# Effect of diagnostic testing on the isolation rate in a compartmental model
with asymptomatic groups ††thanks: The work of Zuzana Chladná was partly
supported by Slovak Grant Agency APVV-0096-12, and the work of Jana Kopfová
was supported by the institutional support for the development of research
organizations IČ 47813059.

Zuzana Chladná  Jana Kopfová  Dmitrii Rachinskii  Pavel Štepánek etc

(Received: date / Accepted: date)

###### Abstract

We present two epidemiological models, which extend the classical SEIR model
by accounting for the effect of indiscriminate quarantining, isolation of
infected individuals based on testing and the presence of asymptomatic
individuals. Given a constraint that limits the maximal number of simultaneous
active cases, we demonstrate that the isolation rate, which enforces this
constraint, decreases with the increasing testing rate. The models predict
that massive testing allows to control the infection spread using a much lower
isolation rate than in the case of indiscriminate quarantining. We also show
that dynamics of infection are more sensitive to the parameters of the
asymptomatic groups than the corresponding parameters of the symptomatic
groups.

##  1 Introduction

As the race for developing effective COVID-19 vaccines and the search for
medications for the treatment of the disease continues, home quarantine
measures remain the main means for controlling the epidemic. Restrictions
imposed by home quarantine policies on businesses and individuals help to
decrease the number of social contacts and thus slow down the spread of the
disease. These restrictions differed in their scale, forms and timings in the
states and countries where they were implemented by the government and health
authorities. However, massive lockdowns are not a sustainable solution for
economic, social and psychological reasons. The situation is complicated by
the asymptomatic COVID-19 infection. The asymptomatic individuals can account
for as many as 40-45% of infections [15] and can transmit the virus to others
for an extended period of time.

Asymptomatic infection, the latent infectious phase of the disease and cases
with mild symptoms can be detected by diagnostic tests. As such, massive
diagnostic testing of the population and isolation of positively tested
individuals can potentially offer an alternative to indiscriminate
quarantining of large population groups [13]. The isolation strategy based on
diagnostic screening was implemented by a number of organizations, business
and government authorities. For example, major airlines such as American
Airlines and Luft Hansa added on-site pre-flight rapid testing facilities at a
number of American and German airports. Duke University comprehensive COVID-19
testing program received results from 16,146 tests administered to students
and faculty from November 7 – 13, 2020. The Chinese city of Qingdao reported
testing its entire population of nine million people for COVID-19 over a
period of five days in October, 2020. Slovakia tested 3.6 million people – two
thirds of its population – in two days on October 31 – November 1, 2020 and
repeated testing the same population in a week’s time in the first attempt of
a large-scale blanket testing campaign in Europe. These massive testing
efforts were assisted by new pulling technologies and the development of
affordable rapid tests such as Abbott’s BinaxNOW.

The health and government authorities are concerned with keeping the number of
active cases below the level dictated by the capacity of the health care
system. Several recent studies have attempted to predict and analyze the
effect of quarantine measures on dynamics of COVID-19 pandemic using
compartmental models of mathematical epidemiology. A variant of the standard
well mixed SEIR model

$\begin{array}[]{l}\dot{S}=-\beta SI,\\\ \dot{E}=\beta SI-\omega E,\\\
\dot{I}=\omega E-\delta I,\\\ \dot{R}=\delta I,\\\ \end{array}$ (1)

involving additionally the transmission from exposed to susceptible
individuals, was used to analyze the effect of social distancing and reducing
the number of contacts [10]. Another implementation of the quarantine policy
was studied in [16], where the authors adapted the SIR model assuming that all
the infected individuals are isolated after the incubation period. Stochastic
age-structured transmission models were applied to explore a range of
intervention scenarios [6]. A model with a two-threshold switching prevention
strategy predicted that flattening the curve can lead to periodic recurrence
of the disease [3]. The authors of all the above mentioned studies conclude
that the interventions are effective in reducing the infection peak; however,
extreme interventions are likely to be required to contain the infection
spread.

Amid the ongoing efforts of bringing the COVID-19 epidemic under control
continues, it is important to explore different scenarios using a variety of
models and modeling assumptions. In this paper, we are interested in the
potential of massive diagnostic testing measures for reducing the quarantining
rate, which enforces a given constraint on the maximum of active cases. A few
earlier studies addressed the role of testing in the trajectory of the
epidemic. In particular, the discrete time adaptation of the SEIR model
proposed in [1] suggests that the isolation based on testing interventions can
make unnecessary costly lockdown measures.

Below we propose two compartmental models, which account for indiscriminate
quarantine measures, targeted detection and isolation of infected individuals
based on diagnostic testing and a possible difference of the epidemiological
parameters of the symptomatic and asymptomatic groups. We compute the
trajectories of the infected and quarantined populations in the case when
indiscriminate quarantining is applied and in the case when targeted
quarantining is facilitated by diagnostic testing. We then evaluate the
relative efficiency of the testing effort by matching the parameters of both
scenarios to ensure that they produce the same infection peak. As the measure
of the efficiency, the time-integral of the total quarantined population over
the duration of the epidemic is used. Finally, we discuss the sensitivity and
robustness of trajectories to the epidemiological parameters of the
symptomatic and asymptomatic groups.

The models are presented in the next section. For simplicity, we do not
consider the possibility of reinfection, even though a few cases of COVID-19
reinfection have been seemingly confirmed. Contact tracing per se is not
included in the models but can be accounted for as a factor increasing the
isolation rates. Two different types of tests are used to detect the presence
of the virus during the illness and the presence of antibodies after the
illness. We consider the former type only. The results are presented in
Section 3. The values of the epidemiological parameters used in the numerical
simulations are based on the average characteristics of COVID-19 that we
distilled from the current literature.

##  2 Models

###  2.1 Basic model

Several recent studies have reported that the coronavirus can be transmitted
during the incubation period before the first symptoms develop. Moreover, a
significant portion of individuals with COVID-19 lack symptoms [4]. These
characteristics of COVID-19 together with the state imposed quarantine and
testing motivate the following extension of the SEIR model:

$\begin{array}[]{l}\dot{S}=-\beta SI_{a}-\beta(1-\rho)S(I_{aQ}+I_{sQ})-\chi
I_{sQ}S,\\\
\dot{S}_{Q}=-\beta(1-\rho)S_{Q}I_{a}-{\beta}{(1-\rho)}^{2}S_{Q}(I_{aQ}+I_{sQ})+\chi
I_{sQ}S,\\\ \dot{E}=\beta SI_{a}+\beta(1-\rho)S(I_{aQ}+I_{sQ})-\omega E-\chi
I_{sQ}E,\\\
\dot{E}_{Q}=\beta(1-\rho)S_{Q}I_{a}+{\beta}{(1-\rho)}^{2}S_{Q}(I_{aQ}+I_{sQ})-\omega
E_{Q}+\chi I_{sQ}E,\\\ \dot{I}_{a}=k\omega E-\chi I_{sQ}I_{a}-\delta
I_{a}-\psi I_{a},\\\ \dot{I}_{aQ}=k\omega E_{Q}+\chi I_{sQ}I_{a}-\delta
I_{aQ}-\psi I_{aQ},\\\
\dot{I}_{sQ}=(1-k)\omega(E+E_{Q})+\psi(I_{a}+I_{aQ})-\delta I_{sQ},\\\
\dot{R}=\delta(I_{a}+I_{sQ})-\chi I_{sQ}R,\\\ \dot{R}_{Q}=\delta I_{aQ}+\chi
I_{sQ}R,\end{array}$ (2)

see Figure 1. Here $S$ denotes the density of susceptible individuals who are
not quarantined, and $S_{Q}$ is the density of susceptible individuals at home
quarantine. A similar labeling convention is adopted for other groups of
individuals, where the subscript $Q$ refers to the subpopulation at home
quarantine. It is assumed that the exposed individuals labeled $E$ and $E_{Q}$
have been infected but are not infectious yet, and show negative test results
if tested for the virus. The compartment $I_{sQ}$ includes infectious
individuals with symptoms and those infectious individuals without symptoms
who tested positively for the virus (which corresponds to the published
statistics such as the graphs provided by WHO [18]). It is therefore the
compartment of detected active cases. The individuals from the compartment
$I_{sQ}$ are all quarantined. Infectious asymptomatic individuals labeled
$I_{a}$ and $I_{aQ}$ show positive test results if tested for the virus, in
which case they are transferred to the compartment labeled $I_{sQ}$. That is,
testing an asymptomatic infectious non-quarantined individual results in their
quarantining. Upon recovery, the individuals from compartments $I_{a}$ and
$I_{sQ}$ are recruited to the compartment $R$, and individuals from
compartment $I_{aQ}$ are recruited to the compartment $R_{Q}$. We exclude
reinfection, although a few cases have been reported. The quarantined
population is composed of the groups $S_{Q},E_{Q},I_{aQ},I_{sQ},R_{Q}$. We
assume a constant total population size, and we scale its density to unity,
hence the equation for the density of the recovered quarantined population
$R_{Q}=1-S-S_{Q}-E-E_{Q}-I_{a}-I_{aQ}-I_{sQ}-R$ is redundant.

We do not consider tests for antibodies which detect that a person had
COVID-19 in the past but now is healthy.

Parameters $\beta$, $\omega$ and $\delta$ have the same interpretation as for
the SEIR model (1), i.e. $\beta$ represents the transmission rate, $\omega$ is
the rate at which an exposed individual becomes infectious and $\delta$ is the
recovery rate. The parameter $\psi$ denotes the testing rate, i.e. the rate of
detection and isolation of asymptomatic individuals. In addition, we assume
that the total quarantining rate is $\chi I_{sQ}(S+I_{a}+E+R),$ where the
latter factor is the proportion of the non-quarantined population in the total
population. In particular, the total quarantining rate is proportional to
$I_{sQ}$ and decreases with the increasing proportion of the total quarantined
population. Assuming indiscriminate quarantining, the rate of quarantining
from a particular compartment is proportional to the density of the population
associated with this compartment. For example, the rate of quarantining from
the $S$-compartment is set to $\chi I_{sQ}(S+I_{a}+E+R)S/(S+I_{a}+E+R)=\chi
I_{sQ}S$. The model allows for the possibility of imperfect isolation with
$\rho\in[0,1]$ representing the isolation effectiveness as in [8]. Finally,
$k$ denotes the fraction of individuas who do not develop symptoms when
infected. Assuming $I_{a}(0)=I_{aQ}(0)=I_{sQ}(0)=0$, the ratio of the
symptomatic population to the asymptomatic population remains $(1-k):k$ at all
positive times.

$S$ $S_{Q}$ $E$ $E_{Q}$ $I_{a}$ v $I_{sQ}$ $I_{aQ}$ $R$ $R_{Q}$
pt$(\beta$,$\beta(1-\rho))$pt$k\omega$pt$(\beta(1-\rho),\beta(1-\rho)^{2})$pt$\chi$pt$\chi$pt$(1-k)\omega$pt$(1-k)\omega$
pt$k\omega$pt$\delta$pt$\chi$pt$\psi$pt$\delta$pt$\psi$pt$\delta$$\chi$$Q$
Figure 1:  Compartmental diagram of model (2). Arrows show the flows (with the
corresponding rate parameters) between the compartments. The dashed rectangle
includes the quarantined populations.

The model exhibits a continuum of equilibrium states for which
$I_{a}=I_{sQ}=I_{aQ}=E_{Q}=E=0$ and $S+S_{Q}+R+R_{Q}=1.$

###  2.2 Model with additional features

In this section, we consider a model with a refined structure of
epidemiological compartments. The symptomatic and asymptomatic groups are
considered separately because the transmission rate, the recovery rate and
other epidemiological parameters can be different for these groups. We also
include additional features of the disease such as the latent infectious
period.

We divide each compartment $S,S_{Q},E,E_{Q},R$ introduced in model (2) into
two compartments containing a subpopulation of individuals showing symptoms
(labeled by the subscript $s$) and a subpopulation of asymptomatic individuals
(labeled by the subscript $a$). For example, $S_{sQ}$ denotes the density of
susceptible quarantined individuals who, if infected, will develop symptoms. A
similar indexing convention is adopted for the other groups. In addition, we
introduce the latent stage of infection and the corresponding groups $L_{s}$,
$L_{sQ}$, $L_{a}$, $L_{aQ}$ of symptomatic and asymptomatic individuals.
Figure 2 presents the diagram of the compartments and the transitions between
them for the following model:

$\displaystyle\dot{S}_{s}$
$\displaystyle=-S_{s}(\beta_{s}L_{s}+\beta_{a}(L_{a}+I_{a}))-S_{s}(1-\rho)(\beta_{s}(L_{sQ}+I_{sQ})$
(3) $\displaystyle+\beta_{a}(L_{aQ}+I_{aQ}))-\frac{S_{s}F}{\Sigma},$
$\displaystyle\dot{S}_{sQ}$
$\displaystyle=-S_{sQ}((1-\rho)\beta_{s}L_{s}+(1-\rho)\beta_{a}(L_{a}+I_{a}))+\frac{S_{s}F}{\Sigma},$
$\displaystyle\dot{E}_{s}$
$\displaystyle=S_{s}(\beta_{s}L_{s}+\beta_{a}(L_{a}+I_{a}))+(1-\rho)S_{s}(\beta_{s}(L_{sQ}+I_{sQ})$
$\displaystyle+\beta_{a}(L_{aQ}+I_{aQ}))-\omega_{s}E_{s}-\frac{E_{s}F}{\Sigma},$
$\displaystyle\dot{E}_{sQ}$
$\displaystyle=(1-\rho)S_{sQ}(\beta_{s}L_{s}+\beta_{a}(L_{a}+I_{a}))-\omega_{s}E_{sQ}+\frac{E_{s}F}{\Sigma},$
$\displaystyle\dot{L}_{s}$
$\displaystyle=\omega_{s}E_{s}-(\lambda_{s}+\psi_{s})L_{s}-\frac{L_{s}F}{\Sigma},$
$\displaystyle\dot{L}_{sQ}$
$\displaystyle=\omega_{s}E_{sQ}-(\lambda_{s}+\psi_{s})L_{sQ}+\frac{L_{s}F}{\Sigma},$
$\displaystyle\dot{I}_{sQ}$
$\displaystyle=(\lambda_{s}+\psi_{s})L_{s}+(\lambda_{s}+\psi_{s})L_{sQ}-\gamma_{s}I_{sQ}$
$\displaystyle+\psi_{a}(L_{a}+L_{aQ})+\psi_{a}(I_{a}+I_{aQ}),$
$\displaystyle\dot{R}_{s}$ $\displaystyle=\gamma_{s}I_{sQ},$
$\displaystyle\dot{S}_{a}$
$\displaystyle=-S_{a}(\beta_{s}L_{s}+\beta_{a}(L_{a}+I_{a}))-(1-\rho)S_{a}(\beta_{s}(L_{sQ}+I_{sQ})$
$\displaystyle+\beta_{a}(L_{aQ}+I_{aQ}))-\frac{S_{a}F}{\Sigma},$
$\displaystyle\dot{S}_{aQ}$
$\displaystyle=-(1-\rho)S_{aQ}(\beta_{s}L_{s}+\beta_{a}(L_{a}+I_{a}))+\frac{S_{a}F}{\Sigma},$
$\displaystyle\dot{E}_{a}$
$\displaystyle=S_{a}(\beta_{s}L_{s}+\beta_{a}(L_{a}+I_{a}))+(1-\rho)S_{a}(\beta_{s}(L_{sQ}+I_{sQ})$
$\displaystyle+\beta_{a}(L_{aQ}+I_{aQ}))-\omega_{a}E_{a}-\frac{E_{a}F}{\Sigma},$
$\displaystyle\dot{E}_{aQ}$
$\displaystyle=(1-\rho)S_{aQ}(\beta_{s}L_{s}+\beta_{a}(L_{a}+I_{a}))-\omega_{a}E_{aQ}+\frac{E_{a}F}{\Sigma},$
$\displaystyle\dot{L}_{a}$
$\displaystyle=\omega_{a}E_{a}-\lambda_{a}L_{a}-\psi_{a}L_{a}-\frac{L_{a}F}{\Sigma},$
$\displaystyle\dot{L}_{aQ}$
$\displaystyle=\omega_{a}E_{aQ}-\lambda_{a}L_{aQ}-\psi_{a}L_{aQ}+\frac{L_{a}F}{\Sigma},$
$\displaystyle\dot{I}_{a}$
$\displaystyle=\lambda_{a}L_{a}-\psi_{a}I_{a}-\gamma_{a}I_{a}-\frac{I_{a}F}{\Sigma},$
$\displaystyle\dot{I}_{aQ}$
$\displaystyle=\lambda_{a}L_{aQ}-\psi_{a}I_{aQ}-\gamma_{a}I_{aQ}+\frac{I_{a}F}{\Sigma},$
$\displaystyle\dot{R}_{a}$
$\displaystyle=\gamma_{a}I_{a}-\frac{R_{a}F}{\Sigma},$
$\displaystyle\dot{R}_{aQ}$
$\displaystyle=\gamma_{a}I_{aQ}+\frac{R_{a}F}{\Sigma},$

where

$\displaystyle F$
$\displaystyle=\lambda_{s}L_{sQ}+\psi_{s}L_{sQ}+\psi_{a}(L_{aQ}+I_{aQ}),$ (4)
$\displaystyle\Sigma$
$\displaystyle=S_{s}+S_{a}+E_{s}+E_{a}+L_{s}+L_{a}+I_{a}+R_{a}.$

The parameters $\beta_{s}$, $\beta_{a}$, $\omega_{s}$, $\omega_{a}$,
$\psi_{s}$, $\psi_{a}$ and $\rho$ have the same meaning as the corresponding
parameters in model (2). The recovery rates $\gamma_{s}$ and $\gamma_{a}$ are
higher than their counterpart $\delta$ in model (2) because the latent phase
of the average duration $1/\lambda_{s}$ and $1/\lambda_{a}$ for the
symptomatic and asymptomatic groups, respectively, is introduced in model (3).

Let us note a few differences between models (2) and (3). In the latter model,
the initial stage of infection (represented by the groups labeled $E$) when
the individuals are not yet infectious is followed by the latent infectious
incubation stage (represented by the groups labeled $L$)111The latency stage
can be similarly introduced into model (2).. This allows us to account for the
fact that individuals can become contagious and spread the disease before
developing symptoms, which is an important characteristic of COVID-19 although
the latency period is relatively short [14]. Further, dividing the
compartments as in model (3), one can use different parameters for symptomatic
and asymptomatic groups (i.e., groups labeled $s$ and $a$), and thus emphasize
that individuals without symptoms (or with mild symptoms) and the symptomatic
cases can possibly show different epidemiological characteristics. For
instance, a difference in the transmission rate and the average duration of
the infectious and latent periods for asymptomatic and symptomatic individuals
can be accounted for if needs be. Moreover, one can consider different values
of the testing rate for different compartments reflecting the fact that the
disease is harder to detect in asymptomatic individuals.

Models (2) and (3) use somewhat different assumptions about the recruitment to
quarantine. Model (2) postulates a constant rate of testing with the
associated isolation of positively testing individuals and, additionally, a
constant rate of isolation unrelated to testing. In model (3), the proportion
of the total quarantined population,
$\Sigma_{Q}=S_{sQ}+E_{sQ}+L_{sQ}+S_{aQ}+E_{aQ}+L_{aQ}+I_{aQ}+R_{aQ},$ is
conserved as we assume that $\Sigma_{Q}$ is maintained at the same level
enforced by the government interventions. The relation $\dot{\Sigma}_{Q}=0$ is
warranted by a feedback loop, which relates the recruitment rate $F$ to the
instantaneous values of the population densities by formula (4).

Contrary to model (2), and as a simplification, we assume zero transmission
from the quarantined individuals because it is relatively small (however, the
corresponding terms proportional to $(1-\rho)^{2}$ can also be included in (3)
as in model (2)).

$S_{s}$ $S_{sQ}$ $S_{aQ}$ $S_{a}$ $E_{s}$ $E_{sQ}$ $E_{aQ}$ $E_{a}$ $L_{s}$
$L_{sQ}$ $L_{aQ}$ $L_{a}$ $I_{sQ}$ $I_{aQ}$ $I_{a}$ $R_{s}$ $R_{aQ}$ $R_{a}$
pt$\beta_{s}(1-\rho)$, $\beta_{s}$$\beta_{a}(1-\rho)$,
$\beta_{a}$pt$\beta_{s}(1-\rho$)$\beta_{a}(1-\rho$)pt$\beta_{s}$,
$\beta_{s}(1-\rho)$$\beta_{a}$,
$\beta_{a}(1-\rho)$pt$\beta_{s}(1-\rho)$$\beta_{a}(1-\rho)$pt$\omega_{s}$pt$\omega_{s}$pt$\omega_{a}$pt$\omega_{a}$pt$F/\Sigma$pt$F/\Sigma$pt$F/\Sigma$pt$F/\Sigma$pt$F/\Sigma$pt$F/\Sigma$pt$F/\Sigma$pt$\lambda_{a}$pt($\lambda_{s}$,$\psi_{s}$)pt($\lambda_{s}$,$\psi_{s}$)pt$\lambda_{a}$pt$F/\Sigma$pt$\gamma_{s}$pt$\gamma_{a}$pt$\psi_{a}$pt$\gamma_{a}$$\psi_{a}$$\psi_{a}$$\psi_{a}$$Q$
Figure 2:  Compartmental diagram of model (3). Arrows show the flows between
the compartments with the corresponding rate parameters. The dashed rectangle
includes the quarantined populations.

Model (3) also exhibits a continuum of equlibrium states for which
$I_{a}=L_{a}=E_{a}=R_{s}=R_{a}=I_{sQ}=I_{aQ}=E_{a}=E_{aQ}=E_{s}=E_{sQ}=S_{s}=L_{s}=L_{sQ}=S_{a}=0$
and $S_{aQ}+S_{sQ}+R_{aQ}=1.$

###  2.3 Epidemiological parameters

The values of the epidemiological parameters used in the numerical simulations
presented below are based on the average characteristics of COVID-19 published
so far.

One key parameter is the average duration of the infectious period. The period
of infectiousness (or period of communicability) is defined as _the time
interval during which an infectious agent may be transferred directly or
indirectly from an infected person to another person_ [7]. According to [14],
COVID-19 can be transmitted by an infected individual before the symptoms
develop. While in model (2) the average infectious period is the reciprocal of
the parameter $\delta$, in model (3) the infectious period is divided into two
stages with the associated populations $L$ and $I$ and is represented by the
parameters $\lambda$ and $\gamma.$ In order to keep our results for models (2)
and (3) comparable, we assume that $1/\delta=1/\lambda+1/\gamma$.

The period of infectiousness is difficult to estimate directly. One of a few
current studies including nine patients with mild symptoms showed that all
patients were non-infectious on the eighth day after the first symptoms
developed [20].

Another possibility to estimate the parameter $\delta$ is through the
estimation of the so-called _serial interval_. The serial interval of an
infectious disease represents the duration between symptom onset of a primary
case and symptom onset of its secondary cases. According to [14], the mean
incubation period of COVID-19 is 6 days. In [2], the serial interval of
COVID-19 was estimated as the weighted mean of the published parameters and
described by the gamma distribution with the mean serial interval of 4.56 days
(credible interval $(2.54,7.36)$) and standard deviation 4.53 days (credible
interval $(4.17-5.05)$).

With a serial interval shorter than the average incubation period ($4.56<6$),
we expect that a significant number of transmissions occur before the index
case has symptoms. Following [6], the period of infectiousness is assumed to
be 5.5 days, 2 days before and 3.5 days after symptom onset. This data
translates into the values $\lambda=1/2,$ $\gamma=1/3.5,$ $\delta=1/5.5,$
$\omega=1/4$ of model parameters. The study [20] indicates that the period of
infectiousness may last longer for patients with symptoms (7 days), hence we
use different values of $\gamma$ for the compartments labeled $s$ and $a$ in
Section 3.3.

Another key parameter is the transmission rate $\beta$, which can be expressed
as the product of the average number of daily contacts which a susceptible
individual has with infected individuals and the probability of transmission
during each contact. The value of this parameter, which is not directly
observable, is inferred from the estimation of the basic reproduction number
$R_{0}$. The value of $R_{0}$ for COVID-19 has been estimated within the range
of $2.24\leq R_{0}\leq 3.58$ [21]. Taking into account the theoretical
relation between $R_{0}$ and $\beta$ in SEIR model, $\beta=\delta R_{0}$, and
the setup of the other model parameters, in our numerical analysis we assume
$\beta=0.5.$ In model (3), while we consider different transmission rates for
symptomatic and asymptomatic individuals, we explore the range of $\beta$ from
the interval $[0.1,0,6].$

The coefficient $(1-\rho)\in[0,1]$ measures the decrease in the transmission
rate due to isolation. For illustrative purposes, we set $\rho=0.5$. The value
of the parameter $k$, which measures the proportion of asymptomatic
individuals among the infected population, is set to $0.5$ in accordance with
the estimate from [19].

##  3 Results

The current epidemiological situation is raising many questions about the
dynamics of the virus spread and the effectiveness of the imposed quarantine
policies. The proposed models (2) and (3) aim to explore some plausible
scenarios depending on the size of initial quarantine, gradual quarantining
rate and the testing rate.

If not explicitly stated otherwise, the values of epidemiological parameters
used in the simulations presented below are as follows: $\beta=0.5$,
$\rho=0.5$, $\omega=\omega_{s}=\omega_{a}=1/4$, $\delta=1/5.5$,
$\lambda_{s}=\lambda_{a}=1/2,$ $\gamma_{s}=\gamma_{a}=1/3.5$ and $k=0.5.$ The
initial conditions are chosen in accordance with the real situation in the
city where the population size $N$ is approximately $500,000$. If not stated
otherwise, simulations of model (2) are initiated with $E(0)=2/N$ and
simulation of model (3) with $E_{s}(0)=E_{a}(0)=1/N$. Moreover, we assume that
originally the rest of the population is susceptible, i.e. the initial size of
all the groups except $E$ and $S$ is zero. The values of parameters $\chi$,
$\psi$, $\psi_{s}$ and $\psi_{a}$ will be specified for each particular
simulation.

(a)

(b)

(c)

(d)

Figure 3:  (a, c) Comparison of the graphs of $I_{sQ}$ and
$I_{a}+I_{aQ}+I_{sQ}$ for model (1) and model (2) with parameters $\chi=0$,
$\psi=0$ and initial conditions $E(0)=2/N$. (b, d) $I_{sQ}$ and the total
infected population ($L_{s}+L_{sQ}+L_{a}+L_{aQ}+I_{a}+I_{aQ}+I_{sQ}$) for
model (3) with $\psi_{s}=\psi_{a}=0$ and initial conditions $E_{s}(0)=2/N$ and
$E_{a}(0)=0$ (green) and $E_{s}(0)=E_{a}(0)=1/N$ (red and blue).

We first compare the classical SEIR model (1) and model (2), see Figure 3 (a).
Model (2) reduces to (1) in the special case when $k=0$, $\rho=0$, $\chi=0$,
$\psi=0$. The green curve in Figure 3 (a) depicts the time evolution of the
infected population for this case. Let us note that the $I_{s}$ compartment in
model (2) fully corresponds to the $I$ compartment in (1) under this
parameters setup. Taking $k=0.5$ with other parameters unchanged in model (2),
we observe the decrease of the infected population by a half because half of
the infected individuals don’t show symptoms — instead of being included in
the $I_{s}$ they are now logged in the compartment $I_{a}$. Setting $\rho=0.5$
leads to a slower infection spread due to quarantine. Similar results for
model (3) are shown in Figure 3 (b), where instead of changing the parameter
$k$ we change the initial proportion of symptomatic and asymptomatic groups.

Let us note that typical COVID-19 statistics such as graphs provided by WHO
[18] record confirmed cases only. However, according to some sources including
[19], the data suggest that up to 80% (we assume 50% in simulations) of
infections can be mild or asymptomatic. Therefore, one can expect that the
propagation of the disease by unregistered cases might play a significant role
in its spread, possibly even more significant than the symptomatic cases
because the latter are usually quarantined and therefore spread the virus
less. It remains a controversial topic whether symptomatic and asymptomatic
cases are equally infectious and in particular whether symptomatic and
asymptomatic individuals produce the same numbers of antibodies, which would
suggest that they spread the disease approximately equally. We assume equal
transmission rates from symptomatic and asymptomatic individuals in model (2)
and allow for different transmission rates in model (3).

Figure 3 shows that the proportion of asymptomatic individuals in the
population can have a significant impact on the height of the maximum of
$I_{sQ}$ (green versus red and blue curves), i.e. the number of infected
symptomatic individuals at the peak of the epidemic. As a result, countries
with different proportions of symptomatic versus asymptomatic individuals
(e.g. due to different age structure of the population or for other immunity
reasons) might show different dynamics of the epidemic, although other
conditions are similar. Introducing a reduction of the transmission due to
quarantine by setting $\rho=0.5$ lowers the infection peak and simultaneously
prolongs the epidemic (red versus blue curve). This prolongation effect is
more pronounced in model (2). In model (3), which extends models (1) and (2)
by introducing the $L$ compartments, we observe a lower peak of reported
cases, $I_{sQ}$, because individuals at the latent phase of the disease are
not counted to this compartment. On the other hand, the total number of
infected individuals is higher than in model (2) because the infected
individuals are not quarantined during the latent phase. Also, the epidemic is
shorter compared to model (2).

###  3.1 Response of the models to variations of the quarantine size

The most effective way to slow down the spread of a disease which is not
vaccine preventable, is probably to impose isolation and quarantine on the
population or a selected subpopulation. We assume that symptomatic and
positively testing individuals are isolated (either at a hospital when
symptoms are severe, or at home). Further, the models assume home quarantining
of part of the untested population, including healthy and asymptomatic
individuals who stay at home due to various state/business restrictions (such
as school closures etc.). We assume that these interventions result in a
decreased transmission rate.

In both models (2) and (3) home quarantine can be controlled in two possible
ways. One is through the initial conditions. The size of the quarantine is
adjusted by setting $S_{Q}(0)>0$ and $S_{sQ}(0)+S_{aQ}(0)>0$ for models (2)
and (3), respectively. This type of quarantine is further referred to as
_abrupt_ quarantining.

Figure 4 shows how the the number of infected symptomatic individuals depends
on the initial quarantine size. As expected, the infection peak of $I_{sQ}$
lowers with increasing $S_{Q}(0)$ ($S_{sQ}(0)+S_{aQ}(0)$, respectively), but
simultaneously the duration of the epidemic increases. This observation is
similar for both models. One can see that the duration of the epidemic is
shorter and the height of the infection peak is lower for model (3), which can
be attributed to the presence of the latent phase $L$.

(a)

(b)

(c)

(d)

Figure 4: Effect of the quarantine size on $I_{sQ}$ and $I_{a}+I_{aQ}+I_{sQ}$.
(a,c) Model (2) with $k=0.5$, $\rho=0.5$, $\psi=0$, $\chi=0$ and $E(0)=2/N$.
(b,d) Model (3) with $\psi_{s}=\psi_{a}=0$ and $E_{s}(0)=E_{a}(0)=1/N$,
$S_{s}(0)=S_{a}(0)$.

Another way to control the home quarantine size in model (2) is by using a
positive isolation rate $\chi$. Below this strategy is referred as the
_gradual_ quarantining. The setup allows $\chi$ to depend on time or to be
controlled by the phase variables such as $I$ through a feedback loop but, for
simplicity, we assume $\chi$ to be constant.

Figure 5 compares the above two approaches to controlling the quarantine size
in model (2). The same infection peak and a similar profile of the function
$I_{sQ}(t)$ can be achieved either with nonzero $\chi$ or with nonzero
$S_{Q}(0),$ see Figure 5 (a). The total quarantined population is twice larger
than $I_{sQ}$ because equal proportions of symptomatic and asymptomatic
individuals are assumed, see Figure 5 (b). Figure 5 (c) shows that the total
quarantined population is initially smaller for the gradual quarantining
strategy than for the abrupt quarantine strategy but eventually has to exceed
the latter to ensure the same infection peak height. The peak appears earlier
with gradual quarantining, and this effect becomes more pronounced with
increasing quarantine size (blue vs. yellow curve).

(a)

(b)

(c)

Figure 5: Comparison of the effect of abrupt and gradual quarantining on
dynamics of (a) $I_{sQ}$ population; (b) $I_{a}+I_{aQ}+I_{sQ}$ population; and
(c) quarantined population in model (2).

In model (3), the quantity $\Sigma=S_{sQ}+L_{sQ}+S_{aQ}+L_{aQ}+I_{aQ}+R_{aQ}$,
which represents the total quarantined population, is conserved (more
realistically, the quarantined population cannot exceed a certain value due to
economic reasons, the necessity of uninterrupted operations of essential
services etc.). Let us note that the isolation rate $F$ is actually a small
quantity depending on the testing rate and the number of people in quarantine.
The effect of the conservation condition represented by the value $F$ is
illustrated in Figure 6. Here we assume that initially
$S_{sQ}(0)+S_{aQ}(0)=0.2$ for the blue and red curves and
$S_{sQ}(0)+S_{aQ}(0)=0.4$ for the yellow and purple curves and $\psi=0.1$. The
blue curve represents the plot of $I_{sQ}$ for model (3) with $F,\Sigma$
defined by (4), while the red curve is obtained with $F=0$ for the same
parameters. The purple and yellow curves were obtained similarly with a
different initial quarantine size. For all the selected setups of parameters,
$F$ does not seem to have a significant effect on dynamics of the number of
reported cases and the total infected population.

(a)

(b)

Figure 6: Effect of the conservation condition on (a) $I_{sQ}$ and (b)
$I_{sQ}+I_{a}+I_{aQ}$ in model (3) for two different sizes of the initial
quarantine and $\psi=0.1$. The value of $F$ is defined by (4) (the size of the
quarantine is conserved) for the red and purple plots and set to zero
(conservation is violated) for the blue and yellow plots.

###  3.2 Response of the models to variations of the testing rate

In both models (2) and (3), by testing we understand the detection and
isolation of infectious asymptomatic individuals. In model (3), this category
includes also individuals during the latent period. We assume that symptomatic
cases are confirmed as positive and isolated (quarantined), hence they all
belong to the $I_{sQ}$ compartment. The parameter $\psi$ can be interpreted as
the rate of successful detection of infectious asymptomatic individuals. We
assume that this rate is the same for all compartments in which testing can
detect infectious individuals. Newly detected cases are recruited from the
compartments $I_{a}$ and $I_{aQ}$ in model (2) and from the compartments
$L_{a}$, $L_{aQ}$, $L_{s}$, $L_{sQ}$, $I_{a}$, $I_{aQ}$ in model (3),
respectively, to $I_{sQ}$. Therefore the compartment $I_{sQ}$ is interpreted
as the compartment of _confirmed active_ cases.

As our simulations show, the time interval from the advent of the epidemic to
the infection peak and the total duration of the epidemic increase with the
increasing testing rate $\psi$, see Figure 7 (a) for model (2) and (b) for
model (3), respectively. Further, a significant increase in the testing rate
decreases the epidemic peak. This effect is more pronounced for model (3).
However, for a relatively low testing rate, for instance if $\psi=0.1$, a
small increase in the number of confirmed active cases is observed, compared
to the case when $\psi=0$ (the red curve versus the blue curve in Figure 7).
Under low detection rate, the decrease of the spread of infection due to
quarantining of positively tested asymptomatic individuals can be masked by
the apparent increase of the $I_{sQ}$ population due to newly detected cases.
However, the total number of cases (i.e. $I_{sQ}+I_{aQ}+I_{a}$ for model (2)
and $I_{sQ}+L_{s}+L_{sQ}+L_{a}+L_{aQ}+I_{a}+I_{aQ}$ for model (3),
respectively) always decreases with increasing testing rate, as Figure 7 (c,d)
shows.

(a)

(b)

(c)

(d)

Figure 7:  Effect of testing on dynamics of $I_{sQ}$ and
$I_{a}+I_{aQ}+I_{sQ}$. (a, c) Model (2) with $\chi=0$ and $E_{Q}(0)=2/N$. (b,
d) Model (3) with $E_{s}(0)=E_{a}(0)=1/N$.

###  3.3 The effect of the difference of parameters of symptomatic and
asymptomatic groups

The composition of model (3) allows us to use different parameters for
symptomatic and asymptomatic cases and thus account for the possibility that
the COVID-19 illness can take different course for these two groups. In
principle, symptomatic and asymptomatic groups can differ in any model
parameters. We tested the sensitivity of the $I_{sQ}$ and
$I_{a}+I_{aQ}+I_{sQ}$ peak heights to the average length of the infectious
stage $1/\gamma_{s}$ and $1/\gamma_{a}$ in the symptomatic and asymptomatic
groups, respectively; the transmission rates $\beta_{s}$ and $\beta_{a}$; and
the testing based quarantining rates $\psi_{s}$ and $\psi_{a}$.

The peak height of the total infected population $I_{a}+I_{aQ}+I_{sQ}$ is more
sensitive to the parameter $1/\gamma_{a}$ than to the parameter
$1/\gamma_{s}$. As can be expected, the difference of the sensitivities with
respect to these parameters decreases with the increasing latency period when
both groups present no symptoms, see Figures 8 (b,d). On the other hand, the
peak height of $I_{sQ}$ is more sensitive to the parameter $1/\gamma_{s}$ than
to the parameter $1/\gamma_{a}$, see Figures 8 (a,c). This can be attributed
to the fact that the number of active symptomatic cases is directly
proportional to the average recovery period $1/\gamma_{s}$, while the
undetected asymptomatic cases are not accounted for by $I_{sQ}$.

It is a controversial topic whether asymptomatic individuals spread the virus
less than symptomatic individuals do. Figure 9 (a,b) shows that the peaks of
$I_{sQ}$ and $I_{a}+I_{aQ}+I_{sQ}$ have higher sensitivity to the parameter
$\beta_{a}$ than to $\beta_{s}$ when the other parameters of the symptomatic
and asymptomatic groups are approximately equal.

(a)

(b)

(c)

(d)

Figure 8:  Height of the peak of (a,c) $I_{sQ}$ and (b,d)
$I_{a}+I_{aQ}+I_{sQ}$ predicted by model (3) as a function of the parameters
$1/\gamma_{s}$ and $1/\gamma_{a}$. The other parameters are
$\psi_{s}=\psi_{a}=0$, $\beta_{s}=\beta_{a}=0.5$, $S_{sQ}=S_{aQ}=0.1$; the
latent period equals $\lambda_{s}=\lambda_{a}=0.5$ on panels (a, b) and
$\lambda_{s}=\lambda_{a}=10$ on panels (c,d).

(a)

(b)

(c)

(d)

Figure 9:  Height of the peak of (a) $I_{sQ}$ and (b) $I_{a}+I_{aQ}+I_{sQ}$ as
a function of $\beta_{s}$ and $\beta_{a}$ for $\gamma_{s}=\gamma_{a}=1/3.5$,
$\psi_{s}=\psi_{a}=0$ and $S_{sQ}=S_{aQ}=0.1$. Height of the peak of (c)
$I_{sQ}$ and (d) $I_{a}+I_{aQ}+I_{sQ}$, predicted by model (3), as a function
of $\psi_{s}$ and $\psi_{a}$ for $\gamma_{s}=\gamma_{a}=1/3.5$,
$\beta_{s}=\beta_{a}=0.5$ and $S_{sQ}=S_{aQ}=0.1$.  Figure 10:  Dependence of
timing and height of the peak of $I_{sQ}$ on $\lambda_{s}$ in model (3) (blue
points) for $1/\lambda_{s}+1/\gamma_{s}=1/\delta=5.5$ and
$1/\lambda_{s}\in(0,5.5).$ Here $\lambda_{s}=\lambda_{a}$,
$\gamma_{s}=\gamma_{a}$, $\psi_{s}=\psi_{a}=0$, $E_{s}(0)=E_{a}(0)=1/N$ and
$S_{sQ}=S_{aQ}=0$. The red point corresponds to model (2).

One can assume that asymptomatic individuals have little reason to test unless
required to do so. Further, according to [12], asymptomatic individuals can
have a weaker immune response to COVID-19 infection, hence symptomatic cases
(even during the latent phase) might be better detectable than asymptomatic
ones. These considerations suggest that $\psi_{a}<\psi_{s}$. Figure 9 (c,d)
shows that in order to ensure that the peak of $I_{sQ}$ and
$I_{sQ}+I_{aQ}+I_{a}$ is low (the dark blue region), it is necessary to
effectively detect and quarantine not only symptomatic individuals in the
latent phase of the disease but also a relatively high proportion of
asymptomatic cases. The peak of $I_{sQ}+I_{aQ}+I_{a}$ is more sensitive to the
testing rate $\psi_{a}$ than to $\psi_{s}$. The non-monotonic dependence of
the peak of $I_{sQ}$ on $\psi_{a}$ in Figure 9 (c) is due to the fact that an
increase of the testing rate (for small values of the testing rate) results in
an increase of the number of reported cases due to a higher detection rate
(the same effect is observed in Figure 7 (a,b)).

The symptomatic cases are detected and quarantined when the symptoms present
themselves after the latent phase. On the other hand, the asymptomatic
infected individuals, if undetected by testing, continue spreading the virus
after the latent phase during the whole infectious stage. This can explain the
higher sensitivity of the peak of the total infected population to the
parameters of the asymptomatic group, as Figures 8, 9 demonstrate for the
transmission, recovery and testing rates. Since the estimates of the
epidemiological parameters are currently based mainly on the data obtained
from symptomatic cases, the above sensitivity analysis suggests that better
estimates of the parameters of the asymptomatic group can play an important
role for improving the predictive power of models of the epidemic dynamics.

The duration of the latent phase can have a significant impact on dynamics of
$I_{sQ}$. Figure 10 shows the timing and height of the peak of $I_{sQ}$ for
model (3) for different values of $\lambda=\lambda_{s}=\lambda_{a}=\lambda$
under the assumption that $\gamma_{s}=\gamma_{a}$ and
$1/\lambda_{s}+1/\gamma_{s}=1/\delta$ with a fixed $\delta$. Both the height
and timing of the peak increase as the duration $1/\lambda$ of the latent
phase decreases. As the length of the latent phase $1/\lambda$ approaches
zero, the dynamics of model (3) converges to the dynamics of model (2).

###  3.4 Indiscriminate quarantining vs massive testing

Due to economic and social reasons, it is clear that lockdowns applied in many
countries cannot last for very long. Therefore there is a demand to find a
more economically sustainable solution to replace massive home quarantine
measures. One reasonable possibility seems to be effective testing. The blue
curve in Figure 11 (a) shows the evolution of the infected population for
model (2) with 13.5% of the total population quarantined at the initial moment
and zero testing rate; the green curve corresponds to the situation where
nobody stays in quarantine initially but 10% of the total population is
screened daily for COVID-19 (more precisely, 10% of infected asymptomatic
individuals are successfully detected daily). Under both setups the total
infected population $I_{a}+I_{aQ}+I_{sQ}$ follows approximately the same
trajectory. However, comparing the dynamics of the quarantined population, we
see that it is significantly smaller at all times when testing is used than in
the case when indiscriminate quarantining is applied, see the green and blue
curves in Figure 11 (b). A similar observation can be made when instead of
abrupt quarantining the gradual quarantining strategy is applied, see the red
curves in Figure 11 (a,b). Again, the quarantined population is significantly
smaller when quarantining is based on testing than in the case when
indiscriminate gradual quarantining is used, while the trajectories of the
infected population are similar for both scenarios. Recall that the number of
active symptomatic cases is proportional to the total infected population and
equals $(1-k)(I_{a}+I_{aQ}+I_{sQ})$.

Figure 11 (c,d) presents similar results for a range of values of the testing
rate. We compare the scenario with a positive testing rate $\psi$, zero
indiscriminate quarantining rate $\chi=0$ and zero initial quarantine size
$S_{Q}(0)=0$ to the scenarios with $\psi=\chi=0$, $S_{Q}(0)>0$ and with
$\psi=0$, $\chi>0$, $S_{Q}(0)=0$. The first scenario corresponds to
quarantining based on testing, while the second and third scenarios correspond
to indiscriminate abrupt and gradual quarantining, respectively. Given a
$\psi>0$ in the first scenario, the $S_{Q}(0)>0$ and $\chi>0$ in the second
and third scenarios are selected in such a way as to ensure that the height of
the infection peak in all the three scenarios is the same (as in Figure 11
(a)). This constraint defines $S_{Q}(0)$ and $\chi$ in the second and third
scenarios, respectively, as functions of the testing rate $\psi$ used in the
first scenario, see the graphs in Figure 11 (c). For each point of these
graphs, we also compute the time-integral of the total quarantined population
over the duration of the epidemic; the latter is technically defined as the
the time interval $[0,t_{e}]$ where $t_{e}$ is the moment after the infection
peak when $I_{sQ}(t_{e})=0.1/N$. Figure 11 (d) shows that the time-integral of
the quarantined population is by an order of magnitude smaller when
quarantining is based on massive testing than in the case when indiscriminate
(abrupt or gradual) quarantining is applied. Moreover, the ratio of the value
of this time-integral in the case of indiscriminate quarantining to its value
in the case of quarantining based on testing increases with the increasing
testing rate.

Similar results were obtained with model (3), see Figure 12.

These observations indicate that effective testing might be a way to replace
massive quarantine measures. Moreover, they may help to explain why some
countries such as South Korea and Singapore, which implemented massive rapid
free testing and an extremely good system of contact tracing, have been most
successful in containing the COVID-19 outbreak.

(a)

(b)

(c)

(d)

Figure 11: Indiscriminate quarantining versus quarantining based on testing in
model (2) with $E_{Q}(0)=2/N.$ (a) Dynamics of the total infected population,
$I_{a}+I_{aQ}+I_{sQ}$, for a selected testing rate $\psi$ and the
corresponding value of $S_{Q}(0)$ and $\chi$ resulting in the same height of
the peak of $I_{a}+I_{aQ}+I_{sQ}$. (b) Dynamics of the total quarantined
population corresponding to the graphs on panel (a); the same color coding is
used. (c) For every point $(\psi_{0},s_{0})$ of the green curve, the testing
rate $\psi=\psi_{0}$ with zero indiscriminate quarantining ($S_{Q}(0)=0$,
$\chi=0$) results in the same peak of infection $I_{a}+I_{aQ}+I_{sQ}$ as the
initial quarantine $S_{Q}(0)=s_{0}$ with zero testing ($\psi=\chi=0$). For
every point $(\psi_{0},\chi_{0})$ of the orange curve, the testing rate
$\psi=\psi_{0}$ with zero indiscriminate quarantining ($S_{Q}(0)=0$, $\chi=0$)
results in the same peak of infection as the indiscriminate quarantining rate
$\chi=\chi_{0}$ with zero testing and zero initial quarantine ($\psi=0,$
$S_{Q}(0)=0$). (d) The ratio of the time-integral of the total quarantined
population in the case of indiscriminate quarantining to the value of this
integral in the case of quarantining based on testing under the constraint
that the infection peak is the same for both scenarios. The green and orange
curves show the dependence of this ratio on $\psi$ and correspond to the
curves of the same colors, respectively, on panel (c).

(a)

(b)

Figure 12: Indiscriminate quarantining versus quarantining based on testing in
model (3) with $S_{sQ}(0)=S_{aQ}(0)=S_{Q}(0)$ and $E_{s}(0)=E_{a}(0)=1/N$ and
$\psi_{a}=\psi_{s}=\psi$. (a) Dynamics of the total infected population,
$I_{a}+I_{aQ}+I_{sQ}$, for a selected testing rate $\psi$ and the
corresponding value of $S_{Q}(0)$ resulting in the same height of the peak of
$I_{a}+I_{aQ}+I_{sQ}$. (b) Dynamics of the total quarantined population
corresponding to the graphs on panel (a); the same color coding is used.

##  4 Conclusions

Several issues involved in modeling the spread of an epidemic in a population
with symptomatic and asymptomatic groups were discussed. We introduced two
models which differ in their complexity and assumptions. Both models account
for two intervention strategies — indiscriminate home quarantining of part of
the population and isolation of positively tested individuals. In the
numerical analysis we used average values of epidemiological parameters
according to the current publications on COVID-19.

The models predict that dynamics of infection are similar for both
intervention strategies. However, massive testing allows to control and
contain the infection spread using a much lower isolation rate than in the
case of indiscriminate quarantining. In particular, given a constraint that
limits the maximal number of simultaneous active cases, the isolation rate
decreases with the increasing testing rate. The models predict that massive
testing measures can help to decrease the size of the quarantine, which
enforces the above constraint, by an order of magnitude and more. Due to a
variety of complicating factors, these findings cannot be considered as a
quantitatively accurate prediction for the coronavirus pandemic. However, they
indicate that massive testing can potentially stand-in for costly quarantine
measures. As such, our findings agree with the earlier predictions based on
different models [1].

We also discussed the role of asymptomatic individuals for the spread of the
epidemic. Our simulations indicate that a possible difference of the
epidemiological parameters of the symptomatic and asymptomatic groups can play
an important role for predicting the infection peak value and duration of the
epidemic. We showed that the peak of the infected population is more sensitive
to the rate of transmission from the asymptomatic group and to the rate of
testing of this group (for higher rates of testing) than to the corresponding
parameters of the symptomatic group. This can be attributed to the fact that
symptomatic cases are detected and quarantined earlier, after the latent phase
of the disease, while the asymptomatic infected individuals, if undetected by
testing, continue spreading the virus during the whole infectious phase.
Therefore more accurate estimates of the epidemiological parameters of
COVID-19, especially those of the asymptomatic groups, can improve predictions
of the epidemic dynamics.

## References

  * [1] Berger D, (2020) An SEIR Infectious Disease model with Testing and Conditional Quarantine. University of Chicago, Becker Friedman Institute for Economics Working Paper No. 2020-25
  * [2] Challen, R.J., Tsaneva, K., Pitt, M., Edwards, T., Gompels, L., Lucasa, L., Brooks-Pollock, E., Martin, C., Griffith, G. and Danon, L., 2020. Estimates of regional infectivity of COVID-19 in the United Kingdom following imposition of social distancing measures. medRxiv.
  * [3] Chladná Z, Kopfová J., Rachinskii D. Rouf S. (2020) Global dynamics of SIR model with switched transmission rate. J. Math. Biology
  * [4] Day M., Covid-19: four fifths of cases are asymptomatic, China figures indicate, BMJ 2020;369:m1375
  * [5] Du, Z., Xu, X., Wu, Y., Wang, L., Cowling, B.J. and Meyers, L.A., 2020. The serial interval of COVID-19 from publicly reported confirmed cases. medRxiv.
  * [6] Davies, N.G., Kucharski, A.J., Eggo, R.M., Gimma, A., Edmunds, W.J. and CMMID COVID-19 Working Group, 2020. The effect of non-pharmaceutical interventions on COVID-19 cases, deaths and demand for hospital services in the UK: a modelling study. medRxiv.
  * [7] European Centre for Disease Prevention and Control, 2016. Systematic review on the incubation and infectiousness/shedding period of communicable diseases in children.
  * [8] Feng Z. (2007) Final and peak epidemic sizes for SEIR models with quarantine and isolation, Math. Biosciences and Engineering, 4(4), 675-686
  * [9] Grassly NC, Fraser C (2006) Seasonal infectious disease epidemiology. Proceedings of the Royal Society of London B: Biological Sciences, 273(1600): 2541-2550.
  * [10] Hou, C., Chen, J., Zhou, Y., Hua, L., Yuan, J., He, S., Guo, Y., Zhang, S., Jia, Q., Zhao, C. and Zhang, J., 2020. The effectiveness of the quarantine of Wuhan city against the Corona Virus Disease 2019 (COVID‐19): well‐mixed SEIR model analysis. Journal of medical virology.
  * [11] Kaddar A (2010) Stability analysis in a delayed SIR epidemic model with a saturated incidence rate. Nonlinear Analysis: modelling and Control, 15(3): 299-306.
  * [12] Long, Q.X., Tang, X.J., Shi, Q.L., Li, Q., Deng, H.J., Yuan, J., Hu, J.L., Xu, W., Zhang, Y., Lv, F.J. and Su, K., 2020. Clinical and immunological assessment of asymptomatic SARS-CoV-2 infections. Nature medicine, pp.1–5.
  * [13] M. J. Mina, R. Parker, D. B. Larremore, Rethinking Covid-19 test sensitivity — A strategy for containment, N. Engl. J. Med. 2020, doi: 10.1056/NEJMp2025631.
  * [14] Siordia Jr, J.A., 2020. Epidemiology and Clinical Features of COVID-19: A Review of Current Literature. Journal of Clinical Virology, p.104357.
  * [15] D. P. Oran, Eric J. Topol, Prevalence of asymptomatic SARS-CoV-2 infection, Ann. Intern. Med. 2020, doi: 10.7326/M20-3012.
  * [16] Volpert, V., Banerjee, M. and Petrovskii, S., 2020. On a quarantine model of coronavirus infection and data analysis. Mathematical modelling of Natural Phenomena, 15, p.24.
  * [17] Wearing HJ, Rohani P, Keeling MJ (2005) Appropriate models for the management of infectious diseases. PLoS medicine, 2(7): e174.
  * [18] WHO Coronavirus Disease (COVID-19) Dashboard, available at:https://covid19.who.int/.
  * [19] WHO Q&A: Influenza and COVID-19: similarities and differences available at:https://www.who.int/emergencies/diseases/novel-coronavirus-2019/.
  * [20] Wölfel, R. et al. Virological assessment of hospitalized patients with COVID-2019. Nature https://doi.org/10.1038/s41586-020- 2196-x (2020).
  * [21] Shi Zhao,Qiany Lin, Jinjun Rand, Salihu S.Musae, Guangpu Yang, Weiming Wang, Yijun Lou, Daozhou Gao, Lin Yang, Daihai He, Maggie H.Wang Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak - Jan. 30, 2020