# Centralized and decentralized isolation strategies and their impact on the
COVID-19 pandemic dynamics

Alexandru Topîrceanu
Department of Computer and Information Technology
Politehnica University Timişoara
Timişoara, 300223, Romania
<EMAIL_ADDRESS>
&Mihai Udrescu
Department of Computer and Information Technology
Politehnica University Timişoara
Timişoara, 300223, Romania
<EMAIL_ADDRESS>
Radu Mărculescu
Department of Electrical and Computer Engineering
The University of Texas at Austin
Austin, TX 78712, USA
<EMAIL_ADDRESS>

###### Abstract

The infectious diseases are spreading due to human interactions enabled by
various social networks. Therefore, when a new pathogen such as SARS-CoV-2
causes an outbreak, the non-pharmaceutical isolation policies (e.g., social
distancing) are the only possible response to disrupt its spreading. Since
there exist several non-pharmaceutical strategies to resort to, we need to
assess their efficiency for thwarting the pandemic impact. To this end, we
introduce the new epidemic model (SICARS) and compare the centralized (C),
decentralized (D), and combined (C+D) social distancing strategies, and
analyze their efficiency to control the dynamics of COVID-19 on heterogeneous
complex networks. Our analysis shows that the centralized social distancing is
necessary to minimize the pandemic spreading. The decentralized policy is
insufficient when used alone, but offers the best results when combined with
the centralized one. Indeed, the (C+D) policy is the most efficient isolation
at mitigating the network superspreaders and reducing the highest node degrees
to less than 10% of their initial values. Our results also indicate that a
moderate social distancing, e.g., cutting 50% of social ties, can reduce the
outbreak impact by 47% for the C isolation, and by 31% for the D isolation. A
stronger social distancing, e.g., cutting 75% of social ties, can reduce the
outbreak by 75% for the C isolation, by 33% for the D isolation, and by 87%
for the (C+D) isolation. Finally, we study the impact of proactive versus
reactive isolation strategies, as well as their delayed enforcement. We find
that the reactive response to the pandemic is less efficient, and delaying the
adoption of isolation measures by over one month (since the outbreak onset in
a region) can have dangerous effects. Thus, our study contributes to an
understanding of the COVID-19 pandemic both in space (i.e., network-centric
analysis of isolation strategies), and time (i.e., delayed application of
isolation strategies). We believe our investigations have a high social
relevance as they provide insights into understanding how different degrees of
social distancing can reduce the peak infection ratio substantially; this can
make the COVID-19 pandemic easier to understand and control over an extended
period.

_K_ eywords COVID-19 $\cdot$ epidemic model $\cdot$ social distancing $\cdot$
complex networks $\cdot$ isolation strategies

##  1 Introduction

The incidence of the new Coronavirus disease (COVID-19), caused by the severe
acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has seen an exponential
rise since the end of 2019, affecting all continents as of March 2020 [1, 2].
The number of cases reported to date is likely an underestimation, owing to
the deficiencies in surveillance policy and diagnostic capacity both in high
and low-income regions [2]. Based on various scientifically relevant criteria,
the WHO has declared a global COVID-19 pandemic [1] in March 2020.

In the absence of an approved pharmaceutical treatment (or vaccine) and in-
depth knowledge of the spreading mechanism, the best strategies against
COVID-19 consist of reducing the interactions between susceptible and infected
individuals, e.g., through early detection and social distancing [3]. Indeed,
such non-pharmaceutical interventions (NPIs) turned out to be very effective
during previous pandemics [4, 5].

The potential effect of social distancing interventions on the COVID-19 has
already been studied in Singapore [6]. Indeed, Singapore was among the first
regions to report imported cases and has so far succeeded in preventing
community spread. However, the scale and severity of the Singapore
interventions are small in comparison with the measures implemented in China
in response to COVID-19. The core Chinese interventions include shutting down
schools and workplaces, closing roads and transit systems, canceling public
gatherings, and imposing a mandatory quarantine on uninfected people (even
those without known exposure to the virus) [7]. Although these actions seem to
be working so far, imposing similar restrictions in other countries represents
an ongoing challenge. To convince people, governments, and public authorities
around the world that such extreme limitations are necessary, we need to back
them up with scientific evidence.

Given the dynamics of COVID-19 spreading, we can assess the efficiency of the control measures for this novel pathogen by using mathematical modeling coupled with computer simulations of infectious spread under various scenarios. To this end, we propose a new agent-based outbreak model called SICARS (Susceptible \- Incubating \- Contagious \- Aware \- Removed | Susceptible), which allows us to assess the impact of the centralized and decentralized isolation strategies on COVID-19 spreading across complex heterogenous networks. Consequently, we run simulations of SICARS and test two fundamentally different strategies, as well as their combined effects:

  1. 1.

Centralized (C) strategy, such as the government-imposed lockdown or
quarantine; this means social distancing by the synchronized removal of a
specified ratio of node social ties from the entire social network.

  2. 2.

Decentralized (D) strategy, such as aware-isolation (DA) and auto-isolation
(DI); this means an individual-level social distancing by asynchronously
removing a specified ratio of personal social ties. More precisely, in DA, the
individuals who become aware of their sickness cut the social links in their
ego-network. In contrast, in DI, the healthy neighbors of sick individuals
isolate themselves from the infected. In both scenarios, the social ties are
removed repeatedly (e.g., daily) based on a probability parameter.

  3. 3.

Hybrid (C+D) strategy, whereby both policies are combined, hence the removal
of the social ties involves both centralized and individual-level decision
mechanisms. To this end, a fraction of social links are synchronously removed
from all nodes in the network, then followed by repeated asynchronous
distancing through self-aware isolation of sick nodes (DA), as well as auto-
isolation of healthy nodes from their sick neighbors (DI).

In contrast to other network approaches like [8, 9, 10], SICARS benefits from an additional state we introduce at individual-level, which allows both synchronous and asynchronous isolation to be analyzed, as well as their delayed or progressive application with an increase in severity of isolation. Our model is inspired by SIR [11], for which there exist many variations including the SI, SIS, MSIR, SEIR, MSEIRS, SI1I2S, SI1|2S, and SI1SI2S models [12]. A modified SEIR model was previously proposed for the Ebola outbreak of 2014 in Western Africa [13], consisting of the states specific to Ebola: Susceptible \- Exposed \- Infected \- Hospitalized \- Buried | Recovered. In principle, the SEIR model can be a candidate for capturing the states of an individual in contact with the COVID-19 outbreak. However, SEIR lacks the Aware state we introduce in SICARS and which we exploit to study the decentralized isolation strategy. We note that, while both Contagious and Aware are infectious states (i.e., the disease is transmitted), it is only when a node becomes aware of being infected that it may isolate itself by applying social distancing. The contagious period during which nodes are unaware of their infectiousness (and thus are unable to protect others) is one of the main characteristics making the COVID-19 pandemic so dangerous and hard to control [3].

We note that there exist several studies based on compartmental models that
present timely conclusions on understanding the COVID-19 pandemic spreading.
For instance, Koo et al. [14] adapt an existing influenza epidemic simulation
model (using data from Singapore) to assess the consequences of social
distancing on the transmission dynamics. They find that the intervention
strategy combining quarantine, school closure, and workplace distancing seems
to be the most effective. Indeed, compared to the no interventions scenario,
the combined intervention strategy reduces the estimated number of infections
in Singapore by 99.3%.

Kucharski et al. [15] merge a stochastic transmission model with data on
coronavirus cases from Wuhan, China; they estimate the early-stage dynamics of
the epidemic and calculate the probability that new cases generate outbreaks
in other areas. The main finding is that the median daily reproduction number
$R_{t}$ declined significantly, from 2.35 to 1.05, after the introduction of
travel restrictions in Wuhan, on Jan 23, 2020.

Prem et al. [16] use synthetic contact patterns, adapted to reduced social
mixing, and employ SEIR to show that the return to work in China should be
further delayed (i.e., by one month). They state that the adoption of physical
distancing measures for a long time may reduce the number of infections by
more than 92% by mid-2020.

Hellewell et al. [3] study a transmission model tailored to the COVID-19
outbreak and present several interesting findings. First, if there would be no
transmission likelihood before symptoms onset, the outbreak containment is
secure; unfortunately, this is not the case of COVID-19. Second, when the
transmission onset or number of initial cases increases, the controllability
of the outbreak decreases significantly. Thus, based on the reproduction
number $R_{t}$ to control the epidemic, 50% to 90% of contacts need to be
traced and isolated. Third, the critical parameter affecting the
controllability of the epidemic is the delay between symptoms onset and start
of isolation [3].

Ferretti et al. [17] warn that the current epidemic spread is too fast to be
contained by manual contact tracing, and suggest the use of digital tools,
like smartphones, to accelerate the contact tracing. Indeed, social behavior
plays a vital role in the transmission dynamics of diseases [18, 19].

Finally, we note that the idea of social distancing, in various forms, is not
new to network science and has been proposed earlier for controlling epidemics
[20]. Glass et al. [21] simulate an influenza model on a community
representative of a small town in the United States and show that social
distancing would be necessary though school closures and home isolation. For
more infectious strains, they found that increased levels of quarantine are
needed. Valdez et al. [22] adopt an intermittent social distancing strategy to
disturb the epidemic spreading process on different complex network
topologies. Specifically, a susceptible individual is allowed to interrupt the
contact with an infected neighbor based on a probability value. Using
percolation theory, the authors find a critical threshold beyond which the
epidemic disappears. Reluga et al. [19] employ game theory to estimate the
social behavior of individuals when adopting social distancing during an
outbreak. They find that the individual adopts a prophylactic behavior only
after the epidemic begins and ceases before the epidemic ends. Additionally,
the reproduction number $R_{t}$ must exceed a certain threshold for
individuals to feel that social distancing is worth the effort. In contrast,
our work studies global, local, and hybrid social distancing strategies to
assess their effectiveness in controlling the COVID-19 dynamics.

Taken together, these related studies agree on the efficacy of a generalized
quarantine and early detection with isolation strategies. We improve the
state-of-the-art with the following contributions:

  * •

In contrast with the COVID-19 epidemic modeling proposed in [14, 15, 3], where
isolation is modeled by reducing the size of the susceptible compartment, our
network-centric approach targets isolation strategies as (local and global)
edge removal mechanisms, hence an emergent and more realistic transmission
dynamics.

  * •

As the differential equations of SEIR do not apply to COVID-19, we use
distributed simulation on well-known complex topologies instead of the
compartmental models based on random uniform contact networks that are
typically used to study epidemics spreading [23, 24, 12, 16, 15].

  * •

Instead of focusing on assessing a specific isolation strategy (_e.g._ ,
Singapore), our study aims at differentiating the efficiency of centralized
(global, government-imposed), decentralized (local, self-imposed), and hybrid
isolation while using the SARS-CoV-2 specific biological parameters.

  * •

We focus on providing a more accurate quantification of the impact of
different levels of social distancing, and explore the realistic scenarios of
delayed and progressive application of isolation, in the context of the
current pandemic.

##  2 Methods

###  2.1 Isolation strategies

We consider a complex interaction network represented as an undirected graph
$G=\\{N,E\\}$, where $N=\\{n_{i}\\}$ is the set of nodes and
$E=\\{e_{ij}\,|\,n_{i},n_{j}\in N\\}$ is the set of edges. The nodes represent
individuals in the society, while edges represent the social interactions
between individuals.

The edge removal ratio $0\leq e\leq 1$ quantifies the ratio of social ties
removed from the network $G$. We define two fundamentally different isolation
strategies, based on the underlying principle of removing social ties, as
follows:

#### Centralized (C):

A ratio $e$ of edges are randomly removed from all nodes $n_{i}\in N$. This
strategy is synchronous, meaning that its application happens simultaneously
across all network nodes during one simulation step (e.g., day); this
translates into a reduction of the number of network edges from $|E|$ to
$(1-e)|E|$.

#### Decentralized (D):

We consider this D strategy in two different variants, namely decentralized
aware-isolation (DA) and decentralized auto-isolation (DI). Both variants are
asynchronous meaning that they are applied on a node-by-node basis, based on a
local decision mechanism. For DA, any node that is aware of being infected
will self-isolate itself by repeatedly removing $e$ edges from its vicinity.
For a high $e$, an infected node can rapidly become disconnected from the
network, thus impeding the pandemic outbreak. For DI, any healthy node that
has an infected neighbor will remove the corresponding social tie with a fixed
probability of $e$. If $e$ is small, the healthy node will likely need several
simulation iterations until successfully removing a specific social tie.

#### Hybrid (C+D):

We also consider the response to a (C+D) isolation strategy by combining all
three strategies C, DI, and DA. More precisely, a centralized strategy is
first applied on all nodes $n_{i}\in N$, by randomly removing a proportion of
$e$ edges from each node $n_{i}$. Then, as on ongoing process, each node
$n_{i}$ that becomes infected and aware will apply self-isolation in its
vicinity $N_{i}$, and each susceptible neighbor $n_{j}\in N_{i}$ will apply
isolation from the sick node $n_{i}$ with probability $e$.

Finally, given the global reaction to the COVID-19 pandemic in early 2020 and
its social and economic long-term implications, we notice a delayed
enforcement of any isolation strategy in various places. Hence, we also
analyze the influence of the response delay ($d=\delta_{Delay}$ days) in
applying any of the enumerated strategies.

###  2.2 SICARS epidemic model

We generalize the popular SIR epidemic model [25, 9] to explicitly enable the
analysis of the isolation strategies that are relevant to the COVID-19
outbreak. More precisely, our model defines five possible states for an
individual: susceptible $S$, incubating $I$, contagious $C$, aware $A$, and
removed $R$, with the caveat that $R$ is equivalent to either recovered or
dead; $R$ can, under specific circumstances, relapse back to the susceptible
state $S$ (see Figure 1). We note that SICARS is an edge removal model, rather
than a node removal model. The reason behind this decision is that removing
edges, in the real context of a pandemic, can be controlled easier through
isolation measures. In contrast, the removal of nodes means total isolation, a
measure that is impractical due to the massive amount of infected patients,
and the social consequences of such an extreme isolation.

We assume that, initially, $s$ individuals act as the outbreak seed set
$N_{s}$ (initially marked as incubating), while the susceptible individuals
are the healthy inactive nodes $N\setminus N_{s}$. Any susceptible node coming
in contact with an incubating node will not necessarily become infected. On
average, only after $\delta_{Incubation}$ days, any incubating node
automatically becomes contagious, and susceptible nodes in the vicinity of a
contagious node become infected with a probability $\lambda_{Infect}$ (see
Figure 1). After some $\delta_{Aware}$ days, the contagious nodes become aware
(i.e., visibly symptomatic), and their infectiousness becomes visible to the
neighboring nodes. From this point on, after a total of $\delta_{Removal}$
days, a node will have transitioned from incubating ($I$), to aware ($A$), and
finally to a removed $R$ or susceptible $S$ state again. A removed node
(either recovered or dead111We assume merging these two states under the
infected nodes gaining immunity assumption, which is likely, but not
ultimately confirmed yet in practice.) will not become susceptible again. At
the end of the $\delta_{Removal}$ recovery period a node becomes dead with
probability $\lambda_{Die}$, susceptible with probability
$\lambda_{Susceptible}$, or recovered with probability
$1-\lambda_{Die}-\lambda_{Susceptible}$.

Figure 1: The states and parameters defining the SICARS model. By splitting
the infectious stage into two sub-states, contagious and aware, the model
becomes unique in its capability of implementing centralized and decentralized
isolation strategies. Algorithm 1 SICARS model algorithm

1:assign randomly $n_{s}\in N_{s}\leftarrow I$ (incubating), where $|N_{s}|=s$

2:assign all other nodes $n_{i}\in N\setminus N_{s}\leftarrow S$ (susceptible)

3:repeat$\triangleright$ for each iteration day $d$

4: if $d\geq\delta_{Delay}$ then

5: if using (C) strategy: remove any edge $e_{ij}\in E$ with probability $e$

6: if using (DA) strategy: for $\forall n_{k}\in A$ (aware): remove any
adjacent edge $e_{kj}\in E$ with probability $e$

7: if using (DI) strategy: for $\forall n_{j}\in S$ (susceptible), where
$n_{j}\in N_{k}$ (neighborhood of an aware node $n_{k}$): remove edge $e_{jk}$
(incident to node $n_{k}$) with probability $e$

8: if using (C+D): apply all corresponding steps (a-c) independently, in this
order

9: end if

10: for $\forall n_{k}\in A$ (aware) for $\geq\delta_{Removal}$ iterations do

11: $n_{k}\leftarrow R(dead)$ with probability $\lambda_{Die}$

12: $n_{k}\leftarrow S(susceptible)$ with probability $\lambda_{Susceptible}$

13: $n_{k}\leftarrow R(removed)$ probability
$1-\lambda_{Die}-\lambda_{Susceptible}$

14: end for

15: for $\forall n_{i}\in C$ (contagious) do

16: for $\forall n_{j}\in S$ (susceptible), where $n_{j}\in N_{i}$:
$n_{j}\leftarrow I$ (incubating) with probability $\lambda_{Infect}$

17: if $n_{i}\in C$ for $\geq\delta_{Aware}$ iterations: $n_{i}\leftarrow A$
(aware)

18: end for

19: for $\forall n_{i}\in I$ (incubating) do

20: if $n_{i}\in I$ for $\geq\delta_{Incubation}$ iterations: $n_{i}\leftarrow
C$ (contagious)

21: end for

22:until $(|I|=0,|C|=0,|A|=0)$, or ($|R|\geq\theta
N(\textit{e.g.},\theta=95\%))$

The SICARS pandemic simulation based on the model in Figure 1 is formally
defined in algorithm 1. We further illustrate how SICARS works in Figure 2,
with a small example network of five connected nodes, which become infected
[3]. The outbreak seed is node A which needs a period of $\delta_{Incubation}$
days to become contagious. After becoming contagious, node A spreads the
disease to its neighbors B and C; these, in turn, become incubating after
several days of contact with A. Node A is still not aware that it became a
disease carrier, nor that it has infected his neighbors. After a delay of
$\delta_{Aware}$ days, node A becomes aware of its infectious condition (or
state). At this point, nodes B and C know that A is a threat, but are unaware
if they have contacted the disease. In the same manner, nodes B and C become
contagious, then aware. Nodes D and E are further infected, and the process
continues similarly. After $\delta_{Removal}$ days (measured individually for
each infected node), every node changes to one of three states: recovered (B,
E), dead (C, D), or susceptible (A). The susceptible node A may start the same
process all over again, going through all the SICARS states.

Figure 2: Example of an outbreak process according to our SICARS model, when
used over a hypothetical contact network with five nodes (A–E), starting with
infected node A. After an incubation period, node A becomes contagious and may
infect neighbors B and C at any time. Likewise, node C enters the incubation
phase and then becomes contagious for nodes D and E. Both contagious and aware
states are infectious states, during which a node may spread the disease. Only
when a node becomes aware may isolation strategies be applied to it. Finally,
all nodes evolve towards one of two states: susceptible or removed (with the
mutually exclusive sub-states recovered or dead).

Furthermore, in Figure 3 we illustrate an intuitive example of applying each
isolation strategy independently on a small contact network of ten nodes. By
setting $e=0.5$, we show that the C isolation removes about 50% of edges in
the entire network, regardless of the node states (hence, all nodes are
colored gray in Figure 3a); this process is applied synchronously over the
entire network. The DA and DI isolation, on the other hand, work
asynchronously, and depend on the (local) development of the disease in the
vicinity of each node. In case of DA, pictured in Figure 3b, only when a node
becomes aware (orange) can it self-isolate by cutting edges in its vicinity
(green nodes). In case of DI, pictured in Figure 3c, a susceptible node
(green) can only isolate itself from infected neighbors once these become
aware (orange), i.e., visibly infected. Finally, in case of the combined C+D
strategy, we simply apply C followed by DA and DI independently on the
network.

Figure 3: Example of applying the C, DA, and DI isolation strategies over a
small example contact network using an edge removal ratio $e=0.5$. (a) For C
isolation, any edge will be removed with a 50% probability, regardless of the
state of any node. (b) For DA isolation, each node that becomes aware will cut
all edges in its vicinity with a probability given by $e$; in this example,
node A removes 3 out of its 6 adjacent edges. (c) For DI isolation, each
susceptible (healthy) node which detects an aware (i.e., sick) neighbor, will
try to detach from it with a given probability $e$; in this example, node S
manages to detach itself from sick nodes $A_{2}$ and $A_{3}$.

With our SICARS model, we start the diffusion process with $s=10$ initial
cases to represent the newly emerging outbreak. The other parameters used in
our model are gathered from the recent literature and summarized in Table 1.

Table 1: Parameter values used with the outbreak model specific to the COVID-19 pandemic. For _investigated_ we experiment with multiple values for the respective parameters. Parameter | Empirical value | Model parameter | Value in model | Reference
---|---|---|---|---
Incubation period | 5.1-5.8 days | $\delta_{Incubation}$ | 5 days | [26, 27, 28]
Delay from contact to onset | 11.5-14 days | $\delta_{Aware}$ | 12 days | [26]
Delay from onset to recovery | 2-8 weeks | $\delta_{Removal}$ | 14 days | [29, 30, 31]
Death rate | 1-3% | $\delta_{Die}$ | 2% | [30, 32]
Susceptibility rate | $<1\%$ | $\delta_{Susceptibility}$ | 1% | assumed
Number of initial cases | unknown | $s$ | 10 |  [33], assumed
Infectious rate | - | $\delta_{Infect}$ | 0.05 |  [9], assumed
Social tie removal rate | - | $e$ | 0–0.9 | investigated

###  2.3 Test data description

Our simulations with the SICARS model use several synthetic network topologies
and real-world network datasets, i.e., a mesh network ($Mesh$), a Watts-
Strogatz small-world network ($SW$), and a scale-free network ($SF$) [34]. It
is essential to analyze these basic topologies because they possess uniquely
distinguishable network properties found in nature [35, 36]. Note that the use
of these topologies is not an attempt to replace the realistic contact
networks, but rather to validate the accuracy and the consistent dynamics of
the SICARS model on several different types of graphs. We also include several
sizable real-world network datasets, which present a variety of heterogeneous
social interaction networks: the Enron email communication network ($Em$)
[37], the Brightkite location-based social network ($Bk$) [38], and the
Epinions trust network ($Ep$) [39]. The reason for choosing these specific
datasets is that from a statistics standpoint, they hold characteristics
representative for the contact network enabling epidemic spreading. In
particular, $Em$ models relevant professional contacts and $Ep$ models
relevant friendship contacts. These connections are the most likely to be
activated in the real-world social networks. $Bk$ is also an appropriate
choice because it models a location-based mesh-like topology, which is a
critical aspect of epidemics spreading [40].

In Table 2, we present the relevant network measurements for our enumerated
test topologies. We include the network size (the numbers of nodes $N$ and
edges $E$), average degree $\langle k\rangle$, maximum degree $max(k)$,
average path length $APL$, average clustering coefficient $ACC$, network
modularity $Mod$, and the network diameter $Dmt$ [34].

Table 2: The relevant parameters of the synthetic and real-world test networks. $Network$ | $N$ | $E$ | $\langle k\rangle$ | $max(k)$ | $APL$ | $ACC$ | $Mod$ | $Dmt$
---|---|---|---|---|---|---|---|---
$Mesh$ | 5000 | 25234 | 5.047 | 36 | 13.942 | 0.185 | 0.855 | 37
$SW$ | 5000 | 19998 | 4.000 | 11 | 10.607 | 0.426 | 0.829 | 22
$SF$ | 5000 | 15762 | 3.152 | 294 | 5.378 | 0.007 | 0.640 | 13
$Em$ | 12625 | 20362 | 3.226 | 576 | 3.811 | 0.577 | 0.684 | 9
$Bk$ | 58228 | 214078 | 7.353 | 1134 | 7.371 | 0.271 | 0.667 | 18
$Ep$ | 75879 | 405739 | 10.694 | 3044 | 11.549 | 0.137 | 0.438 | 14

###  2.4 Analytical modeling of SICARS dynamics

To motivate the importance of simulating SICARS in the context of complex
social interaction networks, we describe the dynamics of the model by
providing the differential equations that capture the time evolution of the
number of _Susceptible_ , _Infectious_ , and _Removed_ individuals ($S(t)$,
$I(t)$, and $R(t)$, respectively). Although relevant for the dynamics of the
network topology (which evolves according to the adopted social distancing
policies), the number of _Contagious_ and _Aware_ individuals, $C(t)$ and
$A(t)$, are simply a delayed version of $I(t)$. Overall, we have
$S(t)+I(t)+C(t)+A(t)+R(t)=N$.

Another critical parameter that changes over time (as a function of the
assumed social distancing policies) is the network density
$\rho\left(t\right)=\frac{2\cdot n_{E}\left(t\right)}{N\left(N-1\right)}$,
where $n_{E}\left(t\right)$ is the number of edges in the network at moment
$t$. Density $\rho(t)$ has a non-trivial evolution due to the heterogeneous
structure of real-world networks, and due to the local decision mechanisms
implemented by isolation (i.e., DA, DI); thus, $\rho(t)$ represents a dynamic
parameter in our analytical evaluation.

The equations system describing SICARS is as follows:

$\begin{cases}\dfrac{dS(t)}{dt}=-\rho(t)\lambda_{Infect}S(t)(I(t)+C(t)+A(t))+\lambda_{Susceptibility}A(t)\\\
\dfrac{dI(t)}{dt}=\rho(t)\lambda_{Infect}S(t)(I(t)+C(t)+A(t))\\\
C(t)=\int_{t-{\delta}_{Incubation}}^{t}I(\tau)d\tau\\\
A(t)=\int_{t-{\delta}_{Incubation}-{\delta}_{Aware}}^{t}I(\tau)d\tau\\\
\dfrac{dR(t)}{dt}=(1-\lambda_{Susceptibility})A(t)\\\ \end{cases}$ (1)

with the initial conditions given by:

$\begin{cases}S(0)=N-s\\\ I(0)=s\\\ C(t)=A(t)=R(t)=0\\\ \end{cases}$ (2)

The isolation strategies determine the dynamics of $\rho(t)$, which, in turn,
interact with the disease dynamics from the equation system 1. In Figure 4a we
present the centralized and decentralized evolution of the infection rate
${I(t)}/{N}$ according to the model described in equations 1 and 2, for an
edge removal ratio of $e=0.5$. After applying the C isolation on the $Em$
network (i.e., removing 50% of network edges), we measure a constant density
$\rho(t)=0.0001$. In case of the D isolation, the density evolution is dynamic
since edge removals are influenced by the spread of disease. We thus obtain
the evolution of density on the $Em$ network, for $e=0-0.75$, as shown in
Figure 4b; the density evolution (for the same $e=0.5$) is best described by a
sigmoid function starting from from $0.00025$ (point A) to $1\cdot 10^{-6}$
(point B), namely $\rho(t)=0.00025\left(1-\frac{1}{1+e^{-0.3t+9}}\right)$. The
analytically determined evolution of the C and D strategies in Figure 4a
suggests that the peak infection ratio is higher and achieved sooner in the
case of D (red line) compared to C (blue line). Specifically, the infection
ratio amplitude is 33% higher for D and occurs 32 days earlier than for C.

Figure 4: (a) Comparison between the C and D social isolation strategies for
the analytical SICARS model with equations 1 and 2. (b) Evolution of network
density $\rho(t)$ in time, on the $Em$ network when applying the DA strategy,
as a function of the edge removal ratio ($e$).

The asymptotic analysis of our model indicates that the centralized policy is
more effective than the decentralized one. However, using the network density
(i.e., an aggregate parameter) does not realistically capture the dynamics of
disease spread, since the structure of real-world social interaction networks
is not homogeneous but rather heterogeneous in nature. Therefore, to test the
interplay between the dynamics of SICARS and the dynamics of the network
topology according to the social distancing policies, we use detailed computer
simulations as described next.

##  3 Results

###  3.1 How to control the epidemic dynamics?

We further analyze how to control the COVID-19 dynamics using social
distancing. To this end, we run comprehensive simulations of our SICARS model,
for edge removal ratios $e=\\{0,0.1,0.25,0.5,0.65,0.75,0.8,0.9\\}$ on all test
networks in Table 2; we present next the results corresponding to the C, DA,
DI, and C+D isolation strategies.

Varying the edge removal ratio $e=0-0.9$ enables us to study a broad spectrum
of the severity degree of social distancing, i.e., from no distancing at all
($e=0$), to mild ($e=0.1-0.25$), to moderate ($e=0.5-0.65$), all the way to
extreme ($e\geq 0.75$) distancing. The effectiveness of these various forms of
isolation is relevant to understand more intuitively how exactly social
distancing works in many regions of the world. For example, in the USA, a
recent poll by Gallup [41] shows that Americans are progressively increasing
their social distancing strictness, from moderate ($e\approx 0.5$, mid-March
2020) to strict ($e\approx 0.75$, end of March 2020). By the end of March
2020, 20% of polled Americans admit to having no contact with people outside
their house, and 44% of having minimal contact outside their households.
Similarly, the strict measures adopted by China [7] and Singapore [6]
correspond to very strict measures within the social distancing spectrum
($e\geq 0.8$).

#### Centralized isolation:

In Figure 5a, we depict the evolution of the infection ratio ($r=\mbox{number
of infected individuals}/\mbox{population size}$) during the simulation of the
SICARS model for the C isolation strategy, using the $Bk$ network as a
representative example. Here we measure a drop in the maximum amplitude of the
outbreak, from an infection rate of 62% to just below 4%, as the ratio of
removed edges increases from $e=0$ (i.e., no isolation at all) to $e=0.9$
(i.e., severe isolation with 90% removed social ties); this means that, for
$e=0.9$, we get a 94% drop in outbreak severity. Additionally, we observe a
delay of the maximum amplitude time, from $d=45$ (point A, for $e=0$) to
$d=116$ (point B, for $e=0.9$), meaning that the duration of the pandemic is
prolonged by about 70 days. In Figure 5b, we display the drop in infection
ratio, averaged over all the test networks in Table 2. These results are
consistent for all test networks, thus suggesting a significant linear drop in
terms of outbreak impact (measured as infection ratio $r$) when a strong
centralized social isolation is applied (region B).

Figure 5: Efficiency of the centralized (C) isolation strategy in reducing the
ratio of infected individuals. (a) The impact of the edge removal ratio $e$ in
reducing and delaying the maximum outbreak amplitude. While no isolation
($e=0$) determines a maximum infection ratio of 62%, the isolation with
$e=0.5$ reduces the peak of the infection ratio to 33%, and isolation with
$e=0.8$ reduces the peak of the infection ratio to just 11%, but also
increases the outbreak duration from roughly 100 days to 180 days. (b) The
achieved outbreak amplitude with an increasing $e$, averaged over all test
networks in Table 2. We highlight the noticeable linear-concave response of
the C strategy in region B (green).

To reduce the infection ratio to 50% of the value corresponding to no
isolation at all, we need to remove slightly more than 50% of social ties.
However, if the number of ties is reduced by 80% ($e=0.8$), we observe a
reduction of 82% in the amplitude of the outbreak.

#### Decentralized isolation:

We run simulations for both DA and DI on all test networks and find only
marginal differences between the two decentralized isolation strategies.
Generally, DI follows the same trend as DA on all test networks, but its
effects are slightly delayed on time (OX axis) by 0-5%. Figure 6a depicts the
evolution of the infected ratio for the decentralized DA isolation strategy,
using the $Bk$ network as a representative example. By increasing the edge
removal ratio from $e=0$ to $e=0.9$, we observe a drop in the maximum
amplitude of the outbreak from 62% to 35%; this translates into a drop of only
43% in outbreak severity. Additionally, we observe a slight acceleration of
the maximum amplitude time, from $d=46$ (for $e=0$) to $d=39$ (for $e=0.9$),
meaning that the duration of the pandemic is shortened by approximately 7
days. In Figure 6b, we display the drop in infection ratio averaged over all
the test networks. The results over the entire set of test networks suggest a
moderate logarithmic drop in outbreak impact, even for a strong decentralized
isolation (region A).

Figure 6: Efficiency of the decentralized (D) isolation in reducing the ratio
of infected individuals. (a) The impact of the edge removal ratio $e$ in
reducing and advancing the maximum outbreak amplitude. While no isolation
($e=0$) determines a maximum infection ratio of 62%, the isolation with
$e=0.5$ reduces the peak infection ratio to 41%, and isolation with $e=0.8$
reduces the peak infection ratio to 35%. The outbreak duration is not affected
in a significant way. (b) The achieved outbreak amplitude with increasing $e$,
averaged over all test networks in Table 2. We highlight the noticeable
logarithmic-convex response of the D strategy in region A (green).

Reducing the number of social ties has an immediate impact, as a reduction of
only 25% ties can reduce the impact of the outbreak by 18%; however, further
reducing the number of ties renders insignificant improvements. Our results
suggest that, by using a decentralized isolation strategy, it is not possible
to reduce the infection ratio by 50% compared to no isolation at all; the best
we can achieve is a reduction of about 26%.

#### Combined C+D:

In comparison to C or D isolation alone, the hybrid C+D gets the best of each
strategy (Figure 7b). We notice that the maximum infection amplitude has a
linear-concave response when using the C isolation (Figure 5b), and a
logarithmic-convex response when using the D isolation (Figure 6b). This
difference means that the effects of the D isolation are seen immediately for
small increases in $e$ (region A in Figure 6b). In contrast, moderate to
significant increases of $e$ is needed to gain the benefit of C isolation
(region B in Figure 5b). In this sense, the C+D strategy shows the same
convexity as D, but with a sustained linear reduction that is specific to C.

Figure 7a depicts the evolution of the infection ratio $r$ for the combined
C+D isolation strategy, using the same $Bk$ network as a representative
example. By increasing the edge removal ratio from $e=0-0.9$, we observe a
significant drop in the maximum amplitude of the outbreak from 62% to just 5%
($e=0.8$), respectively 2% ($e=0.9$). Compared to C and D isolation, with C+D
we measure the highest efficiency in reducing the maximum infection ratio, by
as much as 87% ($e=0.75$), and 98% ($e=0.9$). Additionally, we observe a delay
of the maximum amplitude time, from $d=45$ (point A, for $e=0$) to $d=70$
(point B, for $e=0.8$), meaning that the duration of the pandemic is prolonged
by about 25 days (55%).

Figure 7: Efficiency of the combined (C+D) isolation strategy in reducing the
ratio of infected individuals. (a) The Impact of the edge removal ratio $e$ in
reducing and delaying the maximum outbreak amplitude. While no isolation
($e=0$) determines a maximum infection ratio of 62%, the isolation with
$e=0.5$ reduces the peak of the infection ratio significantly to 25%;
isolation with $e=0.8$ further reduces the peak infection ratio to just 5%.
However, the outbreak duration increases from roughly 100 days to 140 days.
(b) The achieved outbreak amplitude with the increasing $e$, averaged over all
test networks in Table 2. Here, we notice both the initial logarithmic-convex
signature of the D strategy, as well as the final linear-concave signature of
the C strategy combined.

For the hybrid C+D strategy, we need to remove just 25-50% of social ties in
order to reduce the infection ratio to 50%, compared to no isolation at all.
Moreover, if we reduce the number of ties by 75%, we obtain a significant
reduction of 87% in the outbreak amplitude (Figure 7b).

###  3.2 Why isolation strategies work?

After analyzing how to control the outbreak dynamics, we further try to
explain why the presented isolation methods work. To this end, we investigate
a crucial property of network structure, namely the distribution of
superspreaders [42] before and after applying any of the discussed strategies.
Indeed, many real-world networks possess a power-law distribution of nodes
degree, thus favoring the formation of hubs [43, 36]. These hubs have a
significant accelerating effect on the diffusion of innovation, ideas, as well
as epidemics [9, 42, 44]; this is why these hubs are called superspreaders.
During an outbreak, it becomes paramount to reduce the connectivity of these
superspreaders because a single infected hub can exceed the combined adverse
effects of many average or low connected nodes.

Figure 8: Efficiency of the centralized and decentralized strategies in
reducing the maximum degree of the network. (a) The reduction delay
(horizontal time axis, expressed in days $d$) and reduction ratio (vertical
axis); panel a is a snapshot of panel b specifically for $e=0.5$. Each
strategy presents a unique signature: C isolation reduces the maximum degree
moderately from the beginning, DA isolation has a strong but delayed
reduction, DI isolation has a moderate and delayed reduction, and hybrid (C+D)
isolation has an initial moderate reduction, followed by a delayed strong
reduction of maximum degree. (b) The effect of the edge removal ratio $e$ on
the reduction of maximum degree for each isolation strategy.

The simulation results in Figure 8a depict a representative evolution of the
maximum degree reduction, by employing the centralized and decentralized
isolation strategies for $e=0.5$. Based on their ability to mitigate
superspreaders, we notice a new signature for each strategy, which explains
why they are more or less efficient at reducing the outbreak impact.
Specifically, the C strategy (red line) reduces the maximum degree to 50% of
the initial network’s maximum degree (i.e., given by $e=0.5$) from the very
beginning ($d=0$) of the simulation. The DA and DI isolation strategies (blue
lines) have a delay of $\approx d=36-37$ days until the network responds;
after this moment, DI only reduces the maximum degree to about 62%, while DA
reduces it significantly, to only 3%. Finally, the hybrid C+D isolation
carries a signature that indicates the highest efficiency; namely, the
reduction to 50% is obtained at $d=0$, followed by a reduction to 3%, similar
to the DA strategy delay. The observations discussed for $e=0.5$, in Figure
8a, are enforced in Figure 8b, as we provide an extended overview of the
maximum degree reduction for all tested edge removal ratios $e$.

Our experimental results support the idea that the DA and C+D strategies
ensure that no superspreaders remain in the network when applying an edge
removal ratio $e\geq 0.5$. For example, on the $Bk$ network, the C+D strategy
reduces the maximum degree from $k=1134$ to $k=34$; on the $SF$ network, the
C+D strategy reduces the maximum degree from $k=294$ to $k=16$, thus limiting
the impact of hubs in the process of epidemic diffusion.

###  3.3 What may alter the course of the pandemic?

#### Reinfection impact:

An additional parameter worth investigating for our epidemic model is the
susceptibility rate $\lambda_{Susceptibility}$, which is still uncertain for
the SARS-CoV-2 virus at the end of March 2020. Throughout our simulations, we
consider a susceptibility rate of $\lambda_{Susceptibility}=0.01$. However, we
further explore the impact of isolation strategies if the recovered patients
prove to have a much higher chance of becoming reinfected (up to 10-50%).

Figure 9a shows that the pandemic dynamics is not significantly different at
higher susceptibility rates. The moment of the peak infection rate may be
delayed by up to one week, while the amplitude of the infection ratio may
increase by up to 5%. Note that these results are based solely on the
combination of COVID-19 specific parameters, and may not be directly
generalized to other past or future epidemic diseases.

Figure 9: The effect of a higher patient relapse rate ($s=1-50\%$) in terms of
infection ratio. (a) The effect of different relapse rates when using the C
isolation strategy. A higher relapse rate of $s=50\%$ increases the infection
ratio by 5% and delays it by 7 days. (b) The effect of different isolation
strategies when using a relapse rate of $s=50\%$. The combined logarithmic-
convex and linear-concave of the (C+D) hybrid strategy (yellow) explains its
higher effectiveness.

In Figure 9b, we focus on a single relapse rate of $s=50\%$ to show the impact
of each isolation strategy; the response of all isolation strategies is the
same regardless of the relapse rate. The higher efficiency of the C+D
isolation is again underlined in panel 9b. This set of experiments indicates
that even a high patient relapse rate would not alter the effectiveness of the
isolation strategies significantly.

#### Delaying isolation restrictions:

While the WHO praised China’s quick and aggressive response [7], other
countries, such as the UK or USA, have delayed their responses. We analyze
next the efficiency of all isolation strategies in the context of a delayed
implementation of quarantines/lockdowns. To this end, we compare three
scenarios when the same strategies are applied with a delay of
$d=\\{0,20,30,50\\}$ days after the initial outbreak.

Figure 10: Effects of delaying the application of isolation strategies,
relative to the start of the outbreak, measured as the maximum infection ratio
in the network. (a) Detailed comparison on the outbreak amplitude obtained for
delays of $d=\\{0,20,30,50\\}$ days, with $e=0.5$ and $e=0.75$. For $e=0.5$, a
delay of $d=20$ days increases the maximum infection ratio by 8%; a delay of
$d=30$ increases the ratio by 24%; a delay of $d=50$ increases the maximum
ratio by 76%. (b) Overview of the delays’ impact on the maximum infection
ratio with increasing values of $e$. The smaller negative impact of the
shorter delay ($d=20$, orange line) is distinguishable from the impact of the
longer delay ($d=50$, green line).

The simulation results in Figure 10a suggest that governments/authorities
should take an immediate action to enforce the isolation strategies. In
general, for moderate isolation ($e=0.5$), the outbreak amplitude increases
significantly with the delay (Figure 10a); thus, for a delay of $d=20$ days
(yellow line) we observe a +8% higher infection ratio, for $d=30$ days (green
line) we observe a +24% higher infection ratio , and for $d=50$ (blue line)
the results are alarming with +76% more infections than the immediate response
(red line). A more severe isolation ($e=0.75$) further amplifies the effects
of the delaying isolation strategies, compared to the moderate isolation. As
such, for the same delays, we measure increases in the infection ratio of +41%
(yellow dotted line), $3.4\times$ (green dotted line), respectively
$3.88\times$ higher (blue dotted line) compared to an immediate response (red
dotted line). Figure 10b presents an overview of each delay impact on the
maximum infection ratio for the entire range of tested edge removal ratios
$e=0-0.9$. The difference between the shorter delay of $d=20$ days (orange
line) and the long delay of $d=50$ days (green line) is noticeable.

#### Proactive vs. reactive isolation:

Complementary to a delayed application of isolation restrictions discussed in
the previous section, we also investigate the feasibility of a reactive policy
that is being used by some authorities around the world. In other words, some
governments are reacting to the accelerated outbreak spread by adopting
increasingly more severe quarantine measures as time goes by. In contrast to
the proactive isolation (meaning that a fixed $e$ is imposed from a specific
moment in time), we model the reactive policy by progressively increasing the
strength of isolation, from $e=0$ to $e=0.8$, at fixed moments in time. As
such, we analyze two realistic scenarios for adopting a progressive policy.
First, the edge removal ratio is increased as a delayed reaction to the
pandemic spreading, with values $e=\\{0,0.2,0.4,0.6,0.8\\}$, applied at
moments $r_{1}=\\{10,20,40,60\\}$ (faster reaction). Second, the reaction
happens at moments $r_{2}=\\{20,40,60,80\\}$ (slower reaction). This means
that, for example, during the first 20 days of the $r_{2}$ progressive policy,
we have no isolation at all (i.e., $e=0$); after day $d=20$, $e$ is increased
to $e=0.2$, and so on.

Figure 11: Comparison between proactive and reactive responses in applying
isolation measured as the maximum achieved infection ratio. (a) C isolation
strategies (proactive) with $e=0.25$ (mild isolation) and $e=0.65$ (stricter
isolation) are equivalent to reactive policies $r_{2}$ (slower reaction, every
20 days) and $r_{1}$ (faster reaction, every 10 days), respectively. (b) C+D
isolation (proactive) with $e=0.1$ (very mild isolation) and $e=0.5$ (moderate
isolation) are equivalent to reactive policies $r_{2}$ and $r_{1}$,
respectively. The arrows pointing down suggest the reduction in infection
ratio compared to the case of no isolation.

In Figure 11a we compare the reactive policies $r_{1}$ and $r_{2}$ with
roughly equivalent proactive C strategies. In this sense, we find that the
$r_{1}$ policies has an equivalent effect in reducing the impact of the
outbreak, as the C isolation with $e=0.65$ (point B). Additionally, the
$r_{2}$ policies is similar to C isolation with $e=0.25$ (point A). In other
words, if a government reacts slowly, by progressive increases of isolation
severity (e.g., policy $r_{2}$) up to $e=0.8$, the result will be the same as
if the government would have applied a more relaxed C isolation with $e=0.25$
from the very beginning of the outbreak.

In the case of the combined C+D strategy, the impact of the progressive
isolation is less effective than in the centralized context. To that end,
Figure 11b shows that policy $r_{1}$ is roughly similar to C+D isolation
(proactive) applied with $e=0.5$ (point B), respectively policy $r_{2}$ is
similar to C+D isolation with $e=0.1$ (point A). This means that, even though
$r_{2}$ eventually reaches a very strict isolation of $e=0.8$ (after 80 days),
it is not more effective than a very mild C+D isolation with $e=0.1$.

###  3.4 The impact of isolation strategies on small-scale communities

While real contact networks have been studied before in the context of
epidemic spreading [45, 46, 47], their corresponding topologies are
representative only for small-scale communities, or specific micro-
environments (e.g., schools, stadiums, airports, hospitals). However, we
cannot directly extrapolate the topological characteristics of such contact
networks at the level of a metropolis, region, or country, which we are facing
in the case of the COVID-19 pandemic.

Nevertheless, we use two real contact networks to study the impact of the same
isolation strategies. We include the Lyon school network ($LS$) with $N=242$
nodes [46], and a US high school network ($US$) with $N=788$ nodes [45]. Both
networks have been constructed using proximity data (in the range of 1-3m)
gathered from sensors worn by school staff and students.

Figure 12: The impact of the edge removal ratio $e$ in reducing and delaying
the maximum outbreak infection ratio $r$, by applying a combined (C+D)
isolation strategy on two micro-scale contact networks: Lyon school (a) and US
high school (b). On both networks, mild to moderate isolation ($e=0-0.5$) has
no effect in limiting the infectious spreading (points A); only more severe
isolation ($e>0.75$) manages to reduce the peak of the infection ratio, but
only by 4-14% ($e=0.9$; points B). The outbreak duration lengthens by roughly
14 days by increasing $e$.

We find that for such small communities the effect of all isolation strategies
is less obvious. The most significant reduction in infection ratio $r$ is
achieved, again, by the hybrid (C+D) strategy, which we depict in Figure 12a
($LS$ network), and Figure 12b ($US$ network). On both networks, the mild to
moderate isolation ($e=0-0.5$) plays almost no role in reducing the infectious
spreading (points A), with $r\approx 100\%$. A reduction of the peak infection
ratio is only visible for the severe isolation ($e>0.65$); in case of the
maximum isolation with $e=0.9$, the infection ratio drops by 14% ($LS$) and 4%
($US$) (point B). In contrast, the infection reduction on the large scale test
networks, for the same isolation strategy, is 98%, compared to no isolation at
all, similar to Singapore [14]. Furthermore, the outbreak duration on contact
networks $LS$ and $US$ is lengthened by about 14 days when increasing the edge
removal ratio from $e=0$ to $e=0.9$.

##  4 Discussion

According to our SICARS epidemic model, we find that a trade-of exists between
the the COVID-19 outbreak duration and infection rate. Specifically, applying
forms of moderate to strict quarantine can prolong the pandemic duration by
roughly +40% (Hybrid C+D) and +80% (C) days, compared no isolation at all,
where we estimate the duration of the pandemic at $\approx 100$ days. Indeed,
while a more extended quarantine may have complex economic consequences, the
invaluable benefit of applying isolation strategies is the significant
reduction of the peak infection ratio, and implicitly a reduction of
casualties.

The overall difference between the C (centralized), D (decentralized), and
hybrid (C+D) isolation strategies is summarized in Figure 13. Panel 13a
highlights the reduced efficiency of the DA and DI isolation when removing 50%
of social ties. Compared to the no isolation case, DA and DI manage to reduce
the infection amplitude by only 39%. In contrast, the efficiency of the C
isolation grows roughly linearly with $e$, while the hybrid C+D isolation
yields the best results (71% infection reduction). A more strict restriction
might be necessary for COVID-19, meaning a reduction of up to 75% social ties,
as depicted in panel 13b. In this case, the efficiency of the C isolation
remains proportional to $e$ (75%), while the combined C+D isolation offers the
most significant infection amplitude reduction of 87%. Our experimental
results suggest that by further restricting the quarantine to $e=0.9$, we can
achieve a 98% reduction in outbreak impact, comparable to the 99.3% results
obtained on the Singapore population [14].

Figure 13: Efficiency of isolation strategies, in terms of the relative
infection ratio reduction, compared to the baseline case of no isolation
(given as percentage next to each corresponding isolation strategy) when the
isolation strategy removes 50% of social ties from the network (moderate
isolation) (a), 75% of social ties (severe isolation) (b).

By analyzing different parameters which characterize the course of the
COVID-19 pandemic, we find that:

  * •

While an early response to adopting isolation strategies is critical, we find
that a delay of $d=20-30$ days has a significant negative impact (+8-25% more
infected). Higher delay values of $d=50$ days would have a catastrophic
negative impact (+76%, compared to no delay).

  * •

While the real patient relapse rate is not yet fully known (i.e., currently is
estimated at $<1\%$), we find that a higher relapse rate (e.g., 10%) would not
significantly alter the effectiveness of the isolation strategies, neither in
terms of infection amplitude, nor maximum duration.

  * •

From a network structure standpoint, the C isolation strategy mitigates the
superspreaders by immediately reducing the maximum degree in the network
(i.e., proportionally to $e$.) The D isolation also reduces the maximum degree
significantly (as low as $<10\%$), but only after a delay of $d=36-37$ days.

  * •

A reactive isolation policy which implies step-wise delayed increases of $e$,
is less effective than a proactive policy with a specific $e$ imposed from the
outbreak onset. In other words, to match the proactive policy, a reactive
policy would have to use a far more severe social distancing ($e$) over time.

As a comparison between proactive (or early) and reactive (or delayed)
measures applied in real-world settings, we exemplify the cases of China,
Singapore, or South Korea [7, 6] for the former type of response, respectively
the UK [48] or USA [41] for the latter. Indeed, the proactive policy confirms
its significant effectiveness, as both the number of infected cases and deaths
are already declining in the Asian countries mentioned above. In contrast,
current estimations of the COVID-19 progress in Western countries are less
optimistic.

There are several limitations to our analysis. On the one hand, based on
current evidence, we use biological parameters that characterize the SARS-
CoV-2 virus that are not fully confirmed in practice. For instance, we have
explored the unknown susceptibility rate (relapse probability) and found that
its effects on our conclusions are not major; however, one may update these
values as more comprehensive data surfaces. On the other hand, the entire
contact networks of large regions around the world affected by COVID-19 (e.g.,
China, Europe, or the USA) are impossible to reconstruct entirely due to their
size, dynamics, and lack of individual-level information. Similar to other
approaches [9, 10, 49], we base our study on several synthetic and real-world
datasets as reliable proxies of the contact network of individuals. By fitting
the SICARS model to multiple complex network topologies, we aim at making the
best possible assessments on the SARS-CoV-2 transmission dynamics.

Also, our study on the impact of isolation strategies extended to micro-scale
contact networks shows several interesting insights. First, due to the small
scale and high connectivity of these contact networks, the isolation
strategies have little efficiency in defending against the COVID-19 spreading.
We believe that the specific duration from contact to symptom onset,
characterizing the SARS-CoV-2 virus, renders any measures almost useless in
such small communities. The maximum possible reduction of infection ratio of
4-14% is obtained only for the most severe isolation ($e=0.9$). Second, we
observe a lengthening of the outbreak duration (by 14 days), similar to the
one visible on the large scale test networks. Third, we conclude that the
topologies of contact networks are representative only for micro-scale
communities (e.g., the Diamond Princess cruise ship quarantined in Japan, or
the Theodore Roosevelt aircraft carrier docked in Guam), and may not be
directly extrapolated at the level of large cities, regions, or countries, as
is the case of the COVID-19 pandemic.

Of note, our observations come in support of a recent scandal in the U.S. navy
involving the decision to remove infected military personnel from an aircraft
carrier. Indeed, due to the high density of the contact network aboard, and
the limited physical space making social distancing almost impossible, our
results suggest that the captain took the right decision. Indeed, without such
measures the SARS-CoV-2 virus spreading would have been super fast and may
have likely achieved an infection ratio of up to 100% in no time.

Summarizing our simulation results, the main conclusions of this study are:

  * •

The decentralized (D) isolation strategy helps222As it can be considered a
characteristic of antifragile societies that have recently experienced large-
scale emergency situations such as SARS, tsunamis, earthquakes [50, 51] etc.,
but is insufficient on its own. In other words, the decentralized methods are
not effective in mitigating the impact of the superspreaders (i.e., nodes with
a high connectivity, therefore a high contagious potential) in a timely
manner; hence, their potential for reducing the dynamics of pandemics like
COVID-19 is limited.

  * •

The centralized (C) isolation strategy is a must, as it is more efficient than
the decentralized approaches.

  * •

The best response strategy is a hybrid one (C+D). If used too late after the
outbreak of the pandemic, even the combined C+D strategy loses its
effectiveness significantly.

  * •

All isolation strategies applied proactively (at outbreak onset) are more
effective than applied reactively (in response to the epidemic dynamics) even
by progressively increasing the isolation severity.

  * •

A higher patient relapse rate than currently estimated for COVID-19 would not
alter our conclusions on the effectiveness of the isolation strategies
significantly.

With COVID-19 having reached the status of a global pandemic, understanding
the effectiveness of social distancing measures in various settings is crucial
for apprehending the dynamics of the outbreak and being able to contain or
mitigate it adequately.

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