# A Degree Based Approximation of an SIR Model with Contact Tracing and
Isolation

Duan-Shin Lee, Ting-Zhe Liu, Ruhui Zhang, Cheng-Shang Chang D.-S. Lee and
C.-S. Chang are with the Institute of Communications Engineering, National
Tsing Hua University, Hsinchu 30013, Taiwan, R.O.C. D.-S. Lee, R. Zhang and
T.-Z. Liu are with the Institute of Computer Science, National Tsing Hua
University. (Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>peter91015@gmail.com.)This research was supported
in part by the Ministry of Science and Technology, Taiwan, R.O.C., under
Contract 109-2221-E-007-093-MY2.

###### Abstract

In this paper we study a susceptible infectious recovered (SIR) model with
asymptomatic patients, contact tracing and isolation on a configuration
network. Using degree based approximation, we derive a system of differential
equations for this model. This system can not be solved analytically. We
present an early-time analysis for the model. The early-time analysis produces
an epidemic threshold. On one side of the threshold, the disease dies out
quickly. On the other side, a significant fraction of population are infected.
The threshold only depends on the parameters of the disease, the mean access
degree of the network, and the fraction of asymptomatic patients. The
threshold does not depend on the parameter of contact tracing and isolation
policy. We present an approximate analysis which greatly reduces computational
complexity. The nonlinear system derived from the approximation is not almost
linear. We present a stability analysis for this system. We simulate the SIR
model with contact tracing and isolation on five real-world networks.
Simulation results show that contact tracing and isolation are useful to
contain epidemics.

###### Index Terms:

degree based approximation, SIR, contact tracing, isolation, asymptomatic,
stability analysis

##  1 Introduction

In the early stage of an outbreak when there are not many infected
individuals, contact tracing, quarantine and isolation is an effective way to
contain an infectious disease [1, 2, 3, 4, 5, 6, 7]. A succinct example is
Taiwan. Taiwan is in close proximity of China and has a large portion of its
populations residing and working in China. It was expected that Taiwan
suffered from a major epidemic soon after the COVID-19 outbreak started in
December 2019. It turns out that Taiwan had a relatively small number of
infected individuals for a long period of time until April 2022. During this
period of time, Taiwan successfully contained COVID-19 not by extreme measures
such as city lock-downs, but by enforcing regulations on contact tracing,
quarantine and isolation [8].

Contact tracing and isolation as a measure to contain COVID-19 has been
studied by many researchers. Hou et al. [1] studied the effectiveness of
quarantining Wuhan city against COVID-19. They used a mixed “susceptible
exposed infectious recovered” (SEIR) compartmental model, in which some
infected patients are asymptomatic. Hou et al. showed that, by reducing the
contact rate of latent individuals, interventions such as quarantine and
isolation can effectively reduce the potential peak number of COVID‐19
infections and delay the time of peak infection. Aleta and et al. [2] built a
synthetic population network of the Boston metropolitan area in the United
States from mobile devices and census data. Aleta and et al. performed
simulations to show that robust level of testing, contact-tracing and
household quarantine could keep the disease within the capacity of the
healthcare system while enabling the reopening of economic activities after a
period of strict social distancing control of the COVID-19 epidemic. Kucharski
et al. [3] built a social contact graph using BBC Pandemic data from 40162 UK
participants. The authors simulated the effect of a range of different
testing, isolation, tracing, and physical distancing scenarios. Hellewell [4]
established a contact graph and used simulations to quantify the effectiveness
of contact tracing and isolation of cases at controlling COVID-19.

To our knowledge all studies on contact tracing and isolation were based on
empirical, statistical analysis or simulation studies. We consider a
susceptible-infectious-recovered (SIR) model with contact tracing and
isolation on a social contact graph. In this paper we assume that the social
contact graph is a configuration model [9] and apply a degree based
approximation [10, 11, 12, 13] to analyze this model. Suppose that the maximum
degree of vertices in the graph is $K_{\mbox{max}}$. The degree based
approximation leads to a system of differential equations of size
$5(1+K_{\mbox{max}})$. This system can not be solved analytically. We present
an early-time analysis of the system assuming that time is small. We obtain an
epidemic threshold. On one side of the threshold, the disease dies out. On the
other side, a significant fraction of vertices are infected with the disease.
Interestingly, the threshold only depends on the infection rate and recovering
rate of the disease, the mean access degree of the network, and the fraction
of asymptomatic patients. The threshold does not depend on the parameter of
contact tracing and isolation policy. This implies that contact tracing and
isolation can not prevent a disease from spreading widely. However, it
controls the size of the epidemic, if the disease spreads widely. The early-
time analysis is accurate only when the time is small. We also propose an
approximation that reduces the complexity from $5(1+K_{\mbox{max}})$ equations
to five equations. Through numerical studies, we show that the approximation
method works very well. We perform a stability analysis for the nonlinear
system derived from the approximation. This nonlinear system is not almost
linear and the general stability theory for almost linear systems can not be
applied. We present an analysis for the stability of this nonlinear system.

From numerical studies, we show that the contact tracing and isolation is
effective in containing epidemics. However, it comes at a cost. With strict
isolation policies, a significant fraction of susceptible population is
isolated. This can be detrimental to the function of a society, as the work of
isolated people (such as police, fire fighters, garbage collectors and etc.)
must be taken over by someone else. Our numerical results show that a strict
policy can isolate a large fraction of susceptible individuals. The
configuration network that we model the social contact graph is mathematically
convenient. However, it suffers from a few disadvantages. For instance, its
clustering coefficients and degree correlations are very small. It is known
that network clustering has a strong impact to the epidemic [14, 15, 16, 17,
18]. We simulate the SIR model with more realistic contact-tracing and
isolation policies using five real-world networks. We show that contact
tracing and isolation can effectively contain the epidemic.

The outline of this paper is as follows. In Section 2 we introduce an SIR
model with contact tracing and isolation. We derive a set of differential
equations based on degree approximation. In Section 3 we present an early time
analysis of the set of differential equations. In Section 4 we present an
approximate analysis of the set of differential equations. In Section 6 we
present the numerical and simulation results. We present the conclusions of
this paper in Section 7.

##  2 SIR model with contact tracing

We consider a susceptible-infectious-recovered (SIR) model with contact
tracing and isolation. For many diseases, including COVID-19, it is possible
that an infected individual shows no symptoms and yet is capable to transmit
the disease to his/her contacts [19, 20, 21, 22]. Such individuals are called
asymptomatic individuals [23]. An asymptomatic individual has the ability to
transmit the disease to his/her contacts. However, an asymptomatic individual
shows no symptoms and can hardly be detected, unless he or she is traced
because his or her contacts show symptoms. This feature makes the disease
difficult to contain.

Suppose that a susceptible individual is infected with the disease. With
probability $1-\alpha$, this individual is asymptomatic and is in the infected
and asymptomatic state. He or she has the ability to infect his or her
neighbors in the contact graph. On the other hand, with probability $\alpha$
this individual shows symptoms and is isolated right away. In addition, a
fraction $\eta$ of his or her neighbors in the contact graph are quarantined
and isolated. An isolated individual has a rate of $\gamma_{1}$ to be released
from isolation. A susceptible individual has a rate of $\beta$ to contract the
disease from an infected and asymptomatic neighbor. An infected individual,
isolated or not, has a rate of $\gamma$ to enter a recovered state. The
following is a list of states, in which an individual can be.

  * •

Susceptible: An individual is susceptible if the individual is healthy and can
contract the disease if the individual is in touch with an infected
individual.

  * •

Infected and asymptomatic: An individual is infected and shows no symptoms.

  * •

Susceptible and quarantined: A susceptible individual is isolated, because
this individual is in touch with an exposed individual. An isolated and
susceptible individual can not contract the disease, since they are isolated.

  * •

Infected and quarantined: An isolated individual is also infected.

  * •

Recovered: An individual is in the recovered state, if the individual is
recovered from the disease, or is killed by the disease.

A schematic diagram that shows state transitions is displayed in Figure 1.

Figure 1: A transition diagram of an SIR model with contact tracing and
isolation. Spontaneous transitions are shown in solid arrows. Dashed arrows
denote transitions that are caused by spontaneous transitions. Specifically,
as a susceptible individual becomes infected, a fraction $\eta$ of his or her
neighbors are tested. His or her susceptible neighbors (resp. asymptomatic
neighbors) are isolated. This causes a dashed transition from the susceptible
state to the susceptible and quarantined state (resp. infected and quarantined
state). The transition rates of spontaneous transitions are shown besides the
corresponding arrows.

We apply a degree-based approximation to the SIR model with contact tracing
and isolation. Degree-based approximation is a well known technique to study
epidemic spreading in a contact graph [10, 11, 12, 13]. The following is a
list of variables that will be used in the analysis. Note that excess degrees
are defined to be the degree of a vertex reached by traversing an edge. The
excess degree does not include the traversed edge.

  * •

$s_{k}(t)$: The fraction of nodes that are in state Susceptible at time $t$
among all nodes that have excess degree $k$.

  * •

$q^{S}_{k}(t)$: The fraction of nodes that are in state Susceptible and
quarantined at time $t$ among all nodes with excess degree $k$.

  * •

$x_{k}(t)$: The fraction of nodes that are in state Infected and asymptomatic
at time $t$ among all nodes with excess degree $k$.

  * •

$q^{I}_{k}(t)$: The fraction of nodes that are in state Infected and
quarantined at time $t$ among all nodes with excess degree $k$.

  * •

$r_{k}(t)$: The fraction of nodes with excess degree $k$ that are in the
recovered state.

We remark that the quantities above satisfy an equality

$s_{k}(t)+q^{S}_{k}(t)+x_{k}(t)+q^{I}_{k}(t)+r_{k}(t)=1$ (1)

for all $k$ and all $t$.

Let $\\{p_{k},k=0,1,2,\ldots\\}$ be the degree distribution of a randomly
selected vertex. Let $\\{w_{k},k=0,1,2,\ldots\\}$ be the excess degree
distribution of a vertex reached by traversing a randomly selected edge. That
is, if one traverses along a randomly selected edge from one node to a
neighbor of the node, then $w_{k}$ is the probability that this neighbor has
$k$ edges, not including the traversed edge. Let $g_{0}(x)$ and $g_{1}(x)$ be
the probability generating functions of degree distribution $\\{p_{k},k\geq
0\\}$ and excess degree distribution $\\{w_{k},k\geq 0\\}$, respectively. That
is,

$\displaystyle g_{0}(x)$ $\displaystyle=\sum_{k=0}^{\infty}p_{k}x^{k},$ (2)
$\displaystyle g_{1}(x)$ $\displaystyle=\sum_{k=0}^{\infty}w_{k}x^{k}.$ (3)

Let $K_{0}$ and $K_{1}$ denote the expectation of distributions of
$\\{p_{k},k\geq 0\\}$ and $\\{w_{k},k\geq 0\\}$, respectively. That is,

$\displaystyle K_{0}$
$\displaystyle=\sum_{k=0}^{\infty}kp_{k}=g_{0}^{\prime}(1),$ (4)
$\displaystyle K_{1}$
$\displaystyle=\sum_{k=0}^{\infty}kw_{k}=g_{1}^{\prime}(1).$ (5)

Let

$v(t)=\sum_{k=0}^{\infty}w_{k}x_{k}(t).$ (6)

Then, $v(t)$ is the probability that a vertex reached by traversing a randomly
selected edge is infected and not isolated. We now derive a system of
differential equations that link $s_{k}(t)$, $q^{S}_{k}(t)$, $x_{k}(t)$,
$q^{I}_{k}(t)$ and $r_{k}(t)$. First,

$\frac{d}{dt}s_{k}(t)=-\beta
kvs_{k}-\sum_{j=0}^{\infty}\left[\alpha\sum_{\ell=1}^{\infty}\beta\ell
vs_{\ell}p_{\ell}\right]p_{j}j\eta s_{k}+\gamma_{1}q^{S}_{k}.$ (7)

The left hand side of (7) is the rate of change in the fraction of susceptible
nodes with degree $k$. The first term on the right side is the rate of change
in the fraction of susceptible nodes with degree $k$ due to infected but not
isolated neighbors. Each of such a susceptible node has $k$ neighbors. Each
neighbor is infected with probability $v(t)$ which is defined in (6). We thus
have the first term on the right side. The first term also appears in the
degree based approximation of the standard SI, SIR and SIS models. We refer
the reader to Newman [9, p. 659, 665 and 671]. The fraction of vertices that
have degree $k$, are isolated and susceptible is $q^{S}_{k}(t)$. With rate
$\gamma_{1}$, isolated and susceptible individuals are released from isolation
and return to susceptible state. Thus, we have the third term on the right
side of (7). We now explain the second term. The fraction of newly infected
individuals at time $t$ that have degree $k$ is $\beta kvs_{k}$. Thus, the
total fraction of newly infected individuals at time $t$ is

$\sum_{k=1}^{\infty}\beta kvs_{k}p_{k}.$

Among those who are newly infected, a fraction $\alpha$ of individuals show
symptoms and are quarantined and isolated. A fraction $1-\alpha$ of
individuals are asymptomatic and are not isolated. Thus, the fraction of newly
infected individuals who show symptoms is

$\alpha\sum_{k=1}^{\infty}\beta kvs_{k}p_{k}.$

Those who show symptoms will be quarantined and isolated. The fraction of
newly infected individuals who are isolated and have degree $j$ is

$\left(\alpha\sum_{k=1}^{\infty}\beta kvs_{k}p_{k}\right)p_{j}.$

Each individual in the preceding quantity has $j$ neighbors. Each neighbor is
isolated with probability $\eta$. Thus, the fraction of newly isolated
individuals is

$\sum_{j=0}^{\infty}\left(\alpha\sum_{k=1}^{\infty}\beta
kvs_{k}p_{k}\right)p_{j}j\eta.$ (8)

These isolated neighbors are susceptible and have degree $\ell$ with
probability $s_{\ell}(t)$. Thus, the fraction of newly isolated and
susceptible individuals who have degree $\ell$ is

$\sum_{j=0}^{\infty}\left(\alpha\sum_{k=1}^{\infty}\beta
kvs_{k}p_{k}\right)p_{j}j\eta s_{\ell}(t).$ (9)

This is the second term on the right side of (7).

With similar arguments one can derive differential equations for
$q^{S}_{k}(t)$, $x_{k}(t)$, $q^{I}_{k}(t)$ and $r_{k}(t)$. They are listed as
follows.

$\displaystyle\frac{d}{dt}q^{S}_{k}(t)$
$\displaystyle=\sum_{j=0}^{\infty}\left[\alpha\sum_{\ell=1}^{\infty}\beta\ell
vs_{\ell}p_{\ell}\right]p_{j}j\eta s_{k}-\gamma_{1}q^{S}_{k},$ (10)
$\displaystyle\frac{d}{dt}x_{k}(t)$ $\displaystyle=(1-\alpha)\beta
kvs_{k}-\sum_{j=0}^{\infty}\left[\alpha\sum_{\ell=1}^{\infty}\beta\ell
vs_{\ell}p_{\ell}\right]p_{j}j\eta x_{k}$ $\displaystyle\quad-\gamma x_{k},$
(11) $\displaystyle\frac{d}{dt}q^{I}_{k}(t)$ $\displaystyle=\alpha\beta
kvs_{k}+\sum_{j=0}^{\infty}\left[\alpha\sum_{\ell=1}^{\infty}\beta\ell
vs_{\ell}p_{\ell}\right]p_{j}j\eta x_{k}$ $\displaystyle\quad-\gamma
q^{I}_{k},$ (12) $\displaystyle\frac{d}{dt}r_{k}(t)$ $\displaystyle=\gamma
x_{k}+\gamma q^{I}_{k}.$ (13)

The derivation of the second term on the right side of (11) and (12) is
similar. The fraction of newly isolated individuals is given in (8). The
fraction of isolated individuals who are infected (asymptomatically) and have
degree $\ell$ is the quantity in (8) multiplied with $x_{\ell}(t)$. This gives
the second term on the right side of (11) and (12). We can simplify the second
term on the right side of (7) and (11), respectively. Eqs. (7) and (10)-(12)
become

$\displaystyle\frac{d}{dt}s_{k}(t)$ $\displaystyle=-\beta kvs_{k}-\alpha\beta
vK_{0}\eta\left[\sum_{\ell=1}^{\infty}\ell
s_{\ell}p_{\ell}\right]s_{k}+\gamma_{1}q^{S}_{k},$ (14)
$\displaystyle\frac{d}{dt}q^{S}_{k}(t)$ $\displaystyle=\alpha\beta
vK_{0}\eta\left[\sum_{\ell=1}^{\infty}\ell
s_{\ell}p_{\ell}\right]s_{k}-\gamma_{1}q^{S}_{k},$ (15)
$\displaystyle\frac{d}{dt}x_{k}(t)$ $\displaystyle=(1-\alpha)\beta
kvs_{k}-\alpha\beta vK_{0}\eta\left[\sum_{\ell=1}^{\infty}\ell
s_{\ell}p_{\ell}\right]x_{k}-\gamma x_{k},$ (16)
$\displaystyle\frac{d}{dt}q^{I}_{k}(t)$ $\displaystyle=\alpha\beta
kvs_{k}+\alpha\beta vK_{0}\eta\left[\sum_{\ell=1}^{\infty}\ell
s_{\ell}p_{\ell}\right]x_{k}-\gamma q^{I}_{k},$ (17)

where $K_{0}$ defined in (4) denotes the mean degree of a randomly selected
vertex. Unlike the differential equations for the classical SIR model which
can be analytically solved, the presence of the terms such as the second term
on the right side of (14) makes (14)-(17) and (13) very difficult. They can
only be solved numerically. Once $\\{s_{k}(t),0\leq k<\infty\\}$ are obtained,
one can calculate

$s(t)\stackrel{{\scriptstyle\scriptstyle\rm
def}}{{=}}\sum_{k=0}^{\infty}p_{k}s_{k}(t).$ (18)

This is the probability that a randomly selected vertex is susceptible.
Probabilities that a randomly selected vertex is infected and quarantined,
susceptible and quarantined, infected but asymptomatic, or recovered can be
obtained similarly.

In Section 3, we present an early-time analysis. Numerical solution of the
system is also difficult. Suppose that we truncate the excess degree
distribution to $[0,K_{\mbox{\small max}}]$. For each degree value, there are
five differential equations. Thus, the dimension of the system in (7) and
(10)-(13) is $5(1+K_{\mbox{\small max}})$. We propose an approximate analysis
in Section 4 to reduce the numerical complexity.

##  3 Early Time Analysis

In this section we present an early time analysis of the system in (7),
(10)-(13).

We assume that initially there is small number of infected individuals and
that most individuals are susceptible. That is, we assume that

$\displaystyle s_{k}(0)=1-\epsilon,\quad x_{k}(0)=\epsilon,$ $\displaystyle
q^{S}_{k}(0)=q^{I}(0)=r(0)=0,$

for some small number $\epsilon\approx 0$. Substituting $s_{k}(t)\approx
1-\epsilon$ in Eq. (16), multiplying the two sides of (16) with $w_{k}$ and
summing from $k=0$ to infinity, we obtain

$\frac{dv}{dt}=[(1-\alpha)\beta
g_{1}^{\prime}(1)(1-\epsilon)-\gamma]v-\alpha\beta\eta
K_{0}^{2}(1-\epsilon)v^{2}.$

This differential equation is separable and can be rewritten as

$\frac{dv}{v(c_{2}-c_{1}v)}=dt,$ (19)

where

$\displaystyle c_{1}$ $\displaystyle=\alpha\beta\eta K_{0}^{2}(1-\epsilon),$
(20) $\displaystyle c_{2}$ $\displaystyle=(1-\alpha)\beta
K_{1}(1-\epsilon)-\gamma,$ (21)

and $K_{1}=g_{1}^{\prime}(1)$ is defined in (5). The solution of (19) is

$v(t)=\frac{c_{2}D_{1}e^{c_{2}t}}{1+c_{1}D_{1}e^{c_{2}t}},$ (22)

where $D_{1}$ is a constant determined by the initial condition
$v(0)=\epsilon$, i.e.

$D_{1}=\frac{\epsilon}{c_{2}-c_{1}\epsilon}.$

If $c_{2}<0$, $v(t)\to 0$ as $t\to\infty$. In this case, the epidemic dies
down. On the other hand, if $c_{2}>0$, $v(t)\to c_{2}/c_{1}$ as $t\to\infty$.
In this case, a significant fraction of population will be infected with the
disease. Since $\epsilon$ is arbitrary, the condition for the epidemic to die
down is

$(1-\alpha)\beta K_{1}<\gamma$

and the condition for the epidemic to grow significantly is

$(1-\alpha)\beta K_{1}>\gamma.$

Define

$R_{0}=\frac{(1-\alpha)\beta K_{1}}{\gamma}.$ (23)

The epidemic threshold is $R_{0}=1$, which separates the shrinking and growing
regimes of the epidemic. $R_{0}$ is the basic reproduction number of the
disease. It is the expected number of asymptomatic patients passed on by an
infected individual during his/her infection. It is interesting to see that
$R_{0}$ depends only on the parameters of the disease, and not on parameters
$\eta$ and $\gamma_{1}$ of the contact tracing and isolation policy.

##  4 Approximate Analysis

In this section we present an approximation to reduce the dimension of the
system in (7) and (10)-(13). We assume that

$s_{k}(t)=(u(t))^{k}$ (24)

for some unknown function $u(t)$. Note that for the classical SI model and the
SIR model, (24) holds for function $u(t)$ that depends on $v(t)$ and
$\\{r_{k}(t),k\geq 0\\}$, respectively. We refer the reader to (17.60) and
(17.87) in Newman [9]. In our model, (24) is only an approximation. However,
it greatly simplifies the second term on the right side of (14) that was
caused by contact tracing and isolation.

Define

$\displaystyle q^{S}(t)$ $\displaystyle=\sum_{k=0}^{\infty}w_{k}q^{S}_{k}(t),$
(25) $\displaystyle q^{I}(t)$
$\displaystyle=\sum_{k=0}^{\infty}w_{k}q^{I}_{k}(t),$ (26) $\displaystyle
r(t)$ $\displaystyle=\sum_{k=0}^{\infty}w_{k}r_{k}(t).$ (27)

Substituting (24) into (7), one gets

$ku^{k-1}\frac{du}{dt}=-\beta kvu^{k}-\alpha\beta
v\left(\sum_{\ell=1}^{\infty}\ell u^{\ell}p_{\ell}\right)K_{0}\eta
u^{k}+\gamma_{1}q^{S}_{k}.$ (28)

Since

$\displaystyle\sum_{\ell=1}^{\infty}\ell u^{\ell}p_{\ell}$
$\displaystyle=u\sum_{\ell=1}^{\infty}\ell u^{\ell-1}p_{\ell}$
$\displaystyle=u\frac{d\left(\sum_{\ell=1}^{\infty}u^{\ell}p_{\ell}\right)}{du}$
$\displaystyle=ug_{0}^{\prime}(u),$

Eq. (28) becomes

$ku^{k-1}\frac{du}{dt}=-\beta kvu^{k}-\alpha\beta vK_{0}\eta
ug_{0}^{\prime}(u)u^{k}+\gamma_{1}q^{S}_{k}.$

Multiplying the preceding with $w_{k}$, summing from $k=0$ to infinity and
manipulating algebraically, one gets

$\displaystyle\frac{du}{dt}$ $\displaystyle=-\beta vu-\alpha\beta vK_{0}\eta
ug_{0}^{\prime}(u)\frac{g_{1}(u)}{g_{1}^{\prime}(u)}+\gamma_{1}\frac{q^{S}}{g_{1}^{\prime}(u)}.$
(29)

Similarly, we multiply the two sides of (10)-(13) with $w_{k}$ and sum from
$k=0$ to infinity. We obtain

$\displaystyle\frac{dq^{S}}{dt}$ $\displaystyle=\alpha\beta K_{0}\eta
vug_{1}(u)g_{0}^{\prime}(u)-\gamma_{1}q^{S},$ (30)
$\displaystyle\frac{dv}{dt}$ $\displaystyle=(1-\alpha)\beta
vug_{1}^{\prime}(u)-\alpha\beta K_{0}\eta v^{2}ug_{0}^{\prime}(u)-\gamma v,$
(31) $\displaystyle\frac{dq^{I}}{dt}$ $\displaystyle=\alpha\beta
vug_{1}^{\prime}(u)+\alpha\beta K_{0}\eta v^{2}ug_{0}^{\prime}(u)-\gamma
q^{I},$ (32) $\displaystyle\frac{dr}{dt}$ $\displaystyle=\gamma(v+q^{I}).$
(33)

Eqs. (29), (30)-(33) form a nonlinear system of differential equations for
unknown functions $u(t)$, $q^{S}(t)$, $v(t)$, $q^{I}(t)$ and $r(t)$ subject to
initial conditions $u(0),v(0),q^{S}(0),q^{I}(0)$ and $r(0)$. In Section 6 we
numerically solve this system and compare with the numerical solution of the
system in (14)-(17) and (13).

Once functions $u(t)$ and $v(t)$ are obtained, we can determine sequences of
functions $s_{k}(t)$, $q^{S}_{k}(t)$, $x_{k}(t)$, $q^{I}_{k}(t)$ and
$r_{k}(t)$ for $k\geq 0$ in a straight forward manner. First, $s_{k}(t)$ can
be directly obtained in (24) using $u(t)$. Differential equations involving
with $q^{S}_{k}(t)$ and $q^{I}_{k}(t)$ are linear and can be solving using the
integrating factor technique. Function $r_{k}(t)$ can be obtained by a direct
integration.

##  5 Stability Analysis

In this section we present a stability analysis of the nonlinear system in
(29) and (30)-(33). Note that once functions $u(t)$, $q^{S}(t)$ and $v(t)$ are
obtained, functions $q^{I}(t)$ and $r(t)$ can be obtained using the technique
of integrating factors and direct integration, respectively. Thus, one just
needs to consider the nonlinear system in (29), (30) and (31), in which there
are only three unknown functions. In order to facilitate our presentation in
matrix and vector forms, we use symbols $y_{1},y_{2}$, and $y_{3}$ to denote
$u,q^{S}$, and $v$, respectively. Thus, $y_{1}(t),y_{2}(t)$ and $y_{3}(t)$
denote $u(t),q^{S}(t)$ and $v(t)$, respectively. We rewrite (29), (30) and
(31) using new symbols, i.e.

$\displaystyle y_{1}^{\prime}$ $\displaystyle=-\beta y_{3}y_{1}-\alpha\beta
K_{0}\eta
y_{3}y_{1}g_{0}^{\prime}(y_{1})\frac{g_{1}(y_{1})}{g_{1}^{\prime}(y_{1})}+\gamma_{1}\frac{y_{2}}{g_{1}^{\prime}(y_{1})},$
(34) $\displaystyle y_{2}^{\prime}$ $\displaystyle=\alpha\beta K_{0}\eta
y_{3}y_{1}g_{1}(y_{1})g_{0}^{\prime}(y_{1})-\gamma_{1}y_{2},$ $\displaystyle
y_{3}^{\prime}$ $\displaystyle=(1-\alpha)\beta
y_{3}y_{1}g_{1}^{\prime}(y_{1})-\alpha\beta K_{0}\eta
y_{3}^{2}y_{1}g_{0}^{\prime}(y_{1})-\gamma y_{3}.$

Denote the right side of the equations above by $f_{1},f_{2}$ and $f_{3}$,
respectively. We have

$\displaystyle y_{1}^{\prime}$ $\displaystyle=f_{1}(y_{1},y_{2},y_{3}),$
$\displaystyle y_{2}^{\prime}$ $\displaystyle=f_{2}(y_{1},y_{2},y_{3}),$ (35)
$\displaystyle y_{3}^{\prime}$ $\displaystyle=f_{3}(y_{1},y_{2},y_{3}).$

The preceding can be expressed in terms of vectors, i.e.

${\mbox{\boldmath$x$}}^{\prime}={\mbox{\boldmath$f$}}({\mbox{\boldmath$x$}}),$
(36)

where we use boldface letters to denote vectors and matrices. We are
interested in the stability of equilibrium point
$(y_{1},y_{2},y_{3})=(\xi,0,0)$. Recall the definition of the Jacobian matrix
of a system in the form of (36). The $(i,j)$ entry of the Jacobian matrix is
defined to be

$\left({\mbox{\boldmath$J$}}(\xi,0,0)\right)_{i,j}=\left.\frac{\partial
f_{i}}{\partial x_{j}}\right|_{{\mbox{\boldmath\scriptsize$x$}}=(\xi,0,0)}.$

The Jacobian matrix of the functions on the right side of (35) evaluated at
$(\xi,0,0)$ is

$\displaystyle{\mbox{\boldmath$J$}}(\xi,0,0)=$
$\displaystyle\left(\begin{array}[]{ccc}0&\gamma_{1}/g_{1}^{\prime}(\xi)&-\beta\xi-\alpha\beta
K_{0}\eta\xi g_{0}^{\prime}(\xi)g_{1}(\xi)/g_{1}^{\prime}(\xi)\\\
0&-\gamma_{1}&\alpha\beta K_{0}\eta\xi g_{0}^{\prime}(\xi)g_{1}(\xi)\\\
0&0&(1-\alpha)\beta\xi g_{1}^{\prime}(\xi)-\gamma\end{array}\right).$ (40)

Since ${\mbox{\boldmath$J$}}(\xi,0,0)$ is singular, the nonlinear system is
not almost linear in the neighborhood of $(\xi,0,0)$. Thus, the general
stability theory of almost linear systems can not be applied here. In
addition, in almost linear systems equilibrium points are isolated. In our
problem, equilibrium points are not isolated. Any point on the $y_{1}$ axis is
an equilibrium point. These characteristics make each nonlinear system with
singular Jacobian matrices unique. Each problem needs a dedicated analysis. To
see that our nonlinear system with Jacobian matrix in (40) is not almost
linear, note that since the Jacobian matrix is singular, one can apply
elementary row operations to convert all entries in the first row to zero.
Thus, the dominant terms in the first equation are not linear. Rather, the
right side of the first equation is dominated by quadratic terms.

Since probability generating functions are power series, functions
$f_{i},i=1,2,3,$ have continuous derivatives of all orders. Thus, one can
apply Taylor expansion to $f_{i},i=1,2,3,$ around the equilibrium point
${\mbox{\boldmath$x$}}_{e}=(\xi,0,0)^{T}$, where symbol $T$ denotes the
transpose operation. For functions $f_{2}$ and $f_{3}$, we keep only the
linear terms in their Taylor expansions. For function $f_{1}$, we keep both
the linear term and the quadratic term. In matrix form, we have

${\mbox{\boldmath$x$}}^{\prime}={\mbox{\boldmath$J$}}({\mbox{\boldmath$x$}}_{e})({\mbox{\boldmath$x$}}-{\mbox{\boldmath$x$}}_{e})+\left[\begin{array}[]{c}({\mbox{\boldmath$x$}}-{\mbox{\boldmath$x$}}_{e})^{T}{\mbox{\boldmath$H$}}({\mbox{\boldmath$x$}}_{e})({\mbox{\boldmath$x$}}-{\mbox{\boldmath$x$}}_{e})\\\
0\\\ 0\end{array}\right],$ (41)

where ${\mbox{\boldmath$H$}}({\mbox{\boldmath$x$}}_{e})$ is the Hessian matrix
evaluated at point ${\mbox{\boldmath$x$}}_{e}$. The $(i,j)$ entry of
${\mbox{\boldmath$H$}}({\mbox{\boldmath$x$}}_{e})$ is

$\frac{\partial^{2}f_{1}({\mbox{\boldmath$x$}})}{\partial x_{i}\partial
x_{j}},\qquad 1\leq i,j\leq 3.$

It is straight forward to show that the Hessian matrix of the function on the
right side of (34) is

${\mbox{\boldmath$H$}}({\mbox{\boldmath$x$}}_{e})=\left(\begin{array}[]{ccc}0&A&B\\\
A&0&0\\\ B&0&0\end{array}\right),$ (42)

where $A$ and $B$ are defined as

$\displaystyle A$
$\displaystyle=\frac{-\gamma_{1}g_{1}^{\prime\prime}(\xi)}{(g_{1}^{\prime}(\xi))^{2}},$
$\displaystyle B$ $\displaystyle=-\beta-\alpha\beta
K_{0}\eta\left(g_{0}^{\prime}(\xi)\frac{g_{1}(\xi)}{g_{1}^{\prime}(\xi)}+\xi
g_{0}^{\prime\prime}(\xi)\frac{g_{1}(\xi)}{g_{1}^{\prime}(\xi)}+\right.$
$\displaystyle\quad\left.+\xi
g_{0}^{\prime}(\xi)\frac{(-1)g_{1}^{\prime\prime}(\xi)g_{1}(\xi)}{(g_{1}^{\prime}(\xi))^{2}}+\xi
g_{0}^{\prime}(\xi)\right).$

Typically, one considers a “translated system” of a nonlinear system in order
to simplify notations. That is, let

$\tilde{\mbox{\boldmath$x$}}={\mbox{\boldmath$x$}}-{\mbox{\boldmath$x$}}_{e}.$

Rewrite (41) in terms of $\tilde{\mbox{\boldmath$x$}}$, i.e.

$\tilde{\mbox{\boldmath$x$}}^{\prime}={\mbox{\boldmath$J$}}({\mbox{\boldmath$x$}}_{e})\tilde{\mbox{\boldmath$x$}}+\left[\begin{array}[]{c}\tilde{\mbox{\boldmath$x$}}^{T}{\mbox{\boldmath$H$}}({\mbox{\boldmath$x$}}_{e})\tilde{\mbox{\boldmath$x$}}\\\
0\\\ 0\end{array}\right],$ (43)

Note that the equilibrium point ${\mbox{\boldmath$x$}}_{e}$ is translated to
the origin in the translated system.

Now we consider system (43) with initial condition

$(\tilde{y}_{1}(0),\tilde{y}_{2}(0),\tilde{y}_{3}(0))=(\epsilon,\epsilon,\epsilon)$

for some small positive number $\epsilon$. Note the upper triangular form of
the Jacobian matrix in (40). We solve $\tilde{x}_{i}(t)$ successively starting
from $i=3$. From (43) and (40), we have

$\tilde{y}_{3}^{\prime}=a\tilde{y}_{3},$ (44)

where

$a=(1-\alpha)\beta\xi g_{1}^{\prime}(\xi)-\gamma.$ (45)

Thus,

$\tilde{y}_{3}(t)=\epsilon e^{at}.$ (46)

If $a<0$, $\tilde{y}_{3}(t)\to 0$. If $a>0$, $\tilde{y}_{3}(t)\to\infty$. Now
we substitute (46) into the second equation in (43) and obtain

$\tilde{y}_{2}^{\prime}=-\gamma_{1}\tilde{y}_{2}+\alpha\beta K_{0}\eta\xi
g_{0}^{\prime}(\xi)g_{1}(\xi)\epsilon e^{at}.$

This is a linear differential equation and can be solved using the technique
of integrating factors. The solution is

$\tilde{y}_{2}(t)=(he^{at}+(1-h)e^{-\gamma_{1}t})\epsilon,$ (47)

where

$\displaystyle h$ $\displaystyle=\frac{\alpha\beta K_{0}\eta\xi
g_{0}^{\prime}(\xi)g_{1}(\xi)}{a+\gamma_{1}}.$

From (47), if $a<0$, $\tilde{y}_{2}(t)\to 0$. If $a>0$,
$\tilde{y}_{2}(t)\to\infty$. Finally, we substitute (46) and (47) into (41),
algebraically simplify and obtain

$\tilde{y}_{1}^{\prime}=\left(d_{1}e^{at}+d_{2}e^{-\gamma_{1}t}\right)\epsilon\tilde{y}_{1}+\left(d_{3}e^{at}+d_{4}e^{-\gamma_{1}t}\right)\epsilon,$
(48)

where

$\displaystyle d_{1}$ $\displaystyle=Ah+B,$ $\displaystyle d_{2}$
$\displaystyle=A(1-h),$ $\displaystyle d_{3}$
$\displaystyle=\frac{h\gamma_{1}}{g_{1}^{\prime}(\xi)}-\left(\beta\xi+\alpha\beta
K_{0}\eta\xi
g_{0}^{\prime}(\xi)\frac{g_{1}(\xi)}{g_{1}^{\prime}(\xi)}\right),$
$\displaystyle d_{4}$
$\displaystyle=\frac{(1-h)\gamma_{1}}{g_{1}^{\prime}(\xi)}.$

Eq. (48) is linear and its solution is

$\displaystyle\tilde{y}_{1}(t)=\exp\left(\frac{d_{1}}{a}\epsilon
e^{at}-\frac{d_{2}}{\gamma_{1}}\epsilon
e^{-\gamma_{1}t}\right)\Biggl{(}\frac{\epsilon}{\exp\left(\frac{d_{1}\epsilon}{a}-\frac{d_{2}\epsilon}{\gamma_{1}}\right)}+$
$\displaystyle\quad\int_{0}^{t}\exp\left(-\frac{d_{1}\epsilon}{a}e^{ay}+\frac{d_{2}\epsilon}{\gamma_{1}}e^{-\gamma_{1}y}\right)\left(d_{3}\epsilon
e^{ay}+d_{4}\epsilon e^{-\gamma_{1}y}\right)\,dy\Biggr{)}.$ (49)

Clearly, if $a>0$, it follows from (49) that $\tilde{y}_{1}(t)\to\infty$. On
the other hand, if $a<0$, as $t\to\infty$,

$\displaystyle\tilde{y}_{1}(t)$
$\displaystyle\to\epsilon\exp\left(-\frac{d_{1}\epsilon}{a}+\frac{d_{2}\epsilon}{\gamma_{1}}\right)+$
$\displaystyle\quad\int_{0}^{\infty}\exp\left(-\frac{d_{1}\epsilon}{a}e^{ay}+\frac{d_{1}\epsilon}{\gamma_{1}}e^{-\gamma_{1}y}\right)\times$
$\displaystyle\qquad\quad\left(d_{3}\epsilon e^{ay}+d_{4}\epsilon
e^{-\gamma_{1}y}\right)\,dy.$ (50)

We now present an upper bound and a lower bound for the integral on the right
side of (50). Since $\exp(\omega t)$ is increasing with $\omega$ for any $t$,
we have

$\displaystyle e^{at}\leq e^{Mt},\quad e^{-\gamma_{1}t}\leq e^{Mt},$
$\displaystyle e^{at}\geq e^{mt},\quad e^{-\gamma_{1}t}\geq e^{mt},$

where

$\displaystyle M$ $\displaystyle=\max(a,-\gamma_{1}),$ $\displaystyle m$
$\displaystyle=\min(a,-\gamma_{1}).$

The integral on the right side of (50) is bounded above by $U\epsilon$, where

$\displaystyle U$
$\displaystyle=\int_{0}^{\infty}\exp\left(\left|\frac{d_{1}\epsilon}{a}\right|e^{my}+\left|\frac{d_{2}\epsilon}{\gamma_{1}}\right|e^{my}\right)(|d_{3}|+|d_{4}|)e^{my}\,dy$
$\displaystyle=\frac{|d_{3}|+|d_{4}|}{m}\frac{1-\exp\left(\left|\frac{d_{1}\epsilon}{a}\right|+\left|\frac{d_{2}\epsilon}{\gamma_{1}}\right|\right)}{\left(\left|\frac{d_{1}}{a}\right|+\left|\frac{d_{2}}{\gamma_{1}}\right|\right)\epsilon}$
$\displaystyle\approx\frac{|d_{3}|+|d_{4}|}{|m|}.$ (51)

With a similar analysis, we obtain a lower bound $L\epsilon$ of the integral
on the right side of (50), where

$L=-\frac{|d_{3}|+|d_{4}|}{|m|}.$ (52)

The first term on the right side of (50) is approximately equal to $\epsilon$.
Thus, from (51), it follows that

$\tilde{y}_{1}(t)\to\xi^{*}\in(\epsilon(1+L),\epsilon(1+U)).$ (53)

In conclusion, starting initially from $(\epsilon,\epsilon,\epsilon)$, the
solution $(y_{1}(t),y_{2}(t),y_{3}(t))$ converges to $(\xi+\xi^{*},0,0)$,
where $\xi^{*}$ satisfies (53). On the other hand, if $a>0$,
$y_{i}(t)\to\infty$ for $i=1,2,3$, as $t\to\infty$. Note that $a$ depends only
on the parameters of the disease and the generating function of the excess
degree distribution. It does not depend on the parameters of the contact
tracing and isolation policy.

##  6 Numerical and Simulation Results

We present numerical and simulation results in this section. Numerical
solution of Eqs. (13)-(17) is presented in Section 6.1. We simulate the SIR
model with contact tracing and isolation on five real-world networks. The
simulation results are shown in Section 6.2.

###  6.1 Solution of Differential Equations

Given a degree distribution $\\{p_{k},k\geq 0\\}$, we determine its
corresponding excess degree distribution $\\{w_{k},k\geq 0\\}$ according to

$w_{k}=\frac{(k+1)p_{k+1}}{K_{0}}$

for $k\geq 0$ [9]. Recall that $K_{0}$ is the expectation of degree
distribution $\\{p_{k},k\geq 0\\}$. We truncate the two degree distributions
properly such that the error is small. We use Matlab differential equations
solver to numerically solve Eqs. (13)-(17) for Poisson degree distributions
and power law degree distributions. We assume that the initial condition is
$s_{k}(0)=1-10^{-3}$, $x_{k}(0)=10^{-3}$ and
$q^{S}_{k}(0)=q^{I}_{k}(0)=r_{k}(0)=0$. We then apply (18) to compute the
overall probability of recovery. For Poisson degree distributions, we assume
that the mean degree is 25. For power law degree distribution, we assume that
the exponent of the distribution is $-2.5$. We truncate both distributions to
range $[0,1000]$. Values of other parameters are shown in Table I. We show the
probability of recovery for the power law degree distribution in Fig. 2 for
several values of $\eta$. Note that the curve with $\eta=0$ corresponds to the
epidemic without contact tracing and isolation. This figure shows that the
contact tracing and isolation is effective in containing the epidemic. In Fig.
3 we show $q^{S}(t)$ for several values of $\eta$. From this figure, we see
that while contact tracing and isolation is effective in containing the
epidemic, it comes at a cost. The contact tracing and isolation can be
detrimental to the normal function of a society. With a strict isolation rule
corresponding to a large value of $\eta$, we see from Fig. 3 that more than 8%
of the total population are isolated. However, these isolated individuals are
susceptible. We show how $r(t)$ and $q^{S}(t)$ vary with $\alpha$ in Figs. 4
and 5. In the calculation of these two figures, the value of $\eta$ is 0.5.

parameter | $\alpha$ | $\beta$ | $\gamma$ | $\gamma_{1}$ | $\eta$
---|---|---|---|---|---
value | 0.4 | 0.15 | 0.1 | 0.1 | 0.5
TABLE I: Parameter values used in numerical studies.

Figure 2: Probability of recovery as a function of time for several values of
$\eta$.

Figure 3: Probability of isolated individuals who are susceptible as a
function of time for several values of $\eta$.

Figure 4: Probability of recovery as a function of time for several values of
$\alpha$.

Figure 5: Probability of isolated individuals who are susceptible as a
function of time for several values of $\alpha$.

We now present numerical studies of the early-time analysis presented in
Section 3. We calculate a ratio by dividing the result of the early-time
analysis by that of the exact numerical result. The ratios are shown in Fig.
6. As expected, the early time analysis is accurate when time is small and
starts to deviate when time is getting large.

Figure 6: Early time analysis. The ratios of early time analysis divided by
that of the exact numerical result are shown.

We use Matlab to numerically solve Eqs. (29)-(33). We set the initial
condition as follows. Let $v(0)=10^{-3}$, $q^{S}(0)=q^{I}(0)=r(0)=0$, and
$u(0)=g_{1}^{-1}(1-10^{-3})$, where $g_{1}^{-1}$ is the inverse function of
the probability generating function $g_{1}$. We calculate a ratio by dividing
the result of the approximate analysis by that of the exact numerical result.
The ratios are shown in Fig. 7. We see that the approximation method works
quite well. The errors are typically within five percents. We also study the
approximate analysis of the power law degree distribution with exponent
$-2.5$. The ratios are shown in Fig. 8. The errors in this case are higher,
but are within a reasonable range. At late time of the epidemic, the errors
corresponding to $q^{S}(t)$ and $q^{I}(t)$ can be twenty percents. We show the
exact analysis and the approximate analysis of $x(t)$, $q^{S}(t)$ and
$q^{I}(t)$ in Fig. 9. We see that the result of exact analysis and that of the
approximate analysis are very close in early time and mid time. At late time,
the difference between the two analyses are visible. However, the values of
$x(t)$, $q^{S}(t)$ and $q^{I}(t)$ are very small and they are less significant
to the epidemics.

Figure 7: Approximate analysis. The ratios of approximate analysis divided by
that of the exact numerical result are shown. A randomly selected vertex has a
Poisson degree distribution.

Figure 8: Approximate analysis. The ratios of approximate analysis divided by
that of the exact numerical result are shown. A randomly selected vertex has a
power law degree distribution.

Figure 9: Values of $x(t)$, $q^{S}(t)$ and $q^{I}(t)$ obtained by exact
numerical analysis and approximate analysis. Degrees have a power law
distribution with exponent $-2.5$.

###  6.2 Simulation of Real-world Networks

We analyze contact tracing and isolation using configuration network models.
Configuration models are mathematically simple, but lack some important
characteristics such as clustering and degree correlations. It is known that
real-world networks possess these characteristics [9] and it is known that
these characteristics have non-neglectable impacts to many networking problems
[24]. It is well known that a group of individuals with close contacts with
each other, such as members in a household or workmates in an office, are
likely to contract the disease if one in the group does [14, 15]. In this
section we study the SIR model with contact tracing and isolation on real-
world networks by simulation. We simulate a more general contact tracing and
isolation policy. In our simulation model there are two types of edges. The
first type of edges connects two close contacts. The disease has a higher
transmission rate across edges across the first type of edges. The second type
of edges connects two normal contacts. We refer the reader to Fig. 10 for a
graphical illustration. Suppose that vertex $A$ is infected and shows
symptoms. We assume that the contact tracing and isolation policy quarantines
and isolates all $A$’s close contacts, i.e. $B_{1},B_{2},\ldots,B_{i}$. In
addition, the policy isolates a fraction $\eta$ of $A$’s normal contacts.
Specifically, each $C_{k}$, for $1\leq k\leq j$, is isolated with probability
$\eta$.

To the best of our efforts, we have not been able to find datasets of real-
world networks, in which edges are marked according to whether they connect
two close contacts or two normal contacts. Tie strength is a fundamental topic
in sociology [25]. Strong ties generally refer to edges corresponding to
closer friendships or greater frequency of interactions. On the other hand,
weak ties generally refer to edges connecting two acquaintances [26]. Given a
social graph, there have been efforts to mark the strengths of the edges in
the graph [27, 28]. Particularly, Onnela et al. [27] studied a network formed
by communications among cellular phones over an 18-week period. Onnela et al.
found tie strengths and neighborhood overlap are highly correlated. The
neighborhood overlap of an edge connecting vertices $A$ and $B$ is defined as
the ratio

$\frac{|N_{A}\cap N_{B}|}{|N_{A}\cup N_{B}-\\{A,B\\}|},$ (54)

where $N_{A}$ and $N_{B}$ denote the sets of $A$ and $B$’s neighbors,
respectively. The denominator of (54) is the number of $A$’s neighbors and
$B$’s neighbors. However, $A$ and $B$ are excluded. Clearly, the neighbor
overlap defined in (54) is a number between zero and one. In our experiments,
we mark an edge as a type one edge if its neighbor overlap is more than or
equal to a threshold $h$. Otherwise, the edge is marked as a type two edge.

We consider five networks collected in the real world. Their information is
summarized in Table II. The bitcoin and the facebook datasets are available at
the SNAP network datasets site [29]. The dolphin, tvshow and the anybeat
datasets are available at Network Data Repository [30]. We simulate the SIR
model with contact tracing and isolation in these networks. The results are
presented in Table III and Table IV. Quarantine period is the number of time
units that one remains isolated. $S$ (resp. $R$) denotes the fraction of
population that are susceptible (resp. recovered) at the end of the epidemic.
$Q_{max}$ denotes the maximum fraction of population that are isolated and are
susceptible (resp. infected) during the entire process of epidemic. $I_{max}$
is the maximum fraction of asymptomatic individuals who are not isolated
during the epidemic. $t_{q}$ (resp. $t_{i}$) is the amount of time for the
number of quarantined individuals (resp. number of infected individuals) to
reduce to zero. Among all neighbors who are connected by weak ties with an
infected individuals, a fraction of $\eta$ of neighbors are isolated. In Table
III two values of $\eta$ were simulated, and in Table IV two values of $\eta$
were simulated. We draw the following conclusions from these two tables.

  1. 1.

If one tightens the contact tracing and isolation rules by extending the
isolation period or increasing $\eta$, less people are infected with the
disease. Simulation shows that $S$ (resp. $R$) is increasing (resp.
decreasing) with the isolation period.

  2. 2.

With a more stringent isolation policy enforced, there are more susceptible
individuals who are isolated, and there are less asymptomatic individuals who
are not isolated. Simulation shows that $Q_{max}$ (resp. $I_{max}$) is
increasing (resp. decreasing) with isolation period and $\eta$.

  3. 3.

With a more stringent isolation policy enforced, it takes longer to contain
the epidemic. Simulation shows that both $t_{q}$ and $t_{i}$ are increasing
with the isolation period.

  4. 4.

With the same isolation period and the same value of $\eta$, the contract
tracing and isolation policy works more efficiently if a larger fraction $h$
of neighbors are close contacts.

Figure 10: Vertex $A$ has $i$ close contacts denoted by
$B_{1},B_{2},\ldots,B_{i}$ and $j$ normal contacts denoted by
$C_{1},C_{2},\ldots,C_{j}$. The edges between vertex $A$ and
$B_{1},B_{2},\ldots,B_{i}$ are the first type of edges. The edges between
vertex $A$ and $C_{1},C_{2},\ldots,C_{j}$ are the second type of edges. The
edges connecting close contacts are denoted by thick lines.
data set | $n$ | $m$ | $K_{0}$ | $\rho$ | $C$
---|---|---|---|---|---
dolphin | 62 | 159 | 5.13 | -0.0436 | 0.2590
bitcoin | 5881 | 21492 | 7.31 | -0.1648 | 0.1775
tvshow | 3892 | 17262 | 8.87 | 0.5604 | 0.3737
facebook | 4039 | 88234 | 43.69 | 0.0636 | 0.6055
anybeat | 12645 | 49132 | 7.77 | -0.1234 | 0.2037
TABLE II: Five real-world networks. $n$, $m$, $K_{0}$ are the number of vertices, the number of edges and the expected degree of a randomly selected vertex of the network. $\rho$ is the Pearson degree correlation of a randomly selected edge, and $C$ is the clustering coefficient. data set | isolation | $\eta=0.6$ | $\eta=0.9$
---|---|---|---
period | $S$ | $R$ | $Q_{max}$ | $I_{max}$ | $t_{q}$ | $t_{i}$ | $S$ | $R$ | $Q_{max}$ | $I_{max}$ | $t_{q}$ | $t_{i}$
dophin | 3 | 0.734 | 0.266 | 0.087 | 0.090 | 28 | 38 | 0.752 | 0.248 | 0.111 | 0.091 | 27 | 38
7 | 0.737 | 0.263 | 0.104 | 0.090 | 30 | 38 | 0.777 | 0.223 | 0.120 | 0.083 | 27 | 36
14 | 0.755 | 0.246 | 0.114 | 0.085 | 32 | 38 | 0.790 | 0.210 | 0.137 | 0.081 | 30 | 35
bitcoin | 3 | 0.635 | 0.365 | 0.161 | 0.095 | 93 | 101 | 0.657 | 0.343 | 0.210 | 0.090 | 95 | 105
7 | 0.631 | 0.369 | 0.203 | 0.093 | 101 | 109 | 0.657 | 0.343 | 0.257 | 0.086 | 104 | 113
14 | 0.640 | 0.360 | 0.214 | 0.090 | 115 | 123 | 0.663 | 0.337 | 0.263 | 0.085 | 120 | 129
tvshow | 3 | 0.603 | 0.397 | 0.055 | 0.080 | 108 | 119 | 0.658 | 0.342 | 0.066 | 0.071 | 105 | 118
7 | 0.622 | 0.378 | 0.085 | 0.072 | 113 | 124 | 0.681 | 0.319 | 0.099 | 0.062 | 111 | 123
14 | 0.658 | 0.342 | 0.115 | 0.065 | 120 | 128 | 0.720 | 0.280 | 0.129 | 0.057 | 117 | 127
facebook | 3 | 0.202 | 0.798 | 0.144 | 0.319 | 89 | 103 | 0.238 | 0.762 | 0.179 | 0.300 | 91 | 107
7 | 0.212 | 0.788 | 0.196 | 0.309 | 96 | 109 | 0.255 | 0.745 | 0.238 | 0.289 | 99 | 114
14 | 0.238 | 0.762 | 0.227 | 0.307 | 110 | 120 | 0.284 | 0.716 | 0.268 | 0.287 | 114 | 129
anybeat | 3 | 0.673 | 0.327 | 0.190 | 0.115 | 91 | 100 | 0.690 | 0.310 | 0.245 | 0.114 | 90 | 100
7 | 0.673 | 0.327 | 0.209 | 0.117 | 95 | 104 | 0.697 | 0.303 | 0.274 | 0.112 | 95 | 106
14 | 0.679 | 0.321 | 0.213 | 0.115 | 102 | 111 | 0.698 | 0.302 | 0.282 | 0.114 | 105 | 114
TABLE III: Simulation results of the SIR model with contact tracing on five real-world networks. Two values of $\eta$ were simulated. The value of $h$ in this set of simulation is $0.75$. data set | isolation | $h=0.3$ | $h=0.6$
---|---|---|---
period | $S$ | $R$ | $Q_{max}$ | $I_{max}$ | $t_{q}$ | $t_{i}$ | $S$ | $R$ | $Q_{max}$ | $I_{max}$ | $t_{q}$ | $t_{i}$
dolphin | 3 | 0.667 | 0.333 | 0.082 | 0.112 | 32 | 42 | 0.692 | 0.308 | 0.060 | 0.092 | 32 | 41
7 | 0.674 | 0.326 | 0.097 | 0.109 | 32 | 41 | 0.709 | 0.291 | 0.072 | 0.086 | 33 | 40
14 | 0.706 | 0.294 | 0.108 | 0.101 | 34 | 40 | 0.724 | 0.276 | 0.078 | 0.084 | 34 | 39
bitcoin | 3 | 0.628 | 0.372 | 0.124 | 0.092 | 96 | 105 | 0.600 | 0.400 | 0.107 | 0.103 | 94 | 102
7 | 0.627 | 0.373 | 0.156 | 0.088 | 103 | 112 | 0.602 | 0.398 | 0.138 | 0.096 | 98 | 106
14 | 0.634 | 0.366 | 0.164 | 0.085 | 118 | 125 | 0.606 | 0.394 | 0.147 | 0.097 | 107 | 115
tvshow | 3 | 0.546 | 0.454 | 0.041 | 0.096 | 105 | 115 | 0.530 | 0.470 | 0.038 | 0.090 | 109 | 118
7 | 0.562 | 0.438 | 0.064 | 0.088 | 108 | 116 | 0.543 | 0.457 | 0.060 | 0.083 | 112 | 120
14 | 0.590 | 0.410 | 0.090 | 0.084 | 114 | 123 | 0.563 | 0.437 | 0.087 | 0.078 | 119 | 126
facebook | 3 | 0.156 | 0.844 | 0.091 | 0.352 | 86 | 99 | 0.152 | 0.848 | 0.096 | 0.338 | 87 | 99
7 | 0.168 | 0.832 | 0.131 | 0.348 | 89 | 101 | 0.163 | 0.837 | 0.134 | 0.330 | 92 | 102
14 | 0.189 | 0.811 | 0.152 | 0.347 | 97 | 107 | 0.182 | 0.818 | 0.155 | 0.328 | 99 | 109
anybeat | 3 | 0.641 | 0.359 | 0.116 | 0.131 | 89 | 97 | 0.641 | 0.359 | 0.113 | 0.121 | 91 | 99
7 | 0.644 | 0.356 | 0.132 | 0.132 | 90 | 99 | 0.644 | 0.356 | 0.128 | 0.120 | 93 | 101
14 | 0.650 | 0.350 | 0.137 | 0.131 | 95 | 104 | 0.651 | 0.349 | 0.131 | 0.119 | 97 | 105
TABLE IV: Simulation results of the SIR model with contact tracing on five
real-world networks. Two values of $h$ were simulated. The value of $\eta$ in
this set of simulation is $0.6$.

##  7 Conclusions

In this paper we have presented a degree based approximation to the SIR model
with contact tracing and isolation. We proposed an approximation method which
greatly reduced the numerical complexity in solving the differential equations
in the degree based approximation. We have also simulated the SIR model with
contact tracing and isolation on five real-world networks. We showed that
contact tracing and isolation are effective to control epidemics.

## References

  * [1] Can Hou and et al., “The effectiveness of quarantine of Wuhan city against the corona virus disease 2019 (COVID-19): A well-mixed SEIR model analysis,” _Journal of Medical Virology_ , vol. 92, no. 7, pp. 841–848, apr 2020.
  * [2] Alberto Aleta and et al., “Modelling the impact of testing, contact tracing and household quarantine on second waves of COVID-19,” _Nature Human Behaviour_ , vol. 4, no. 9, pp. 964–971, aug 2020.
  * [3] Adam Kucharski and et al., “Effectiveness of isolation, testing, contact tracing, and physical distancing on reducing transmission of SARS-CoV-2 in different settings: a mathematical modelling study,” _Lancet Infect Dis._ , vol. 20, no. 10, pp. 1151–1160, Oct. 2020.
  * [4] Joel Hellewell and et al., “Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts,” _The Lancet Global Health_ , vol. 8, no. 4, 2020.
  * [5] X. Yan and Y. Zou, “Control of epidemics by quarantine and isolation strategies in highly mobile populations,” _International J. of Information and Systems Sciences_ , vol. 5, pp. 271–286, 2009.
  * [6] C. Castillo-Chavez, C. W. Castillo-Garsow, and A.-A. Yakubu, “Mathematical Models of Isolation and Quarantine,” _JAMA_ , vol. 290, no. 21, pp. 2876–2877, 12 2003. [Online]. Available: https://doi.org/10.1001/jama.290.21.2876
  * [7] Abba B. Gumel and et al., “Modelling strategies for controlling sars outbreaks,” _Proceedings Biological Sciences_ , vol. 271(1554), pp. 2223–2232, 2004.
  * [8] C. J. Wang, C. Y. Ng, and R. H. Brook, “Response to COVID-19 in Taiwan: Big Data Analytics, New Technology, and Proactive Testing,” _JAMA_ , vol. 323, no. 14, pp. 1341–1342, 04 2020. [Online]. Available: https://doi.org/10.1001/jama.2020.3151
  * [9] M. Newman, _Networks: An Introduction_. New York: Oxford University Press, 2010.
  * [10] R. Pastor-Satorras and A. Vespignani, “Epidemic dynamics and endemic states in complex networks,” _Physical Review E_ , vol. 63, p. 066117, 2001.
  * [11] ——, “Epidemic spreading in scale-free networks,” _Phys. Rev. Letter_ , vol. 86, pp. 3200–3203, 2001.
  * [12] M. Barthélemy, A. Barrat, R. P. Satorras, and A. Vespignani, “Velocity and Hierarchical Spread of Epidemic Outbreaks in Scale-Free Networks,” _Physical Review Letters_ , vol. 92, no. 17, pp. 178 701+, Apr. 2004. [Online]. Available: http://dx.doi.org/10.1103/physrevlett.92.178701
  * [13] M. Barthelemy, A. Barrat, R. Pastor-Satorras, and A. Vespignani, “Dynamical patterns of epidemic outbreaks in complex heterogeneous networks,” _Journal of Theoretical Biology_ , vol. 235, pp. 275–88, 08 2005.
  * [14] T. House and M. Keeling, “Deterministic epidemic models with explicit household structure,” _Math. Biosci._ , vol. 213, no. 1, pp. 29–39, May 2008\.
  * [15] ——, “Household structure and infectious disease transmission,” _Epidemiol. Infect._ , vol. 137, no. 5, pp. 654–661, May 2009.
  * [16] M. Boguñá and R. Pastor-Satorras, “Epidemic spreading in correlated complex networks,” _Physical Review E_ , vol. 66, p. 047104, 2002.
  * [17] M. Boguñá”, R. Pastor-Satorras, and A. Vespignani, “Absence of epidemic threshold in scale-free networks with degree correlations,” _Physical Review Letters_ , vol. 90, p. 028701, 2003.
  * [18] V. M. Eguíluz and K. Klemm, “Epidemic threshold in structured scale-free networks,” _Physical Review Letters_ , vol. 89, no. 10, 108701, 2002.
  * [19] Barry Rockx and et. al., “Comparative pathogenesis of COVID-19, MERS, and SARS in a nonhuman primate model,” _Science_ , vol. 368, pp. 1012–1015, 5 2020.
  * [20] Z. Du, X. Xu, Y. Wu, L. Wang, B. Cowling, and L. A. Meyers, “The serial interval of covid-19 from publicly reported confirmed cases,” _preprint, medRxiv_ , May 2020.
  * [21] A. Kronbickler and et al., “Asymptomatic patients as a source of COVID-19 infections: a systematic review and meta-analysis,” _International Journal of Infectious Diseases_ , vol. 98, pp. 180–186, September 2020.
  * [22] Z. Gao and et al., “A systematic review of asymptomatic infections with COVID-19,” _Journal of Microbiology, Immunology and Infection_ , vol. 54, pp. 12–16, February 2021.
  * [23] S. Mwalili and et al., “Seir model for covid-19 dynamics incorporating the environment and social distancing,” _BMC Research Notes_ , vol. 13, p. 352, 2020.
  * [24] R. Zhang, D.-S. Lee, and C.-S. Chang, “A configuration model with triadic closure,” _arXiv preprint https://arxiv.org/abs/2105.11688_ , 2022.
  * [25] M. Granovetter, _Getting a Job: A Study of Contacts and Careers_. University of Chicago Press, 1975.
  * [26] D. Easley and J. Kleinberg, _Networks, crowds and markets reasoning about a highly connected world_. Cambridge University Press, 2010.
  * [27] J.-P. Onnela and et al., “Structure and tie strengths in mobile communication networks,” _Proc. Natl. Acad. Sci. USA_ , vol. 104, pp. 7332–7336, 2007.
  * [28] P. V. Marsden and K. E. Campbell, “Measuring tie strength,” _Social Forces_ , vol. 63, pp. 482–501, December 1984.
  * [29] J. Leskovec and A. Krevl, “SNAP Datasets: Stanford large network dataset collection,” http://snap.stanford.edu/data, Jun. 2014.
  * [30] R. A. Rossi and N. K. Ahmed, “The network data repository with interactive graph analytics and visualization,” in _AAAI_ , 2015. [Online]. Available: https://networkrepository.com