$, the total prevalence is easily
shown to be an increasing function of $\alpha_{\mathrm{i}}$, while setting
$\alpha_{\mathrm{i}}=A/\alpha_{\mathrm{o}}$ leads to a total prevalence that
decreases with $\alpha_{\mathrm{o}}$. In other words, while using more
effective SELF interventions (i.e., reducing $\alpha_{\mathrm{i}}$) decreases
$R_{\infty}$, imposing more effective OTHER interventions (i.e reducing
$\alpha_{\mathrm{o}}$) creates the opposite result, an increase of
$R_{\infty}$. This signals that SELF interventions, protecting the adopter,
are more effective than OTHER interventions, protecting the contacts, at
reducing the overall spreading of the disease at the community level. This is
confirmed by integrating numerically the homogeneous mean field equations
(52)-(58), for generic values of $\alpha_{\mathrm{i}}$ and
$\alpha_{\mathrm{o}}$ (see Fig. 1(a)).

Figure 1: SELF interventions are more effective than OTHER interventions at
the population level. (a) Difference between the final prevalence $R_{\infty}$
for the OTHER and the SELF scenario
($R_{\infty}(\mathrm{OTHER})-R_{\infty}(\mathrm{SELF})$), as a function of
$\alpha_{\mathrm{i}}$ and $\alpha_{\mathrm{o}}$ of the individual PI. Notice
that $R_{\infty}(\mathrm{OTHER})$ depends only on $\alpha_{\mathrm{o}}$, and
$R_{\infty}(\mathrm{SELF})$ depends only on $\alpha_{\mathrm{i}}$. The solid
black line indicates the zero value. (b) Difference between the final
prevalence $R_{\infty}$ for the OTHER and the SELF scenario as a function of
the fraction $f$ of adopters and the efficacy $\alpha$ of the individual PI.
The solid yellow line denotes where $\lambda_{c}(\alpha)=\lambda$. Above it,
the system is subcritical and
$R_{\infty}(\mathrm{OTHER})-R_{\infty}(\mathrm{SELF})=0$. (c) Difference
between the maximum value of the incidence $I_{m}$ for the OTHER and the SELF
scenario ($I_{m}(\mathrm{OTHER})-I_{m}(\mathrm{SELF})$), as a function of
$\alpha_{\mathrm{i}}$ and $\alpha_{\mathrm{o}}$ of the individual PI. Notice
that $I_{m}(\mathrm{OTHER})$ depends only on $\alpha_{\mathrm{o}}$, and
$I_{m}(\mathrm{SELF})$ depends only on $\alpha_{\mathrm{i}}$. The solid black
line indicates the zero value. (d) Difference between the maximum value of the
incidence $I_{m}$ for the OTHER and the SELF scenario as a function of the
fraction $f$ of adopters and the efficacy $\alpha$ of the individual PI. The
solid yellow line denotes where $\lambda_{c}(\alpha)=\lambda$. In all plots
quantities greater than zero indicate that SELF interventions perform better.
Solution of homogeneous MF equations for a network of degree $K=7$ and for
$\lambda=0.5$. In panels (a) and (c) $f=0.5$. For these parameters, in panels
(a) and (c) both SELF and OTHER strategies operate above their respective
thresholds.

Most of the parameter space results in
$R_{\infty}(\mathrm{OTHER})>R_{\infty}(\mathrm{SELF})$, i.e. a larger
circulation of the disease when OTHER interventions are adopted.

The comparison of SELF and OTHER scenarios provides remarkable insight also at
the level of the individual. As shown in Methods IV.1.3, the relation between
the asymptotic prevalence $R_{1}(\infty)$ for adopting individuals and the
prevalence $R_{0}(\infty)$ for nonadopting ones is

$\displaystyle R_{1}(\infty)=1-[1-R_{0}(\infty)]^{\alpha_{\mathrm{i}}},$ (4)

which does not depend on the protection level $\alpha_{\mathrm{o}}$ offered to
contacts. This means that adopting interventions offering more protection to
contacts (i.e. reducing $\alpha_{\mathrm{o}}$) has exactly the same effect on
the prevalence of adopting and nonadopting individuals. In particular, if
OTHER PIs are used, since $\alpha_{\mathrm{i}}=1$ then
$R_{1}(\infty)=R_{0}(\infty)$, for any value of $\alpha_{\mathrm{o}}$. In such
a case, from the point of view of a given individual, adopting an intervention
is perfectly equivalent to not adopting it, as the probability to be
eventually infected is exactly the same. With the benefit of hindsight this
result is not so odd: If $\alpha_{\mathrm{i}}=1$, PIs do not provide any
protection against the transmission from others to the adopter, hence for the
individual adopter the risk of being infected cannot be smaller than for a
non-adopter. However, Eq. (4) reveals that, while adopting a PI always implies
some degree of inconvenience (think for example at the bother of wearing a
mask), no immediate utility exists for the individual adopting an OTHER PI,
while such a utility exists for SELF PIs. See the concluding section for
further discussion.

From now on we mostly focus on the comparison of the effects at the population
level of SELF and OTHER interventions with exactly the same efficacy $\alpha$
($\alpha_{\mathrm{i}}=\alpha$ for the SELF case, $\alpha_{\mathrm{o}}=\alpha$
for the OTHER case). For reference we consider also the case NONE, where no
intervention is adopted, corresponding to
$\alpha_{\mathrm{i}}=\alpha_{\mathrm{o}}=1$, and the case BOTH, given by
$\alpha_{\mathrm{i}}=\alpha_{\mathrm{o}}=\alpha$, and representing equally
protecting performance in both directions. In this last case (i.e., along the
diagonal in Fig. 1(a)) SELF PIs outperform OTHER PIs for any value of the
fraction $f$ of adopters. This is checked by considering
$R_{\infty}(\mathrm{OTHER})-R_{\infty}(\mathrm{SELF})$ as a function of
$\alpha$ and $f$. The values plotted in Fig. 1(b), which are nonnegative for
any value of $\alpha$ and $f$, demonstrate that in all cases SELF
interventions are better than or equal to OTHER interventions in curbing
disease diffusion. These conclusions remain true for other values of the
infectivity $\lambda>\lambda_{c}$ as we can see from Supplementary Figures SF
1 to SF 4, where we present figures analogous to Fig. 1 and 2 for
$\lambda=1.0$ and $\lambda=2.0$. Interestingly, from Fig. 1(b) it turns out
that the difference between the two types of PI is maximal for very small
values of $\alpha$ (as expected) and in general, for intermediate,
$\alpha$-dependent values of the adoption fraction $f$.

The numerical solution of MF equations also allows us to determine the height
$I_{m}$ of the peak of incidence over time. This is a crucial quantity that
must be kept as low as possible in order to avoid the saturation of health
care systems by an inflow of too many seriously ill individuals. Fig. 1(c)
shows clearly that also in this respect SELF PIs perform better than the OTHER
counterparts, as witnessed by the asymmetry of the plot with respect to the
diagonal. The same conclusion is confirmed by Fig. 1(d):
$I_{m}(\mathrm{OTHER})-I_{m}(\mathrm{SELF})\geq 0$ for any value of $\alpha$
and $f$. Adopting SELF interventions “flattens the curve” more effectively
than the adoption of OTHER PIs does. A comparison between panels (b) and (d)
in Fig. 1 reveals also that PI adoption affects differently the total number
of infected individuals with respect to the peak incidence. In particular, the
improved performance of SELF PIs is maximized for $\alpha\to 0$ and $f\approx
0.3$ for what concerns the flattening of the curve, while $f\approx 0.45$ for
the total size of the outbreak.

Figure 2: OTHER interventions flatten the curve but barely reduce prevalence
with respect to no intervention. (a) Difference between the final prevalence
$R$ for the NONE and the OTHER scenario as a function of the fraction $f$ of
adopters and the efficacy $\alpha$ of the individual PI. (b) Difference
between the maximum value of the incidence $I_{m}$ for the NONE and the OTHER
scenario as a function of the fraction $f$ of adopters and the efficacy
$\alpha$ of the individual PI. Solution of MF equations for a homogeneous
network of degree $K=7$ for $\lambda=0.5$. The solid yellow lines are where
$\lambda_{c}(\alpha)=\lambda$ for the OTHER scenario.

Unanticipated effects can also be observed by comparing the performance of
OTHER PIs with a scenario with no interventions at all, and given by
$\alpha_{\mathrm{i}}=\alpha_{\mathrm{o}}=1$, see Fig. 2. While it is expected
that the effect of PIs grows with efficacy (small $\alpha$) and widespread PI
adoption ($f$ close to 1) it is surprising to see that in a large domain of
the parameter space $R_{\infty}(\mathrm{NONE})-R_{\infty}(\mathrm{OTHER})$ is
very close to zero (PIs do not significantly change the number of people
eventually infected) while $I_{m}(\mathrm{NONE})-I_{m}(\mathrm{OTHER})$ is
rather large: OTHER PIs do not lead to an overall reduction of the outbreak
size but indeed substantially flatten the curve.

Figure 3: Numerical simulations confirm analytical results. Results of
numerical simulations on a random regular network of degree $K=7$. The various
curves are for the different scenarios of PI deployment. In all cases $f=0.5$
and $\alpha=0.2$. SIR dynamics: a) Value of the average size $R_{\infty}$ of
extensive outbreaks as a function of $\lambda$; network size $N=10^{4}$,
single random infected seed. b) Temporal evolution of the incidence $I(t)$ for
a single realization of the dynamics; network size $N=10^{5}$, 50 initial
random infected seeds, $\lambda=0.5$. The exponential growth is the same for
SELF and OTHER scenarios, but it lasts longer in the OTHER case. c) Value of
the probability $P_{s}$ to observe an extensive outbreak as a function of
$\lambda$; network size $N=10^{3}$, single random infected seed. SIS dynamics:
d) Value of the stationary density $I$ of infected nodes as a function of
$\lambda$. Network size $N=10^{4}$.

The previous predictions have been obtained within the mean-field framework.
The comparison with numerical simulations, to check whether the picture
derived above remains true, requires to take into account the phenomenology of
the stochastic SIR model in finite systems (see Methods IV.2), but reveals new
interesting features. Fig. 3(a) shows that, also in numerical simulations, the
better efficacy of SELF interventions at the population level remains valid
for any value of the infectivity $\lambda$. Indeed the average size
$R_{\infty}$ of extensive outbreaks is smaller when SELF PIs, rather than
OTHER PIs, are adopted.

Different values of the asymptotic prevalence for long times $R_{\infty}$
reflect a different temporal evolution. The mean-field approach predicts (see
Methods IV.1.4) that the initial growth is exponential, with a characteristic
time scale depending only on the threshold value, and thus on the product
$\alpha_{\mathrm{o}}\alpha_{\mathrm{i}}$, and hence is equal for both SELF and
OTHER scenarios with the same
$\alpha_{\mathrm{i}}=\alpha_{\mathrm{o}}=\alpha$. The difference in the value
of $R_{\infty}$ stems from the fact that this exponential growth ends later in
the OTHER case, thus corresponding to a higher value of the peak incidence
$I_{m}$. Notice that this is different with what happens for the NONE
($\alpha_{\mathrm{i}}=1$, $\alpha_{\mathrm{o}}=1$) or BOTH
($\alpha_{\mathrm{i}}=\alpha_{\mathrm{o}}<1$) scenarios. In these other cases
already the exponential growth rate is different. Performing simulations of
the stochastic SIR model (see Methods IV.2) and analyzing the temporal
evolution of the number of infected individuals (see Fig. 3(b)), these
differences are clearly observed. The maximum value of the incidence in the
SELF case is in general lower than in the OTHER case: adopter-protecting
interventions are more effective than contact-protecting interventions in
“flattening the curve” and thus reduce the maximum instantaneous pressure on
health care systems.

SELF interventions, however, are not the silver bullet against epidemics.
Indeed, while they reduce the overall prevalence of the infectious disease,
they are worse than OTHER interventions at preventing the emergence of a
macroscopic outbreak. As Fig. 3(c) shows, the probability that a macroscopic
outbreak emerges is larger for the SELF prescription than for the OTHER one.
This means that, if people adopt SELF PIs, it is more likely that a single
infected individual can give rise to a macroscopic outbreak. However, as shown
above, if such an outbreak occurs, or if the disease is introduced in
different nodes (for example because several infected individuals arrive
simultaneously in a community) the outbreak will affect, on average, less
people if SELF PIs are adopted.

###  II.3 Recurrent diseases

The paradigmatic model for recurrent diseases, leading to a steady endemic
state, is the Susceptible-Infected-Susceptible (SIS) model Keeling and Rohani
(2007). In this case the epidemic transition separates values of the parameter
$\lambda=\beta/\mu$ for which the disease quickly goes extinct
($\lambda\leq\lambda_{c}$), from values such that an endemic state is reached
($\lambda>\lambda_{c}$), with a steady fraction of infected individuals. Also
in this case it is possible to investigate the effect of different types of
interventions via a mean-field approach for homogeneous networks (see Methods
IV.3), obtaning for the threshold

$\lambda_{c}=\frac{1}{K}\frac{1}{f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+1-f},$
(5)

while the stationary density of infected nodes in the vicinity of this
threshold is

$I\simeq\frac{\Delta}{\lambda_{c}^{2}K}\frac{f\alpha_{\mathrm{i}}+1-f}{f\alpha_{\mathrm{o}}\alpha_{\mathrm{i}}^{2}+1-f}.$
(6)

Hence the phenomenology of the SIS model is analogous to what is found for
SIR: The onset of the endemic state is a symmetric function of the PI
efficacies. The prevalence about it, however, is, for a constant
$\alpha_{\mathrm{o}}\alpha_{\mathrm{i}}$, an increasing function of
$\alpha_{\mathrm{i}}$ and a decreasing one of $\alpha_{\mathrm{o}}$.
Interventions protecting the adopter (small $\alpha_{\mathrm{i}}$,
$\alpha_{\mathrm{o}}=1$) result in lower disease circulation at the population
level than interventions protecting contacts of the adopter (small
$\alpha_{\mathrm{o}}$, $\alpha_{\mathrm{i}}=1$). These predictions are
verified numerically to be true for all values of $\lambda>\lambda_{c}$, not
only close to the transition (see Fig. 3(d)).

###  II.4 An application to COVID-19 diffusion

While we have considered so far extremely simple epidemic models on idealized
networks, the qualitative conclusions extend to more complicated epidemic
models and more general types of networks. For example, the extension to
heterogeneous networks with a general degree distribution $P(k)$, presented in
the Supplementary Information, leads to results perfectly analogous to those
discussed above. To give a more realistic example, here we consider a
Susceptible-Exposed-Infected-Recovered (SEIR) dynamics on a large weighted
network describing interactions among people in Portland, OR Hartnett  _et
al._ (2021). This network has been inferred using mobile device data before
social distancing measures were enacted during the COVID-19 pandemic. SEIR
model parameters were calibrated to describe the first wave of COVID-19 in
2020 (see Methods IV.4 for more details). The size of the network is
$N=214,393$, which is a substantial fraction of the census population of
Portland city in 2020, namely $N_{P}=652,503$ pop (2022).

Figure 4: SEIR dynamics on the Portland network. a) $I$ vs $t$ for a single
run; b) Distribution of the fraction $R$ of recovered people in the asymptotic
state for 100 runs in each of the 4 scenarios; c) Distribution of the maximum
fraction $I_{m}$ of simultaneously infected people for 100 runs in each of the
4 scenarios.

In this framework, we tested the four scenarios for PI efficacy, assuming
again $f=0.5$. Figure 4 shows that also in this case SELF interventions
flatten the curve of the number of infected people more effectively than OTHER
interventions with the same individual efficacy. With respect to the final
prevalence $R_{\infty}$, adopting SELF PI reduces the value in the absence of
interventions $R_{\infty}(\mathrm{NONE})=0.595$ to
$R_{\infty}(\mathrm{SELF})=0.487$, which is significantly (beyond stochastic
fluctuations) smaller than $R_{\infty}(\mathrm{OTHER})=0.510$, while for
interventions with both SELF and OTHER efficacy the prevalence is
$R_{\infty}(\mathrm{BOTH})=0.406$. On the scale of the city, extrapolating
these values for a population of $N_{P}=652,503$, the adoption of SELF PI
instead of OTHER PI would imply a reduction of around $15,000$ in the total
number of cases. For the maximum number of simultaneously infected people, we
obtain the values $I_{m}(\mathrm{NONE})=0.238$, $I_{m}(\mathrm{SELF})=0.166$,
$I_{m}(\mathrm{OTHER})=0.180$, and $I_{m}(\mathrm{BOTH})=0.130$, which on the
city scale implies a reduction of around $9,000$ concurrently infected
individuals when SELF interventions are used instead of OTHER interventions.

Although we do not claim these numbers to be directly applicable to a real
world scenario, the results clearly show that the collective effects leading
to a better performance of SELF interventions with respect to OTHER
interventions at the population level are at work also for more complicated
epidemic models on realistic networks.

##  III Discussion

A vast range of protecting interventions of different type, both
pharmaceutical and non-pharmaceutical, can be used to combat the diffusion of
an infectious disease. Each intervention may in general affect differently the
infectivity and the susceptibility of the single individual.

Here we have considered, in a simple formalism based on disease propagation on
networks, the effects of generic PIs on the reduction of the total number of
infected individuals and of the maximum number of individuals concurrently
infected, the so-called goal of “flattening the curve”, to prevent the
saturation of health care systems. In the simple setting adopted, PIs are
characterized by two parameters which, in the case of masks, gauge the level
of viral particle filtration offered from the mask wearer to the environment
($\alpha_{\mathrm{o}}$), and in the opposite direction
($\alpha_{\mathrm{i}}$).

The results presented above demonstrate that collective effects have
surprising consequences for what concerns the level of global protection
guaranteed by different kinds of intervention. In agreement with many other
publications, we find, of course, that no matter how imperfect, the adoption
of PIs reduces the circulation of an infectious disease with respect to the
case of no adoption, and that a higher efficiency at the individual level
guarantees a higher protection for the whole community. Surprisingly, however,
we find that interventions protecting the adopter (SELF PIs) or interventions
protecting mostly contacts of the adopter (OTHER PIs) have not the same
effect, even for equal efficacy of the individual intervention. SELF PIs turn
out to reduce more the overall incidence of the infection and to “flatten the
curve” more efficiently, i.e., they reduce more drastically the peak value of
the number of individuals simultaneously infected. Although we have focused on
purely SELF ($\alpha_{\mathrm{o}}=1$, $\alpha_{\mathrm{i}}<1$) and purely
OTHER ($\alpha_{\mathrm{o}}<1$, $\alpha_{\mathrm{i}}=1$) interventions, the
same picture applies when SELF (OTHER) PIs provide some small level of
protection also in the other direction, i.e.
$\alpha_{\mathrm{i}}\ll\alpha_{\mathrm{o}}$
($\alpha_{\mathrm{i}}\gg\alpha_{\mathrm{o}}$), respectively. Our results are
presented in terms of a mean-field theory for the simple SIR and SIS models of
disease propagation in homogeneous networks, that can be readily extended to
heterogeneous contact patterns (see Supplementary Information), and are backed
up with numerical simulations. Additionally, we consider a SEIR model, often
used for COVID-19 analyses and estimations, on an empirical network
representing contact patterns in the city of Portland. This last analysis
allows us to confirm the expected effect of the different kinds of PIs in a
realistic scenario. Our results can be interpreted in the framework of the
general theory for heterogeneous agents put forward by Miller Miller (2007),
predicting that control strategies having a heterogeneous impact on
susceptibilities are more effective in reducing epidemic size while strategies
with heterogeneous impact on infectivities reduce more the probability of
large outbreaks.

What causes the difference between the performance of the SELF and OTHER
strategies? To investigate this issue we considered a system where the choice
of the PI is not individual-based, but contact-based: in other words,
$\beta_{ij}$ is still a quenched random variable given by (1), but the
variables $m_{i}$ and $m_{j}$ are extracted independently for each edge.
Hence, considering the example of mask wearing, an individual $i$ can use a
mask when interacting with contact $j_{1}$ and not use it when interacting
with contact $j_{2}$. In such a case (see Supplementary Figure SF5) SELF and
OTHER scenarios give exactly the same results. Therefore asymmetry arises at
the community level only if each individual consistently adopts (or does not
adopt) the protecting intervention in every interaction.

Concerning the example of masks, our results place a strong question mark on
policies adopted to curb the diffusion of COVID-19. During the initial stages
of the pandemic, concerns were raised about the use of valved masks, as it was
argued that they do not protect contacts Staymates (2020). This led to advice
against their use by the World Health Organization (WHO) World Health
Organization (2020) and the Centers for Disease Control and Prevention (CDC)
Centers for Disease Control and Prevention (2019), and to outright bans in
some cities Kelleher (2020) and in major US airlines Wolford (2020). At the
same time the usage of masks protecting contacts of the wearer has been
encouraged. As we have shown, unless effective masks providing protection in
both directions can be widely used, it is better to adopt SELF (valved) masks
to maximally reduce the impact of the disease at the population level.

This global effect is not the only argument pushing to reconsider this type of
policies. Another compelling reason has to do with the utility of single
individuals. The usage of PIs can be imposed by ordinances issued by the
government, but the adoption of individual PIs, even if policed by
authorities, always involves a personal choice. As shown above, the adoption
of an intervention that protects only the contacts of the adopter
($\alpha_{\mathrm{i}}=1$) does not imply any direct utility for the adopter:
The probability to be eventually infected is exactly the same for adopting and
nonadopting persons. On the other hand, adopting an intervention (for example
wearing a mask) is a burden, so that the net direct utility for adopting an
OTHER PI is negative. A positive overall payoff for the individual adopting an
OTHER PI can be obtained only if many people cooperate so that the common good
is achieved, overcoming the inconvenience implied by the adoption. Thus OTHER
PIs are both less effective at the population level and less likely to be
adopted at the individual one. No such barrier to adoption exists for self-
protecting (SELF) PIs: In such a case it is not only beneficial at the
collective level, but also in the selfish interest of each individual to adopt
a protecting intervention. Hence, we conjecture that the overall higher
effectiveness of self-protecting interventions at the community level will be
further enhanced by the fact that they are more likely to be spontaneously
adopted.

##  IV Methods

###  IV.1 Mean-field theory for the SIR model

In the case of a homogeneous network, with all nodes sharing the same degree
$k=K$, a mean-field approximation considers that all nodes with the same PI
state are equivalent, and therefore we can characterize the dynamics by the
probability that a node in a given PI state (adopter, nonadopter) is in the
compartment $S$, $I$, or $R$. Under these conditions, the dynamics of the
network is defined by the set of probabilities

$\displaystyle I_{a},\quad S_{a},\quad R_{a},$ (7)

that an individual in the PI state $a$, taking values $a=1$ (adopter) and
$a=0$ (nonadopter), is infected, susceptible or recovered, respectively. In a
population of fixed size, these sets of probabilities are subject to the
normalization condition

$I_{a}+S_{a}+R_{a}=1.$ (8)

The mean-field equations for the system are constructed by considering the
contacts that can change the number of infected individuals in each PI state.
Thus, the rate equation for the number of removed individuals takes the form

$\frac{dR_{a}}{dt}=\mu I_{a},$ (9)

for all PI states. For the density of infected adopting individuals, we can
write Keeling and Rohani (2007); Pastor-Satorras  _et al._ (2015)

$\frac{dI_{1}}{dt}=-\mu
I_{1}+(K-1)S_{1}\left[fI_{1}\beta\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+(1-f)I_{0}\beta\alpha_{\mathrm{i}}\right].$
(10)

In this equation, the first term considers the decay to the recovered state
with rate $\mu$. The second term considers the probability of a new infection
along a given edge, that is proportional to the density of susceptible
adopting individuals, and the probability that the edge points to an adopting
infected individual (with probability $f$), which will lead to an effective
infection rate $\beta\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}$, or to an
infected nonadopting individual (with probability $1-f$), associated to an
effective infection rate $\beta\alpha_{\mathrm{i}}$. The term $K-1$ takes into
account that infected individuals have at most only $K-1$ edges available to
propagate the infection, since one is used up by the neighbor that induced
their infection. Rescaling time by $\mu$, we can write

$\frac{dI_{1}}{dt}=-I_{1}+\lambda\alpha_{\mathrm{i}}(K-1)S_{1}\theta,$ (11)

where $\lambda=\beta/\mu$ and we have defined

$\theta=f\alpha_{\mathrm{o}}I_{1}+(1-f)I_{0}.$ (12)

From here, the rate equation for adopting susceptible individuals is

$\frac{dS_{1}}{dt}=-\lambda\alpha_{\mathrm{i}}(K-1)S_{1}\theta.$ (13)

Finally, for nonadopting individuals, we can write

$\displaystyle\frac{dI_{0}}{dt}$ $\displaystyle=$ $\displaystyle-
I_{0}+\lambda(K-1)S_{0}\theta,$ (14) $\displaystyle\frac{dS_{0}}{dt}$
$\displaystyle=$ $\displaystyle-\lambda(K-1)S_{0}\theta.$ (15)

From the general Eq. (63), considering that $\mu$ is absorbed into the time
rescaling, we have

$R_{a}(t)=\int_{0}^{t}I_{a}(t^{\prime})dt^{\prime},$ (16)

and from Eqs. (62) and (58),

$S_{1}(t)=e^{-\lambda(K-1)\alpha_{\mathrm{i}}\phi(t)},\quad
S_{0}(t)=e^{-\lambda(K-1)\phi(t)},$ (17)

where $\phi(t)=\int_{0}^{t}\theta(t^{\prime})dt^{\prime}$ can be written as

$\phi(t)=f\alpha_{\mathrm{o}}R_{1}(t)+(1-f)R_{0}(t),$ (18)

and where we assume an initial condition given by a vanishing fraction of
infected individuals.

####  IV.1.1 Threshold evaluation

To estimate the value of the threshold, we consider the final outbreak size,
given by the number of removed individuals at time $t\to\infty$. Since
$I_{a}(\infty)=0$, we have from the normalization condition, Eq. (8),
$R_{a}(\infty)=1-S_{a}(\infty)$. In this infinite time limit, defining
$\phi_{\infty}\equiv\phi(\infty)$,

$\displaystyle\phi_{\infty}$ $\displaystyle=$ $\displaystyle
f\alpha_{\mathrm{o}}R_{1}(\infty)+(1-f)R_{0}(\infty)$ (19) $\displaystyle=$
$\displaystyle
f\alpha_{\mathrm{o}}[1-e^{-\lambda(K-1)\alpha_{\mathrm{i}}\phi_{\infty}}]$
(20) $\displaystyle+$
$\displaystyle(1-f)[1-e^{-\lambda(K-1)\phi_{\infty}}]\equiv\Psi(\phi_{\infty}).$
(21)

A non-zero solution is obtained when
$\left.\frac{d\Psi(\phi_{\infty})}{d\phi_{\infty}}\right|_{\phi_{\infty}=0}\geq
1$, leading to the threshold condition $\lambda>\lambda_{c}$ to observe a
finite outbreak, with

$\lambda_{c}=\frac{1}{(K-1)\left[f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+1-f\right]}.$
(22)

####  IV.1.2 Behavior close to the threshold

In the limit $\lambda\to\lambda_{c}^{+}$, we have that $R_{a}(\infty)$ and
$\phi_{\infty}$ are both small. From Eq. (19), performing an expansion for
small $\phi_{\infty}$ up to second order, we can solve it and obtain

$\phi_{\infty}\simeq\frac{2\Delta}{\lambda_{c}^{3}(K-1)^{2}}\frac{1}{f\alpha_{\mathrm{o}}\alpha_{\mathrm{i}}^{2}+1-f},$
(23)

where $\Delta=\lambda-\lambda_{c}$ is the distance to the critical threshold.
For the total prevalence, $R_{\infty}=fR_{1}(\infty)+(1-f)R_{0}(\infty)$,
using Eq. (17) and the normalization condition, we can write

$\displaystyle R_{\infty}$ $\displaystyle=$ $\displaystyle
f[1-e^{-\lambda(K-1)\alpha_{\mathrm{i}}\phi_{\infty}}]+(1-f)[1-e^{-\lambda(K-1)\phi_{\infty}}]$
(24) $\displaystyle\simeq$
$\displaystyle\lambda(K-1)[f\alpha_{\mathrm{i}}+1-f]\phi_{\infty},$

where the have expanded the exponential factors to leading order. Combining
Eqs. (24) and (67), we finally obtain the total prevalence close to the
threshold

$R_{\infty}\simeq\frac{2\Delta}{\lambda_{c}^{2}(K-1)}\frac{f\alpha_{\mathrm{i}}+1-f}{f\alpha_{\mathrm{i}}^{2}\alpha_{\mathrm{o}}+1-f}.$
(25)

####  IV.1.3 Relation between $R_{0}(\infty)$ and $R_{1}(\infty)$

Taking the limit $t\to\infty$ in (17) and eliminating $\phi_{\infty}$ we
obtain $S_{1}=S_{0}^{\alpha_{\mathrm{i}}}$, from which, using the condition
$R_{a}(\infty)=1-S_{a}(\infty)$, we arrive at

$R_{1}(\infty)=1-[1-R_{0}(\infty)]^{\alpha_{\mathrm{i}}}.$ (26)

####  IV.1.4 Initial time behavior

To estimate the initial time behavior of the epidemic outbreak, we consider
the limit of a very small density of infected individuals. Linearizing the
corresponding equations, we have

$\displaystyle\dot{I}_{1}$ $\displaystyle\simeq$ $\displaystyle-
I_{1}+\lambda(K-1)\alpha_{\mathrm{i}}\theta,$ (27) $\displaystyle\dot{I}_{0}$
$\displaystyle\simeq$ $\displaystyle-I_{0}+\lambda(K-1)\theta.$ (28)

Using these expressions in Eq. (53), we have

$\dot{\theta}=f\alpha_{\mathrm{o}}\dot{I}_{1}+(1-f)\dot{I}_{0}\simeq-\theta+\frac{\lambda}{\lambda_{c}}\theta.$
(29)

Defining the characteristic time scale

$\tau=\frac{\lambda_{c}}{\lambda-\lambda_{c}},$ (30)

the solution for $\theta$ is

$\theta(t)=\theta(0)e^{t/\tau}.$ (31)

Using these expressions, we can integrate Eqs. (27) and (28) to obtain

$\displaystyle I_{1}(t)$ $\displaystyle\simeq$
$\displaystyle\lambda_{c}(K-1)\alpha_{\mathrm{i}}\theta(0)e^{t/\tau}$ (32)
$\displaystyle I_{0}(t)$ $\displaystyle\simeq$
$\displaystyle\lambda_{c}(K-1)\theta(0)e^{t/\tau},$ (33)

while the total incidence $I(t)=fI_{1}(t)+(1-f)I_{0}(t)$ has the form

$I(t)\simeq\lambda_{c}(K-1)\left[f\alpha_{\mathrm{i}}+1-f\right]\theta(0)e^{t/\tau}.$
(34)

For the initial condition, let us consider that a randomly chosen node is
initially infected. In this case, $I_{m}(0)=I_{0}(0)=1/N$, and therefore

$\theta(0)=\frac{1}{N}\left[f\alpha_{\mathrm{o}}+1-f\right].$ (35)

Therefore

$I(t)\simeq\frac{\lambda_{c}(K-1)}{N}\left[f\alpha_{\mathrm{i}}+1-f\right]\left[f\alpha_{\mathrm{o}}+1-f\right]e^{t/\tau}.$
(36)

###  IV.2 Phenomenology of the SIR model in finite networks

####  IV.2.1 Two types of outbreaks

Starting from a single infected node, outbreaks can develop in two
qualitatively different ways Kiss  _et al._ (2017b). Some of them die after a
few infection events; others instead survive much longer and affect an
extensive fraction of the individuals. This is reflected in the distribution
of outbreak sizes, which has two components. For small values of $\lambda$
only the small component exists, made up by short-lived small outbreaks and
weakly depending on $N$. For large values of $\lambda$ also the second
component exists, peaked around a size proportional to $N$. The epidemic
threshold marks the birth of the large, extensive component. This distinction
is exactly defined only in the thermodynamic limit (indeed the threshold is
properly defined only in this limit), but it is operatively meaningful also
for finite, but large, size $N$. The fraction of outbreaks belonging to the
large component is the probability $P_{s}$ that an extensive outbreak emerges.
It is zero below the threshold and grows with $\lambda>\lambda_{c}$ above it.
The average size of extensive outbreaks $R_{\infty}$ (i.e. the average value
calculated only for the large component) is another observable that is zero up
to $\lambda_{c}$ and grows with $\lambda$. For epidemics on undirected
networks the two quantities coincide Kiss  _et al._ (2017b). On directed
networks (such as the effective networks over which the epidemic spreads if
PIs are used) they are in general different Kenah and Robins (2007); Miller
(2007).

By construction, the mean-field approach on networks Pastor-Satorras  _et al._
(2015); Kiss  _et al._ (2017b) deals only with extensive outbreaks in an
infinite size system. Hence the quantity $R_{\infty}$ computed above has to be
compared with the average size of macroscopic outbreaks, which must be
defined, in simulations, as outbreaks larger than a given fraction of the
whole system. We take this fraction to be $0.01$. Clearly only for
$N\to\infty$ and sufficiently far from the transition point, results are
independent from the choice of this fraction.

####  IV.2.2 Temporal evolution

For simulations in finite systems starting with a single infected node, one
must take into account that the exponential growth is preceded by a regime
dominated by stochastic fluctuations, whose duration has quite large
variations depending on the realization of the process. Only when a
sufficiently large fluctuation in the number of infected nodes $I(t)$ is
generated, exponential growth is triggered. Averaging at fixed time the value
of $I(t)$ over many such realizations leads to a spurious bending as a
function of time. Moreover, the duration of the initial stochastic regime (and
hence the size of the spurious bending) is different in the various scenarios,
further complicating the analysis. To overcome these difficulties, as it is
customary, we start from a larger number of initial infected nodes, namely
$50$, so that the initial regime dominated by fluctuations is strongly
suppressed.

###  IV.3 Mean-field theory for the SIS model

The mean-field equations for the SIS model in a homogeneous network of degree
$k_{i}=K$ depend only on the densities of infected adopting and nonadopting
individuals $I_{a}$, and take the form Keeling and Rohani (2007); Pastor-
Satorras  _et al._ (2015)

$\displaystyle\dot{I_{1}}$ $\displaystyle=$ $\displaystyle-
I_{1}+(1-I_{1})\lambda K\alpha_{\mathrm{i}}\theta$ (37)
$\displaystyle\dot{I_{0}}$ $\displaystyle=$ $\displaystyle-
I_{0}+(1-I_{0})\lambda K\theta,$ (38)

with

$\theta=f\alpha_{\mathrm{o}}I_{1}+(1-f)I_{0}.$ (39)

####  IV.3.1 Threshold evaluation

We can compute the threshold $\lambda_{c}$ by performing a linear stability
analysis around the solution $I_{a}=0$, corresponding to the healthy, non-
endemic state. The Jacobian matrix of Eqs. (68) and (38), evaluated at the
origin, is

$\displaystyle J=\begin{pmatrix}-1+\lambda
Kf\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}&\lambda K(1-f)\alpha_{\mathrm{i}}\\\
\lambda Kf\alpha_{\mathrm{o}}&\lambda K(1-f)\end{pmatrix},$ (40)

whose associated eigenvalues are $\Lambda_{1}=-1$ and $\Lambda_{2}=-1+\lambda
K(f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+1-f)$. The healthy state becomes
unstable when the largest eigenvalue becomes positive, that is, $-1+\lambda
K(f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+1-f)>0$. This defines the threshold
$\lambda_{c}$, given by

$\lambda_{c}=\frac{1}{K}\frac{1}{f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+1-f},$
(41)

such that for $\lambda>\lambda_{c}$ there is a steady infected state with a
non-zero prevalence.

####  IV.3.2 Behavior close to the threshold

For homogeneous networks of degree $K$, the steady state condition
$\dot{I}_{a}=0$ translates into the relations

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle\frac{\lambda
K\alpha_{\mathrm{i}}\theta}{1+\lambda K\alpha_{\mathrm{i}}\theta},$ (42)
$\displaystyle I_{0}$ $\displaystyle=$ $\displaystyle\frac{\lambda
K\theta}{1+\lambda K\theta}.$ (43)

Inserting these into the definition of $\theta$, Eq. (70), we obtain the self-
consistent equation

$\theta=\frac{\lambda
Kf\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}\theta}{1+\lambda
K\alpha_{\mathrm{i}}\theta}+\frac{\lambda K(1-f)\theta}{1+\lambda K\theta}.$
(44)

Eq. (44) can be solved close to the threshold, in the limit of small $\theta$,
performing a Taylor expansion up to second order, which leads to the solution

$\theta\simeq\frac{\Delta}{\lambda_{c}^{3}K^{2}}\frac{1}{f\alpha_{\mathrm{o}}\alpha_{\mathrm{i}}^{2}+1-f}.$
(45)

Using the lowest order approximation $I_{1}\simeq\lambda
K\alpha_{\mathrm{i}}\theta$ and $I_{0}\simeq\lambda K\theta$, the final steady
state prevalence $I=fI_{1}+(1-f)I_{0}$ takes the form, close to the threshold

$I\simeq\frac{\Delta}{\lambda_{c}^{2}K}\frac{f\alpha_{\mathrm{i}}+1-f}{f\alpha_{\mathrm{o}}\alpha_{\mathrm{i}}^{2}+1-f},$
(46)

in close analogy to the SIR result in Eq. (25).

####  IV.3.3 Initial time behavior

It is easy to see that the initial time behavior of the SIS model is described
by the same equations as the SIR case, simply replacing the factor $K-1$ by
$K$. Therefore, the time initial time evolution of an outbreak initiated by a
randomly chosen node takes the form, as Eq. (36),

$I(t)\simeq\frac{\lambda_{c}K}{N}\left[f\alpha_{\mathrm{i}}+1-f\right]\left[f\alpha_{\mathrm{o}}+1-f\right]e^{t/\tau}.$
(47)

###  IV.4 Details on the network and the epidemic model for COVID-19

We consider as the contact pattern for the SEIR dynamics the static contact
network determined in Ref. Hartnett  _et al._ (2021) for the city of Portland,
Oregon, before social distancing measures were enacted. The network has
$N=214393$ nodes and $M=1538092$ edges, so that the average degree is
$\left\langle{k}\right\rangle=14.4$. Further details on network statistics can
be found in the original publication. The network is undirected and weighted,
with each contact weighted according to its duration.

On top of this network we perform simulations of the SEIR epidemic model in
continuous time, using the Gillespie algorithm. The model is characterized by
three parameters. An exposed node E spontaneously becomes infectious (I) at a
rate $\alpha$, while the transition between the infectious state E to the
recovered state R occurs spontaneously at rate $\mu$. In the absence of PI, an
infectious node $i$ transmits the infection to a susceptible neighbor $j$ at a
rate $\beta w_{ij}$ where $w_{ij}$ is the weight associated to the network
edge. In the presence of PIs this quantity is further modified as in Eq. (1).
The values of the parameters are the same calibrated for COVID-19 in Ref.
Hartnett  _et al._ (2021): $\alpha=1/3$, $\nu=1/14$, $\beta=1.337$. The
initial condition for simulations is that a fraction $10^{-3}$ of the
individuals is infected and the rest is susceptible.

## Acknowledgments

We acknowledge financial support from the Spanish
MCIN/AEI/10.13039/501100011033, under Project No. PID2019-106290GB-C21

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## Supplementary Information

##  V Heterogeneous mean-field theory for the SIR model

In the case of heterogeneous networks with a non-trivial degree distribution,
heterogeneous mean-field (HMF) theory Pastor-Satorras  _et al._ (2015)
considers that all nodes in the network with the same degree share the same
dynamical state, and therefore we can characterize the state of the epidemics
in terms of the probability that a node of degree $k$ is in the state $S$,
$I$, or $R$, taking the same value for all nodes with the same degree. In the
case of a population with a fraction $f$ of protecting intervention (PI)
adopters we must consider the independent probabilities for adopting and
nonadopting individuals, in such a way that the state of the network is fully
described by the set of probabilities

$I_{k}^{1},\quad S_{k}^{1},\quad R_{k}^{1}$ (48)

that an adopting individual of degree $k$ is infected, susceptible or
recovered, respectively, and the corresponding probabilities for nonadopting
individuals

$I_{k}^{0},\quad S_{k}^{0},\quad R_{k}^{0}.$ (49)

Both sets of probabilities are subject to the normalization condition

$I_{k}^{1}+S_{k}^{1}+R_{k}^{1}=I_{k}^{0}+S_{k}^{0}+R_{k}^{0}=1.$ (50)

In order to construct the rate equations for the time evolution of these
variables we need to consider the probability that a randomly chosen link
points to an infected node. From a mean field point of view, considering that
any individual has a probability $f$ to adopt a PI, and assuming that the
contact networks are undirected and uncorrelated Pastor-Satorras  _et al._
(2001); Dorogovtsev  _et al._ (2008) we can write the rate equation for the
density of infected adopting individuals of degree $k$ as Moreno  _et al._
(2002); Boguñá  _et al._ (2004); Pastor-Satorras  _et al._ (2015)

$\frac{dI_{k}^{1}}{dt}=-\mu
I_{k}^{1}+S_{k}^{1}k\sum_{k^{\prime}}\frac{(k^{\prime}-1)P(k^{\prime})}{\left\langle{k}\right\rangle}\left[fI_{k^{\prime}}^{1}\beta\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+(1-f)I_{k^{\prime}}^{0}\beta\alpha_{\mathrm{i}}\right].$
(51)

In this equation, the second factor considers the probability that a link
emanating from any node points to a node of degree $k^{\prime}$ (taking into
account in the $-1$ term that an infected node is necessarily connected to at
least another infected node that transmitted the disease to it and is thus not
capable of further transmission), and the probability that this node of degree
$k^{\prime}$ either adopts a PI (with probability $f$) and the effective
infection rate is $\beta\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}$, or he/she
does not adopt a PI (with probability $1-f$), with an effective infection rate
$\beta\alpha_{\mathrm{i}}$. Performing a time rescaling $t\to t\mu$, Eq. (51)
can be written as

$\frac{dI_{k}^{1}}{dt}=-I_{k}^{1}+\lambda\alpha_{\mathrm{i}}S_{k}^{1}k\theta,$
(52)

where $\lambda=\beta/\mu$ and we have defined

$\theta=f\alpha_{\mathrm{o}}\sum_{k^{\prime}}\frac{(k^{\prime}-1)P(k^{\prime})}{\left\langle{k}\right\rangle}I_{k^{\prime}}^{1}+(1-f)\sum_{k^{\prime}}\frac{(k^{\prime}-1)P(k^{\prime})}{\left\langle{k}\right\rangle}I_{k^{\prime}}^{0}.$
(53)

From here, we can write the rate equations for the rest of variables related
to adopting individuals as

$\displaystyle\frac{dS_{k}^{1}}{dt}$ $\displaystyle=$
$\displaystyle-\lambda\alpha_{\mathrm{i}}S_{k}^{1}k\theta,$ (54)
$\displaystyle\frac{dR_{k}^{1}}{dt}$ $\displaystyle=$ $\displaystyle
I_{k}^{1},$ (55)

while the variables affecting nonadopting individuals fulfill the rate
equations

$\displaystyle\frac{dI_{k}^{0}}{dt}$ $\displaystyle=$ $\displaystyle-
I_{k}^{0}+\lambda S_{k}^{0}k\theta,$ (56) $\displaystyle\frac{dS_{k}^{0}}{dt}$
$\displaystyle=$ $\displaystyle-\lambda S_{k}^{0}k\theta$ (57)
$\displaystyle\frac{dR_{k}^{0}}{dt}$ $\displaystyle=$ $\displaystyle
I_{k}^{0}.$ (58)

From these equations, we immediately have

$R_{k}^{1}(t)=\int_{0}^{t}I_{k}^{1}(t^{\prime})dt^{\prime},\quad
R_{k}^{0}(t)=\int_{0}^{t}I_{k}^{0}(t^{\prime})dt^{\prime}$ (59)

and

$S_{k}^{1}(t)=e^{-\lambda k\alpha_{\mathrm{i}}\phi(t)},\quad
S_{k}^{0}(t)=e^{-\lambda k\phi(t)},$ (60)

where $\phi(t)=\int_{0}^{t}\theta(t^{\prime})dt^{\prime}$ can be written as

$\phi(t)=\sum_{k^{\prime}}\frac{(k^{\prime}-1)P(k^{\prime})}{\left\langle{k}\right\rangle}\left[f\alpha_{\mathrm{o}}R_{k^{\prime}}^{m}(t)+(1-f)R_{k^{\prime}}^{0}(t)\right]$
(61)

and we assume that the initial condition consists in a very small fraction of
infected individuals and thus that almost all individuals are susceptible.

In order to compute the epidemic threshold, we consider the state of the
system in the infinite time limit, where
$I_{k}^{1}(\infty)=I_{k}^{0}(\infty)=0$. In this regime, the final prevalence
(density of infected individuals) is given by

$\displaystyle R_{k}^{1}(\infty)$ $\displaystyle=$ $\displaystyle
1-S_{k}^{1}(\infty)=1-e^{-\lambda k\alpha_{\mathrm{i}}\phi_{\infty}}$ (62)
$\displaystyle R_{k}^{0}(\infty)$ $\displaystyle=$ $\displaystyle
1-S_{k}^{0}(\infty)=1-e^{-\lambda k\phi_{\infty}},$ (63)

with

$\displaystyle\phi_{\infty}$ $\displaystyle=$
$\displaystyle\sum_{k^{\prime}}\frac{(k^{\prime}-1)P(k^{\prime})}{\left\langle{k}\right\rangle}\left[f\alpha_{\mathrm{o}}R_{k^{\prime}}^{1}(\infty)+(1-f)R_{k^{\prime}}^{0}(\infty)\right]$
(64) $\displaystyle=$
$\displaystyle\frac{\left\langle{k}\right\rangle-1}{\left\langle{k}\right\rangle}[f\alpha_{\mathrm{i}}+1-f]-\sum_{k^{\prime}}\frac{(k^{\prime}-1)P(k^{\prime})}{\left\langle{k}\right\rangle}\left[f\alpha_{\mathrm{o}}e^{-\lambda
k\alpha_{\mathrm{i}}\phi_{\infty}}+(1-f)e^{-\lambda
k\phi_{\infty}}\right]\equiv\Psi(\phi_{\infty}).$

A non-zero solution $\phi_{\infty}\neq 0$ is obtained when the condition
$\left.\frac{d\Psi(\phi_{\infty})}{d\phi_{\infty}}\right|_{\phi_{\infty}=0}\geq
1$ is fulfilled, leading to

$\lambda\frac{\left\langle{k^{2}}\right\rangle-\left\langle{k}\right\rangle}{\left\langle{k}\right\rangle}\left[f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+1-f\right]\geq
1,$ (65)

which leads to the threshold value

$\lambda_{c}=\frac{\left\langle{k}\right\rangle}{\left\langle{k^{2}}\right\rangle-\left\langle{k}\right\rangle}\frac{1}{f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+1-f}.$
(66)

From Eqs. (62) and (63) we can recover a relation between $R_{k}^{1}(\infty)$
and $R_{k}^{0}(\infty)$, namely

$\displaystyle
R_{k}^{1}(\infty)=1-\left[1-R_{k}^{0}(\infty)\right]^{\alpha_{\mathrm{i}}},$
(67)

analogous to the one obtained for homogeneous networks.

##  VI Heterogeneous mean-field theory for the SIS model

The HMF equations for the SIS model Pastor-Satorras and Vespignani (2001) in a
general network can be written in terms of the densities of infected adopting
and nonadopting individuals of degree $k$, $I_{k}^{1}$ and $I_{k}^{0}$,
respectively, that in this case take the form

$\displaystyle\dot{I_{k}^{1}}$ $\displaystyle=$ $\displaystyle-
I_{k}^{1}+(1-I_{k}^{1})\lambda k\alpha_{\mathrm{i}}\theta$ (68)
$\displaystyle\dot{I_{k}^{0}}$ $\displaystyle=$ $\displaystyle-
I_{k}^{0}+(1-I_{k}^{0})\lambda k\theta,$ (69)

where $\theta$, in random uncorrelated networks, is defined by

$\theta=\sum_{k^{\prime}}\frac{k^{\prime}P(k^{\prime})}{\left\langle{k}\right\rangle}\left[f\alpha_{\mathrm{o}}I_{k^{\prime}}^{1}+(1-f)I_{k^{\prime}}^{0}\right].$
(70)

The steady state condition, $\dot{I}_{k}^{1}=\dot{I}_{k}^{0}=0$, corresponding
to the large time behavior, leads to the equations

$I_{k}^{1}=\frac{\lambda\alpha_{\mathrm{i}}k\theta}{1+\lambda\alpha_{\mathrm{i}}k\theta},\qquad
I_{k}^{0}=\frac{\lambda k\theta}{1+\lambda k\theta}.$ (71)

Inserting these expressions into the definition of $\theta$, we obtain the
self-consistent equation

$\theta=\sum_{k^{\prime}}\frac{k^{\prime}P(k^{\prime})}{\left\langle{k}\right\rangle}\left[\frac{\lambda
f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}k^{\prime}\theta}{1+\lambda\alpha_{\mathrm{i}}k^{\prime}\theta}+\frac{(1-f)\lambda
k^{\prime}\theta}{1+\lambda k^{\prime}\theta}\right]\equiv\Psi(\theta).$ (72)

The non-zero solution, corresponding to a finite prevalence, appears when the
condition $\left.\frac{d\Psi(\theta)}{d\theta}\right|_{\theta=0}\geq 1$ is
fulfilled, which leads to

$\lambda\frac{\left\langle{k^{2}}\right\rangle}{\left\langle{k}\right\rangle}\left[f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}1-f\right]\geq
1,$ (73)

and to the threshold

$\lambda_{c}=\frac{\left\langle{k}\right\rangle}{\left\langle{k^{2}}\right\rangle}\frac{1}{f\alpha_{\mathrm{i}}\alpha_{\mathrm{o}}+1-f}.$
(74)

##  VII Supplementary figures

Figure SM-1: SELF interventions are more effective than OTHER interventions at
the population level. (a) Difference between the final prevalence $R_{\infty}$
for the OTHER and the SELF scenario
($R_{\infty}(\mathrm{OTHER})-R_{\infty}(\mathrm{SELF})$), as a function of
$\alpha_{\mathrm{i}}$ and $Appl.\leavevmode\nobreak\ Opt.\leavevmode\nobreak\
$ of the individual PI. Notice that $R_{\infty}(\mathrm{OTHER})$ depends only
on $Appl.\leavevmode\nobreak\ Opt.\leavevmode\nobreak\ $, and
$R_{\infty}(\mathrm{SELF})$ depends only on $\alpha_{\mathrm{i}}$.. The solid
black line indicates the zero value. (b) Difference between the final
prevalence $R_{\infty}$ for the OTHER and the SELF scenario as a function of
the fraction $f$ of adopters and the efficacy $\alpha$ of the individual PI.
The solid yellow line denotes where $\lambda_{c}(\alpha)=\lambda$. Above it,
the system is subcritical and
$R_{\infty}(\mathrm{OTHER})-R_{\infty}(\mathrm{SELF})=0$. (c) Difference
between the maximum value of the incidence $I_{m}$ for the OTHER and the SELF
scenario ($I_{m}(\mathrm{OTHER})-I_{m}(\mathrm{SELF})$), as a function of
$\alpha_{\mathrm{i}}$ and $Appl.\leavevmode\nobreak\ Opt.\leavevmode\nobreak\
$ of the individual PI. Notice that $I_{m}(\mathrm{OTHER})$ depends only on
$Appl.\leavevmode\nobreak\ Opt.\leavevmode\nobreak\ $, and
$I_{m}(\mathrm{SELF})$ depends only on $\alpha_{\mathrm{i}}$. The solid black
line indicates the zero value. (d) Difference between the maximum value of the
incidence $I_{m}$ for the OTHER and the SELF scenario as a function of the
fraction $f$ of adopters and the efficacy $\alpha$ of the individual PI. The
solid yellow line denotes where $\lambda_{c}(\alpha)=\lambda$. In all plots
quantities greater than zero indicate that SELF interventions perform better.
Solution of homogeneous MF equations for a network of degree $K=7$ and for
$\lambda=1.0$. In panels (a) and (c) $f=0.5$. For these parameters, in panels
(a) and (c) both SELF and OTHER strategies operate above their respective
thresholds. Figure SM-2: SELF interventions are more effective than OTHER
interventions at the population level. (a) Difference between the final
prevalence $R_{\infty}$ for the OTHER and the SELF scenario
($R_{\infty}(\mathrm{OTHER})-R_{\infty}(\mathrm{SELF})$), as a function of
$\alpha_{\mathrm{i}}$ and $Appl.\leavevmode\nobreak\ Opt.\leavevmode\nobreak\
$ of the individual PI. Notice that $R_{\infty}(\mathrm{OTHER})$ depends only
on $Appl.\leavevmode\nobreak\ Opt.\leavevmode\nobreak\ $, and
$R_{\infty}(\mathrm{SELF})$ depends only on $\alpha_{\mathrm{i}}$ The solid
black line indicates the zero value. (b) Difference between the final
prevalence $R_{\infty}$ for the OTHER and the SELF scenario as a function of
the fraction $f$ of adopters and the efficacy $\alpha$ of the individual PI.
The solid yellow line denotes where $\lambda_{c}(\alpha)=\lambda$. Above it,
the system is subcritical and
$R_{\infty}(\mathrm{OTHER})-R_{\infty}(\mathrm{SELF})=0$. (c) Difference
between the maximum value of the incidence $I_{m}$ for the OTHER and the SELF
scenario ($I_{m}(\mathrm{OTHER})-I_{m}(\mathrm{SELF})$), as a function of
$\alpha_{\mathrm{i}}$ and $Appl.\leavevmode\nobreak\ Opt.\leavevmode\nobreak\
$ of the individual PI. Notice that $I_{m}(\mathrm{OTHER})$ depends only on
$Appl.\leavevmode\nobreak\ Opt.\leavevmode\nobreak\ $, and
$I_{m}(\mathrm{SELF})$ depends only on $\alpha_{\mathrm{i}}$. The solid black
line indicates the zero value. (d) Difference between the maximum value of the
incidence $I_{m}$ for the OTHER and the SELF scenario as a function of the
fraction $f$ of adopters and the efficacy $\alpha$ of the individual PI. The
solid yellow line denotes where $\lambda_{c}(\alpha)=\lambda$. In all plots
quantities greater than zero indicate that SELF interventions perform better.
Solution of homogeneous MF equations for a network of degree $K=7$ and for
$\lambda=2$. In panels (a) and (c) $f=0.5$. For these parameters, in panels
(a) and (c) both SELF and OTHER strategies operate above their respective
thresholds. Figure SM-3: OTHER interventions flatten the curve but barely
reduce prevalence with respect to no intervention. (a) Difference between the
final prevalence $R$ for the NONE and the OTHER scenario as a function of the
fraction $f$ of adopters and the efficacy $\alpha$ of the individual PI. (b)
Difference between the maximum value of the incidence $I_{m}$ for the NONE and
the OTHER scenario as a function of the fraction $f$ of adopters and the
efficacy $\alpha$ of the individual PI. Solution of MF equations for a
homogeneous network of degree $K=7$ for $\lambda=1$. The solid yellow lines
are where $\lambda_{c}(\alpha)=\lambda$ for the OTHER scenario. Figure SM-4:
OTHER interventions flatten the curve but barely reduce prevalence with
respect to no intervention. (a) Difference between the final prevalence $R$
for the NONE and the OTHER scenario as a function of the fraction $f$ of
adopters and the efficacy $\alpha$ of the individual PI. (b) Difference
between the maximum value of the incidence $I_{m}$ for the NONE and the OTHER
scenario as a function of the fraction $f$ of adopters and the efficacy
$\alpha$ of the individual PI. Solution of MF equations for a homogeneous
network of degree $K=7$ for $\lambda=2$. The solid yellow lines are where
$\lambda_{c}(\alpha)=\lambda$ for the OTHER scenario. Figure SM-5: No
difference between SELF and OTHER scenarios for contact-based protecting
behavior. Simulations on a random regular graph with $K=7$, $N=10000$,
$f=0.5$, $\alpha=0.2$. (a) Final prevalence $R_{\infty}$ of extensive
outbreaks as a function of $\lambda$, for the SELF and the OTHER scenarios.
(b) Probability $P_{s}$ to observe an extensive outbreak as a function of
$\lambda$ for the SELF and the OTHER scenarios. (c) Temporal evolution of the
incidence $I(t)$ for a single realization of the dynamics; $N=10^{5}$, 50
initial random infected seeds.