# Path-dependent course of epidemic: are two phases of quarantine better than
one?

Varun Nimmagadda  Novi High School, Novi, MI 48375, USA  Department of
Physics, Oakland University, Rochester, MI 48309, USA  Oleg Kogan  Physics
Department, California Polytechnic State University, San Luis Obispo,
California 93407, USA  Evgeniy Khain<EMAIL_ADDRESS>Department of Physics,
Oakland University, Rochester, MI 48309, USA

###### Abstract

The importance of a strict quarantine has been widely debated during the
COVID-19 epidemic even from the purely epidemiological point of view. One
argument against strict lockdown measures is that once the strict quarantine
is lifted, the epidemic comes back, and so the cumulative number of infected
individuals during the entire epidemic will stay the same. We consider an SIR
model on a network and follow the disease dynamics, modeling the phases of
quarantine by changing the node degree distribution. We show that the system
reaches different steady states based on the history: the outcome of the
epidemic is path-dependent despite the same final node degree distribution.
The results indicate that two-phase route to the final node degree
distribution (a strict phase followed by a soft phase) are always better than
one phase (the same soft one) unless all the individuals have the same number
of connections at the end (the same degree); in the latter case, the overall
number of infected is indeed history-independent. The modeling also suggests
that the optimal procedure of lifting the quarantine consists of releasing
nodes in the order of their degree - highest first.

###### pacs:

87.19.xd, 05.40.-a, 05.10.-a

##  I Introduction

Investigating disease dynamics on networks has received much recent attention
Newman book ; Another book ; Keeling_and_Rohani ; Newman PRE ;
Kenah_and_robbins ; Recent_RMP ; When_networks ; Community_structure ;
Community_structure2 . Network paradigm has been used in the analysis of
epidemics of HIV Keeling_and_Rohani , SARS SARS , COVID-19 COVID1 , and flu
FLU . In many cases in which disease is studied on networks, each node in a
network represents a single individual and edges between the nodes represent
the contacts between people. The degree of a node is just the number of edges
connected to this node; in general, a network has nodes of various degrees,
since there are individuals with a high number of contacts and individuals
with a low number of contacts. Once the node degree distribution is specified,
one can analyze the spread of the disease by employing one of the
compartmental epidemiological models, for example, an SIR model mathbiology .
In this model, the node can be in one of three possible states: susceptible
(S), infected (I) and recovered (R). If an infected node is connected to a
susceptible node, the latter becomes infected with a certain probability (per
unit time). Using an algorithm called a configuration model, it is possible to
create a network with any desired degree distribution Newman book , Newman PRE
. Important results for the spread of an epidemic on such networks were
obtained in the framework of the SIR model Newman book . In particular, there
is an exact mapping between the spread of disease on networks and bond
percolation. Therefore, there exists a threshold: the epidemic spreads if the
transmission rate exceeds a certain value and decays otherwise. In addition,
once the epidemic spreads, one can compute the overall number of people who
catch the disease in the course of the epidemic Newman book .

This theory assumes that the network itself does not change with time, so that
people have the same number of contacts no matter how serious the epidemic
situation is. Recently, a two-way coupling between the network topology and
disease dynamics has been studied Adaptive1 , Adaptive2 . However, during the
epidemic such as COVID-19, another temporal variation of the network structure
has been crucial - one that results from from the various quarantine measures
that were introduced (and are currently in effect) in many countries across
the globe. A quarantine aims at reducing the number of contacts, and
therefore, the node degree distribution (1) differs from the normal life
situation Quarantine1  and (2) changes once one phase of the quarantine (say,
the strict quarantine) is replaced by a second phase of the quarantine (the
softer one). Recently, several authors incorporated quarantine into COVID-19
modeling Quarantine2 ; Quarantine3 ; Quarantine4 . However, these models are
not network-based. In addition, they do not address the question of the
optimal strategy of easing the quarantine in order to minimize the net number
of infected individuals - one of the central questions in the present paper.

Looking from a strictly epidemiological point of view, the necessity of a
strict quarantine has been called into question Quarantine_critique1 ;
Quarantine_critique2 ; Quarantine_critique3 . One argument against strict
lockdown measures is that once the strict quarantine is lifted, the epidemic
spread returns to full effect, resulting in the total number of infected
individuals during the entire epidemic staying the same. Some argue,
therefore, that the way Sweden handled the epidemic Sweden  might be better.
Indeed, if the overall number of infected is the same, softer quarantine
measures are more appealing rather than the two phases (a strict quarantine
followed by a soft one) from the economic and social considerations. Is this
argument correct from a purely epidemiological point of view? We test this
idea by running simulations of the SIR epidemic on a network where the node
degree distribution changes with time, following administrative decisions to
move from a strict quarantine to a soft one. We focus on the final number of
susceptible individuals (or the overall number of infected individuals
throughout the epidemic) and test different strategies of lifting the
quarantine.

##  II Equal number of connections

Let us first discuss the argument in favor of a single-phase soft quarantine.
One can argue that some quarantine is certainly needed to “flatten the curve”.
Indeed, without any quarantine measures, the instantaneous number of infected
might be so high that it would exceed the medical system capacity. However,
one continues, there is no need for a quarantine stricter than that needed to
avoid medical collapse. Since eventually the strict quarantine needs to be
lifted, the disease will be back, and the total number of people who catch the
disease in the course of epidemic remains the same. To support this argument,
one can consider the standard SIR model mathbiology  of an epidemic. The
equation for the fraction of infected individuals $I(t)$ reads

$\frac{dI}{dt}=r\,I\,S-\alpha\,I,$ (1)

where $S(t)$ is a fraction of susceptible individuals, the parameter $r$
characterises the transmission rate and the parameter $\alpha$ is the rate of
recovery. The number of infected increases if a susceptible person becomes
infected (see the first term on the right-hand side) and decreases if an
infected person recovers (see the second term on the right-hand side). From
this equation, the number of infected starts declining when $dI/dt$ becomes
negative, which happens only after the fraction of susceptible individuals
becomes smaller than some threshold $S^{*}$ given by $S^{*}=\alpha/r$. This
means that the epidemic starts decaying only after a certain fraction of
population $1-S^{*}$ (a rather big fraction in the case of COVID-19) catch the
disease. Therefore, any quarantine measures can only delay this “herd
immunity”, since after lifting the quarantine, we can not avoid infecting less
than $1-S^{*}$ fraction of the population, which often amounts to $70$ or even
$80$ percent of the population (depending on the estimates of $r$ and
$\alpha$).

Let us now test this argument, considering the SIR model on a network. In
general, modeling the spread of the disease on a network is a substantial
improvement Newman book  of the standard continuum (mean field) modeling in
two respects. The first advantage is related to the rate of recovery of an
infected individual. The continuum modeling implies a Poisson process, which
means an exponential distribution of individual disease duration. This is in
contrast to observations showing that the distribution is peaked around an
average disease duration. Network modeling can easily account for this peaked
distribution of individual disease durations. The second drawback of the
continuum modeling (that can be eliminated by using networks) is the
assumption of an equal number of contacts for each individual, i.e. ignoring
the underlying microscopic structure of the social network Newman book .

Let us first eliminate only the first drawback and consider a network where
once the quarantine is lifted, all of the individuals have an identical number
of connections. To be specific, let us consider a somewhat extreme example.
Suppose that during the strict quarantine only $10$ percent of the population
are “essential workers” and maintain a high number of contacts, $20$ per day.
The remaining population works from home, where each individual has a minimal
number of contacts, only $1$ per day. Specifying the transmission probability
(taken to be $0.01$) and the duration of the disease (taken to be $14.5$
days), we can follow the number of susceptible as a function of time. To run
the simulations, first a network with $10000$ nodes was constructed using the
configuration model. Then one node was randomly infected and the SIR model was
simulated using the standard Monte-Carlo method. The entire procedure was
repeated many times (of the order of $100$ runs) and the average values for
$S(t)$ and $I(t)$ were computed taking into account only the runs where an
outbreak did occur.

Simulation results are shown in Figure 1. The blue dash-dotted curve shows the
fraction of susceptible individuals as a function of time. After several
months (once the epidemic is almost over), the quarantine is lifted, so that
each node has the same degree $k=20$. The inset shows the fraction of infected
as a function of time, and the corresponding blue dash-dotted line describes
two waves of the epidemic: the smaller one during the quarantine and the
larger one after the quarantine is lifted. Suppose now, we handle the epidemic
differently and do not introduce the strict quarantine at all. The node degree
distribution is the same as the final node degree distribution above: each
node has the same degree $k=20$. The black solid curve shows the fraction of
susceptible individuals, while the red dotted line shows the theoretical
prediction for the final fraction of susceptible for the SIR epidemic on a
configuration model of a network (the mathematics is discussed in the next
section). One can see that not only does the epidemic end much earlier, but
also the final fraction of susceptible is almost the same as with the strict
quarantine! So, the strict quarantine did not change the total number of
people who caught the disease in the course of the epidemic. This result is in
agreement with the argument based on the continuum SIR model. Does it mean
that the strict quarantine is not needed at all? The next section shows that
this result is generally wrong and is related to the unrealistic assumption of
an equal number of contacts across the population.

Figure 1: The fraction of susceptible individuals as a function of time with
and without strict quarantine. Shown are the simulation of the SIR model on a
network (the configuration model) with $10000$ nodes. The black solid line
corresponds to the no-quarantine case, where all the nodes have the same
degree $k=20$. The blue dash-dotted line corresponds to the two-phase
scenario. In the first phase (the quarantine), $10$ percent of nodes have
degree $k=20$ and $90$ percent of nodes have degree $k=1$. After several
months (once the epidemic is almost over), the quarantine is lifted (the
second phase), and each node has the same degree $k=20$ as in the no-
quarantine scenario. The inset shows the fraction of infected as a function of
time for both scenarios. The dotted red line shows a theoretical prediction
for the final fraction of susceptible individuals for a single phase (without
the strict quarantine), see text.

##  III Distribution of the number of connections

Now we are going to eliminate the second main drawback of standard continuum
(mean-field) models and consider a network where different nodes have a
different number of connections. Again we will test the results of two ways of
handling the epidemic; importantly, the final node degree distribution is the
same in both cases. The first scenario does not imply a strict quarantine:
$90$ percent of nodes have degree $14$ and $10$ percent of nodes have degree
$50$. The second scenario has two phases, each with its own node degree
distribution. The first phase is a strict quarantine: $90$ percent of nodes
have degree $2$ and $10$ percent of nodes have degree $50$. These number of
contacts are similar to the numbers of casual contacts suggested in the
literature Marissa . The second phase corresponds to lifting the strict
quarantine: $90$ percent of nodes have degree $14$ and $10$ percent of nodes
have degree $50$ as in the first scenario. What will be the final fraction of
susceptible individuals in these two scenarios?

Figure 2: The fraction of susceptible individuals as a function of time with
and without quarantine. Shown are the simulation of the SIR model on a network
(the configuration model) with $10000$ nodes. The black solid line corresponds
to the no-quarantine scenario, where $10$ percent of nodes have a high degree
($k=50$) and $90$ percent of nodes have a moderate degree ($k=14$). The blue
dash-dotted line corresponds to the two-phase scenario. In the first phase
(the strict quarantine), $90$ percent of nodes have a low degree $k=2$ and
$10$ percent of nodes have a high degree $k=50$. In the second phase, the
quarantine is lifted, and the node degree distribution matches this of the
first scenario: $10$ percent of nodes have a high degree ($k=50$) and $90$
percent of nodes have a moderate degree ($k=14$). The inset shows the fraction
of infected as a function of time for both cases.

Figure 2 shows that the results are completely different. While the final
fraction of susceptible individuals without the strict quarantine is
$S_{f}\approx 0.16$, the final fraction of susceptible in the scenario with
quarantine is $S_{fq}\approx 0.4$, more than two times larger. This means that
the quarantine helped more than $20$ percent of population, who avoided
catching the highly unpleasant disease. This also illustrates a very
interesting point: the final steady state depends not only on the final node
degree distribution (which is the same in both scenarios), but also strongly
depends on the history.

The final fraction of susceptible individuals for a single phase (without
quarantine) can be computed theoretically. We present here only a summary of
results, and the reader is referred to Ref. Newman book  for more detail. The
calculation uses the concept of a generating function for the node degree
distribution defined as $g_{0}(z)=\sum_{k=0}^{k=\infty}p_{k}z^{k}$, where
$p_{k}$ is a fraction of nodes with degree $k$. In our case, the generating
function is given by $g_{0}(z)=0.9\,z^{m}+0.1\,z^{50}$. The generating
function for the excess degree distribution is
$g_{1}(z)=(1/\bar{k})dg_{0}/dz=(0.9\,m\,z^{m-1}+5\,z^{49})/(0.9\,m+5)$, where
$\bar{k}=0.9\,m+5$ is the average degree. Solving numerically the equation
$x=1-\phi+\phi g_{1}(x)$, we obtain $x$ and compute the final fraction of
susceptible individuals for a single phase, $S_{f}=g_{0}(x)$. Here
$\phi=1-exp(-\beta\tau)$, where we considered the following transmission rate
and disease duration: $\beta=0.01$ and $\tau=14.5$ days. The computed value of
$S_{f}$ (for $m=14$, $S_{f}=0.1624$) is shown in Figure 2 by the dotted red
line. A similar calculation for $S_{f}$ was performed for the node degree
distribution presented in Figure 1 (see the dotted red line). It is possible
to perform a theoretical calculation for the fraction of susceptible $S_{k}$
and infected $I_{k}$ nodes of a certain degree $k$ as a function of time.
Using a degree-based approximation of the SIR model Newman book , one can
write down rate equations for $S_{k}$ and $I_{k}$. These equations however, do
not accurately reproduce the results of Monte-Carlo simulations on the network
(shown in Figures 1 and 2) since they assume exponential distribution of
disease duration times delay .

Figure 3: The relative importance of the strict quarantine. Shown is
$\eta=(1-S_{f})/(1-S_{fq})$ as a function of the lower degree $m$, where the
final node degree distribution is the following: $90$ percent of nodes have a
low degree $m$ and $10$ percent of nodes have a high degree $50$.

The relative importance of the strict quarantine can be characterised by
comparing the final (steady state) fraction of susceptible individuals:
$\eta=(1-S_{f})/(1-S_{fq})$. Considering the final node degree distribution
generated by $g_{0}(z)=0.9\,z^{m}+0.1\,z^{50}$, we computed $\eta$ as a
function of the degree $m$. Figure 3 shows these numerical results. As
expected, in the limit of low $m$, $\eta$ tends to $1$, since in this limit,
phase $2$ does not differ from phase $1$. As we have shown in the previous
section, $\eta$ tends to $1$ also in the limit of high $m$, since the
quarantine has no effect for the degree distribution where all individuals
have the same number of contacts. The maximum of $\eta$ for intermediate
values of $m$ is related to the fact that lifting the quarantine does not lead
to an outbreak as most of the high degree nodes caught the disease during
phase $1$ and are immune during phase $2$. Therefore, as $m$ increases (from
$m=2$ to $m=10$), $S_{f}$ decreases, but $S_{fq}$ does not change. For larger
$m$, the second wave of epidemic becomes stronger leading to a decrease in
$S_{fq}$ and smaller values of $\eta$.

Figure 4: The fraction of susceptible individuals (the upper panel) and the
fraction of infected individuals (the bottom panel) as a function of time for
the two-phase dynamics: the first phase is a strict quarantine, and the second
phase is a softer quarantine. See text for the corresponding node degree
distributions. The quarantine is lifted when a fraction $f_{i}$ of the
population is still infected. Different curves in both panels correspond to
different timing of lifting the quarantine (different values of $f_{i}$). The
inset in the lower panel shows the final fraction of susceptible individuals
as a function of $f_{i}$, demonstrating the effect of timing.

We have also checked that the effect persists for more complicated node degree
distributions. Let us say that during the strict quarantine $90$ percent of
nodes have a low degree $2$ and $10$ percent of nodes have a high degree $50$
as before. These high-degree nodes can represent cashiers in big grocery
stores Quarantine1 . Lifting the quarantine means allowing gatherings (for
example, $70$ percent of the population increase the number of connections
from $2$ to $6$) and opening smaller stores and restaurants etc (for example,
$20$ percent of the population increase the number of connections from $2$ to
$25$). The numbers are of course quite arbitrary, but they represent the
situation qualitatively. Figure 4 shows results similar to the ones on Figure
2: the quarantine helps. In addition, we investigated the effect of timing:
when it is best to lift the quarantine. Reaching the steady state takes time,
and it is difficult to keep the population in a strict quarantine for a long
time. What happens if we lift the quarantine earlier? Suppose, the quarantine
is lifted when a certain (small) fraction of the population (let us denote it
by $f_{i}$) is still infected. The upper panel (as well as the inset in the
lower panel) of Figure 4 shows that the second wave of the epidemic is
independent of $f_{i}$ for small values of $f_{i}$ and only after a while
starts growing with $f_{i}$.

The explanation of this effect can be seen in the lower panel of Figure 4. The
strength of the second wave seems to crucially depend on the fraction of the
remaining high degree nodes. As depicted by the dotted yellow line in the
lower panel of Figure 4, the epidemic in the population of high degree nodes
reaches the steady state significantly faster (since these nodes are catching
the disease earlier). Once the steady state is reached, the final number of
susceptible individuals $S_{fq}$ stops changing (see the inset).

##  IV Summary and discussion

This work discusses the importance of a strict quarantine from the purely
epidemiological point of view. One argument in favor of it is clear: it is
called “flatten the curve” and says that some quarantine measures are needed
so that the instantaneous number of infected individuals would not exceed the
medical system capacity. However, should we really go beyond that and apply
stricter measures?

We have shown that in the case when each individual has the same number of
contacts, the strict quarantine is not necessary: the same total number of
people catch the disease during the entire duration of epidemic. However, in
reality, the node degree distribution is not strongly peaked around a certain
degree, and there are individuals with low number of connections (low degree
nodes) and individuals with a high number of connections (high degree nodes).
In this case, the two phase (or a multi-phase) procedure of the strict
quarantine followed by a softer quarantine can save a substantial fraction of
population from being infected. The effect results from the fact that
essential workers (people who still go to work during the strict quarantine)
have a high number of connections (cashiers in the big grocery stores or
medical staff are the common examples). A large number of these high degree
nodes become infected and then immune by the time the strict quarantine is
lifted, decreasing the average degree of the remaining network.

This results suggest an optimal degree-based procedure for lifting the
quarantine: high degrees go first. In practice, when the state lifts the
strict quarantine (or moves from one phase of quarantine to the next phase),
there is always a choice. One can open smaller stores (where cashiers are
high-degree nodes) or/and one can allow gatherings (which typically consist of
low degree nodes). The model suggests that the stores need to be opened first:
this way we can save many individuals (mostly low degree nodes) from being
infected.

Clearly, while qualitatively representative of real-life, these results were
obtained using a toy model with simplified node degree distributions. To make
quantitative predictions, one needs to have data on realistic node degree
distributions during different stages of quarantine. Although some attempts to
measure the network of contacts in small group experiments are known
nodedegree , to our knowledge, the real life node degree distributions on a
scale of a city are not available. Finally, in this work we did not take into
account the spatial degrees of freedom, although the travel patterns are
certainly important and are known to be affected by the epidemic travel . We
have checked however, that the main effect persists in the spatial model of
neighborhoods on the lattice Quarantine1 , and the results obtained with that
model (not shown) are similar to the ones presented in Figure 2.

###### Acknowledgements.

One of us (E.K.) is grateful to Tali Khain for fruitful discussions.

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