[figure]capposition=bottom,capbesideposition=top,left,floatwidth=

# Complexity of Government response to Covid-19 pandemic: A perspective of
coupled dynamics on information heterogeneity and epidemic outbreak

Xiaoqi Zhang Email<EMAIL_ADDRESS>Jie Fu Email<EMAIL_ADDRESS>National
School of Development, Southeast University, Nanjing, China  Sheng Hua Email:
<EMAIL_ADDRESS>National School of Development, Southeast University,
Nanjing, China  Han Liang Email<EMAIL_ADDRESS>Dong Fureng Institute of
Economic and Social Development, Wuhan University, Wuhan, China  Zi-Ke Zhang
Email<EMAIL_ADDRESS>Corresponding Author. College of Media and
International Culture, Zhejiang University, Hangzhou, China

###### Abstract

This study aims at modeling the universal failure in preventing the outbreak
of COVID-19 via real-world data from the perspective of complexity and network
science. Through formalizing information heterogeneity and government
intervention in the coupled dynamics of epidemic and infodemic spreading;
first, we find that information heterogeneity and its induced variation in
human responses significantly increase the complexity of the government
intervention decision. The complexity results in a dilemma between the
socially optimal intervention that is risky for the government and the
privately optimal intervention that is safer for the government but harmful to
the social welfare. Second, via counterfactual analysis against the COVID-19
crisis in Wuhan, 2020, we find that the intervention dilemma becomes even
worse if the initial decision time and the decision horizon vary. In the short
horizon, both socially and privately optimal interventions agree with each
other and require blocking the spread of all COVID-19-related information,
leading to a negligible infection ratio 30 days after the initial reporting
time. However, if the time horizon is prolonged to 180 days, only the
privately optimal intervention requires information blocking, which would
induce a catastrophically higher infection ratio than that in the counter-
factual world where the socially optimal intervention encourages early-stage
information spread. These findings contribute to the literature by revealing
the complexity incurred by the coupled infodemic-epidemic dynamics and
information heterogeneity to the governmental intervention decision, which
also sheds insight into the design of an effective early warning system
against the epidemic crisis in the future.
Keywords: COVID-19, coupled-dynamics, government response dilemma, information
heterogeneity, information injection, dinformation blocking

##  1 Introduction

Since the outbreak of COVID-19, a variety of major national governments have
failed to manipulate preventive strategies against the pandemic one after
another. As shown in the Fig. 1, no matter whether the countries made a fast
official warning to the public, whether their first domestic case of COVID-19
was reported before or after the government’s warning statement, whether or
not the government had a strict restriction on its domestic information
spreading media, almost all countries suffered from the failure of controlling
the pandemics of COVID-19 [1]. Fig. 1 suggests a counter-example to the theory
that free information spreading and fast warning are helpful in containing the
outbreak of an epidemic crisis. We believe that the worldwide failure in front
of the COVID-19 crisis reflects some systematic incapability of tackling the
unknown infectious diseases by modern government; one source of the
incapability is the policy dilemma: remedy or overkill.

Figure 1: Timeline of COVID-19 outbreak and government responses in 21
countries

Timelines of COVID-19 outbreak in the top 20 countries ranked according to the
number of their cumulative infectious cases of COVID-19 by Jun. 30, 2020, and
China, the country with the first confirmed infectious case of COVID-19. The
event time data comes from Wikipedia [2], and the infection number comes from
the COVID-19 dashboard at the John Hopkins University [3]. Using the blue and
red color bar, we mark three milestones on the timeline of each country that
are the time when the first confirmed case was reported, the time when the
central government first published an official statement regarding COVID-19,
and the time when the country first seriously executed non-pharmaceutical
intervention. The left half of the annular (with blue background) corresponds
to those countries that have their first official statement made ahead of the
first domestic infectious case, and the right half of the annular (with red
background) corresponds to the countries that have their first domestic case
reported ahead of the official statement. The time difference (days) from the
first domestic infectious case to the next event (the first official statement
or the first execution of NPIs) is measured by the length of the red bar,
while the time difference from the first official statement to the next event
(the first domestic cases or the first NPIs) is measured by the length of the
blue bar. The two-time differences measure the speed of government response
toward the outbreak of COVID-19. For each country, the exact occurrence date
of every event is marked at the ends of the blue and red bars with the date
marked toward the center of the annular representing date of the earlier
event. We also sketch the number of cumulative infectious cases at the time of
the first NPIs with the black bar and mark the exact number on the top of the
black bar. The comparison of the relative length of three color bars across
countries provides a measure for the relative speed of COVID-19 spreading
versus the government response.

On the one hand, the government may choose an inactive move (remedy), such as
blocking information spreading in the early stage of an epidemic crisis so as
to keep the public away from panic, by which the infection ratio may not grow
too fast in the early stage while the government suffers the least public
pressure. On the other hand, the government can choose a socially optimal
strategy (overkill), by which the government encourages information spreading
and even discloses as much classified information to the public as possible.
By doing so, a fast accumulation of infectious cases is witnessed by the
public, which may effectively send warning signals but also increase public
pressure, therefore, is not optimal from the government’s angle. The conflict
between the government’s private optimal and social optimal makes it almost
unlikely for the government to make the right decision for the public, which
triggers the outbreak of the public health crisis. Therefore, we need to
understand how this conflict arises during an epidemic crisis, and whether and
how it is avoidable. To this end, information is one key factor as it is
believed to be critical for the early-stage control of any infectious disease.

It is demonstrated that the key factor in controlling the spread of epidemics
successfully is to understand the complex two-way interaction between disease
dynamics and human social behavior[4]. First, the epidemic spreads can
stimulate information spreading about the disease, leading to people’s
awareness of the crisis[5, 6, 7]. Some people will take preventive measures to
protect themselves from the disease[8, 9]. From this perspective, the epidemic
dynamics experience an endogenous negative feedback[10, 11, 12, 13, 14], by
which the disease might be self-contained. Second, information propagation may
not always induce preventive behavior; it can oppositely stimulate more risky
behavior and accelerate the spread of disease[15]. The positive feedback
exists because information does not only come from people’s own observations
and experiences (source information), but also mingles unreliable second-hand
information from social media[16, 17, 18], where people can get information
and interactive views[19, 20]. On social media, there is a spontaneous trend
of turning information into rumors with strong emotional tendency[21, 22],
which increases anxiety feelings and magnifies irrational behavior[23, 24, 25,
26]. As an example, it is observed that 41.3% of diagnosed patients of
COVID-19 have contracted the disease during hospital visits in Wuhan [27].

As the main body of social governance, the government is also the most
critical information node in the entire social network. Although the
government is often asked to disclose the “truth” to the public as soon as
possible, it shares the same kind of uncertainty as the public when facing
such an unprecedented virus as COVID-19, there is no “truth” to disclose at
least at the early stage. Restricted by this real constraint, the
effectiveness of any government responses may be questionable. Therefore, this
study aims to theorize the role of government intervention within a coupled
infodemic-epidemic dynamic model where both positive and negative feedback are
allowed. We further conducted the counterfactual analysis on both the
synthetic settings and the parameters calibrated from real-world pandemic data
in order to reveal the heterogeneous impact of information on epidemic
spreading and its incurred decision complexity faced by government. We believe
they are critical to the outbreak of the COVID-19 pandemic, hence should be
paid more attention in the future design of warning systems for a public
health crisis.

The remaining sections are organized as the following. Section 2 introduces
the coupled dynamic model of information and epidemic spreading and formalizes
the government intervention decision problem through the model. Section 3
analyzes the mechanism that leads to the government response dilemma by
numerical experiments on both the synthetic model setting and the calibrated
setting derived from the early-stage epidemic data of Wuhan, China. We also
briefly discuss the potential solution to the government response dilemma from
the perspective of an information network. Section 4 concludes the findings
and discusses the direction of future research.

##  2 Literature Review

###  2.1 Infodemic and Epidemic Spreading

The current study is closely related to the literature on the co-evolution
dynamics of epidemics and infodemics [28, 29, 30]. It has been widely
acknowledged that the complex two-way interaction between disease dynamics and
human social behavior is critical to the outbreak of an epidemic crisis and
its prevention. The information on the epidemic may induce fear effect [31]
and intrigue self-protection behavior, which helps contain the outbreak of
epidemic crisis [31, 32]. Despite the positive effect, studies [33, 34, 35,
36] found that information may not always be kept real, mis-information is
inevitable and will reduce the scientific trust of the public, which incurs
negative impact on the containment of epidemic crisis. More recent studies
jump out of the dichotomy between information and mis-information, the
heterogeneity in information literacy [32], sources [29] channels by which
information is propagated [30], and their impact on the epidemic spreading are
thoroughly investigated.

Despite the persistent discussions on the co-evolution of infodemic and
epidemic, the focus of the literature was inclined to theoretically model the
co-evolution process itself and/or empirically estimate the extent of
infodemic-epidemic interaction. It was rarely studied whether and how the co-
evolution process can be intervened and/or utilized for governance purposes.
Particularly, whether or not the control of information propagation by the
government can be supported by the coupled infodemic-epidemic dynamics [37]
and facilitate the containment of the outbreak of the epidemic crisis is still
an open question, to which we attempt to give a formal discussion.

###  2.2 Prevention and Control

Our study is also connected to the literature on the optimal control and
prevention strategies against epidemic crisis [38, 39, 40, 41, 42]. Existing
studies have extensively discussed a long list of the preventive measures,
including but not limited to the social distancing [41], traffic jam and
lockdown [38, 43], vaccination [39], mask-use [44], and the effectiveness and
potential trade-offs incurred by them. It is worthwhile noting that these
discussions focus almost exclusively on the late-stage of the epidemic crisis.
Unless the outbreak of infectious diseases has persisted for long or the
spreading cannot be contained, it is not really needed to adopt any social
distancing measure [45]. On the the other hand, during the early-stage of an
epidemic outbreak, the lack of knowledge and awareness on infectious diseases,
such as COVID-19 [46], makes it also reluctant to adopt any prevention measure
in the aforementioned list. From this perspective, information might be the
only way in the early-stage epidemic crisis that the government can intervene
to affect and guide the public to protect themselves [46]. However, to the
information intervention and its effectiveness and trade-offs in the early-
stage coupled dynamics of infodemic and epidemic crisis, the existing studies
paid very little attention, to which a formal investigation will be carried
out in this study.

From the brief review, we identify our main contribution to the literature as
bridging the two branches of research on the coupled infodemic-epidemic
dynamics and the optimal intervention, which is critical to the early-stage
warning and prevention against an epidemic crisis.

##  3 Methods

###  3.1 SI-NLH model: set-up and properties

To capture the heterogeneous response to different types of information and
the induced impact on disease dynamics, in this section, we propose an SI-NLH
dynamic model where the SI component is the classical Susceptible-Infection
model capturing the disease dynamics. The NLH component reflects the dynamic
transition of individuals among three information statuses: No information,
with Low-quality information and High-quality information.

Compared with the SIR/SIS model, the SI model excludes the recovery dynamics
and is, therefore, more appropriate to model the early-stage disease dynamics
in the real world, such as the pandemic of SARS and COVID-19. As in the early
stage of epidemic spreading, the number of recovered cases is completely
negligible in relation to the newly infected cases[47]. In fact, a rigorous
mathematical analysis[48] shows that when the infection/recovery ratio, i.e.,
the ratio of infected/recovered cases versus the population size, is very
small, the infection dynamics derived from a SI model and a SIR model are
equivalent. At the early stage of the epidemic crisis, the small
infection/recovery ratio always holds (for instance, in the COVID-19 pandemic
in China, even in the epicenter, Wuhan, the greatest infection/recovery ratio
has never exceeded 0.8%); therefore, it is safe to consider the simpler SI
model for the epidemic spreading. (In the supplementary material to this
study, we conduct a simulation study similar to the one discussed in the
following sections, but based on a SIR-NLH model where the SI component is
replaced with the SIR model, the result is similar, the details can be found
in Supplementary Fig. 1.)

The differentiation of information quality in the NLH component is critical,
as the quality often determines the response behavior of individuals toward
infectious diseases. High-quality information consists of that which is
officially issued by truth holders, such as the government or specialists in
infectious diseases. The contents of this information are correct and the
description is complete, therefore they can intrigue self-protection behavior
and lower the infection risk. In contrast, low-quality information is rumors
and/or that whose contents are not completely wrong but the description is
incomplete. This information can easily trigger incorrect interpretations and
be converted to rumors in later-stage propagation. Consequently, low-quality
information is likely to intrigue panic and irrational behaviors that rise up
the infection risk[49]. However, the existing literature on coupled infodemic-
epidemic dynamics focuses on the positive side of the information [4, 50, 51],
the low-quality information and the induced higher infection risk are often
missed. Although some recent studies start to pay attention to the negative
side of information[30, 46], they mainly focus on empirical and/or simulation
studies on the epidemic consequences of rumor spreading rather than
theoretically modeling the underlying mechanism. In the SI-NLH model, we
integrate both the positive and negative sides of information and their
interaction mechanism, which provides a more unified view of the coupled
infodemic-epidemic dynamics.

According to the joint of the information and disease status, the population
in the SI-NLH model is divided into six groups, namely the susceptible without
information ($S_{N}$), the susceptible with low-quality information ($S_{L}$)
and the susceptible with high-quality information ($S_{H}$), the infected
without information ($I_{N}$), the infected with low-quality information
($I_{L}$) and the infected with high-quality information ($I_{H}$). The
fraction of the six groups and their dynamic transitions are formally
described via the following ordinary differential equation system and
illustrated in Fig. 2a,

$\begin{cases}S_{N}^{\prime}=&-\gamma_{1}S_{N}L(1-I)-\gamma_{2}S_{N}H(1-I)-b_{m}S_{N}I\\\
S_{L}^{\prime}=&\gamma_{1}S_{N}L\cdot(1-I)+S_{H}(1-I)(\beta_{1}L+\beta_{2})\\\
&-b_{h}S_{L}I(I+\theta S_{L})-b_{m}S_{L}I(1-I-\theta S_{L})\\\
S_{H}^{\prime}=&-S_{H}(1-I)(\beta_{1}L+\beta_{2})+\gamma_{2}S_{N}H(1-I),\\\
I_{N}^{\prime}=&-\gamma_{1}I_{N}L-\gamma_{2}I_{N}H+b_{m}S_{N}I\\\
I_{L}^{\prime}=&\gamma_{1}I_{N}L+I_{H}(\beta_{1}L+\beta_{2})+b_{h}S_{L}I(I+\theta
S_{L})\\\ &+b_{m}S_{L}I(1-I-\theta S_{L})\\\
I_{H}^{\prime}=&-I_{H}(\beta_{1}L+\beta_{2})+\gamma_{2}I_{N}H\end{cases}$
(3.1)

where the fractions $L$, $H$, $I$ and $S$ satisfy the relations that
$L=I_{L}+S_{L}$, $H=I_{H}+S_{H}$, $I=I_{N}+I_{L}+I_{H}$, $S=S_{N}+S_{L}+S_{H}$
and $I+S_{N}+S_{L}+S_{H}\equiv 1$. Based on Eq. (3.1) and Fig. 2a, there are
three classes of parameters governing the SI-NLH dynamics, which are i)
information parameters: $\gamma=(\gamma_{1},\gamma_{2})$, characterizing the
rate of transition from No information to Low- and High-information, and
$\beta=(\beta_{1},\beta_{2})$, the rate of transition between L- and High-
information; ii) the transmission rates to disease $b=(b_{m},b_{h})$ for
susceptible people under unawareness ($b_{m}$) and panic ($b_{h}$),
respectively; and iii) the panic parameter $\theta$ quantifying the
susceptibility to panic. For the convenience of reading, a full list of the
notations and meanings for our model parameters and endogenous variables are
presented in Appendix Table 1.

Figure 2: Coupled SI-NLH model and trajectories of infection ratios generated
under different information parameters

(a) Illustration of the six compartments and their interconnection within the
SI-NLH model. (b) Evolution of infection ratio for a variety of information
parameters and initial conditions. Given the benchmark trajectory of the
infection ratio growth (bold red), trajectories are plotted against a variety
of modifications on the model set-up, including i) a decrease in the initial
infection ratio $I(0)$ (dashed red); ii) decrease in the information
transition rate $\gamma_{1}$ (black), $\gamma_{2}$ (purple) and the decay rate
$\beta_{2}$ (light blue); iii) high-quality information dominates the
information dynamics, $\beta_{1}=0$ (green) and $\beta_{1}<0$ (blue).

For the three classes of parameters, we made the following assumptions in
order to reflect the real-world interaction mechanism between infodemic and
epidemic spreading. First, for information parameters, we suppose that
information spreading is irreversible, which means the rate
$\gamma_{1}$/$\gamma_{2}$ of $N$-type individuals transmitting to
$L$-/$H$-type is nonnegative. For the transmission between $L$-type and
$H$-type, we let the sign of $\beta_{1}$ characterize which direction of
transmission is dominant, a positive $\beta_{1}$ implies the transmission from
$H$ to $L$ is prevalent, the negative $\beta_{1}$ implies the contrary. To
capture the natural trend of the information decay during propagation, we set
a nonnegative rate $\beta_{2}$ to capture the transmission from $H$ to $L$ due
to quality decay. Second, for disease dynamics, we assume that individuals
with different types of information would react differently to the disease,
which leads to different transmission rates. $S_{H}$ owns high-quality
information and tends to self-protect which leads to a low transmission rate,
which is set to $0$ for the sake of simplicity (this assumption does not
significantly affect the result of this paper, we relax it in the
supplementary material and report the simulation results in supplementary Fig.
2 in which $S_{H}$ is allowed be infected in a positive rate $b_{l}$ that is
lower than the rate of the $S$ individuals with the other types of
information). Because of no information, $S_{N}$ is unaware of the disease and
likely to maintain their regular behavioral patterns, which yields a middle-
level transmission rate $b_{m}>0$. $S_{L}$ behaves more complicatedly. On one
side, they intend to ignore their information due to its low quality. In this
case, they behave as if being unaware of the disease, yielding the
transmission rate $b_{m}$. On the other side, the low quality might stimulate
panic or other kinds of irrational behaviors for $S_{L}$, such as gathering in
hospitals and supermarkets, which leads to a much higher infection risk
$b_{h}\gg b_{m}$. Finally, panic during an epidemic crisis always results from
the increasing infectious cases and the self-exciting property that the more
people already in a panic mood, the more likely they are to entail panic among
the remaining people. Therefore, we assume $S_{L}$ has the probability
$I+\theta S_{L}$ of turning to panic, where $\theta$ is the panic parameter
measuring the self-exciting degree of panic.

Based on the aforementioned model set-up, the disease propagation following a
SI model implies that as $t\rightarrow\infty$ the infection ratio $I(t)$ would
always converge, while the limit ratio depends on the choice of information
parameters $\beta$. As shown in Fig. 2b, the ultimate infection ratio would
almost always converge to 1, i.e. the entire population will be infected,
provided that $\beta_{1}\geq-\beta_{2}$, no matter where the dynamic system
starts. On the contrary, when $\beta_{1}<-\beta_{2}$, the limit infection
ratio is possible to be strictly less than $1$, the limit depends on both the
initial condition and the parameter $\beta_{1}$ and $\beta_{2}$. Especially,
when the configuration is fixed, the infection ratio increases with both
$\beta_{2}$ and $\beta_{1}$. The remaining information parameter $\gamma$, the
transmission rate, and the panic parameter have no impact on the ultimate
infection ratio, but would significantly impact the convergence speed as shown
in Fig. 2b. In particular, it can be proved that the temporal infection ratio
$I(t)$ is monotonically increasing with the parameter $\gamma_{1}$,
$\beta_{1}$, $\beta_{2}$ and decreasing with parameter $\gamma_{2}$ when all
others keep constant (Fig. 2b shows a few examples of the general relation).

Fig. 2b shows a different perspective for thinking of information governance
during the early stage of epidemic spreading. Once the heterogeneity of
information is introduced, it is no longer correct as claimed in the
literature [10, 11, 12, 13, 14] that more information can help better contain
disease outbreaks. The low-quality information cannot help mitigate the
infection risk, but even increase the risk exposure of susceptible
individuals. The heterogeneity of information is neither static nor external
to the disease dynamics, but co-evolves with it. This fact makes the
underlying structure of the information market extraordinarily important
because it shapes the propagation process and determines which type of
information will be prevalent and which type will be thrown away. The
information filtered by propagation will feed back again to affect disease
dynamics. In the situation that $\beta_{1}>0$ (the solid red line in Fig. 2b)
and/or $\beta_{1}=0$ meanwhile $\beta_{2}>0$ (solid green line in Fig. 2b), an
information market admitting free propagation of all information types would
intrinsically squeeze out the high-quality information and gradually make all
people exposed to the rumors and diseases. In this situation, government
intervention in the information market will become indispensable.

###  3.2 Government intervention in SI-NLP model: classification and trade off

During the disease pandemic in the real world, such as the pandemic of
COVID-19, it is often the case that information about the disease passes
through a hierarchical network in the direction from the bottom to the top. As
to be discussed in later sections, such a network structure and information
flow direction makes information quality decay inevitable which further
stimulates the outbreak of disease. Therefore, external intervention from an
independent third-party is indispensable. Government can intervene in the
infodemic-epidemic co-evolution process to achieve a low ultimate infection
ratio. However, it is not so easy for the government to figure out the correct
time and the correct intervention methods. To better interpret the trade-off
and dilemma faced by the government, we theorize the role of government
intervention in the SI-NLH model.

To be more concentrated, we focus only on the intervention measures on
information dynamics, which means the government can only have an impact on
the values of the information parameters $\gamma$ and $\beta$. This assumption
is reasonable as the transmission rates and the panic parameter are determined
essentially by the virus and the cultural background of a society that is out
of control by the government. Without loss of generality, we concentrate on
two classes of intervention strategies: (i) information injection strategy[52,
53]; and (ii) the blocking strategy[54, 55].

Figure 3: Effect of blocking and information injection.

(a,b) Illustration of the mechanism that blocking and information injection
influence the coupled infodemic-epidemic dynamics. Plot a shows their direct
impact on information parameters. Plot b illustrates their ultimate impact on
the trajectory of infection ratio. (c,d) Trade-offs incurred by delayed
information injection and blocking. Plot c reveals the decomposition of
positive ($A_{2}$) and negative ($A_{1}$) impact of delayed information
injection on suppressing infection ratio, plot d decomposes the positive
($B_{1}$) and negative ($B_{2}$) impact induced by increasing the degree of
blocking.

Information injection and blocking impact the coupled dynamics in different
ways, which are summarized in Fig.3a. By injecting more high-quality
information into the public, the government can accelerate the conversion of
$S_{N}$ and $S_{L}$ to $S_{H}$ and avoid the decay of high-quality information
$H$ to $L$, which lowers the population-level transmission rate into
infection, resulting in that the infection ratio trajectory decreases from the
green-line level to the blue-line level in Fig. 3b. However, before the
government can provide high-quality information for the public, it takes time
to collect and preprocess tons of noisy information. The easier the government
can collect highly accurate information about the disease, the more effective
the injection can be, which implies the execution and effectiveness of
information injection rely on time and accumulation of information and
particularly the high-quality information in the system. In contrast, by
information blocking, government controls the speed of information spreading
in the system, while does not have to provide any processed information for
the system. Therefore, blocking can slow down the rumor spreading which helps
lower the population-level transmission rate to infection, resulting in the
infection ratio trajectory decreasing from the red-line level to the green-
line level in Fig. 3b. However, on the other hand, without extra information
being provided, blocking cannot stop the decay of high-quality information,
while its effectiveness does not depend on the execution time and the
accumulation of information in the system.

To formalize the difference between information injection and blocking in the
SI-NLH model, we assume that the epidemic crisis breaks out at time $t^{\ast}$
since then the government decides whether or not and to what degree to inject
and block information. The government made the intervention decision in an
attempt to minimize the cumulative impact of the epidemic crisis by a decision
time horizon $T^{\ast}$ with $T^{\ast}>t^{\ast}$. To reflect the time-
dependence of information injection, we assume it takes effect since a given
injection time $\tau\in[t^{\ast},T^{\ast})$ and induces the increase in rate
$\gamma_{1}$ and decrease in rate $\gamma_{2}$, $\beta_{1}$ and $\beta_{2}$
during the time interval $(\tau,T^{\ast}]$ by the degree
$g_{\tau}=g(H_{\tau},L_{\tau}+H_{\tau})$ as shown in Fig. 3a. To reflect the
information-dependence, we let $g_{\tau}=g(H_{\tau},L_{\tau}+H_{\tau})$
monotonically increase in both the ratio of high-quality information
$H_{\tau}$ and the ratio of all informed individuals $L_{\tau}+H_{\tau}$.
Since the impact of information injection depends on the injection time $\tau$
via the degree function $g_{\tau}$, the government needs to select the
injection time so as to optimize the intervention effect.

For information blocking, we suppose that it induces the decrease in the
absolute value of parameter $\gamma_{1}$, $\gamma_{2}$ and $\beta_{1}$ to the
degree $s$ since the very beginning $t^{\ast}$ of the epidemic crisis, but
cannot affect the decay rate $\beta_{2}$, as shown in Fig. 3a. In addition, we
suppose that blocking can take effect only if it can be maintained for a
while, frequent changes in the blocking degree $s$ would interrupt the normal
order in the information market and are forbidden by assumption. Without loss
of generality, we assume the degree $s$ of the information block is held
constant for the entire period from $t^{\ast}$ to $T^{\ast}$. To generalize
the scope of analysis, we allow the blocking degree $s$ to take zero and even
negative values, which corresponds to no blocking strategy the government does
not block any information at all and maintains the current information
spreading speed, and the “negative” blocking strategies by which government
attempts to accelerate information spreading among the public.

Both information injection and blocking can generate positive and negative
impacts on the control of the final infection ratio, $I(T^{\ast})$, at the
decision horizon $T^{\ast}$. Fig. 3c and Fig. 3d sketch those conflicted
impacts in detail. In Fig. 3d, the effect of information injection depends on
the injection time, a delayed injection from $\tau$ to $\tau^{\prime}>\tau$
firstly induces a positive structural effect associating with a greater
$g_{\tau}$ which leads to a faster transition from $S_{L}/S_{N}$ to $S_{H}$
and a lower ultimate infection ratio, which is represented as the gap term
$A_{1}$. On the other hand, as shown in Fig. 3c, the later injection makes the
infection ratio grow in its original speed during the period
$[\tau,\tau^{\prime})$ that is faster than the speed induced by injection at
$\tau$. Although the faster growth of infectious cases is temporal, it
increases the cumulative infection ratio by time $\tau^{\prime}$, which
further increases the transmissibility of the disease after $\tau^{\prime}$
due to a greater initial infectious population at $\tau^{\prime}$. This
negative temporal effect is represented as the term $A_{2}$ as shown in Fig.
3c.

In Fig. 3d, the impact of blocking on $I(T^{\ast})$ is also decomposed into
two conflicted parts. First, a positive structural effect that is purely
attributable to the change of information parameters, represented by the gap
$B_{1}$, which slows down the transition to $S_{L}$ from $S_{H}$ and $S_{N}$,
therefore suppresses the panic-induced infection risk and lowers the overall
infection ratio in the end. Second, the negative injection effect, $B_{2}$, by
which the decrease in information spreading speed would reduce the portion of
informed people, represented as $H_{\tau}$ and $L_{\tau}+H_{\tau}$, for every
time $\tau$ after the initial decision time $t^{\ast}$ for blocking that
further reduces the effectiveness of information injection, $g_{\tau}$, and
facilitate the growth of ultimate infection ratio.

Note that the four marginal effects, $A_{1}$, $A_{2}$ and $B_{1}$, $B_{2}$,
are not separable in practice, combining them together would disturb the
coupled infodemic-epidemic dynamics in a complicated way that induces a joint
trade-off effect for the government when facing strategy selection.

###  3.3 Set-ups for synthetic analysis and model calibration

We will numerically analyze the effect of both intervention strategies on the
final infection ratio for a wide range of synthetic settings. For fixed
initial decision time $t^{\ast}$ and decision horizon $T^{\ast}$, we will
calculate the variation trend of the intervention effect (measured by
$I(T^{\ast})$, the infection ratio at the end of the decision horizon) along
with the joint adjustment of injection time $\tau$ and blocking degree $s$. We
let $\tau$ range within $(t^{\ast},T^{\ast})$ and let $s$ vary within $[0,2]$
such that the three information parameters are determined via
$\gamma_{1}=\gamma_{2}=\beta_{1}=s$. We also test the sensitivity of our
analytic results against the change of both $t^{\ast}$ and $T^{\ast}$, for
which the benchmark setting is represented by the initial distribution of
population within the six groups $(S_{N},S_{L},S_{H},\\\
I_{N},I_{L},I_{H})=(0.99,0,0.009,0.001,0,0)$ and the decision horizon
$T^{\ast}=4$ weeks (or 28 days). To capture the impact of delayed $t^{\ast}$,
we consider the delayed-initial setting by which we change the initial
distribution to
$(S_{N},S_{L},S_{H},I_{N},I_{L},I_{H})=(0.93,0.4,0.02,0.01,0,0)$ wi-
th a higher ratio of the infectious cases and both low- and high-quality
information holders. The worse initial always associates with a delayed
government intervention (a greater $t^{\ast}$), as under delayed intervention
the infodemic-epidemic dynamics will be kept on their original track to evolve
for a longer time, which leads to a greater infection ratio and a greater
proportion of both low- and high-quality information holders. To capture the
impact of a shorter decision horizon, we consider the shorter-horizon setting
where we shrink $T^{\ast}$ from 4 weeks in the benchmark setting to 1 week (or
7 days) in the comparison setting. Throughout all synthetic set-ups above, the
remaining model parameters are kept constant.

To align our synthetic analysis with the real-world infodemic-epidemic
dynamics, we calibrate our model with the COVID-19 infection data reported in
the earliest epicenter, Wuhan, China. The data collection period is from Jan.
10 2020, when the Wuhan government and the CDC in China started the daily
report of COVID-19 infectious cases in Wuhan, to Jan. 24, 2020, when Wuhan was
officially locked down. There was not any external intervention conducted
within Wuhan during these two weeks, therefore, the epidemic spreading of
COVID-19 can be approximately viewed as occurring at its natural speed within
this period. Counterfactual analysis based on the model parameters trained
during this period can better reflect reality. Since the entire population is
susceptible to COVID-19[56, 57], $N=11$ million, the official population size
in Wuhan, is taken as the base number to calculate the infection ratio.
Without loss of generality, we suppose that at the initial time, all the
initially reported 41 infectious cases had no information regarding COVID-19,
i.e. $I_{L}(0)=I_{H}(0)=0$. Then, the initial ratio of low- and high-quality
information holders, $S_{L}(0)$ and $S_{H}(0)$, together with four information
parameters $\gamma$ and $\beta$, transmission rate $b_{m}$ and $b_{h}$ and the
panic parameter $\theta$ are unknown and need to be calibrated with the
infection data. To calibrate the unknowns, we suppose the reported infection
number at day $t$ is a random number generated from the Poisson distribution
with mean parameter $N\cdot I(t)$, then the unknowns can be trained as the
minimum of the square-sum loss function:

$\displaystyle loss$
$\displaystyle\left(S_{L}(0),S_{H}(0),\gamma,\beta,b_{m},b_{h},\theta\right)=$
$\displaystyle\sum_{t}$
$\displaystyle\left(\mathcal{I}(t)-I\left(t;S_{L}(0),S_{H}(0),\gamma,\beta,b_{m},b_{h},\theta\right)\cdot
N\right)^{2}$ (3.2)

where $\mathcal{I}(t)$ is the reported infection number at day $t$ in the
data.
$I\left(\cdot;S_{L}(0),S_{H}(0),\gamma,\beta,b_{m},b_{h},\theta\right)$ is the
infection ratio trajectory that solves the ordinary differential equation
system subject to the given set of parameters.

Based on the calibrated model parameters, a set of counterfactual analyses is
carried out to detect whether there exists an alternative information
intervention strategy that can help better contain the outbreak of COVID-19 in
China. In the counterfactual world, we still focus on the two classes of
strategies, information injection, and information blocking. Similar to the
synthetic setting, we consider the joint adjustment of alternative injection
time $\tau$ within the range $(t^{\ast},T^{\ast})$ and alternative blocking
degree $s$ ranging from $0$ to $2$ such that the information parameters
adjusted by $s$ are given by $\gamma^{s}_{1}=\hat{\gamma}_{1}\cdot
s,\,\gamma^{s}_{2}=\hat{\gamma}_{2}\cdot s$ and
$\beta_{1}^{s}=sign(\hat{\beta}_{1})|\hat{\beta}_{1}|\cdot s$ where $\hat{x}$
denotes the estimated value of parameter $x$. For the adjustment of blocking,
to avoid the scale effect, we re-scale the blocking degree $s$ by the absolute
value of the estimated information parameters. Due to the potential negativity
of $\beta_{1}$, we let $sign(a)$ represent the sign of $a$ and use the
multiplier $sign(\hat{\beta}_{1})|\hat{\beta}_{1}|$ to preserve both scale and
the sign of $\hat{\beta}_{1}$. Also analogous to the synthetic setting, we
consider the impact of different decision time parameters,
$(t^{\ast},T^{\ast})=(0,30),\,(0,180),\,(30,210)$, where the unit of time is
day and day $0$ is naturally identified with the initial reporting time, Jan.
10, 2020.

For both synthetic analysis and the counterfactual analysis based on model
calibration, the functional form of $g_{\tau}=g(H_{\tau},H_{\tau}+L_{\tau})$
is fixed as the following

$g_{\tau}=\begin{cases}\scalebox{0.85}{$\left(|\gamma_{1}|,|\gamma_{2}|,|\beta_{1}|,|\beta_{2}|\right)\cdot
H_{\tau}(L_{\tau}+H_{\tau})$},&L_{\tau}+H_{\tau}>0.1\\\ 0,&else\end{cases}$
(3.3)

where $\cdot$ stands for the product between constant and vector. The
discontinuity in (3.3) captures the threshold effect that without a sufficient
amount of information permeable in the system, the government even cannot
notice the existence of viral infection, nor is possible to offer higher-
quality information.

##  4 Results

###  4.1 Government decision complexity and intervention dilemma

Fig. 4 presents the results of our synthetic analysis. In Fig. 4 the
complexity behind the government’s decision on the optimal intervention
strategy emerges, from which we witness quite a few interesting but counter-
intuitive phenomena as below.

Figure 4: Joint effects of government intervention.

(a-f) Variation trend of final infection ratio $I(T^{\ast})$ in response to
the joint adjustment of blocking $s$ and injection time $\tau$. The variation
trend is characterized through the gradient field of $I(T^{\ast})$ with
respect to $s$ and $\tau$ under fixed initial conditions. In a-f the gradient
field $\vec{\nabla}=-\left(\frac{\partial
I(T^{\ast})}{\partial\tau},\frac{\partial I(T^{\ast})}{\partial s}\right)$ is
represented as the background arrows. Plots a,d, b,e and c,f associate with
the simulation result for the benchmark, delayed-initial, shorter-horizon
setting, respectively. Throughout a-c, we give the heat maps of $I(T^{\ast})$,
marking the value of $I(T^{\ast})$ at each pair $(\tau,s)$ through the
darkness of background color. Throughout d-f, we mark, by dark dots(curves),
those $(\tau,s)$ satisfying $\frac{\partial I(T^{\ast})}{\partial s}=0$ and
$\frac{\partial I(T^{\ast})}{\partial\tau}\geq 0$. They are the steady states
under the forward-looking gradient field
$\overrightarrow{\nabla}_{f}=-\left(\frac{\partial
I(T^{\ast})}{\partial\tau}\cdot\mathbf{1}\left(\frac{\partial
I(T^{\ast})}{\partial\tau}\geq 0\right),\frac{\partial I(T^{\ast})}{\partial
s}\right)$ by which the injection time can only be adjusted forwardly, the
backward adjustment is not allowed. Restricting the adjustment direction of
injection time reflects the fact that the decision on injection time is
irreversible in time. (g-i) Aggregated infection ratios over different
injection time $\tau$ against fixed blocking degree $s$. The aggregated
$I(T^{\ast})$ is considered as the pay-off for the government adopting action
$s$ at the initial decision time $t^{\ast}$ in the dynamical game discussed in
the main text. The variation of this pay-off against $s$ is plotted in g-i
with respect to various aggregation functions, including the “max”, “min”,
“mean”, “median” and “75-quantile” functions. g(/h/i) shares the same set of
initial conditions and initial decision time $t^{\ast}$ with a,d(/b,e/c,f).
Finally, in alignment with each aggregated pay-off function in g/h/i, the
induced Nash-equilibrium (NE) combinations of blocking degree and injection
time is plotted a,d/b,e/c,f.

Waiting period exists between the time when government becomes capable of
information injection and the optimal injection time. The set of all steady
states under the forward-looking gradient field of $I(T^{\ast})$ (dark dots in
Fig. 4d-4f) are naturally clustered around two separated curves, where the
horizontally straight line on the left is locally attractive via the gradient
field (gray arrows in Fig. 4a-4f). By the discontinuous design of $g_{\tau}$
in Sec. Method, the injection time $\tau_{1}^{\ast}$ associated with the right
end of the horizontal dark line is the earliest time when information
injection can effectively impact the disease dynamics. Apparently from Fig.
4d-4f, a gap exists between $\tau_{1}^{\ast}$ and the optimal time
$\tau_{2}^{\ast}$ associated with the red “X” in Fig. 4a-4c (for easiness of
comparison, the red “X” is also marked at the same location in Fig. 4d-4f),
where the red “X” represents the globally optimal combination of blocking
degree and information injection time in terms of minimizing the final
infection ratio. Also remarkably in Fig. 4d, the gradient field, projected to
the dimension of injection time, points away from $\tau_{1}^{\ast}$ to the
right at $\tau_{1}^{\ast}$ and points toward $\tau_{2}^{\ast}$ from the left
at $\tau_{2}^{\ast}$. This observation implies government should wait and hold
the information for a gap period $\tau^{\ast}_{2}-\tau^{\ast}_{1}$, rather
than release them to the public immediately since releasing becomes feasible.
The existence of a waiting period contradicts to the common belief that the
best strategy for the government is to publish high-quality information as
soon as possible. Although counter-intuitive, the existence of a waiting
period is deeply rooted in the trade-off expressed in Fig. 3c and 3d. No
waiting period suggests an earlier information injection which leads to an
immediate drop-down of the current infection ratio $I(\tau+\Delta\tau)$ during
the short time after injection (the gap between the green and red line in the
interval from $\tau$ to $\tau^{\prime}$ in Fig. 3c). A lower infection ratio
at the later injection time $\tau^{\prime}$ would also shrink the risk of a
further outbreak, measured by the gap between the green line and red line in
the horizon time $T^{\ast}$ in Fig. 3c. Therefore, the delayed injection
induces a negative $A_{1}$ and a higher final infection ratio. From this
perspective, we can confirm the positive immediate effect of early injection
on containing virus outbreaks. However, on the other hand, compared to
injection after waiting, earlier information injection cannot generate a
substantial structural impact on the information propagation process, hence
has limited restriction on the virus outbreak speed in the later time, leading
to a positive $A_{2}$ in Fig. 3c and a greater final infection ratio. Such a
negative structural effect offsets the positive immediate effect of early
injection and leads to an overall negative impact on the final infection. The
existing literature often neglects the negative structural impact induced by
earlier information injection, leading to an over-optimistic attitude toward
the functionality of quick information release.

Government faces the dilemma between the optimal strategy that minimizes the
cost of its own mistakes and the socially optimal strategy that minimizes the
final infection ratio. Fig. 4a also indicates a temporal inconsistency in the
joint decision of blocking degree and injection time. In fact, the global
optimality (the red “X”) in Fig. 4a can be reached if and only if the
following latent assumption holds: i) the decision of blocking degree is made
simultaneously with that of information injection; ii) the government have
complete information on the form of injection function $g_{\tau}$. However,
none of the two conditions can really hold in practice. The decision of
blocking degree is always made ahead of the injection time. Meanwhile, at the
decision time of blocking degree ($t^{\ast}$), the information in Fig. 4a
regarding how the degree would affect the effectiveness of information
injection at different future injection times is completely absent for the
government. The incomplete information induces a conflict. On one hand, the
optimal injection time can reach its optimal containment effect if and only if
the blocking can hold on a fairly low level. On the other hand, given such a
low-level blocking, the infection ratio could be much worse than that on a
higher blocking level in case the real injection time deviates from its
theoretical optimum. This uncertainty leads to a choice dilemma that can be
perfectly reformulated as a two-player dynamical game with incomplete
information, in which the first player is the government at the decision time
of blocking, $t^{\ast}$, while the second player is the future government
facing the decision of injection time. The pay-off matrix is given by Fig. 4a,
the vertical axis (blocking degree) corresponds to all feasible actions of
player 1, and the horizontal axis (injection time) corresponds to all feasible
actions of player 2. Player 2 has complete information on the blocking degree
taken by player 1 at its decision time, therefore its pay-off is exactly equal
to the infection ratio colored in Fig. 4a. While due to incomplete information
regarding the actual injection time of player 2 at $t^{\ast}$, for every
blocking degree $s$, the pay-off for player 1 is only an aggregation of
infection ratios for all different $\tau$s and the fixed $s$. If both the two
government players behave rationally, the final decision on blocking degree
$s$ and injection time $\tau$ should achieve a Nash equilibrium (NE). In Fig.
4g-4i, we plot the aggregated pay-off for government at $t^{\ast}$ against the
blocking degree $s$ under a variety of aggregation principles. The resulting
Nash-equilibrium joint decision on $\tau$ and $s$ are plotted in Fig. 4a-4d.
This game-theoretical interpretation of the government decision dilemma
provides a different way to think of the failure of government in the early-
stage fighting with infectious diseases. Fig. 4d shows if government at
$t^{\ast}$ is prudent and takes the max aggregation, i.e. following the well-
known “mini-max” decision principle[58, 59], the optimal blocking degree
exactly coincides with the degree under the local steady states of the
gradient field for small injection time (the horizontal dark line in Fig. 4d),
which is distant from the global optimal blocking degree associated with the
red “X” in Fig. 4a and 4d. In real-world catastrophes, such as the pandemic of
SARS, H1N1, and COVID-19, the “mini-max” principle is more likely to be
adopted for government decisions. Given a blocking degree $s$, the maximally
aggregated infection ratio provides an upper bound for the government’s
mistakes in selecting injection time. The “mini-max” principle attempts to
minimize this upper bound, which is equivalent to minimizing the
responsibility of government. In contrast, the global optimal intervention
strategy is equivalent to the Nash-equilibrium under the “mini-min” principle,
which might be optimal for the entire society as it minimizes the final
infection ratio. But once the government makes mistakes in the second-stage
decision for injection time, the potential loss will be giant. For instance,
in Fig. 4g, at the blocking degree $s^{\ast}$ associated with the minimum of
the “min” curve, the difference in the infection ratio between government
correctly and mistakenly selecting the injection time can exceed 70% in the
worst case (measured by the difference between the “max” and “min” curves at
$s^{\ast}$ in Fig. 4g). Such a huge difference in infection ratio means a
giant cost for the government’s mistake, which is not affordable for the
government in the initial decision timing $t^{\ast}$. Consequently, the
optimal intervention strategy in the views of government has the natural
tendency to deviate from the global optimal strategy, which partially explains
the universal loss of control of the COVID-19 pandemic by the central
government in most major countries around the world, such as China, US, Italy,
Great Britain and the like.

Delayed initial decision time $t^{\ast}$ relieves government dilemma but leads
to a much higher infection ratio. Compared to Fig. 4a, 4d and 4g, government
decision under a delayed initial time $t^{\ast}$ is plotted in Fig. 4b, 4e and
4h. Due to the delay, there are more infectious cases and more low-quality
information holders in society at the beginning. In the real world, it always
takes time to figure out the occurrence of the crisis, hence government can
never be in a decision position until the outbreak has lasted for a while.
Fig. 4b, 4e and 4h capture the challenges in the delayed decision setting.
Comparing Fig. 4b and 4e with Fig. 4a and 4d, several structural changes of
the gradient fields can be observed. First, the waiting period is even
prolonged under the delayed $t^{\ast}$, reflecting that given more infections
and more low-quality information holders, the structural impact of information
injection captured by $A_{2}$ in Fig. 3c becomes even more prevalent compared
to the effect of $A_{1}$ on the immediate reduction of infection number. On
the other hand, the temporal inconsistency between the decision of blocking
degree and injection time gets weakened in the delayed case which reflects as
the difference shrinks between the pay-off for government at $t^{\ast}$ under
different aggregation methods (in Fig. 4h), as well as the fact that the red
“X” in Fig. 4b and 4e is getting closer to the Nash-equilibrium strategies
under the other pay-off aggregation methods, especially under “mini-max”
principle. The decreasing temporal inconsistency reduces the uncertainty and
the cost of making mistakes faced by government and makes the decision on
blocking degree much easier. But the subsequent side effect is significant as
well, the minimal infection ratio in Fig. 4b is much higher than that in Fig.
4a, reflecting by the different scale of the color bar associated with Fig. 4a
and 4b, respectively. In Fig. 4a, the global minimal infection ratio can be
maintained below 20%, while in Fig. 4b, the global minimal infection ratio
exceeds 90% which is 70% higher than that in the former case. As the worse
final infection ratio results from the corresponding initial condition induced
by the delayed $t^{\ast}$, the comparison between the first and second column
of Fig. 4 reveals a social dilemma faced by government. On one hand, a delayed
$t^{\ast}$ can help reduce the inconsistency between the optimal strategy for
government measured by minimizing the cost of making mistakes and the optimal
strategy for the entire society measured by minimizing the final infection
ratio, hence government may prefer to delay so as to reduce the difficulty of
making the correct decision. However, on the other hand, a delayed $t^{\ast}$
would make the “consistently optimal” strategy no longer “good” via
significantly rising up the final infection ratio and causing more social
welfare loss.

Shortening decision horizon $T^{\ast}$ relieves government dilemma at the cost
of a much higher infection ratio. Similarly to delaying $t^{\ast}$, Fig. 4c,
4f and 4i show that the variation on the decision horizon $T^{\ast}$ can also
significantly impact the trade-off between the difficulty of government
decision and social optimal infection ratio. Compared to Fig. 4a, the decision
horizon $T^{\ast}$ in Fig. 4c is compressed from 4 weeks to only 1 week. The
severity of the dilemma is also reduced in terms of the difference in optimal
blocking degree under different aggregation principles. In Fig. 4f, the
difference in the optimal blocking degree under the “mini-min” aggregation
principle and under the “mini-max” aggregation principle is 0.35, which is
significantly lower than the difference (0.54) in Fig. 4d. Meanwhile, the
optimal blocking degrees in Fig. 4f under different aggregations are
universally lower than those in Fig. 4d, reflecting that the horizontal dark
line in Fig. 4f is uniformly below that in Fig. 4d. In the other words, viewed
in a shorter horizon, a lower final infection ratio can be achieved via a much
freer information environment, which agrees with the widely held belief on
that free information propagation suppresses disease spreading[51]. But if we
compare the relative position of two red “X”s and the associated infection
ratio in Fig. 4a, a novel temporal inconsistency appears. The global optimal
strategy viewed in a shorter decision horizon $T^{\ast}$ leads to an
extraordinarily high infection ratio ($>80\%$) when the horizon is extended.
In fact, a freer information environment helps sharply rise up the proportion
of high-quality information holders in the short run, which reduces the
infection ratio within this group. But in a longer horizon, if the trend of
information decay is not reversed, “holding high-quality information” is only
a temporal status. As time passes away, the initially high-quality information
holders will ultimately decay into low-quality information holders, which
significantly enlarges their infection risk. This observation implies that the
decision horizon is critical to the effectiveness of optimal intervention
strategies. If the government is myopic and puts overweight on containing the
disease outbreak in the short run, it is likely to be misled by the myopic
preference and make the wrong decision that causes huge social welfare loss
viewed in a relatively longer time horizon.

###  4.2 Case study of COVID-19 pandemic in mainland China

The model calibration result and the counterfactual analysis based on
calibration are reported in Fig. 5, which provides us with some new insights
into the dilem-ma that are not covered in the synthetic settings but the
government has to face in the real world.

Figure 5: Response dilemma during real-world pandemics.

(a-c) Heat maps of the final infection ratio $I(T^{\ast})$. The heat map
sketches the joint trade-offs faced by the Chinese government when dealing
with information control during the COVID-19 pandemic. (a) Benchmark setting
where the blocking decision timing $t^{\ast}$ is set as the initial time of
the data, i.e. Jan. 10, 2020, the decision horizon $T^{\ast}$ is 30 (days),
i.e. one month. (b) Decision under a longer horizon with
$(t^{\ast},T^{\ast})=(0,180)$. (c) Decision under delayed initial decision
time and longer horizon with $(t^{\ast},T^{\ast})=(15,195)$. Plot b,c reveal
the impact of the variation of $t^{\ast}$ and $T^{\ast}$ on government choice
dilemma. Plot d sketches the evolutionary trajectories of the fractions $I$,
$S_{N}$, $S_{L}$, and $S_{H}$ during the first 180 days under calibrated model
parameters and under the globally optimal intervention strategies in a and b
respectively. All the counterfactual results in a-d are calculated on the
basis of the calibrated model parameters. The estimated information parameters
$\hat{\gamma_{1}}=1.76$, $\hat{\gamma}_{2}=0.63$, $\hat{\beta}_{1}=-2.16$ and
$\hat{\beta}_{2}=1.23$, the transmission rate $\hat{b}_{m}=0.02$,
$\hat{b}_{h}=2.94$, the panic parameter $\hat{\theta}=0.97$ and the initial
$\hat{S}_{L}(0)=5\times 10^{-6}$, $\hat{S}_{H}(0)=1.3\times 10^{-3}$. Based on
calibration results and the counterfactual set-ups, the government intervenes
in the infodemic-epidemic dynamics via information blocking and injection, and
the execution of intervention strategies is identical to that in the synthetic
case discussed in Fig. 4.

Completely blocking information spreading is the universally optimal strategy
for the Wuhan government under myopic decision horizon. Surprisingly from Fig.
5a, the Chinese government didn’t face to the dilemma if there is no delay and
the decision horizon $T^{\ast}$ is set to 30 days. The optimal blocking should
be implemented to its strongest degree, i.e. cut off all contacts that spread
information regarding COVID-19 no matter whether or not the information
content is high-quality. Meanwhile, the optimal injection time is zero, i.e.
no information injection at all. Remarkably, such a combination of blocking
degree and injection time is universally optimal for all different ways of
aggregating the intervention pay-off. Especially, the “mini-min” and “min-max”
principles agree with each other. This surprising result can be explained
through the difference between the transmission rate under different
information statuses ( $\hat{b}_{h}/\hat{b}_{m}\approx 300$), and the high
conversion rate to low-quality information from no information state
($\hat{\gamma}_{1}=1.76$), as well as from information decay
($\hat{\beta}_{2}=1.22$). In fact, the high ratio $\hat{b}_{h}/\hat{b}_{m}$
implies panic-induced infection is the dominant driving force for the
accumulation of infectious cases. Since panic happens to and only to low-
quality information holders, then a high-conversion rate from the other
information status to low-quality status makes the whole population suffer
from a high risk of panic-induced infection. Comparatively, the infection risk
due to no awareness is relatively low. Consequently, the optimal strategy for
the government is to keep everyone unaware of COVID-19, in exchange for a low
transmission rate $\hat{b}_{m}$.

Government dilemma appears as the polarized optimal intervention strategies
under different decision horizons. The “keeping unawareness” strategy is only
temporally optimal. If the decision horizon $T^{\ast}$ is extended 180 days
(Fig. 5b) and/or the initial decision timing $t^{\ast}$ is delayed (Fig. 5c),
a restriction-free information environment should be maintained under the
optimal, and the optimality is concurred by all different aggregation
principles in Fig. 5b and 5c, especially by both the “mini-max” and “mini-min”
principles. Also remarkably, if the optimal blocking under a 30-day horizon is
actually executed, the final infection ratio viewed under the 180-day horizon
is terribly high ($>$60%) which is more than 60% higher than the infection
ratio reached by the optimal strategy under the 180-day horizon, no matter
whether the initial decision is delayed. These observations suggest that in
the real world, the choice dilemma that the Chinese government faces no longer
comes from the inconsistent temporal decision induced by different aggregation
principles and different risk preference behind government decision, but arise
from the temporally inconsistent effect of the same set of strategy viewed
from different time horizons. The temporally inconsistent containment effect
is deeply rooted in the complexity of the coupled infodemic-epidemic dynamics.
Comparing the half-a-year horizon with the one-month horizon, the difference
in the optimal blocking degree comes mainly from that in the one-month case,
due to the low initial amount of infectious cases, the disease dynamics are
mainly driven by the high transmission rate $\hat{b}_{h}$ for low-quality
information holders. But as time pass away, as shown in Fig. 5d, the
infectious cases keep accumulating meanwhile the number of healthy low-
information holders are decreasing (reflected as the decrease of the blue
lines in a later time in Fig. 5d), in the second stage of disease dynamics,
the infection will be mainly driven by the infection of healthy but unaware
individuals (reflecting as the sharp decrease of the dash yellow line
accompanied with the sharp increase of the dash red line in Fig. 5d). The
switch of driving force increases the effectiveness of free information
propagation and information injection while suppressing the effect of
blocking, which makes government once again face a dilemma between myopic
decisions versus long-horizon decisions. As documented in the literature of
behavioral economics and psychology, human beings are naturally inclined to
over-weigh the current loss and pretend to be blind to the loss that happened
in the longer future[60, 61, 62]. On the other hand, making decisions for a
longer horizon is a much more complicated task that needs more information to
overcome the increased uncertainty. At the initial decision time, information
is too limited to support such a longer-horizon decision. Therefore, driven by
both the myopic preference and initial information limitation, government when
facing the contradicted optimal strategy in Fig. 5a and Fig. 5c is more likely
to select the short-run optimal strategy in Fig. 5a, rather than the long-run
optimal in Fig. 5c, which makes the society suffer from a much higher
infection ratio. In the real world, the initial reaction of the government in
the epicenter, Wuhan, followed exactly the myopic optimal strategy, it did not
disclose any information to the public, meanwhile hid all key information
regarding the transmissibility of COVID-19 and prohibited the public
discussion on the internet. As known, this short-run optimal reaction leads to
the outbreak of COVID-19 in China. The evidence of Wuhan demonstrates that the
myopic decision mode and the induced inaccurate evaluation on the long-term
infection ratio form a driving force to mislead the government’s choice of the
correct intervention strategy and cause severe social welfare loss. The same
logic applies as well to understand the failure of many other countries in
selecting the correct intervention strategy during the early-stage containment
of COVID-19, such as the US.

###  4.3 Fighting with government dilemma from the view of network

In the aforementioned analysis, the dilemma faced by the government relies on
information quality decay, i.e. $\beta_{1}>0$ and/or $\beta_{2}>0$. To verify
this premise, in this section, we present an example network structure through
which fast information decay is unavoidable. Meanwhile, we discuss the
potential re-structure of the network that helps relieve the government from
the dilemma. We let the information dynamics occur through a hierarchical
network, the network topology is illustrated in Fig. 6a. The key assumption
regarding the information dynamics is that every node, when receiving
information from the others, updates his/her own quality level via the
weighted sum of the quality of all received information with the weight of
each delivery node determined by their out-degree (a rigorous mathematical
formulation of propagation mechanism is provided in the Appendix). The out-
degree of every node provides a measure of its influence on the information
propagation process, which is a widely used assumption in the literature[63,
64] of propagation dynamics, such as the virtual dynamics behind the
calculation of the renowned Page Rank [65]. The hierarchical network topology
is quite realistic in describing information propagation during the pandemic
because, unlike the propagation of general news, the government always plays a
central role in the propagation of disease-related information. The
extraordinary influence of government on the propagation process naturally
leads to an inequality between itself and the public, resulting in the
hierarchical structure of the network. We show in Fig. 6c and 6d that in a
hierarchical network, the overall quality level of the system is hyper-
sensitive to both the number of layers in the hierarchy and the distribution
of the source information holders. If the source information holders are on
the top level, i.e. the government itself holds the source information, the
quality level of the whole system can be maintained at a relatively high level
which corresponds to a low infection ratio in the end. But the increasing
number of layers lying between the government and the bottom-layer public
would offset the advantage of the government holding source information as
shown in Fig. 6c, reflecting the importance of a flattened management
structure during a disease pandemic. On the contrary, if the source
information is held by some nodes at the bottom layers (i.e. the individuals),
the quality level would decay very fast, as shown in Fig. 6d. This result
suggests that in a hierarchical society where information propagation channels
are by and large shaped by the hierarchical social structure, the coupled
dynamics of information quality decay and infectious disease outbreak would be
accelerated. This finding reveals a deep connection between the structure of a
society and its vulnerability to infectious diseases.

Figure 6: Information dynamics on a hierarchical network and the effect of
deloop.

(a) Example of the hierarchical network structure (with three layers). The
number of nodes in each layer increases along with a rank decrease of the
layer within the hierarchy. Nodes in the top 2 layers are extraordinarily
influential to those in the next-level layers (reflecting as the thick arrows
pointed downward from the high to the lower layers in a), but nodes in the
bottom layer only have weak influence on each other and a much weaker
influence to the nodes in the top 2 layers (reflecting as the thin arrows
within the third layer and pointed upward from the third layer in a). Such a
hierarchical network structure captures the information propagation mechanism
among the central government (top layer), local government (intermediate
layer), and the public (bottom layer). At the bottom layer, the red dot
represents the source information holder at the initial time. Due to the lack
of a direct link from the source to the central government, it takes much more
time and more quality loss before the information can be delivered to the
central government and passed to the public. The time and quality loss
facilitates the rumor spreading via the low-layer links and loops within the
bottom layer. (b) Illustration of how direct warning system and deloop can
reshape the network structure and affect the decay process. The direct warning
system works as adding an arrow pointing toward the top from the information
source, while the deloop works as cutting off all links that facilitate loop
formation within the bottom layers. Deloop together with a direct warning
system makes the resulting network a four-layers hierarchical network, where
the top layer is replaced with the source information holder. The renewed
network structure stops the decay as shown in e. (c-d) Trend of quality decay
and infection growth within a hierarchical information network with a
different number of layers and different initial assignments of source
information. The number of nodes in c and d is fixed to 1000, the number of
layers varies from 3 to 7, and the link structure of the network is given by a
random weighted adjacency matrix with the out-degree weight distribution
determined by the real out-degree distribution of the follower-ship network on
Weibo, the biggest Chinese online social media. In c the source information is
always assigned to the unique top-layer node, while in d assigned randomly to
one node in the bottom layer. (e) Effect of deloop on mitigating the trend of
quality decay and infection. To measure the degree of deloop, we set a
parameter $d\in[0,1]$. In e, deloop works as changing the entry weight of the
adjacency matrix $W$ by multiplying $d$ with $W_{ij}$ for those $ij$s that
form loops in the network. $d=1$ means no deloop, $d=0$ implies complete
deloop described in b. For robustness, the simulation result in c-e is
averaged over 100 realizations for each fixed setting.

To restrict the negative effect of the hierarchical network on the infodemic-
epidemic dynamics, we suggest as in Fig. 6b that i) strengthening the links
from source information holders to the top-layer government; ii) cutting off
those links that do not affect the connectivity of the entire network
meanwhile can reduce the number of loops in the network (deloop). Combining i)
and ii) can significantly decelerate the coupled dynamics of information decay
and disease outbreak (see Fig. 6e). Method i) is equivalent to setting a
direct warning system for risky infectious diseases, which helps flatten the
hierarchical structure and reduce the quality decay during propagation from
the bottom to the top. Although such a warning system has been established in
most major countries around the world, as we have seen during the pandemic of
COVID-19, it fails in most countries to send an effective warning signal to
the central government. The failure can be partially explained by the absence
of the second method, deloop. In fact, the existence of loops speeds up the
propagation of low-quality information. Without a deloop, a direct warning
system can only guarantee that the government can receive information with
very high quality in the very beginning, it has nothing to do with the
information quality decay via rumors circulating around loops. Consequently,
the information decay might be delayed to some extent as shown in Fig. 6e, but
it is ultimately unavoidable together with a very high infection ratio at the
end. This suggests the importance and complexity of information management
during a public health crisis, such as the pandemic of COVID-19.

The effectiveness of deloop and the connection between information circulating
over loops and its quality decay also inspire us to re-think the relationship
between critical public health events, such as the COVID-19 pandemic, and the
booming of social media in the current era. In fact, the development of online
social media, such as Twitter and Facebook, extremely extends the freedom of
individuals to join the information propagation process, making it possible to
freely announce personal opinions to the whole world for every single person.
Social media adds tremendous loops to the underlying information network,
these added loops might be good for society without concerning the pandemic of
infectious disease. But in the special time of the COVID-19 pandemic, they may
enlarge the risk of information decay and rumor prevailing, which ultimately
facilitates the spreading of COVID-19. Therefore, the discussion in this
section points out a severe conflict between the increasingly free information
environment and the enlarged risk of loop-forming and the COVID-19 pandemic,
which is rarely mentioned in the literature and should be put more attention
in future studies.

##  5 Conclusion & Discussion

### Summary of major findings

This study sophisticates the ”remedy and overkill” response dilemma by
modeling the effectiveness of an intervention that is restricted by
information dynamics. To quickly release high-quality information to the
public and maintain the overall quality level of information among the public,
the government needs a fast-propagation information market to accelerate
information collection and pre-processing, while that increases the risk of
faster quality decay and rumor prevalence. A more restricted information
market can definitely reduce the risk of rumor-induced panic, but it also
reduces the space for high-quality information which will make it less likely
for the government to gather and release high-quality information. The trade-
off on the information propagation speed traps the government into a choice
dilemma, a restriction-free information market may lead to a higher risk of
panic-induced-infection in a short time, while a restricted information market
compresses the chance of the government releasing high-quality information and
increase the risk of long-run infection.

Facing this choice dilemma, the government has the motive to delay the
intervention decision and pretend to be “blind” of the diseases pandemic, as
the severity of the dilemma is declined as the infection ratio has already
reached a fairly high level meanwhile the overall information quality level in
the system becomes stable. The less severity of choice inconsistency means
that delayed intervention will incur fewer criticisms which might be optimal
for the government but deviate away from the social optimal as it induces a
significant rise-up in the long-term infection ratio. Therefore, one extra
social dilemma arises for the government between the politically optimal and
socially optimal intervention timing.

Similar to delayed intervention, a myopic decision pattern can also reduce the
severity of the choice dilemma faced by the government, because the disease
dynamics might be driven by completely different forces in a relatively
shorter horizon in contrast to in a longer horizon. Then, the reduced decision
difficulty leads to “mistake” intervention strategy and a much greater final
infection ratio and social welfare loss. We remark that compared to delayed
intervention, a myopic decision might be riskier for the treatment of a public
health crisis, such as the pandemic of COVID-19. It has been widely studied in
psychological literature [60, 61, 62] that people are more likely to put
overweight to the current and/or short-period loss than the long-period yield
when facing cross-period decision under uncertainty. Therefore, the government
is naturally inclined to be a myopic decision maker when facing the pandemic
of viral infection and prefers the intervention strategy that is optimal for a
short horizon, despite its detrimental ex-post loss viewed in the long
horizon. This fact also explains the real-world failure of early-stage
containment of viral infections, such as SRAR, H1N1, and the most recent
COVID-19, for most major countries around the world.

In summary, we proposed a novel model to explain the deep systematic failure
rooted in the information/b- ehavioral heterogeneity and its induced dilemma
behind government intervention decisions on COVID-19. The study highlights the
complexity, information heterogeneity, and disease dynamics, and thus calls
for more collaborations from social science and global health experts to help
countries in designing more effective crisis management systems.

### Limitations and future studies

This paper has some limitations. First, we only focus on the information
intervention. In the real world, there exist many other non-pharmaceutical
intervention (NPI) strategies, such as social distancing and conditional
quarantine [38], which are not considered in this study. To this point, we
claim that most parts of the decision dilemma discussed in this study still
apply to the other NPI strategies because the effectiveness of NPI strategies
is also time-dependent and/or information dependent. In fact, both the NPIs
and information injection attempt to convert high-susceptible individuals
$S_{N}/S_{L}$ to low-susceptible $S_{H}$ so as to lower the population-level
transmission rate into infection[66]. Information injection requires the
government to own high-quality information before announcement which takes
time and positively depends on the ratio of informed people $L+H$ and high-
quality information $H$. NPI strategies are effective only if the public
chooses to cooperate [67, 68], the cooperativeness of the public also takes
time and increases with the information amount ($L+H$) and quality level $H$.
Hence, NPIs can be identified with information injection as a special type of
information intervention in our framework, the choice dilemma government faces
between blocking and injection holds for NPIs as well. Second, the findings of
this study apply to the early stage of an epidemic crisis. As time passes
away, the public will ultimately get sufficient information about the disease
when the difference between the different information groups and their
behavioral responses would disappear. At that stage, the infodemic may no
longer be a great concern. However, on the other hand, we want to highlight
that even though the trade-off of information intervention matters only for a
relatively short period, its induced social-economic impact could be long-
lasting, therefore its importance should not be neglected.

The current study can also be extended in the following directions. First, we
focus only on the single direction that low-quality information could induce
panic and higher infection risk. But reversely, the panic and fear of the
disease can also impact the information propagation process and change the
relative speed of propagation for different types of information. This kind of
two-way feedback between information and disease dynamics induced by panic
should be added into consideration. Second, combining the coupled dynamics of
information and disease with an adaptive network has attracted wide attention
[4] where nodes can select their local link structure of the information
network or the infection network in adaptive to the change of information or
infection status, which should be a promising direction to extend the current
discussion on the government response dilemma.

## Appendix A: Notation List

Table 1: Notations in SI-NLH model

Types | Notations | Meaning
---|---|---
Endogeneous Variables | $S_{N}$ |  the group of susceptible individuals without information
$S_{L}$ |  the group of susceptible individuals with low-quality information
$S_{H}$ | the group of susceptible individuals with high-quality information
$I_{N}$ | the group of infected individuals without information
$I_{L}$ | the group of infected individuals with low-quality information
$I_{H}$ | the group of infected individuals with high-quality information
$I$ | the group of infected individuals
$S$ | the group of susceptible individuals
$N$ | the group of individuals without information
$L$ | the group of individuals with low-quality information
$H$ | the group of individuals with high-quality information
Parameters | $\gamma$ |  $:=(\gamma_{1},\gamma_{2})$ the rate of transition from group $N$ to $L$ and $H$, respectively
$\beta$ |  $:=(\beta_{1},\beta_{2})$ the rate of transition from between group $L$ and $H$
$b$ |  $:=(b_{m},b_{h})$ the rate of transmission to disease for susceptible individuals under unawareness ($b_{m}$) and panic ($b_{h}$)
$\theta$ | proportion of individuals being panic within the group of susceptible

## Appendix B: Information propagation and decay in network

The information propagation (decay) dynamics considered in this study is
essentially the virtual dynamics behind the calculation of Page Rank [65]. To
describe the dynamics on a given network, we firstly define $\zeta\in[0,1]$ as
the initial quality level of information for source information holders,
$\delta\in(0,1)$ as the one-period decay rate of information quality, and
$\eta\in[0,1]$ as the quality threshold for individuals such that the $i$th
individual will be classified into $H$ at time $t$ iff at $t$ its information
quality level $\zeta_{t,i}>\eta$, classified into $L$ iff
$0<\zeta_{t,i}\leq\eta$, and classified into $N$ if $\zeta_{t,i}=0$. Given a
network represented by its binary adjacency matrix $W$, the evolution of
information group size $N$, $L$, and $H$ is governed by the following
propagation dynamics on network $W$, where at time $t=0$, randomly select a
set of individuals $\mathcal{I}_{0}^{\ast}\subset\\{1,\dots,n\\}$ as the
source information holders and assign them with the information with quality
$\zeta$, which yields an $n$ dimension vector $\zeta\cdot I_{0}$ where $I_{0}$
is a binary-valued vector with all entry $j\in\mathcal{I}_{0}^{\ast}$ taking
the value $1$ and the other entries taking the value $0$. Then, for each time
$t>0$, the information quality $\zeta_{t,i}$ for the $i$th individual is
determined inductively via the following updating rule:

$\zeta_{t,i}=\frac{\delta}{n}\cdot\left\langle
rand\left(\frac{W_{i,\cdot}}{\langle\tilde{W}_{i,\cdot},\zeta_{t-1}^{+}\rangle}\right),\zeta_{t-1}\right\rangle$
(.1)

where $\langle\cdot,\cdot\rangle$ is the inner product of two vectors,
$\zeta_{t}$ is the vector formed by $(\zeta_{t,i},\dots,\zeta_{t,n})$,
$\zeta_{t}^{+}$ is the binary vector derived from $\zeta_{t}$ in which all the
entry $j$ with $\zeta_{t,j}>0$ takes value $1$, the other entries are 0,
$W_{i,\cdot}$ is the $i$th row vector of matrix $W$. $rand(V)$ is a random
binary vector drawn from the probability vector $V=(V_{1},\dots,V_{n})$
($V_{i}\geq 0$ and $\sum_{i}V_{i}=1$) where the $i$th entry take value $1$ in
the probability $V_{i}$. $\tilde{W}$ is the matrix derived from $W$ via
$\tilde{W}=diag(W\cdot\mathbf{1}_{n})^{-1}W$ where $\mathbf{1}_{n}$ is the
$n$-dimensional vector with all entries taking value $1$, $diag(\cdot)$ is the
operation that converts an $n$-dimensional vector to a diagonal matrix with
the diagonal entries identified with the input vector. The constructions of
$\tilde{W}$ and $\zeta_{t}^{+}$ imply that under the information propagation
dynamics (.1) the information quality of individual $i$ at time $t$ is the
natural decay rate $\delta$ times a weighted average of information quality of
its network neighbors according to the relative network influence of each
neighbor measured by their out-degree. The influence-based information
propagation mechanism is widely used in literature [63, 64]. From $\zeta^{t}$,
the fraction of $N$, $L$ and $H$ can be calculated easily via comparing each
$\zeta_{t,i}$ and the threshold $\eta$ as discussed before.

Given the information dynamics on a subtly structured network, the disease
dynamics are still simulated based on the assumption of a well-mixed
population. In the other words, the conversion from $S_{N}$, $S_{L}$, and
$S_{H}$ to $I$ still follows equation (3.1). In the agent-based simulation,
this conversion is equivalent to updating a binary vector $I_{t}$ that stores
the infection status of every node for every time $t$, and can be realized via
the following steps given the initial $I_{0}$:

  * Step 1:

Given node $i$, if $i$ has been infected, directly continue to node $i+1$;
otherwise, randomly match $i$ with another node $j\in\\{1,\dots,n\\}$ and
$j\not=i$.

  * Step 2:

If $j$ has not been infected, directly continue to node $i+1$; otherwise,
determine the information status of $i$ according to $\zeta_{t,i}$ and $\eta$.

  * Step 3:

if $i$ is in $S_{H}$, directly continue to node $i+1$; if $i$ is in $S_{N}$,
update $i$’s infection status to $I_{t,i}=1$ by the probability $b_{m}$; if
$i$ is in $S_{L}$, randomly assign $b_{h}$ to $i$ in the probability of
$\frac{\sum{l=1}^{n}I_{t-1,l}+\theta\sum_{l=1}^{n}\mathbf{1}(\zeta_{t,l}<\eta)}{n}$;
if $b_{h}$ is assigned to $i$, update $i$’s infection status to $I_{t,i}=1$ by
$b_{h}$, otherwise, update $i$’s infection status to $I_{t,i}=1$ by $b_{m}$.

The three-step algorithm will be used to generate the infection ratio in Fig.
6c - 6e.

## acknowledgements

This work was partially supported by the National Natural Science Foundation
of China (Grant Nos. 72101268 and 61673151), the Natural Science Foundation of
Zhejiang Province (Grant Nos. LR18A050001), the Major Project of The National
Social Science Fund of China (Grant No. 19ZDA324), the China Postdoctoral
Science Foundation (Grant No. 2021M703569), and the Fundamental Research Funds
for the Central Universities (ZJU and Grant No. 224220S30024 at SEU).

## Conflict of interest

We declare no competing interests.

## References

  * [1] Xiu-Xiu Zhan, Kaiyue Zhang, Lun Ge, Junming Huang, Zinan Zhang, Lu Wei, Gui-Quan Sun, Chuang Liu, and Zi-Ke Zhang.  Exploring the effect of social media and spatial characteristics during the covid-19 pandemic in china.  IEEE Transactions on Network Science and Engineering, 10(1):553–564, 2023.
  * [2] https://en.wikipedia.org/wiki/Timeline_of_the_COVID-19_pandemic, Accessed on Jul. 13, 2020.
  * [3] Center for Systems Science and Engineering (CSSE) at John Hopkins University.  Covid-19 dashboard, 2020.  https://coronavirus.jhu.edu/map.html, Accessed on Jul. 13, 2020.
  * [4] Zhen Wang, Michael A Andrews, Zhi-Xi Wu, Lin Wang, and Chris T Bauch.  Coupled disease–behavior dynamics on complex networks: A review.  Physics of life reviews, 15:1–29, 2015.
  * [5] Aaron Lynch.  Thought contagion as abstract evolution.  Journal of Ideas, 2(1):3–10, 1991.
  * [6] Zixue Tai and Tao Sun.  Media dependencies in a changing media environment: The case of the 2003 sars epidemic in china.  New Media & Society, 9(6):987–1009, 2007.
  * [7] Romualdo Pastor-Satorras, Claudio Castellano, Piet Van Mieghem, and Alessandro Vespignani.  Epidemic processes in complex networks.  Reviews of modern physics, 87(3):925, 2015.
  * [8] Neil Ferguson.  Capturing human behaviour.  Nature, 446(7137):733–733, 2007.
  * [9] Zhongyuan Ruan, Ming Tang, and Zonghua Liu.  Epidemic spreading with information-driven vaccination.  Physical Review E, 86(3):036117, 2012.
  * [10] Clara Granell, Sergio Gómez, and Alex Arenas.  Dynamical interplay between awareness and epidemic spreading in multiplex networks.  Physical review letters, 111(12):128701, 2013.
  * [11] Sebastian Funk, Erez Gilad, Chris Watkins, and Vincent AA Jansen.  The spread of awareness and its impact on epidemic outbreaks.  Proceedings of the National Academy of Sciences, 106(16):6872–6877, 2009.
  * [12] Sebastian Funk, Marcel Salathe, and Vincent AA Jansen.  Modelling the influence of human behaviour on the spread of infectious diseases: a review.  Journal of the Royal Society Interface, 7(50):1247–1256, 2010.
  * [13] Faryad Darabi Sahneh, Fahmida N Chowdhury, and Caterina M Scoglio.  On the existence of a threshold for preventive behavioral responses to suppress epidemic spreading.  Scientific reports, 2:632, 2012.
  * [14] A Lima, M De Domenico, V Pejovic, and Mirco Musolesi.  Disease containment strategies based on mobility and information dissemination.  Scientific reports, 5:10650, 2015.
  * [15] Roger E Kasperson, Ortwin Renn, Paul Slovic, Halina Szejnwald Brown, Jacque L Emel, Robert Goble, Jeanne X Kasperson, and Samuel J Ratick.  The social amplification of risk: A conceptual framework.  Risk Analysis, 8(2):177–187, 1988.
  * [16] Kate Starbird and Leysia Palen.  Pass it on?: Retweeting in mass emergency.  International Community on Information Systems for Crisis Response and …, 2010.
  * [17] Akiko Murakami and Tetsuya Nasukawa.  Tweeting about the tsunami? mining twitter for information on the tohoku earthquake and tsunami.  In Proceedings of the 21st International Conference on World Wide Web, pages 709–710, 2012.
  * [18] Kate Starbird, Jim Maddock, Mania Orand, Peg Achterman, and Robert M Mason.  Rumors, false flags, and digital vigilantes: Misinformation on twitter after the 2013 boston marathon bombing.  IConference 2014 Proceedings, 2014.
  * [19] Pete Burnap, Matthew L Williams, Luke Sloan, Omer Rana, William Housley, Adam Edwards, Vincent Knight, Rob Procter, and Alex Voss.  Tweeting the terror: modelling the social media reaction to the woolwich terrorist attack.  Social Network Analysis and Mining, 4(1):206, 2014.
  * [20] Sarah Vieweg, Amanda L Hughes, Kate Starbird, and Leysia Palen.  Microblogging during two natural hazards events: what twitter may contribute to situational awareness.  In Proceedings of the SIGCHI conference on human factors in computing systems, pages 1079–1088, 2010.
  * [21] Jessica T Davis, Nicola Perra, Qian Zhang, Yamir Moreno, and Alessandro Vespignani.  Phase transitions in information spreading on structured populations.  Nature Physics, 16(5):590–596, 2020.
  * [22] Nickolas M Jones, Rebecca R Thompson, Christine Dunkel Schetter, and Roxane Cohen Silver.  Distress and rumor exposure on social media during a campus lockdown.  Proceedings of the National Academy of Sciences of the United States of America, 114(44):11663–11668, 2017.
  * [23] Sheena Aislinn Taha, Kimberly Matheson, and Hymie Anisman.  H1n1 was not all that scary: Uncertainty and stressor appraisals predict anxiety related to a coming viral threat.  Stress and Health, 30(2):149–157, 2014.
  * [24] Prashant Bordia and Ralph L Rosnow.  Rumor rest stops on the information highway: Transmission patterns in a computer-mediated rumor chain.  Human Communication Research, 25(2):163–179, 1998.
  * [25] Yu Jie Ng, Z Janet Yang, and Arun Vishwanath.  To fear or not to fear? applying the social amplification of risk framework on two environmental health risks in singapore.  Journal of Risk Research, 21(12):1487–1501, 2018.
  * [26] Paul Slovic.  Perception of risk.  Science, 236(4799):280–285, 1987.
  * [27] Dawei Wang, Bo Hu, Chang Hu, Fangfang Zhu, Xing Liu, Jing Zhang, Binbin Wang, Hui Xiang, Zhenshun Cheng, Yong Xiong, Yan Zhao, Yirong Li, Xinghuan Wang, and Zhiyong Peng.  Clinical Characteristics of 138 Hospitalized Patients With 2019 Novel Coronavirus–Infected Pneumonia in Wuhan, China.  JAMA, 323(11):1061–1069, 03 2020.
  * [28] Wenyao Li, Meng Cai, Xiaoni Zhong, Yanbing Liu, Tao Lin, and Wei Wang.  Coevolution of epidemic and infodemic on higher-order networks.  Chaos, Solitons & Fractals, 168:113102, 2023.
  * [29] Xiao Hong, Yuexing Han, Gouhei Tanaka, and Bing Wang.  Co-evolution dynamics of epidemic and information under dynamical multi-source information and behavioral responses.  Knowledge-Based Systems, 252:109413, 2022.
  * [30] Xiaoqi Zhang, Zi-Ke Zhang, Wenbo Wang, Donglin Hou, Jiajing Xu, Xinyue Ye, and Shengwen Li.  Multiplex network reconstruction for the coupled spatial diffusion of infodemic and pandemic of covid-19.  International Journal of Digital Earth, 14(4):401–423, 2021.
  * [31] Surajit Debnath, Prahlad Majumdar, Susmita Sarkar, and Uttam Ghosh.  Memory effect on prey–predator dynamics: Exploring the role of fear effect, additional food and anti-predator behaviour of prey.  Journal of Computational Science, 66:101929, 2023.
  * [32] Jiang Wu, Renxian Zuo, Chaocheng He, Hang Xiong, Kang Zhao, and Zhongyi Hu.  The effect of information literacy heterogeneity on epidemic spreading in information and epidemic coupled multiplex networks.  Physica A: Statistical Mechanics and its Applications, 596:127119, 2022.
  * [33] Sarah E Kreps and Douglas L Kriner.  Model uncertainty, political contestation, and public trust in science: Evidence from the covid-19 pandemic.  Science advances, 6(43):eabd4563, 2020.
  * [34] Matteo Cinelli, Walter Quattrociocchi, Alessandro Galeazzi, Carlo Michele Valensise, Emanuele Brugnoli, Ana Lucia Schmidt, Paola Zola, Fabiana Zollo, and Antonio Scala.  The covid-19 social media infodemic.  Scientific reports, 10(1):1–10, 2020.
  * [35] Chao Gu and Yi Feng.  Influence of public engagement with science on scientific information literacy during the covid-19 pandemic: Empirical evidence from college students in china.  Science & education, 31(3):619–633, 2022.
  * [36] Michael A Gisondi, Rachel Barber, Jemery Samuel Faust, Ali Raja, Matthew C Strehlow, Lauren M Westafer, and Michael Gottlieb.  A deadly infodemic: Social media and the power of covid-19 misinformation.  Journal of medical Internet research, 24(2):e35552, 2022.
  * [37] Xiu-Xiu Zhan, Chuang Liu, Ge Zhou, Zi-Ke Zhang, Gui-Quan Sun, Jonathan JH Zhu, and Zhen Jin.  Coupling dynamics of epidemic spreading and information diffusion on complex networks.  Applied Mathematics and Computation, 332:437–448, 2018.
  * [38] Xiaoqi Zhang, Zheng Ji, Yanqiao Zheng, Xinyue Ye, and Dong Li.  Evaluating the effect of city lock-down on controlling covid-19 propagation through deep learning and network science models.  Cities, 107:102869, 2020.
  * [39] Jayanta Kumar Ghosh, Prahlad Majumdar, and Uttam Ghosh.  Qualitative analysis and optimal control of an sir model with logistic growth, non-monotonic incidence and saturated treatment.  Mathematical Modelling of Natural Phenomena, 16:13, 2021.
  * [40] Jayanta Mondal and Subhas Khajanchi.  Mathematical modeling and optimal intervention strategies of the covid-19 outbreak.  Nonlinear Dynamics, 109(1):177–202, 2022.
  * [41] Jayanta Kumar Ghosh, Sudhanshu Kumar Biswas, Susmita Sarkar, and Uttam Ghosh.  Mathematical modelling of covid-19: A case study of italy.  Mathematics and Computers in Simulation, 194:1–18, 2022.
  * [42] Prahlad Majumdar, Bapin Mondal, Surajit Debnath, and Uttam Ghosh.  Controlling of periodicity and chaos in a three dimensional prey predator model introducing the memory effect.  Chaos, Solitons & Fractals, 164:112585, 2022.
  * [43] Wenyuan Zhou, Xiaoqi Zhang, Yanqiao Zheng, Tutiantian Gao, Xiaobei Liu, and Han Liang.  Psychological impact of covid-19 lockdown and its evolution: A case study based on internet searching data during the lockdown of wuhan 2020 and shanghai 2022.  In Healthcare, volume 11, page 289. Multidisciplinary Digital Publishing Institute, 2023.
  * [44] Sarah M Bartsch, Kelly J O’Shea, Kevin L Chin, Ulrich Strych, Marie C Ferguson, Maria Elena Bottazzi, Patrick T Wedlock, Sarah N Cox, Sheryl S Siegmund, Peter J Hotez, et al.  Maintaining face mask use before and after achieving different covid-19 vaccination coverage levels: a modelling study.  The Lancet Public Health, 7(4):e356–e365, 2022.
  * [45] Kai Sing Sun, Terence See Man Lau, Eng Kiong Yeoh, Vincent Chi Ho Chung, Yin Shan Leung, Carrie Ho Kwan Yam, and Chi Tim Hung.  Effectiveness of different types and levels of social distancing measures: a scoping review of global evidence from earlier stage of covid-19 pandemic.  BMJ open, 12(4):e053938, 2022.
  * [46] Yao Lu, Zheng Ji, Xiaoqi Zhang, Yanqiao Zheng, and Han Liang.  Re-thinking the role of government information intervention in the covid-19 pandemic: An agent-based modeling analysis.  International journal of environmental research and public health, 18(1):147, 2021.
  * [47] Tao Zhou, Jian-Guo Liu, Wen-Jie Bai, Guanrong Chen, and Bing-Hong Wang.  Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity.  Physical Review E, 74(5):056109, 2006.
  * [48] Andrea L Bertozzi, Elisa Franco, George Mohler, Martin B Short, and Daniel Sledge.  The challenges of modeling and forecasting the spread of covid-19.  Proceedings of the National Academy of Sciences, 117(29):16732–16738, 2020.
  * [49] Riccardo Gallotti, Francesco Valle, Nicola Castaldo, Pierluigi Sacco, and Manlio De Domenico.  Assessing the risks of ‘infodemics’ in response to covid-19 epidemics.  Nature Human Behaviour, 4(12):1285–1293, 2020.
  * [50] Wei Wang, Quanhui Liu, Shimin Cai, Ming Tang, Lidia A Braunstein, and H Eugene Stanley.  Suppressing disease spreading by using information diffusion on multiplex networks.  Scientific Reports, 6(1):29259–29259, 2016.
  * [51] Louis Kim, Shannon M Fast, and Natasha Markuzon.  Incorporating media data into a model of infectious disease transmission.  PloS one, 14(2):e0197646, 2019.
  * [52] Teun Terpstra, Michael K Lindell, and Jan M Gutteling.  Does communicating (flood) risk affect (flood) risk perceptions? results of a quasi-experimental study.  Risk Analysis, 29(8):1141–1155, 2009.
  * [53] Dana Rose Garfin, Roxane Cohen Silver, and E Alison Holman.  The novel coronavirus (covid-2019) outbreak: Amplification of public health consequences by media exposure.  Health Psychology, 2020.
  * [54] Liang-an Huo, Peiqing Huang, and Xing Fang.  An interplay model for authorities’ actions and rumor spreading in emergency event.  Physica A: Statistical mechanics and its applications, 390(20):3267–3274, 2011.
  * [55] Alivelu Mukkamala and Roman Beck.  The role of social media for collective behavior development in response to natural disasters.  2018\.
  * [56] Lirong Zou, Feng Ruan, Mingxing Huang, Lijun Liang, Huitao Huang, Zhongsi Hong, Jianxiang Yu, Min Kang, Yingchao Song, Jinyu Xia, et al.  Sars-cov-2 viral load in upper respiratory specimens of infected patients.  New England Journal of Medicine, 382(12):1177–1179, 2020.
  * [57] Joseph T Wu, Kathy Leung, and Gabriel M Leung.  Nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study.  The Lancet, 395(10225):689–697, 2020.
  * [58] Lucia Cabulea and Mihaela Aldea.  Making a decision when dealing with uncertain conditions.  Acta Universitatis Apulensis Mathematics-Informatics, 7:85–92, 2004\.
  * [59] Karen Page Winterich, Andrea C Morales, and Vikas Mittal.  Disgusted or happy, it is not so bad: Emotional mini-max in unethical judgments.  Journal of business ethics, 130(2):343–360, 2015.
  * [60] Shlomo Benartzi and Richard H Thaler.  Myopic loss aversion and the equity premium puzzle.  The quarterly journal of Economics, 110(1):73–92, 1995.
  * [61] Amos Tversky and Daniel Kahneman.  Judgment under uncertainty: Heuristics and biases.  science, 185(4157):1124–1131, 1974.
  * [62] Daniel Kahneman and Amos Tversky.  Prospect theory: An analysis of decision under risk.  In Handbook of the fundamentals of financial decision making: Part I, pages 99–127. World Scientific, 2013.
  * [63] Yu Liu, Bai Wang, Bin Wu, Suiming Shang, Yunlei Zhang, and Chuan Shi.  Characterizing super-spreading in microblog: An epidemic-based information propagation model.  Physica A: Statistical Mechanics and its Applications, 463:202–218, 2016.
  * [64] Meeyoung Cha, Alan Mislove, and Krishna P Gummadi.  A measurement-driven analysis of information propagation in the flickr social network.  In Proceedings of the 18th international conference on World wide web, pages 721–730, 2009.
  * [65] Lawrence Page, Sergey Brin, Rajeev Motwani, and Terry Winograd.  The pagerank citation ranking: Bringing order to the web.  Technical report, Stanford InfoLab, 1999.
  * [66] Hellewell Joel, Abbott Sam, Gimma Amy, Bosse I Nikos, Jarvis I Christopher, Russell W Timothy, Munday D James, Adam Kucharski J, and Edmunds W John.  Feasibility of controlling covid-19 outbreaks by isolation of cases and contacts.  The Lancet Global Health, 8(4):e488–e496, 2020.
  * [67] Timothy C Reluga.  Game theory of social distancing in response to an epidemic.  PLoS Comput Biol, 6(5):e1000793, 2010.
  * [68] Frederick Chen.  A mathematical analysis of public avoidance behavior during epidemics using game theory.  Journal of theoretical biology, 302:18–28, 2012.