# Modeling the amplification of epidemic spread by misinformed populations

Matthew R. DeVerna1†, Francesco Pierri2, Yong-Yeol Ahn1,
Santo Fortunato1, Alessandro Flammini1, Filippo Menczer1
1Luddy School of Informatics, Computing, and Engineering, Indiana University,
Bloomington, USA
2Department of Electronics, Information and Bioengineering, Politecnico di
Milano, Milano, Italy
†Corresponding author. Email<EMAIL_ADDRESS>

( This preprint has not yet undergone peer review.
)

###### Abstract

Understanding how misinformation affects the spread of disease is crucial for
public health, especially given recent research indicating that misinformation
can increase vaccine hesitancy and discourage vaccine uptake. However, it is
difficult to investigate the interaction between misinformation and epidemic
outcomes due to the dearth of data-informed holistic epidemic models. Here, we
propose an epidemic model that incorporates a large, mobility-informed
physical contact network as well as the distribution of misinformed
individuals across counties derived from social media data. Our model allows
us to simulate and estimate various scenarios to understand the impact of
misinformation on epidemic spreading. Using this model, we estimate that
misinformation could have led to 47 million additional COVID-19 infections in
the U.S. in a worst-case scenario.

Social factors, such as information sharing, play a crucial role in shaping
the dynamics and epidemiology of infectious diseases12, 5. For instance, a
population’s willingness to adopt public health measures (or lack thereof)
largely determines their successes or failures30, 4. A population’s behavioral
response to outbreaks can be influenced by mass media, as witnessed during the
2009 H1N1 influenza pandemic36, or by social media and the anti-vaccination
movement1, 22, 10, 13.

In particular, misinformation spreading on social networks has been linked to
poor compliance with COVID-19 public health guidance41. Greater exposure to
unreliable news articles about COVID-19 vaccines has been linked to an
increase in vaccine hesitancy and a decrease in vaccination rates at both
state and county levels in the United States38, 35. Exposure to online
misinformation has also been shown to increase vaccine hesitancy in laboratory
experiments29. This is particularly detrimental during vaccination campaigns
as clusters of individuals adopting anti-vaccination opinions can make it
challenging for a population to reach herd immunity16, 43. Proper management
of epidemic crises in the modern age thus requires the understanding of the
complex relationship between the spread of (mis)information through online
social networks and the spread of disease through physical contact networks
(Fig. 1).

Figure 1: The spread of misinformation affects the transmission of disease.
(A) Schematic illustration of the misinformation and contact networks. Online
social networks foster misinformation dissemination while physical contact
networks, such as those that connect co-workers in an office or pupils in a
school, facilitate disease transmission. Dotted links indicate that the same
people participate in both networks, which have different topologies; e.g.,
the information network tends to have stronger political homophily while the
contact network tends to have stronger geographic homophily. We focus on the
impact of misinformation spread on disease transmission (downward arrow),
while the opposite effect (upward arrow, e.g., individuals ceasing to share
misinformation due to illness) falls outside the scope of this investigation.
(B) A contact network based on 0.01% county population samples. Nodes are
sized based on degree (number of contacts). In a scenario with limited spread
of misinformation (black nodes in C), the simulations of disease spread leads
to a number of infected individuals (red nodes in D). In a scenario where the
misinformation spreads more widely (purple nodes in E), more individuals get
infected (yellow nodes in F).

While previous agent-based simulations have shown that misinformation may
impede the suppression of epidemics in various ways46, 32, 37, 9, there is a
growing need to integrate real-world data to strengthen the connections
between simulation results and real-world outcomes. Social media can provide
valuable data sources for this purpose6, 47.

We address this challenge by proposing a data-informed epidemic model that
takes both the distribution of misinformed individuals and a physical mobility
network into account. Using this data, we augment the Susceptible-Infected-
Recovered (SIR) model to account for a subpopulation of “misinformed”
individuals, who do not heed expert public health guidance. We explore how
this group can affect the larger, ordinary population using both a mean-field
approximation, which assumes all individuals have an equal chance of
interacting, and a multi-level agent-based simulation based on two large,
data-informed network: a social network where misinformation spreads and a
contact network where the disease can propagate. A contact network of
approximately 20 million nodes is constructed by leveraging large-scale
Twitter data, county-level voting records, and cellphone mobility data. We
incorporate theoretically extreme values of the parameter responsible for the
epidemic transmission to evaluate best- and worst-case scenarios about the
impact of misinformed individuals on the spread of disease and obtain
quantitative bounds on the harm caused by misinformation.

### Results

In our model, as in the standard SIR modeling framework2, a parameter $\beta$
describes the average number of infected individuals generated by an infected
individual in a time unit. Infected individuals recover with rate $\gamma$. We
extend this epidemic model to account for misinformed and ordinary
subpopulations. We utilize six compartments to capture the three stages of SIR
for each subpopulation. The fractions of individuals in the susceptible,
infected, and recovered compartments are denoted by $S_{O}$, $I_{O}$, and
$R_{O}$ respectively for the ordinary group; and by $S_{M}$, $I_{M}$, and
$R_{M}$ for the misinformed group.

Ordinary individuals are considered to be well-informed about public health
guidelines, while misinformed individuals, having been exposed to
untrustworthy information, are assumed to be less likely to follow recommended
behaviors, such as social distancing, mask-wearing, and vaccination, thereby
increasing the risk of infection for themselves and others. We therefore
define two distinct parameters $\beta_{O}$ and $\beta_{M}$ for ordinary and
misinformed susceptibles, respectively. We model the risky behavior of the
misinformed group by setting $\beta_{M}\geq\beta_{O}$. The recovery rate
$\gamma$ is identical for both subpopulations. We refer to this as the
Susceptible Misinformed Infected Recovered (SMIR) model.

#### Mean-field simulations

The mean-field approach assumes that an infected individual can transmit
disease to any susceptible individual. With this assumption, we derive the
model’s mean-field system of equations (see Methods) and explore parameter
values for which epidemic spread occurs. We denote the proportions of ordinary
and misinformed individuals as $\mu$ and $1-\mu$, respectively. An initial
proportion of the population $\epsilon$ is infected, split evenly between the
ordinary and misinformed groups. Thus, we initialize the compartments with
$I_{O}=\frac{\epsilon}{2}$, $I_{M}=\frac{\epsilon}{2}$,
$S_{O}=\mu-\frac{\epsilon}{2}$, $S_{M}=1-\mu-\frac{\epsilon}{2}$, and
$R_{O}=R_{M}=0$.

The effects of varying the transmission and recovery parameters are
predictable (see Supplementary Information): lower $\beta$ and higher $\gamma$
“flatten the curve” and reduce the negative outcomes of an infection. Based on
these explorations we set $\beta_{O}=0.3$ and $\gamma=0.2$ such that the basic
reproduction number $R_{0}=\beta/\gamma>1$ to ensure epidemic spread. Next we
focus on the effects of varying the misinformed subpopulation’s level of risk
and the degree of homophily within the network.

Figure 2: Increasing $\lambda=\beta_{M}/\beta_{O}$ accelerates and amplifies
the infection. We use $\beta_{O}=0.3$, $\gamma=0.2$, and $\mu=0.5$. (A)
Proportion of the population infected on each day. (B) Extra proportion of the
total population that is infected as a function of $\lambda$.

To explore the effects of risky behaviors by misinformed individuals, let us
assume two equally-sized subpopulations ($\mu=1/2$) and introduce the scaling
factor $\lambda=\beta_{M}/\beta_{O}\geq 1$. Fig. 2 illustrates the increasing
negative impact of the misinformed subpopulation on the disease-spreading
dynamics as $\lambda$ becomes larger. The social cost associated with the more
risky behaviors by the misinformed group is passed on to the whole network.
For example, when $\lambda=3.0$, peak infection for the entire population is
reached 38 days earlier than in the $\lambda=1$ case (day 22 vs. 60; Fig. 2A),
leading to an additional 29.2% of the population becoming infected (Fig. 2B).

Let us explore the effect of homophily among the ordinary and misinformed
subpopulation networks. Homophily means that infected individuals are more
likely to interact with (and infect) susceptibles from the same subpopulation
(ordinary or misinformed) than the other group. The degree of homophily is
modeled by a parameter $\alpha$. When $\alpha=0.5$, individuals are equally
likely to interact within and across groups (no homophily), whereas $\alpha=1$
is the case when homophily is strongest and the subpopulations do not interact
with each other (see Methods for details).

Figure 3: Homophily in the contact network worsens the infection among
misinformed individuals, especially for lower transmission rates. The combined
effects of transmission and homophily parameters, $\beta_{O}$ and $\alpha$,
are examined with the mean-field approximation when $\lambda=3$, $\gamma=0.2$,
and $\mu=0.5$. We plot the proportions of infected individuals in (A) the
ordinary population, (B) the misinformed population, and (C) the overall
population. The maximum proportion of the overall population infected for each
$\beta_{O}$ is marked with a black dot. When the transmission rate is
sufficiently high, homophily benefits the entire population but harms the
misinformed group.

Fig. 3 illustrates the effects of homophily ($0.5\leq\alpha\leq 1$) for
different levels of ordinary transmission ($0.1\leq\beta_{O}\leq 0.4$). For
less infectious disease (low $\beta_{O}$), increasing homophily significantly
harms the misinformed group (Fig. 3B): the infection remains confined within
this group. There is no discernible effect on the ordinary population as long
as the two groups interact; when they do not ($\alpha=1$), we observe a sharp
reduction in infections (Fig. 3A). For $0.12<\beta_{O}<0.16$, peak infection
scenarios coincide with intermediate homophily levels, as indicated by the
black dots in Fig. 3C 33. Under these conditions, while increased homophily
decreases infections in the general populace, it significantly worsens
outcomes for the misinformed group (compare Fig. 3B and C). As $\beta_{O}$
increases further, while nearly the entire misinformed population becomes
infected regardless of $\alpha$ (Fig. 3B), high homophily shields the full
population (Fig. 3C): ordinary individuals have a lower risk of becoming
infected through interactions with misinformed individuals. In summary,
homophily offers greater protection to the ordinary group by isolating
misinformed communities, which suffer a greater disease burden, exacerbating
health disparities 42, 45.

#### Agent-based simulations

To examine the influence of misinformation within a more realistic epidemic
framework, we utilize a multi-level model. Our approach combines an
empirically derived information network with a contact network calibrated with
real-world data, as illustrated in Fig. 4. Information diffusion is modeled by
leveraging a large set of users of a popular social media platform and
epidemic simulations are subsequently conducted on contact networks populated
with misinformed individuals.

Figure 4: An idealized example of our multi-level modeling framework. (A)
Spread of misinformation through an information network (dashed lines). Colors
represent ideological homophily. Nodes with bold borders are misinformed about
the epidemic. The misinformation spreads through a complex contagion (linear
threshold) model; two scenarios show that a lower threshold $\phi$ leads to
more misinformed nodes. (B) Construction of the contact network (solid lines)
for counties with sufficient information diffusion data (in black) to provide
reasonable estimates about the fraction of misinformed individuals. Note that
these counties account for 63.52% of U.S. voters. Each location’s population
size and ideological mix are based on empirical data, and misinformed
individuals are based on the information diffusion model. Links among
individuals within and between locations are based on empirical mobility data.
(C) The infection spreads through the contact network (black nodes), according
to the SMIR model.

We leverage a large collection of English-language discussions taking place on
Twitter about COVID-19 vaccines19. From approximately nine months of this data
(Jan. 4–Sep. 30, 2021), we geolocate over 2 million U.S. users who shared
almost 26 million tweets and focus on accounts in 341 U.S. counties containing
more than 200 Twitter users each. We also infer an account’s political
alignment and whether they shared any likely misinformation (see Methods).
Twitter is not representative of the U.S. population, and people also access
information in other ways, such as traditional media and word of mouth.
However, this empirical social network serves as a data-informed proxy for a
large, realistic network through which people share information about the
disease.

With this data, we build a directed and weighted information diffusion
network, in which an edge $(i\rightarrow j,w)$ indicates that $j$ retweeted
$i$ $w$ times. We simulate the spread of misinformation on this network, as
illustrated in Fig. 4A. Accounts that share or reshare posts containing
misinformation are considered misinformed. These accounts function as the
initial _seeds_ from which misinformation proliferates to the wider network.
Not all users actively participate in content sharing, with only about half of
U.S. Twitter users, for instance, engaging in this activity34. But even
without active sharing, exposure to misinformation can still influence
individual behavior29.

To account for users who may be misinformed through exposure, we employ a
single-step linear threshold opinion-spreading process23. This is a simple way
to model complex contagion15, 49, 31, according to which individuals require
multiple exposures to misinformation before they become misinformed
themselves. Let a linear threshold $\phi$ represent the minimum number of
misinformed friends needed for an ordinary node to become misinformed. If the
total number of misinformed friends of $i$ is greater than or equal to $\phi$,
$i$ is marked as misinformed ($M$). The remaining nodes are marked as ordinary
susceptibles ($O$). We can interpret $\phi$ as a measure of “resilience” to
misinformation; as it increases, individuals require more exposure to
misinformation to be converted to the misinformed group. Note that since we
explore the full range of $\phi$ values, the following results are unaffected
whether the threshold is defined based on the number of users or the number of
retweets.

Fig. 5A shows how $\phi$ influences the number of misinformed individuals
within the retweet network. With strong resilience ($\phi>10$), exposure to
misinformation does not have much effect and few nodes are converted to the
misinformed group. Conversely, when resilience to misinformation is very low
(as in the simple contagion case $\phi=1$), all nodes exposed to a
misinformation-containing post are converted to the misinformed group. Through
this process, empirically observed misinformation-sharing behavior leads to
information networks with misinformed subpopulations of varying sizes based on
different $\phi$ values.

Figure 5: More misinformed individuals lead to a larger portion of the network
becoming infected. Decreasing the resilience $\phi$ (A) increases the size of
the misinformed subpopulation, leading to (B) faster infection spreading and
(C) a greater cumulative number of infections. In panels (B, C), lines and
corresponding shaded regions represent the mean and standard deviation across
simulations, respectively.

We generate contact networks for different thresholds ($1\leq\phi\leq 20$) to
compare the impact of misinformed subpopulations of different sizes. Given a
threshold $\phi$ and the corresponding information network, we aim to
construct a physical contact network containing empirically calibrated
misinformed subpopulations (Fig. 4B). The process begins by selecting a sample
of individuals from each county within the information network. As party
affiliation has been identified as a risk factor associated with excess
mortality during the COVID-19 pandemic25, county samples are constructed to
match the percentage of Republicans and Democrats who voted in the 2020 U.S.
presidential election. For each county, we add the sampled nodes to the
physical network marked as misinformed ($M$) or ordinary susceptible ($O$),
based on their label within the retweet network. Sampling with replacement
allows us to select individuals such that the overall proportions of
Republicans and Democrats match the voting records. A 10% sample leads to
$N\approx 20$ million nodes. A network based on a much smaller sample is
illustrated in Fig. 1B. This process captures empirical measurements of the
ideological split, relative population size, and quantity of misinformed
individuals in each county. We add contact network edges by leveraging cell
phone mobility data that provides the probability of an individual traveling
within and between counties. See Methods for details.

Disease-spreading dynamics on a contact network are simulated using an agent-
based version of SMIR (Fig. 4C). Two critical parameters affecting these
dynamics are the density of the contact network, captured by its average
degree $\bar{k}$, and the transmission probability $p$ (note that
$p\bar{k}=\beta$). Mitigation measures such as social distancing decrease
$\bar{k}$, while those such as masking and vaccination decrease $p$. A simple
way to model the combined effects of misinformation on these behaviors is to
set $\bar{k}=25$, a high value corresponding to the average number of daily
contacts prior to the COVID-19 pandemic28, and explore a wide range of
transmission probabilities to capture worst- and best-case scenarios. To this
end, we model the refusal of any mitigation measures by selecting the maximum
value $p_{M}=1$ for misinformed individuals. In contrast, we model the
adoption of several mitigation measures by selecting an extremely small value
$p_{O}=0.01$ for ordinary individuals. These values illustrate two
theoretically extreme scenarios. The former portrays a realistic number of
interactions during non-pandemic times, accompanied by high transmission rates
due to the absence of preventive measures, such as social distancing, mask-
wearing, or vaccinations. The latter demonstrates decreased daily interactions
and reduced transmission rates resulting from the implementation of these
preventive measures. Using the empirically calibrated contact networks in
conjunction with these extreme parameters, the simulation approach allows us
to bound the best- and worst-case scenarios in a data-informed manner (see
Methods for more information).

The worst-case effect of the misinformed subpopulation size on the daily
incidence of infection (illustrated in Fig. 1C-F on a small network) is
quantified in Fig. 5B on a large network (10% sample). A heavily misinformed
population ($\phi=1$) corresponds to an additional 9% of the population being
infected at peak time (a six-fold increase) compared to a population following
expert guidance in the best-case scenario ($\phi=20$). The peak also occurs
approximately two weeks earlier. The cumulative effect of a heavily
misinformed population is also significant, with an additional 14% of the
population infected over the course of the epidemic — a 32% relative increase
(Fig. 5C). We repeated this analysis for different sample sizes and found that
the main results are robust (see Supplementary Information).

### Discussion

This work presents large-scale simulations based on an epidemic model that
accounts for a subpopulation of misinformed individuals, informed by real-
world data. Such an approach answers calls from the literature to anchor
models in empirical data6, 47, allowing us to explore realistic potential
outcomes. While we know that exposure to online health misinformation is
associated with risky behaviors such as vaccine hesitancy and refusal35, our
model further links these behaviors by a misinformed population to an
accelerated spread of disease.

By deriving the mean-field system of equations for the novel SMIR model and
exploring its dynamics, we demonstrate how the risky behaviors of misinformed
individuals can adversely impact those following public health guidelines,
worsening outcomes for the entire population. Additionally, increasing
homophily can benefit the overall population by protecting ordinary citizens;
however, it may also lead to higher infection rates within the misinformed
subpopulation.

Agent-based simulations of the SMIR model let us study the epidemic on
empirically calibrated contact networks, which provide more realistic
scenarios compared to the mean-field approach. The model estimates that, in
the worst-case scenario, a heavily misinformed population can amplify the peak
of the infection by a factor of six and accelerate it by two weeks. This would
result in an additional 14% of the population becoming infected — nearly 47
million Americans based on recent U.S. Census data48. The corresponding price
tag of vaccine misinformation would be over $143B, using estimated health care
costs associated with COVID-19 in the U.S.3.

These results provide an alarming quantitative bound on the harm of exposure
to online vaccine misinformation. They should provide public health
authorities as well as social media platforms with heightened motivation to
curb vaccine misinformation, despite the difficulties posed by social media
design11.

We acknowledge certain limitations in our approach. While individual beliefs
and behaviors may vary over time, our model simplifies the scenario by
dichotomizing individuals into misinformed and ordinary subpopulations and
assuming constant transmission rates. We also assume uniform resilience to
misinformation for all individuals during the information diffusion process,
although this attribute likely differs across individuals. Future directions
could involve more sophisticated models to account for these heterogeneities.
For instance, cognitive models of misinformation acceptance7 could be
incorporated into the simulation with misinformation exposure data collected
from social media. Such integration would enable the transition of individuals
from ordinary to misinformed susceptible states throughout the simulation,
allowing for a simultaneous examination of opinion and disease dynamics. Some
theoretical models have already explored similar approaches and obtained
results that align with our findings32, 46. Additional extensions of the model
could account for a feedback loop whereby witnessing local infections could
drive changes in behaviors equivalent to the transition of individuals out of
the misinformed population.

### Methods

##### SMIR model.

For both the ordinary and misinformed subpopulations, the Susceptible
Misinformed Infected Recovered (SMIR) model replicates the standard SIR
compartments, denoted as $S_{O}$/$I_{O}$/$R_{O}$ and $S_{M}$/$I_{M}$/$R_{M}$,
respectively. While SMIR adopts distinct transmission parameters for the
misinformed and ordinary groups ($\beta_{M}$ and $\beta_{O}$, respectively),
the mean-field approach assumes that infected individuals from either group
($I_{O}+I_{M}$) can potentially infect anyone in the susceptible populations.
The mean-field model is governed by the following system of equations:

$\left\\{\begin{array}[]{@{}l@{}}\frac{dS_{O}}{dt}=-\beta_{O}S_{O}(I_{O}+I_{M}),\;\;\frac{dI_{O}}{dt}=\beta_{O}S_{O}(I_{O}+I_{M})-\gamma
I_{O},\;\;\frac{dR_{O}}{dt}=\gamma I_{O}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\frac{dS_{M}}{dt}=-\beta_{M}S_{M}(I_{O}+I_{M}),\;\;\frac{dI_{M}}{dt}=\beta_{M}S_{M}(I_{O}+I_{M})-\gamma
I_{M},\;\;\frac{dR_{M}}{dt}=\gamma I_{M}.\end{array}\right.$ (1)

To account for homophily, we modify the term $I_{O}+I_{M}$ in Eq. 1 to account
for increased (decreased) contacts within (across) groups, according to the
parameter $\alpha\geq 0.5$. When homophily does not play a role
($\alpha=0.5$), there is an equal probability of interacting with either
subpopulation’s infected group. We thus obtain:

$\left\\{\begin{array}[]{@{}l@{}}\frac{dS_{O}}{dt}=-2\beta_{O}S_{O}(I_{O}\alpha+I_{M}(1-\alpha)),\;\;\frac{dI_{O}}{dt}=2\beta_{O}S_{O}(I_{O}\alpha+I_{M}(1-\alpha))-\gamma
I_{O},\;\;\frac{dR_{O}}{dt}=\gamma I_{O}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\frac{dS_{M}}{dt}=-2\beta_{M}S_{M}(I_{O}(1-\alpha)+I_{M}\alpha),\;\;\frac{dI_{M}}{dt}=2\beta_{M}S_{M}(I_{O}(1-\alpha)+I_{M}\alpha)-\gamma
I_{M},\;\;\frac{dR_{M}}{dt}=\gamma I_{M}.\end{array}\right.$ (2)

We initialize the simulation by randomly marking $\frac{\epsilon}{2}$ nodes as
infected in both the misinformed and ordinary groups, where $\epsilon=0.001$.

##### Twitter and derived data.

Twitter posts in the CoVaxxy dataset19 were collected in real time via the
stream/filter endpoint of the Twitter Application Programming Interface (API).
To capture the online discourse surrounding COVID-19 vaccines in English, a
comprehensive set of English-language keywords was carefully curated.
Beginning with the initial seeds of “covid” and “vaccine,” a snowball sampling
technique20 was used to identify co-occurring relevant keywords in December
202019. The resulting list contained almost 80 keywords, available online18.
To confirm the relevance of the collected tweets to the topic of vaccines, we
examined the coverage obtained by incrementally adding keywords, starting with
the most common ones. Over 90% of the tweets in 2021 contained at least one of
the three most common keywords: “vaccine,” “vaccination,” or “vaccinate.” To
infer the location of accounts, we used the Carmen Python library21 that
leverages self-reported location metadata within user profiles (embedded in
tweets). As an account’s location may change over time (captured across
multiple tweets), we utilize the most recent location. We geolocate 2,047,800
users residing in all 50 U.S. states, who shared a total of 25,806,856 tweets
by mapping self-reported locations to U.S. counties. The information network
is constructed from accounts in 341 counties that contain more than 200
Twitter users each. Political alignment is estimated using a third-party list
of annotated news sources40, 39. It is averaged across all the sources shared
by each account. Nodes with an estimated alignment greater (smaller) than zero
are considered Republican (Democrat). We infer the political alignment of some
additional accounts, who did not share links to news sources, using a label-
propagation algorithm17 on the retweet network. If all of a node’s neighbors
have political alignment scores, its score is estimated using the weighted
average of its neighbors, with weights based on retweets. The process is
iterated until each node without a score has at least one neighbor without a
score. Misinformation is defined at the source level. Tweets containing links
to articles from a list of low-credibility sources compiled by NewsGuard
(score below 60) are labeled as spreading misinformation. This approach is
common practice and has been validated in the literature35, 24, 8, 27, 44.

##### Contact network edges.

To construct edges in our contact network, we utilize SafeGraph cell-phone
mobility data50, which contains information on the number of people residing
in over 200K Census-Block-Groups (CBG) who visited 4.3M Points-of-Interest
(POI) in the United States. This data has been widely employed to study human
mobility patterns during the COVID pandemic. We used the average daily number
of individuals moving during 2019, as a reference for business-as-usual
mobility, and aggregated all CBGs and POIs at the county level. This
aggregation results in a county-by-county matrix $L$, where each element
$L_{xy}$ represents the average daily number of individuals in county $x$
moving to county $y$ or vice versa. We then normalized $L_{xy}$ to obtain the
average probability of individuals in counties $x$ and $y$ coming into
contact, and multiplied by the total number of edges to obtain the expected
number of connections between individuals in counties $x$ and $y$:
$E_{xy}=\frac{L_{xy}}{\sum_{x^{\prime},y^{\prime}}L_{x^{\prime}y^{\prime}}}\frac{\bar{k}N}{2}$
where the sum is over all county pairs and $\frac{\bar{k}N}{2}$ is the total
number of edges. Next, we create a physical contact network with $N$ nodes by
following a procedure akin to a stochastic block model26 used to generate
networks with localized communities. For each pair of distinct locations $x$
and $y$, we draw $E_{xy}$ edges between random pairs of nodes in $x$ and $y$.
Additionally, we draw $E_{xx}$ edges among random pairs of individuals within
the same location $x$, representing homogeneous mixing within each county. At
the end of the process, the network has the target average degree $\bar{k}$.
We use $\bar{k}=25$ and show how this parameter affects the infections in
Supplementary Information.

##### Simulation details.

Agent-based SMIR simulations are initiated by randomly selecting 100
misinformed nodes and designating them as infected. The disease spreading
dynamics are then simulated for 100 steps, which correspond to days. To align
with COVID-19 dynamics, we utilize the CDC’s recommended quarantine period of
5 days as our recovery period14 ($\gamma=0.2$). Each simulation is repeated
ten times, and the average outcome is reported.

##### Data and code availability.

Code and data are available in a public repository (github.iu.edu/NaN-
team/bounding-misinfo-impact-on-disease-spread).

### Acknowledgments

We are grateful to Yuan Yuan, Marco Ajelli, Alessio Brina, and Brea Perry for
helpful discussions. This work was supported in part by the Swiss National
Science Foundation (grant 209250), the National Science Foundation (grants
1927425 and 1927418), the Army Research Office (contract W911NF-21-1-0194),
the European Union (NextGenerationEU project PNRR-PE-AI FAIR), the Italian
Ministry of Education (PRIN PNRR grant CODE prot. P2022AKRZ9 and PRIN grant
DEMON prot. 2022BAXSPY), Knight Foundation, and Craig Newmark Philanthropies.

### Ethics

This study, focusing on public data, poses minimal risk to human subjects.
Consequently, the Indiana University Institutional Review Board has exempted
it from review (protocol number 1102004860). All data collection and analysis
adhered to Twitter’s terms of service.

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## Supplementary information

###  Appendix 1 Mean-field SMIR model

Let us denote the proportions of misinformed and ordinary individuals as $\mu$
and $1-\mu$, respectively. A proportion $\epsilon=0.001$ of the population is
initially infected, split evenly between the ordinary and misinformed groups.
Thus, during the initial state, we have initial values for each compartment:
$R_{O}=R_{M}=0$, $S_{O}=\mu-\frac{\epsilon}{2}$,
$S_{M}=1-\mu-\frac{\epsilon}{2}$, $I_{M}=\frac{\epsilon}{2}$, and
$I_{O}=\frac{\epsilon}{2}$.

Figure S1: Reducing the transmission parameter $\beta_{O}$ decreases the
severity of the epidemic. We plot (a) the proportion of the population
infected each day, (b) $R_{0}$ values, and (c) the total proportion of the
population infected as $\beta_{O}$ varies. Here we do not consider the role of
misinformation or homophily.

To identify a suitable base value for the transmission rate among ordinary
susceptibles, we begin by exploring the scenario with no misinformed
individuals ($\mu=1$), setting $\gamma=0.2$ and varying the transmission
parameter in the range $0.02\leq\beta_{O}\leq 1$. As is typical of SIR
dynamics, Fig. S1 shows that lower $\beta_{O}$ values delay and lower the
infection peak — the so-called “flattening of the curve.” Lower $\beta_{O}$
also decreases the total proportion of the population that becomes infected at
any point during the epidemic, while higher $\beta_{O}$ values increase this
proportion. These dynamics are tied to the basic reproduction number
$R_{0}=\beta/\gamma$: the disease spreading dynamics only reach epidemic
levels when $R_{0}>1$, such that an infected individual infects more than one
other person on average. This happens when $\beta_{O}>0.2$. As $R_{0}$
increases, the infection spreads more quickly, the peak infection day occurs
sooner, and the proportion of the population that is ultimately infected
increases.

Figure S2: Effects of varying the recovery rate. We plot (a) the proportion of
the population infected each day, (b) $R_{0}$ values, and (c) the total
proportion of the population infected as a function of the number of days to
recover, $\tau$. Here we do not consider the role of misinformation or
homophily.

We now explore the effect of the recovery rate, again in the scenario with no
misinformation or homophily, by setting $\beta_{O}=0.3$ and varying the
recovery period $\tau=1/\gamma$ between 1 and 20 days. Fig. S2 shows that when
$\tau<4$, $R_{0}<1$ and the disease does not reach epidemic proportions. At
this level, the epidemic takes a long time to reach its peak ($\approx$80
days). Increasing $\tau$ means that individuals remain infected longer, so the
population gets infected faster and the peak infection is reached more
rapidly. We set $\tau=5$ ($\gamma=0.2$) in the main text to align with
quarantine recommendations from the CDC14. With this value and
$\beta_{O}=0.3$, the dynamics are at epidemic levels ($R_{0}=1.5$).

###  Appendix 2 Robustness analyses

To test the robustness of the main results (Fig. 5C) with respect to the
sample size used to construct the contact network, all simulations were rerun
after generating contact networks based on the different sampling percentages
between 0.01% and 10%. Fig. S3 shows the relative increase in the percentage
of the population that becomes infected as a function of the linear threshold
$\phi$, using the best-case scenario in which the fewest nodes in the network
are misinformed as the baseline. We observe a substantial decrease in the
effect of misinformation as the sampling size grows to 1%. However, sample
sizes above 1% return nearly identical results. We conclude that using a
sample size of 10% (as reported in the main text) is sufficient to rule out
any size-induced bias.

Figure S3: Relative increase in the mean total population infected as a
function of the sampling size utilized in the contact network creation
process. The $\phi=20$ scenario, in which the fewest nodes in the network are
misinformed, is utilized as the baseline.

We also investigate the impact of the contact network density ($\bar{k}$) on
infection dynamics, as illustrated in Fig. S4 for $5\leq\bar{k}\leq 25$. We
consider this range because $\bar{k}=25$ represents pre-pandemic daily social
contacts while $\bar{k}=5$ represents COVID-19 lockdown conditions28. As
expected, Fig. S4(a) demonstrates that higher $\bar{k}$ leads to increased
infections through the population, since the higher contact density provides
more opportunities for transmission. But while a larger percentage of the
overall population is infected, the relative effect of misinformed individuals
decreases. This is because, at higher $\bar{k}$ values, the infected
population is already substantial even in the low-misinformation ($\phi=20$)
baseline. The combined effect of these two opposing trends, as shown in Fig.
S4(b), is that the additional percentage of infected individuals relative to
the $\phi=20$ scenario reaches a maximum for some intermediate $\bar{k}$. Fig.
S4(b) also shows that, consistent with our primary findings, larger $\phi$
(misinformed resilience) results in a smaller infected population. In our main
analysis we focus on $\bar{k}=25$ and model an effective reduction of
contacts, resulting for example from social distancing or lockdowns, by
decreasing the $p$ parameter.

Figure S4: Effects of average contact network degree $\bar{k}$ on infection
dynamics. (a) Infected individuals ($\phi=1$) as a percentage of the overall
population and relative to the baseline condition $\phi=20$, in which the
fewest nodes in the network are misinformed. (b) Additional percentages of
infected population relative to the baseline condition $\phi=20$.