# Asymptotically FRW black holes

J.T. Firouzjaee  Department of Physics, Sharif University of Technology,
Tehran, Iran<EMAIL_ADDRESS>Reza Mansouri  Department of
Physics, Sharif University of Technology, Tehran, Iran and
School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran, Iran<EMAIL_ADDRESS>
###### Abstract

Special solutions of the LTB family representing collapsing over-dense regions
extending asymptotically to an expanding closed, open, or flat FRW model are
found. These solutions may be considered as representing dynamical mass
condensations leading to black holes immersed in a FRW universe. We study the
dynamics of the collapsing region, and its density profile. The question of
the strength of the central singularity and its nakedness, as well as the
existence of an apparent horizon and an event horizon is dealt with in detail,
shedding light to the notion of cosmological black holes. Differences to the
Schwarzschild black hole are addressed.

###### pacs:

95.30.Sf,98.80.-k, 98.62.Js, 98.65.-r

##  I introduction

Let us use the term cosmological black hole for any solution of Einstein
equations representing a collapsing overdensity region in a cosmological
background leading to an infinite density at its center sultana . There have
been different attempts to construct solutions of Einstein equations
representing such a collapsing central mass. Gluing of a Schwarzschild
manifold to an expanding FRW manifold is one of the first attempts to
construct such a cosmological black hole, as done first by Einstein and Straus
Einstein Straus . Depending of the way the model is constructed, one is led to
un-physical behavior of the trajectories Baker .

Models not based on a cut-and-paste technology is much more interesting giving
more information on the behavior of the mass condensation within a FRW
universe model. The first attempt is due to McVittie McVittie  introducing a
spacetime metric that represents a point mass embedded in a Friedmann-
Robertson-Walker (FRW) universe. There have been many other attempts to
construct cosmological black holes such as Nolan interior solution nolan m ,
and Sultana-Dyer solutionsultana , each of them contrasting some of the
features one expect from theory or observation.

The interest for cosmological black holes in the past has been mainly from the
theoretical side to understand concepts like black hole, singularity, horizon,
and thermodynamics of black holes haw-bh . Indeed, the conventional definition
of black holes implies an asymptotically flat space-time and a global
definition of the event horizon. In practice, however, the universe is not
asymptotically flat. The need for local definition of black holes and their
horizons has led to concepts such as Hayward’s trapping horizon Hayward94 ,
Ashtekar’s isolated horizon ashtekar99 , Ashtekar and Krishnan’s dynamical
horizon ashtekar02 , and Booth and Fairhurst’s slowly evolving horizon booth04
.
There are cases where both apparent and event horizon maybe possibly defined.
For example, for dynamical black holes one may define the event horizon as the
very last ray to reach future null infinity or the light ray that divides
those observer who cannot escape the future singularity from those that can
kra-hel-BH . Eardley proposed the conjecture that in such cases trapped
surfaces can be deformed to get arbitrarily close to the event horizon
Eardley98 . Numerical evidence was provided in Krishnan05  and later proved
analytically for the Vaidya metric Ben-Dov .

The precision cosmology has opened a new arena for questions like cosmological
black holes and their behavior. New observation of our galactic center allow
to resolve phenomena near the Schwarzschild horizon of the central black hole
doeleman . It is therefore desirable to have black hole models embedded in
cosmological environment to see if there may be considerable differences to
the familiar Schwarzschild black hole. There have been also increasing
interest in the gravitational lensing by a cosmological mass condensation such
as a cluster of galaxies in a cosmological background. The simplest cases are
the Kottler and the Einstein-Straus model rindler . The more complex situation
is lensing by a mass condensation within a dynamical background.

Now, a widely used metric to describe the gravitational collapse of a
spherically symmetric dust cloud is the so-called Tolman-Bondi-Lemai tre (LTB)
metric LTB . These models have been extensively studied for the validity of
the cosmic censorship conjecture cencorship ,joshi  and joshi-cell . In
particular, we know already initial condition  that, depending upon the
initial conditions defined in terms of the initial density and velocity
profiles from which the collapse develops, the central shell-focusing
singularity at $r=0$ can be either a black hole or a locally or globally naked
singularity. We may note however, that in all these papers a compact LTB
region is glued to the Schwarzschild metric or the FRW outer universe mansouri
. Therefore, the results have to be taken cautiously: any principally existent
event horizon is cut off by the outer static or homogeneous space-time. The
statement may still be correct that in a dynamic spacetime the cosmic
censorship hypotheses is valid, as discussed in wald98 . It is also possible
to glue two different LTB metrics to study the structure formation out of an
initial mass condensation or the formation of a galaxy with a central black
hole kra-hel-sf  and kra-hel-BH . Here again the structure of the metric
outside the mass condensation is defined by hand to match with a specific
galaxy or cluster feature. Faraoni et al have tried to change McVitte metric
so that it resemble a collapsing mass condensation. Their solution, however,
represents a singularity within a horizon embedded in a universe filled with a
non-perfect fluid where the change of the mass is not because of the in-
falling matter but the heat flow faraoni2 . This metric gives us no clue
whatsoever about the dynamics of a possible collapsed mass condensation.
Harada et al, being interested in the behavior of primordial black holes
within cosmological models with a varying gravitational constant, use a LTB
solution to study the evolution of a background scalar field when a black hole
forms from the collapse of dust in a flat Friedmann universe probing the
gravitational memory harada .

Our goal is to look for a model of a cosmological black hole, i.e. a mass
condensation leading to a singularity within a FRW universe universe. In this
paper we propose the models for closed-, open-, and flat FRW universe studying
their density profiles, singularities, and horizon behaviors. There are many
nontrivial questions to be answered before understanding in detail the
differences of these cosmological black holes to the familiar Schwarzschild
ones, which are beyond the scope of this paper and are to be dealt with in
future publications.

The question of singularities and the definition of a black hole in such a
dynamical environment has been subject of different studies in the last 15
years. We review very shortly different definitions of horizons in section II
as a reference to the properties of model solutions we propose. Some initial
attempts to model black holes within a FRW universe is introduced in section
III. Section IV is devoted to the LTB metric as the generic solution
representing a spherically symmetric ideal fluid. Section V is devoted to
different models of cosmological black holes, their dynamics, density profile,
apparent and event horizons,and singularities. The question of strength and
the nakedness of singularities are dealt with in section VI. We then conclude
in section VII. Throughout the paper we assume $8\pi G=c=1$.

##  II local definitions of black holes

Standard definition of black holes haw-bh  needs some global assumptions such
as regular predictability and asymptotic flatness. In the cosmological context
concepts of asymptotic flatness and regular predictability have no
application. This has already been noticed by Demianski and Lasota na73
stressing the fact that in the cosmological context the standard global
definition of black holes using event horizons may not be used any more.
Tipler tipler77  also present a definition of black hole in non asymptotically
flat space time, but these definition did not have comprehensive property of
black hole such as thermodynamic laws. In the last decade the interest in a
local definition of black holes has led to four different concepts based
primarily on the concept of the apparent horizon.

Let us start by assuming a spacelike two surface $S$ with two normal null
vectors $\ell^{a}$ and $n^{a}$ on it.The corresponding expansions are then
defined as $\theta_{(\ell)}$,$\theta_{(n)}$.

Definition 1 Hayward94 . A _trapping horizon_ $H$ is a hypersurface in a
4-dimensional spacetime that is foliated by 2-surfaces such that
$\theta_{(\ell)}\mid_{H}=0$, $\theta_{(n)}\mid_{H}\neq 0$, and
$\pounds_{n}\theta_{(\ell)}\mid_{H}\neq 0$. A trapping horizon is called
_outer_ if $\pounds_{n}\theta_{(\ell)}\mid_{H}<0$, _inner_ if
$\pounds_{n}\theta_{(\ell)}\mid_{H}>0$, _future_ if $\theta_{(n)}\mid_{H}<0$,
and _past_ if $\theta_{(n)}\mid_{H}>0$. The most relevant case in the context
of black holes is the _future outer trapping horizon_(FOTH).

Definition 2 ashtekar99 . A _weakly isolated horizon_ is a three-surface H
such that :

1\. H is null;

2\. The expansion $\theta_{(\ell)}\mid_{H}=0$ where $\ell^{a}$, being null and
normal to the foliations $S$ of $H$;

3.$-T^{b}_{a}\ell^{a}$ is future directed and causal;

4\. $\pounds_{\ell}\omega_{a}=0$, where
$\omega_{a}=-n_{b}\nabla_{\underleftarrow{a}}\ell^{b}$, and the arrow
indicates a pull-back to H.

Weakly isolated horizon is a useful term to be used for characterization of
black holes not interacting with their surroundings, and corresponds to
isolated equilibrium states in thermodynamics. These definition do not apply
to cosmological mass condensations because of their dynamical behavior.

Definition3 ashtekar02 . A _marginally trapped tube_ T (MTT) is a hypersurface
in a 4-dimensional spacetime that is foliated by two-surfaces $S$, called
_marginally trapped surfaces_ , such that $\theta_{(n)}|_{T}<0$ and
$\theta_{(\ell)}|_{T}=0$. MTTs have no restriction on their signature, which
is allowed to vary over the hypersurface. This is a generalization of the
familiar concept of the apparent horizon ashtekar02 . If a MTT is everywhere
spacelike it is referred to as a _dynamical horizon_. If it is everywhere
timelike it is called a timelike membrane (TLM). In case it is everywhere null
and non-expanding then we have an isolated horizon. The apparent horizons
evolving in the our proposed models will not be everywhere spacelike and will
have a complex behavior.

Irrespective of different concepts related to the apparent horizon we may
still compromise on a definition of event horizon differing principally from
the apparent horizon. We follow the definition of kra-hel-BH  as the very last
ray to reach future null infinity or the light ray that divides those observer
who cannot escape the future singularity from those that can. We will see in
the next sections that cosmological black holes may have distinct apparent and
even horizons, in contrast to the Schwarzschild black hole.

##  III Existing metrics representing over-densities within a cosmological
background and their deficiencies

###  III.1 McVittie s solutions

In 1933, McVittie McVittie  found an exact solution of Einstein s equations
for a perfect fluid mimicking a black hole embedded in a cosmological
background. McVittie s solutions can be written in the form

$ds^{2}=-(\frac{1-\frac{M}{2N}}{1+\frac{M}{2N}})^{2}dt^{2}+e^{\beta(t)}(1+\frac{M}{2N})^{4}(dr^{2}+h^{2}d\Omega^{2}),$
(1)

where $M=me^{\beta(t)/2}$ and $m$ is a constant. Functions $h(r)$ and $N(r)$
depend on a constant $k$, and are given, respectively, by

$\mbox{$h(r)=$}~{}\begin{cases}&\sinh(r)~{}~{}~{}k=1{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
&r~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}k=0\\\ &\sin(r)~{}~{}~{}~{}k=-1\end{cases}\\\
\mbox{$N(r)=$}~{}\begin{cases}&2\sinh(r/2)~{}~{}~{}k=1{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
&r~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}k=0\\\ &2\sin(r/2)~{}~{}~{}~{}k=-1\end{cases}$
(2)

This metric represents a point mass embedded into an isotropic universe. It
possesses a curvature singularity at proper radius $R=2m$, in contrast to the
Schwarzschild metric where there is a coordinate singularity. It has been
shown that this singularity is space-like and weaknolan . The interpretation
of the metric in the region $R<2m$ is also not clear nolan . Therefore, the
McVittie’s metric is not a suitable solution of Einstein equations to
represent the collapse of a spherical mass distribution with over-density
within a cosmological setting.

###  III.2 Sultana-Dyer solution

Recently Sultana and Dyer sultana  found an exact solution representing a
primordial cosmological black hole. It describes an expanding event horizon in
the asymptotic background of the Einstein-de Sitter universe. The black hole
is primordial in the sense that it forms ab initio with the big bang
singularity and therefore does not represent the gravitational collapse of a
matter distribution.
This metric is given by

$ds^{2}=t^{4}[(1-\frac{2m}{r})dt^{2}-\frac{4m}{r}dtdr-(1-\frac{2m}{r})dr^{2}-r^{2}d\Omega^{2}].$
(3)

Though the metric has the same causal characteristics as the Schwarzschild
spacetime, there are significant differences for timelike geodesics. In
particular an increase in the perihelion precession and the non-existence of
circular timelike orbits should be mentioned. The matter content is described
by a non-comoving two-fluid source, one of which is a dust and the other is a
null fluid. At late times the dust becomes superluminal near horizon violating
the energy condition.

##  IV Introducing realistic Models of Cosmological Mass Condensation

There maybe different ways of constructing solutions of Einstein equations
representing a collapsing mass concentration in a FRW background, as the
preceding sections show. We choose the direct way of a cosmological spherical
symmetric isotropic solution, and look for an overdensity mass distribution
within the model universe undergoing a collapse to see if and how a
singularity representing a black hole emerges. To begin with, we choose a so-
called flat LTB metric. This is the simplest spherically symmetric solution of
Einstein equations representing an inhomogeneous dust distribution LTB .

###  IV.1 LTB metric

The LTB metric may be written in synchronous coordinates as

$ds^{2}=dt^{2}-\frac{R^{\prime 2}}{1+f(r)}dr^{2}-R(t,r)^{2}d\Omega^{2}.$ (4)

It represents a pressure-less perfect fluid satisfying

$\displaystyle\rho(r,t)=\frac{2M^{\prime}(r)}{R^{2}R^{\prime}},\hskip
22.76228pt\dot{R}^{2}=f+\frac{2M}{R}.$ (5)

Here dot and prime denote partial derivatives with respect to the parameters
$t$ and $r$ respectively. The angular distance $R$, depending on the value of
$f$, is given by

$\displaystyle R=-\frac{M}{f}(1-\cos\eta(r,t)),$ $\displaystyle\hskip
22.76228pt\eta-\sin\eta=\frac{(-f)^{3/2}}{M}(t-t_{n}(r)),$ (6)
$\displaystyle\dot{R}=(-f)^{1/2}\frac{sin(\eta)}{1-cos\eta},$ (7)

for $f<0$, and

$R=(\frac{9}{4}M)^{\frac{1}{3}}(t-t_{n})^{\frac{2}{3}},$ (8)

for $f=0$, and

$\displaystyle R=\frac{M}{f}(\cosh\eta(r,t)-1),$ $\displaystyle\hskip
22.76228pt\sinh\eta-\eta=\frac{f^{3/2}}{M}(t-t_{n}(r)),$ (9)

for $f>0$.
The metric is covariant under the rescaling $r\rightarrow\tilde{r}(r)$.
Therefore, one can fix one of the three free parameter of the metric, i.e.
$t_{n}(r)$, $f(r)$, and $M(r)$. The function $M(r)$ corresponds to the Misner-
Sharp mass in general relativity, as shown in the general case of spherically
symmetric solutions of Einstein equationsmis-sharp .
There are two generic singularities of this metric: the shell focusing
singularity at $R(t,r)=0$, and the shell crossing one at $R^{\prime}(t,r)=0$.
To get rid of the complexity of the shell focusing singularity, corresponding
to a non-simultaneous big bang singularity, we will assume $t_{n}(r)=0$. This
will enable us to concentrate on the behavior of the collapse of an
overdensity region in an expanding universe without interfering with the
complexity of the inherent bang singularity of the metric.
Now, an expanding universe means generally $\dot{R}>0$. However, in a region
around the center it may happen that $\dot{R}<0$, corresponding to the
collapsing region. It is then easy to show that in this collapsing region
$\theta_{(\ell)}\propto(1-\frac{\sqrt{\frac{2M}{R}+f}}{\sqrt{1+f}})$,
$\theta_{(n)}\propto(-1-\frac{\sqrt{\frac{2M}{R}+f}}{\sqrt{1+f}})<0$.
Therefore, $R=2M$, is obviously a _marginally trapped tube_ , as defined in
section 2, representing an apparent horizon according to the familiar
definitionshaw-bh ; ashtekar02 . It will turn out that this apparent horizon
is not always spacelike and can have a complicated behavior for different $r$,
as was first seen in booth-mtt .

###  IV.2 Behavior of the curvature function $f(r)$

Now, we are interested in an expanding universe, meaning generally
$\dot{R}>0$. However, in a region around the center we expect to have a late
time behavior $\dot{R}<0$ corresponding to the collapse phase of the
overdensity region. From equations (IV.1), (8), and (IV.1) we infer that to
have a collapsing region one has to ask for $f(r)<0$ in that region. In
contrast, the universe outside the collapsing region being expanding leave us
to choose $f(r)>0$, $f(r)=0$, or $f(r)<0$ depending on the model. We may have
an asymptotically flat FRW universe, however, with $f(r)>0$ or $f(r)<0$
tending to zero for large $r$.

Now, we have still to make a choice for $f(r)$ at the center $r=0$. Expecting
the mass $M$ to be zero at $r=0$ to avoid a central singularity, we see from
(IV.1) and (IV.1) that $\frac{f^{3/2}}{M}|_{r=0}=const$, or $f(r=0)=0$ origin
con . This give us different possibilities of the function $f(r)$ to behave as
shown in Fig.1.

Figure 1:  Different behaviors of the curvature function $f(r)$.

##  V Construction of Models

We have now the necessary prerequisites to construct our models of mass
condensation immersed in FRW models leading to singularities and representing
cosmological black holes. Our cosmological black hole solutions evolve from
mass condensations within closed, open, or flat FRW universes, leading to
singularities having different horizons, and providing us examples of
collapsed regions behaving differently to known Schwarzschild ones.

###  V.1 Example I: $f<0$: asymptotically closed LTB metric

As mentioned before, we are free to choose one of the three parameters of the
LTB metric. Assuming a negative $f(r)$, we may choose $r$ such that
$f(r)=-M(r)/r$ cencorship . Now, let us choose the mass function $M$ such that

$M(r)=2^{3}a^{2}r^{3}\frac{\alpha+r^{3}}{1+r^{3}},$

where $a$ and $\alpha$ are constants to be defined properly. We then obtain
from (IV.1)

$\displaystyle R=r(1-cos\eta(r,t))$ $\displaystyle\hskip 22.76228pt\eta-
sin(\eta)=\sqrt{\frac{7.2+r^{3}}{1+r^{3}}}2^{3/2}at.$ (10)

We are free to fix $a$ and $\alpha$ such that for the present time, $t_{0}$,
the region around the center of the overdensity, $r=0$, is collapsing while
far from the center the universe expands. Note that in contrast to the
familiar FRW universe, where the scale factor as a function of time, $t$, is
an explicit function having a straightforward behavior. In the LTB case,
$R(t)$ playing the role of the scale factor is an implicit function of time
and comoving coordinate $r$ given by (IV.1). We now fix $a$ and $\alpha$ such
that $r=0$ corresponds to $\eta=\frac{3\pi}{2}$, and $r\gg 1$ corresponds to
$\eta=\frac{5\pi}{6}$ (Fig. 2). We then find $a\simeq\frac{0.75}{t_{0}}$ with
$t_{0}$ being the present time, and $\alpha\simeq 7$.
Now, the expansion phase of the model is given by $\dot{R}$ (7). We then see
from (7) that the region around $r=0$, corresponding to
$\eta\sim\frac{3\pi}{2}$, is always collapsing for any time $t$, while the
regions far from the center, $r>>0$, at the present time, corresponding to
$\eta\sim\frac{5\pi}{6}$, are expanding. Note that this bound LTB model,
similar to the closed FRW one, has a maximum comoving radius corresponding to
$f(r)=-1$.

Figure 2:  Evolution of the Cauchy surfaces.

The density evolution and the causal structure of the model is shown in Fig.
3. We see clearly how the central overdensity region collapses to a
singularity at $r=0$, while the universe is expanding. Note also how the slope
of outgoing null geodesics tend to infinity in the vicinity of the
singularity, i.e. $R^{\prime}\rightarrow+\infty$ at $R=0$.

Figure 3:  The case of the asymptotically closed universe: in the central
region the density increases with time indefinitely while far from the center
the density is decreasing with time. The apparent horizon and the trapped
region is shown in the lower diagram.

###  V.2 Example II: $f<0$, $\lim_{r\rightarrow\infty}f(r)\rightarrow 0$;
asymptotically flat LTB metric 1

Our favorite choice is a solution representing a collapsing overdensity region
at the center and a flat FRW far from the overdensity region. Of course the
overdensity region may take part in the expansion of the universe at early
times but gradually reversing the expansion and start collapsing. To achieve
this, we require $f(r)<0$ and $f(r)\rightarrow 0$ when $r\rightarrow\infty$.
This choice give us trivially $M(0)=0$.
Let us now make the ansatz $f(r)=-re^{-r}$ leading to

$M(r)=\frac{1}{a}r^{3/2}(1+r^{3/2}),$

where $a$ is a constant having the dimension $[a]=[L]^{-2}$. We fix $a$ by
$at_{0}=3\pi/2$. Similar to our previous model I, this value of $a$
corresponds to the collapsing mass condensation around $r=0$ starting in the
expanding phase of the bound LTB model.

Equation (IV.1), (8) then leads to

$\displaystyle R=\frac{\sqrt{r}(1+r^{3/2})}{ae^{-r}}(1-\cos\eta(r,t)),$
$\displaystyle\hskip 22.76228pt\eta-
sin(\eta)=\frac{e^{-\frac{3}{2}r}}{(1+r^{3/2})}at.$ (11)

We have plotted the density evolution and casual structure of this model in
Fig.4.

Figure 4:  The density profile for the cosmic black hole within a closed but
asymptotically flat universe. The causal structure is shown below. Note the
behavior of the event horizon for arbitrary large but finite $t$.

As a result of $R^{\prime}\rightarrow+\infty$ near the singularity, the slope
of the outgoing null geodesics becomes infinite at the central singularity.
Again we see clearly how the collapse of the central region and the evolution
of the apparent horizon separates the overdense region from the expanding
universe.
The negativity of the curvature function $f(r)$ means that, although the
universe is asymptotically flat, waiting enough, every slice $r=constant$ will
collapse to the central region. We may , however, define an event horizon
according to the definition of section 2 for any large but finite time, as
shown in Fig.4.

###  V.3 Example III: $f>0$, $f(r)\rightarrow 0$ when $r\rightarrow\infty$;
asymptotically flat LTB metric 2

What would happen if we choose the curvature function $f(r)$ such that it
tends to zero for large $r$ while it is positive? We still have a model which
tends to a flat FRW at large distances from the center, but having a density
less than the critical one.
Let us make the ansatz $f(r)=-r(e^{-r}-\frac{1}{r^{n}+c})$ with $n=2$ and
$c=20000$, leading to

$M(r)=\frac{1}{a}r^{3/2}(1+r^{3/2}),$

where $a$ is a constant having the dimension $[a]=[L]^{-2}$. We fix $a$ by
requiring $at_{0}=3\pi/2$. Equation (IV.1), (8) then leads to

$\displaystyle
R=\frac{\sqrt{r}(1+r^{3/2})}{a(e^{-r}-\frac{1}{r^{2}+20000})}(1-\cos\eta(r,t)),$
$\displaystyle\hskip 22.76228pt\eta-
sin\eta=\frac{(e^{-r}-\frac{1}{r^{2}+20000})^{1.5}}{(1+r^{3/2})}at.$ (12)

and for $f>0$ region,

$\displaystyle
R=\frac{\sqrt{r}(1+r^{3/2})}{a(\frac{1}{r^{2}+20000}-e^{-r})}(\cosh\eta(r,t)-1),$
$\displaystyle\hskip 22.76228pt\eta-
sinh\eta=\frac{(\frac{1}{r^{2}+20000}-e^{-r})^{1.5}}{(1+r^{3/2})}at.$ (13)

The solution is continuous at $r=1$, as can be checked by evaluating
$\dot{R}$, $R^{\prime}$, $\dot{R}^{\prime}$, $\ddot{R}$, and
$\ddot{R}^{\prime}$ at $r=1$(see the appendix).

We have plotted the density evolution and casual structure of this model in
Fig.5.

Figure 5:  Evolution of the cosmic black hole within an open but
asymptotically flat universe is similar to the closed case. The causal
structure, however, is significantly different, as seen from the lower
diagram. Result of the numerical calculation of the locations of the event
horizon, apparent horizon and the singularity is also shown.

The term $\frac{1}{r^{n}+c}$ is responsible for $f(r)$ being positive and
tending to zero for large $r$ given $n\geq 2$ and $c>>1$. Let us check if this
may cause shell crossing in the region where $f^{\prime}(r)<0$ while $f>0$.
Using (IV.1) we obtain

$\frac{R^{\prime}}{R}=\frac{M^{\prime}}{M}(1-\phi_{4})+\frac{f^{\prime}}{f}(\frac{3}{2}\phi_{4}-1),$
(14)

where
$\frac{2}{3}\leq\phi_{4}=\frac{sinh\eta(sinh\eta-\eta)}{(cosh\eta-1)^{2}}\leq
1$. The condition for no shell crossing singularity is then $\frac{M\mid
f^{\prime}\mid}{fM^{\prime}}<\frac{1-\phi_{4}}{\frac{3}{2}\phi_{4}-1}$. For
$\phi_{4}\sim 1$, corresponding to $\eta>>1$ or $t>>1$ the inequality breaks
down leading to a shell crossing singularity. The shell crossing, however, can
be shifted to arbitrary large $t$ by choosing $f^{\prime}<<1$ corresponding to
$n>>1$ and $c>>1$ hell-shell . Therefore, for the model we are proposing the
shell crossing will happen out of the range of applicability of it.
As a result of $R^{\prime}\rightarrow+\infty$ near the singularity, the slope
of the outgoing null geodesics become infinite at the central singularity.
Again we see clearly how the collapse of the central region and the evolution
of the apparent horizon separates the overdense region from the expanding
universe. There is an event horizon defined by the very last ray to reach
future null infinity and separates those observer who can not scape the future
singularity from those that can. A fixed $r=r_{0}$ value, being the non-
trivial root of $f(r)=0$, divides the absolute collapsing region from the
absolute expanding region. We may be living in a region inside the event
horizon but outside the apparent one without noticing it soon!

This solution represents a collapsing mass within an asymptotically flat FRW
universe. The collapsed region is dynamical in the sense that its mass is not
constant. In fact the rate of change of the Misner-Sharp energy is given by
$\frac{dM(r)}{dt}|_{R=const}=\frac{dM(r)}{dr}\frac{dr}{dt}|_{R=const}>0$
because $\frac{dM(r)}{dr}>0$, $R^{\prime}dr+\dot{R}dt=0$, $R^{\prime}>0$, and
$\dot{R}<0$ for collapsing region, so $\frac{dr}{dt}|_{R=const}>0$. Therefore,
it is clear that concepts such as isolated horizon and slowly evolving horizon
do not apply to this case.

###  V.4 Example IV: $f>0$: asymptotically open FRW metric

Now we look for a solution which goes to an open FRW metric at distances far
from the center. At the same time one should take care of the conditions
$M(0)=0$ and $\frac{f(0)^{3/2}}{M(0)}\neq\infty$. Let us choose

$f(r)=-r(1-r),$

and

$M(r)=\frac{1}{a}r^{3/2}(1+r^{3/2}),$

where $a$ is a constant, which may be fixed by assuming $r=0$ at the present
time $t_{0}$ corresponding to $\eta=\frac{3\pi}{2}$. This leads to
$at_{0}=3\pi/2+1$ . We then obtain from (IV.1)

$\displaystyle R=\frac{\sqrt{r}(1+r^{3/2})}{a(1-r)}(1-\cos\eta(r,t)),$
$\displaystyle\hskip 22.76228pt\eta-
sin(\eta)=\frac{(1-r)^{3/2}}{(1+r^{3/2})}at,$ (15)

for $r<1$, and

$\displaystyle R=\frac{\sqrt{r}(1+r^{3/2})}{a(r-1)}(cosh\eta(r,t)-1)$
$\displaystyle\hskip
22.76228ptsinh\eta-\eta=\frac{(r-1)^{3/2}}{(1+r^{3/2})}at,$ (16)

for $r>1$. The solution is again continuous at $r=1$, as can be checked by
evaluating $\dot{R}$, $R^{\prime}$, $\dot{R}^{\prime}$, $\ddot{R}$, and
$\ddot{R}^{\prime}$ at $r=1$(see the appendix).

The resulting density profile and the causal structure is plotted in Fig.6.
Obviously a singularity at the origin forms gradually while the universe is
expanding. The causal structure is also similar to the open but asymptotically
flat case.

Figure 6:  The case of asymptotically open model: the density profile is
similar to the previous open but asymptotically flat case, except for the less
mass concentrated in the central region. The causal structure is also similar.
The locations of the event horizon, apparent horizon and the singularity are
also calculated numerically.The separation between the singularity and the
apparent horizon is not clear here due to the scale chosen.

This solution represents a collapsing mass within an open FRW universe. The
collapsed region is again dynamical in the sense that its mass is not
constant, and the rate of change of the Misner-Sharp energy is given by the
same amount as the previous model. Therefore, concepts of isolated horizon and
slowly evolving horizon do not apply to this case.

##  VI Characteristics of singularities of proposed models

We have avoided in the models proposed the shell crossing singularities except
example III with a late time shell crossing singularity.

The shell focusing singularities, however, are unavoidable and in fact it is
what we are looking for to study characteristics of cosmological black holes.
An important aspect of such a singularity is its gravitational strength
strength , which is an important differentiating feature of black hole.

###  VI.1 Strength of the shell focusing singularities

Heuristically, a singularity is termed gravitationally strong, or simply
strong, if it destroys by crushing or stretching any object which falls into
it. The prototype of such a singularity is the Schwarzschild one: a radially
infalling object is infinitely stretched in the radial direction and crushed
in the tangential directions, with the net result of crushing to zero volume.
Otherwise a singularity is termed weak where no object falling into it is
destroyed. To check the strength of singularities of our models we use the
criteria defined by Clarke strength .
Let $k^{\mu}$ be the tangent vector to the ingoing null geodesic, and
$\lambda$ the corresponding affine parameter being zero at the center.
$R_{\mu\nu}$ being the Ricci tensor, the singularity is said to be strong if

$\Psi=lim_{r\rightarrow 0}\lambda^{2}k^{\mu}k^{\nu}R_{\mu\nu}\neq 0.$ (17)

For a general LTB metric one obtains easily
$k^{\mu}k^{\nu}R_{\mu\nu}=2(k^{t})^{2}\frac{M^{\prime}}{R^{2}R^{\prime}}$. For
the three interesting cases of cosmological black holes in flat and open LTB
models we have done the calculation along the lines of the joshi  using
appropriate coordinates near singularity. For our cases (V.2-V.3-V.4) we
obtain after some calculation $\Psi=0$ for $r\rightarrow 0$. Therefore, shell
focusing singularities occurring in the center of the models we are proposing
are week. This is in contrast to the Schwarzschild singularity which is a
strong one. We leave it to future studies if this weakness is generic of any
cosmological black holes.

###  VI.2 Nakedness of singularities

We know already from Oppenheimer-Snyder collapse of a homogeneous dust
distribution how the shells become singular at the same time, and thus none of
them crosses. In the case of spherically symmetric inhomogeneous matter
configurations, however, the proper time of collapse depends on the comoving
radius $r$. Thus the piling up of neighboring matter shells at finite proper
radius can occur, thereby producing two-dimensional caustics where the energy
density and some curvature components diverge. These singularities can be
locally naked, but they are gravitationally weak cencorship ; nolan-sc-w ,
i.e. curvature invariants and tidal forces remain finite. It has also been
shown that analytic continuations of the metric, in a distributional sense,
can always be found in the neighborhood of the singularity shell-c .
Models proposed in this paper are, however, free from shell crossing
singularities. The shell crossing singularity of example III at late times
does not influence the following argumentation. Conditions for the absence of
shell crossing singularities have been studied in detail in hell-shell . In
our case these conditions are equivalent to $M^{\prime}(r)>0$ and
$R^{\prime}>0$, which are satisfied by the models discussed above. We may then
conclude that

$\frac{\frac{dt}{dr}|_{AH}}{\frac{dt}{dr}|_{null}}=\left(1-\frac{2M^{\prime}}{R^{\prime}}\right)<1.$
(18)

Therefore, the condition for the apparent horizon $R=2M$ to be spacelike is,
i.e. $-1<\frac{\frac{dt}{dr}|_{AH}}{\frac{dt}{dr}|_{null}}<1$, leads to the
condition $R^{\prime}-M^{\prime}>0$, which is not everywhere satisfied in our
model. As a result we notice that apparent horizons of the models proposed
here are not spacelike everywhere. Such a behavior has already been discussed
inbooth-mtt .
The case of shell focusing singularities is, however, a different one.
Irrespective of the behavior of the apparent and event horizons, it is then a
relevant question if the shell focusing singularity could be a naked one. We
notice that the slope of the outgoing null geodesics at the singularity are
greater than the slope of the singularity itself. Therefore, the singularity
is spacelike and no timelike or null geodesic can come out of the singularity.
We then conclude that the singularities we are facing can not be naked.

##  VII Discussion and conclusions

Unlike models discussed so far in the literature, we have constructed models
of mass condensation within the FRW universe leading to cosmological black
holes without having the usual pathologies we know from other models: the
cosmic fluid is dust and ideal producing a singularity at the center in the
course of time. The central singularity is spacelike and not naked. In the
case of flat or open universe models the singularity is weak and has distinct
apparent and event horizons. The apparent horizons are not everywhere
spacelike, to be compared with the Schwarzschild one which is null everywhere.
This has already been noticed in a general context by booth-mtt . While the
apparent horizon is defined by the surfaces $R=2M$, similar to the
Schwarzschild horizon, the even horizon is further away. Models we have
proposed show that one has to expect new effects while considering dynamical
cosmic black holes. The simple Schwarzschild static model may not reflect all
the phenomena one may expect in observational cosmology, and the black hole
thermodynamics. Even the simple concept of mass is not a trivial one in such a
dynamical environment. the answer to these questions are beyond the scope of
these paper and will be deal with in future publications.

## Appendix A

The curvature function $f(r)$ has a zero point where it changes sign for
models III and IV, corresponding to two different solutions. Therefore, we
have to take care of joining two solutions across the hypersurface defined by
$f(r)=0$ to be continuous. This is done by looking at the metric functions and
their derivatives to be continuous.

Let us first look at the model IV. There we have to look at the metric
function $R$ and its derivatives, $R$, $R^{\prime}$, $\dot{R}$, $\ddot{R}$ and
$\ddot{R}^{\prime}$, at the point $r=1$ where $f$ vanishes. From the following
relations derived from the Einstein equations (5)

$\ddot{R}=-\frac{M}{R^{2}},$ (19)

$\dot{R}^{\prime}=\frac{M^{\prime}}{R\dot{R}}-\frac{MR^{\prime}}{\dot{R}R^{2}}+\frac{f^{\prime}}{2\dot{R}},$
(20)

and

$\ddot{R}^{\prime}=-\frac{M^{\prime}}{R^{2}}+\frac{2MR^{\prime}}{R^{3}},$ (21)

we infer that these second derivatives relevant for the Einstein equations to
be continuous on the hypersurface $f(r)=0$ are continuous if the $f$, $R$,
$R^{\prime}$, $\dot{R}$, $M^{\prime}$, and $M$ are continuous. Now, because of
the continuity of $f$, $M^{\prime}$, and $M$, we just have to prove the
continuity of $R$, $\dot{R}$, and $R^{\prime}$.

Let us look first at $R$ and its derivative $R^{\prime}$. In the case of $r<1$
we have

$\displaystyle R=\frac{a(r)}{1-r}(1-cos\eta),$ $\displaystyle\eta-
sin\eta=\frac{(1-r)^{1.5}}{b(r)}t,$ (22)

where $a(r)=\sqrt{r}+r^{2}$, $b(r)=1+r^{1.5}$, and $a(1)=2$,
$a^{\prime}(1)=2.5$, $b(1)=2$, $b^{\prime}(1)=1.5$, and

$\dot{R}=\frac{a\sqrt{1-r}}{b}\frac{sin\eta}{1-cos\eta}.$ (23) $\displaystyle
R^{\prime}=\frac{a^{\prime}(1-r)+a}{(1-r)^{2}}(1-cos\eta)-\frac{a}{1-r}$
$\displaystyle\frac{sin\eta}{1-cos\eta}\frac{1.5(1-r)^{0.5}b+b^{\prime}(1-r)^{1.5}}{b^{2}}t.$
(24)

Defining $1-r=x$, we have

$\eta-sin\eta=\frac{\eta^{3}}{6}-O(\eta^{5})=\frac{x^{3}}{2}t.$ (25)

Therefore, to first order in $\eta$ we have $\eta=\sqrt[3]{3t}\sqrt{x}$. Now
taking the limit $x\rightarrow 0^{-}$ we obtain

$\displaystyle\lim_{x\rightarrow 0^{-}}R(x)=\lim_{x\rightarrow
0^{-}}\frac{2}{x}(1-cos\eta)=\lim_{x\rightarrow
0^{-}}(\frac{\eta^{2}}{x}-O(\eta^{4})/x)$
$\displaystyle=(3t)^{2/3}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(26)

which is a well defined quantity.

In the case of $r>1$ we have

$\displaystyle R=\frac{a(r)}{r-1}(cosh\eta-1),$ $\displaystyle
sinh\eta-\eta=\frac{(r-1)^{1.5}}{b(r)}t,$ (27)

$\dot{R}=\frac{a\sqrt{r-1}}{b}\frac{sinh\eta}{cosh\eta-1},$ (28)

and

$\displaystyle
R^{\prime}=\frac{a^{\prime}(r-1)-a}{(r-1)^{2}}(cosh\eta-1)+\frac{a}{r-1}$
$\displaystyle\frac{sinh\eta}{cosh\eta-1}\frac{1.5(r-1)^{0.5}b-b^{\prime}(r-1)^{1.5}}{b^{2}}t.$
(29)

Now, defining $r-1=x$, and noting that

$sinh\eta-\eta=\frac{\eta^{3}}{6}+O(\eta^{5})=\frac{x^{3}}{2}t,$ (30)

we obtain to first order of $\eta$ the relation $\eta=\sqrt[3]{3t}\sqrt{x}$.
Therefore,

$\displaystyle\lim_{x\rightarrow 0^{+}}R(x)=\lim_{x\rightarrow
0^{+}}\frac{2}{x}(cosh\eta-1)=\lim_{x\rightarrow
0^{+}}\frac{\eta^{2}}{x}+O(\eta^{4})/x$
$\displaystyle=(3t)^{2/3}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}.$
(31)

Therefore the continuity of $R$ across $r=1$ is established.
Similar calculation for the first derivatives shows the continuity of
$R^{\prime}$ and $\dot{R}$ having well defined values on both sides of the
$r=r_{0}$ hypersurface:

$R^{\prime}(1)=2.5(3t)^{2/3}-\frac{3}{4(3t)^{1/3}}t,$ (32)

and

$\dot{R}(1)=\frac{2}{(3t)^{1/3}}.$ (33)

The case of model III is similar except for the hypersurface defined by
$g(r)=\frac{f(r)}{r}=e^{-r}-\frac{1}{r^{2}+20000}=0$ with the root of
$e^{-r_{0}}-\frac{1}{r_{0}^{2}+20000}=0$ being at a point $r=r_{0}$ different
from $r=1$. It is easy to see that $g(r)$ is an analytic function at
$r=r_{0}$, and can be approximated by $g(r)\approx
g^{\prime}(r_{0})(r-r_{0})+\frac{g^{\prime\prime}(r_{0})}{2}(r-r_{0})^{2}+...$.
Similar calculations verify the continuity of the metric function $R$ and its
relevant derivatives across the hypersurface $r=r_{0}$.

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# Effective containment explains sub-exponential growth in confirmed cases of
recent COVID-19 outbreak in Mainland China

Benjamin F. Maier<EMAIL_ADDRESS>Robert Koch Institute, Nordufer
20, D-13353 Berlin, Germany  Dirk Brockmann  Robert Koch Institute, Nordufer
20, D-13353 Berlin, Germany  Institute for Theoretical Biology, Humboldt-
University of Berlin, Philippstr. 13, D-10115 Berlin

(August 28, 2024)

###### Abstract

The recent outbreak of COVID-19 in Mainland China is characterized by a
distinctive algebraic, sub-exponential increase of confirmed cases with time
during the early phase of the epidemic, contrasting an initial exponential
growth expected for an unconstrained outbreak with sufficiently large
reproduction rate. Although case counts vary significantly between affected
provinces in Mainland China, the scaling law $t^{\mu}$ is surprisingly
universal, with a range of exponents $\mu=2.1\pm 0.3$. The universality of
this behavior indicates that, in spite of social, regional, demographical,
geographical, and socio-economical heterogeneities of affected Chinese
provinces, this outbreak is dominated by fundamental mechanisms that are not
captured by standard epidemiological models. We show that the observed scaling
law is a direct consequence of containment policies that effectively deplete
the susceptible population. To this end we introduce a parsimonious model that
captures both, quarantine of symptomatic infected individuals as well as
population wide isolation in response to mitigation policies or behavioral
changes. For a wide range of parameters, the model reproduces the observed
scaling law in confirmed cases and explains the observed exponents.
Quantitative fits to empirical data permit the identification of peak times in
the number of asymptomatic or oligo-symptomatic, unidentified infected
individuals, as well as estimates of local variations in the basic
reproduction number. The model implies that the observed scaling law in
confirmed cases is a direct signature of effective contaiment strategies
and/or systematic behavioral changes that affect a substantial fraction of the
susceptible population. These insights may aid the implementation of
containment strategies in potential export induced COVID-19 secondary
outbreaks elsewhere or similar future outbreaks of other emergent infectious
diseases.

COVID-19, 2019-nCoV, coronavirus, epidemics, outbreaks, scaling laws,
epidemiology, modelling

##  I Introduction

The current outbreak of the new coronavirus in Mainland China (COVID-19,
previously named 2019-nCoV) is closely monitored by governments, researchers,
and the public alike cohen_scientists_2020 ; hsu_heres_2020 ;
lewis_chinas_2020 ; who_novel_2020 ; chen_epidemiological_2020 ;
zhao_preliminary_2020 ; cdc_2019_2020 . The rapid increase of positively
diagnosed cases in Mainland China and subsequent exportation and confirmation
of cases in more than 20 countries worldwide raised concern on an
international scale. The World Health Organization (WHO) therefore announced
the COVID-19 outbreak a Public Health Emergency of International Concern
who_novel_2020 .

Confirmed cases in Mainland China increased from approx. 330 on Jan. 21st,
2020 to more than 17,000 on Feb. 2nd, 2020 in a matter of two weeks
center_for_systems_science_and_engineering_johns_hopkins_university_cssegisanddatacovid-19_2020
. In Hubei Province, the epicenter of the COVID-2019 outbreak, confirmed cases
rose from 270 to 11,000 in this period, in all other Chinese provinces the
cumulated case count increased from 60 to 6,000 in the same period.

An initial exponential growth of confirmed cases is generically expected for
an uncontrolled outbreak and in most cases mitigated with a time delay by
effective containment strategies and policies that reduce transmission and
effective reproduction of the virus, commonly yielding a saturation in the
cumulative case count and an exponential decay in the number of new
infections. Although in Hubei the number of cases was observed to grow
exponentially in early January li_early_2020 , the subsequent rise followed a
sub-exponential, super-linear, algebraic scaling law $t^{\mu}$ with an
exponent $\mu=2.3$ (between Jan. 24th and Feb. 9th), cf. Fig. 1A. For the
majority of the affected Chinese provinces of Mainland China, however, this
type of algebraic rise occured from the beginning, lacking an initial
exponential growth altogether. Surprisingly, the exponent $\mu$ does not vary
substantially with a typical value of $\mu=2.1\pm 0.3$ for the confirmed case
curves in other substantially affected provinces (confirmed case counts larger
than 500 on Feb. 12th), despite geographical, socio-economical differences,
differences in containment strategies, and heterogeneties that may have
variable impacts on how the local epidemic unfolds, cf. Fig. 1B and C.

Figure 1: A: Confirmed case numbers in Hubei. The increase in cases follows a
scaling law $t^{\mu}$ with an exponent $\mu=2.32$ after a short initial
exponential growth phase. On Feb 9th the case count starts deviating towards
lower values. B: Aggregated confirmed cases in all other affected provinces
except Hubei. $C(t)$ follows a scaling law with exponent $\mu=1.92$ until Feb.
2nd when case counts deviate to lower values. The insets in A and B depict
$C(t)$ on a log-log scale. C: Confirmed cases as a function of time for the 8
most affected provinces in China. The curves follow a scaling law with
exponents $\mu\approx 2$ with the exception of Chongqing Province ($\mu=1.45$)
and Jiangxi Province ($\mu=2.75)$.

The universality of the scaling relation $t^{\mu}$ with similar exponents
$\mu$ is evidence that this aspect of the dynamics is determined by
fundamental principles that are at work and robust with respect to variation
of other parameters that typically shape the temporal evolution of epidemic
processes. Three questions immediately arise, (i) what may be the reason for
this functional dependency, (ii) are provinces other than Hubei mostly driven
by export cases from Hubei and therefore follow a similar functional form in
case counts as suggested by preliminary studies prasse_network-based_2020 ;
sanche_novel_2020 ; chen_time_2020 , or, alternatively, (iii) is the scaling
law a consequence of endogeneous and basic epidemiological processes and a
consequence of a balance between transmission events and containment factors.

Here we propose a parsimonious epidemiological model that incorporates
quarantine measures and containment policies. In addition to standard
epidemiological parameters such as transmission and recovery rate, the model
introduces effective quarantine measures that act on _both_ , symptomatic
infected individuals as well as susceptible individuals. We show that
depleting both “ends” of the transmission interactions that involve an
infectious and a susceptible individual naturally yields a sub-exponential,
algebraic increase in confirmed cases. This behavior is qualitatively
different from the behavior expected in a scenario in which only infecteds are
targets of containment strategies to reduce transmission. In this case one
expects either a stable exponential growth at a lower rate or an exponential
decay if mitigation is sufficiently effective.

The model predicts the observed growth of case numbers exceptionally well for
almost all affected provinces for epidemiological parameters estimated by
preliminary studies who_novel_2020-1 ; cdc_2019_2020 ; sanche_novel_2020 ;
zhao_preliminary_2020 . Furthermore, the model permits an indirect assessment
of the peak time in the number of undetected infectious individuals
(asymptomatic and oligosymptomatic cases) that preceeds the depletion of
confirmed cases. This indicates that the number of unidentified infectious
indviduals peaked in early February.

##  II Situation assessment

###  II.1 Case numbers in Hubei and other provinces

We rely on case number data provided by the Systems Science and Engineering
group of Johns Hopkins University who currently give up to two updates per day
on the number of laboratory-confirmed cases globally
center_for_systems_science_and_engineering_johns_hopkins_university_cssegisanddatacovid-19_2020
. For the time period discussed in this paper, the group integrated and
curated case data manually collected at four sources: the World Health
Organization (WHO), the US Centers for Disease Control and Prevention (CDC),
the European Centre for Disease Prevention and Control (ECDC), the Chinese
physician’s platform DXY.cn, and the National Health Commission of the
People’s Republic of China (NHC). The published data comprises total confirmed
cases, total deaths, and total recovered cases as a function of time for each
affected location. In the following we will focus on the number of total
confirmed cases $C(t)$.

In Hubei, we find that the initital increase in confirmed cases is followed by
an algebraic scaling with time, i.e. $C(t)\propto t^{\mu}$, with a scaling
exponent $\mu\approx 2.3$ that persisted until Feb. 9th, see Fig. 1A. On Feb.
12th, the case definition was changed by Chinese authorities which labeled a
large number of previously unconfirmed cases as confirmed, leading to a
discontinuity in the curves. We will therefore only consider data prior to
Feb. 12th, 6am UTC.

Fig. 1B illustrates the cumulated case count in all affected provinces except
Hubei province. In the period Jan. 21st until Feb. 2nd the curve follows an
algebraic scaling law $t^{\mu}$ with $\mu\approx 1.9$ lacking the initial
exponential phase observed in Hubei province. Starting on Feb. 2nd, the
observed case count starts deviating towards lower case counts. This poses the
question whether the observed behavior is an averaging affect introduced by
accumulating case counts across provinces. Interestingly, Fig. 1C provides
evidence that this is not so. The confirmed case counts in the most affected
provinces all exhibit a scaling law with exponents close to $\mu=2$. Among the
most affected provinces only Chongqing Province exhibits a significantly lower
exponent. Furthermore, all provincial case count curves start deviating from
the algebraic curve around Feb. 2nd.

###  II.2 Implemented policies

The Chinese government put several mitigation policies in place to contain the
spread of the epidemic. In particular, positively diagnosed cases were either
quarantined or put under a form of self-quarantine at home gan_over_2020 .
Similarly, suspicious cases were confined in monitored house arrest, e.g.
individuals who arrived from Hubei before all traffic from its capital Wuhan
was effectively restricted fahrion_how_2020 ; rosales_wuhan_2020 . These
measures aimed at the removal of infectious individuals from the transmission
process.

Additionally, measures were implemented that aimed at the protection of the
susceptible population by the partial shutdown of public life. For instance,
universities remained closed, many businesses closed down, and people were
asked to remain in their homes for as much time as possible fahrion_how_2020 ;
rosales_wuhan_2020 ; gan_over_2020 . These actions that affect both,
susceptibles and non-symptomatic infectious individuals, not only protect
susceptibles from acquiring the infection but also remove a substantial
fraction of the entire pool of susceptibles from the transmission process and
thus indirectly mitigate the proliferation of the virus in the population in
much the same way that herd immunity is effective in the context of vaccine
preventable diseases.

##  III Epidemiological modeling with quarantine and isolation

On a very basic level, an outbreak as the one in Hubei is captured by SIR
dynamics keeling_modeling_2008 . The population is devided into three
compartments that differentiate the state of invididuals with respect to the
contagion process: (I)nfected, (S)usceptible to infection, and (R)emoved (i.e.
not taking part in the transmission process). The corresponding variables $S$,
$I$, and $R$ quantify the respective compartments’ fraction of the total
population such that $S+I+R=1$. The temporal evolution of the number of cases
is governed by two processes: The infection that describes the transmission
from an infectious to a susceptible individual with basic reproduction number
$R_{0}$ and the recovery of an infected after an infectious period of average
length $T_{I}$. The basic reproduction number $R_{0}$ captures the average
number of secondary infections an infected will cause before he or she
recovers or is effectively removed from the population.

Initially, a small fraction of infecteds yields an exponential growth if the
basic reproduction number is larger than unity. When the supply of
susceptibles is depleted, the epidemic reaches a maximum and the infecteds
decline. A simple reduction of contacts caused by isolation policies could be
associated with a reduction in the effective reproduction number, which would,
however, still yield an exponential growth in the fraction of infecteds as
long as $R_{0}>1$, inconsistent with the observed scaling law $t^{\mu}$
discussed above. To test the hypothesis that the observed growth behavior can
be caused by isolation policies that apply to both, infected and susceptible
individuals, by effective public shutdown policies, we extend the SIR model by
two additional mechanisms one of which can be interpreted as a process of
removing susceptibles from the transmission process. First, we assume that
general public containment policies or individual behavioral changes in
response to the epidemic effectively remove individuals from the interaction
dynamics or significantly reduce their participation in the transmission
dynamics. Secondly, we account for the removal of symptomatic infected
individuals by quarantine procedures. The dynamics is governed by the system
of ordinary differential equations:

$\displaystyle\partial_{t}S$ $\displaystyle=-\alpha SI-\kappa_{0}S$ (1)
$\displaystyle\partial_{t}I$ $\displaystyle=\alpha SI-\beta
I-\kappa_{0}I-\kappa I$ (2) $\displaystyle\partial_{t}X$
$\displaystyle=(\kappa+\kappa_{0})I,$ (3)

a generalization of the standard SIR model, henceforth referred to as the
SIR-X model. The rate parameters $\alpha$ and $\beta$ quantify the
transmission rate and the recovery rate of the standard SIR model,
respectively. Additionally, the impact of public containment is captured by
the terms proportional to the containment rate $\kappa_{0}$ that is effective
in both $I$ and $S$ populations. Infected individuals are removed at rate
$\kappa$ corresponding to quarantine measures that only affect symptomatic
infecteds. The new compartment $X$ quantifies symptomatic, quarantined
infecteds. Here we assume that the fraction $X(t)$ is proportional to the
empirical confirmed and reported cases. The case $\kappa_{0}=0$ corresponds to
a scenario in which the general population is unaffected by policies or does
not commit behavioral changes in response to an epidemic. The case $\kappa=0$
corresponds to a scenario in which symptomatic infecteds are not isolated
specifically. Note that infecteds are always removed from their compartment
$I$ at a higher rate than susceptibles as
$\beta+\kappa+\kappa_{0}>\kappa_{0}$.

In the basic SIR model that captures unconstrained, free spread of the
disease, the basic reproduction number $R_{0}$ is related to transmission and
recovery rate by $R_{0}\equiv R_{0,\mathrm{free}}=\alpha/\beta$ because
$\beta^{-1}=T_{I}$ is the average time an infected individual remains
infectious before recovery or removal. Here, the time period that an infected
individual remains infectious is
$T_{I,\mathrm{eff}}=(\beta+\kappa_{0}+\text{$\kappa$})^{-1}$ such that the
effective, or “observed” reproduction number $R_{0,\mathrm{eff}}=\alpha
T_{I,\mathrm{eff}}$ is smaller than $R_{0,\mathrm{free}}$ since both
$\kappa_{0}>0$ and $\kappa>0$.

We introduce _public containment leverage_

$P=\frac{\kappa_{0}}{\kappa_{0}+\kappa}$

that reflects how strong isolation measures affect the general public in
comparison to quarantine measures imposed on symptomatic infecteds alone. We
further define

$Q=\frac{\kappa_{0}+\kappa}{\beta+\kappa_{0}+\kappa}$

as the _quarantine probability_ , i.e. the probability that an infected is
identified and quarantined either in specialized hospital wards or at home.

The key mechanism at work in the model defined by Eqs. (1)-(3) is the
exponentially fast depletion of susceptibles in addition to isolation of
infecteds. This effect is sufficient to account for the observed scaling law
in the number of confirmed cases for a plausible range of rate parameters as
discussed below.

Figure 2: Case numbers in Hubei compared to model predictions. The quarantined
compartment $X(t)$ and the unidentified infectious compartment $I(t)$ are
obtained from least-squares fits of the model defined by Eqs. (1)-(3), cf.
App. A. All fits were performed with fixed values of $R_{0,\mathrm{free}}=6.2$
and $T_{I}=8\mathrm{d}$. Note that the effective, observed basic reproduction
number $R_{0,\mathrm{eff}}$ has lower values and varies for each of the
affected provinces as discussed in Sec. IV and shown in Tab. 1. A: In Hubei,
the model captures both, the initial rise of confirmed cases as well as the
subsequent algebraic growth. The confirmed cases are predicted to saturate at
$C=51,000$. The model also predicts the time-course of the number of
unidentified infectious individuals $I(t)$ which peaks on Feb. 7th and
declines exponentially afterwards. While the order of $I(t)$ is associated
with rather large fluctuations depending on the fitting procedure, the
predicted peak time is robust, consistently around Feb. 7th. B: Model
prediction for case numbers aggregated over all affected provinces other than
Hubei. The case numbers’ algebraic growth is well reflected and predicted to
saturate at $C=12,600$. In contrast to Hubei, the fraction of unidentified
infecteds peaks around Feb. 1st, approximately a week earlier. The insets in A
and B depict both data and fits on a log-log scale. C: Fits for confirmed
cases as a function of time for the remaining 8 most affected provinces in
China. All curves are well captured by the model fits that predict similar
values for the peak time of unidentified infecteds.

##  IV Results

We assume that a small number of infected individuals travelled from Hubei to
each of the other affected provinces before traffic restrictions were
effective but at a time when containment measures were just being implemented.
Fig. 2 illustrates the degree to which the case count for Hubei province and
the aggregated case count for all other provinces is captured by the SIR-X
model as defined by Eqs. (1)-(3).

Province | $N/10^{6}$ | $Q$ | $P$ | $I_{0}/X_{0}$ | $R_{0,\mathrm{eff}}$
---|---|---|---|---|---
Hubei | 57.1 | 0.47 | 0.66 | 2.55 | 3.28
All exc. Hubei | 1277.8 | 0.69 | 0.21 | 6.98 | 1.93
Guangdong | 104.3 | 0.51 | 0.75 | 3.66 | 3.02
Henan | 94.3 | 0.61 | 0.38 | 41.00 | 2.41
Zhejiang | 51.2 | 0.50 | 0.98 | 18.83 | 3.10
Hunan | 67.0 | 0.47 | 1.00 | 58.34 | 3.28
Anhui | 64.6 | 0.73 | 0.12 | 27.46 | 1.70
Jiangxi | 44.0 | 0.44 | 1.00 | 19.41 | 3.46
Jiangsu | 76.8 | 0.72 | 0.15 | 19.10 | 1.75
Chongqing | 28.4 | 0.78 | 0.07 | 5.17 | 1.36
Table 1: Fit parameters as described in Sec. III, fixed population size $N$,
and resulting effective basic reproduction number $R_{0,\mathrm{eff}}$ for
Hubei and the remaining majorly affected provinces discussed in the main text,
decreasingly ordered by largest case number. The infectious period $T_{I}$ and
the basic reproduction number $R_{0,\mathrm{free}}=\alpha T_{I}$ were fixed to
values $T_{I}=8\,\mathrm{d}$ and $R_{0,\mathrm{free}}=6.2$, respectively.

We find that for a wide range of model parameters, the case count is well
reproduced by the model. The model reproduces the scaling law $t^{\mu}$ as
observed in the data for a significant period of time before saturating to a
finite level. Remarkably, the model is able to reproduce both growth behaviors
observed in the data: The model predicts the expected initial growth of case
numbers in Hubei Province followed by an algebraic growth episode for $\approx
11$ days until the saturation sets in, a consequence of the decay of
unidentified infected individuals after a peak time around Feb. 7th (see Fig.
2A). Furthermore, the model also captures the immediate sub-exponential growth
observed in the remaining most affected provinces (Fig. 2B-C). Again,
saturation is induced by a decay of unidentified infecteds after peaks that
occur several days before peak time in Hubei, ranging from Jan. 31st to Feb.
5th. For all provinces, following their respective peaks the number of
unidentified infecteds $I(t)$ decays over a time period that is longer than
the reported estimation of maximum incubation period of $14$ days
cdc_2019_2020 ; who_novel_2020-1 . It is important to note that due to the
uncertainty in the population size, the numerical value of unidentified
infecteds is sensitive to parameter variations—the general shape of $I(t)$,
however, is robust for a wide choice of parameters, as discussed in App. A.

Parameter choices for best fits were a fixed basic reproduction number of
$R_{0,\mathrm{free}}=6.2$ (note that this reproduction number corresponds to
an unconstrained epidemic) and a fixed mean infection duration of
$T_{I}=8\,\mathrm{d}$ consistent with previous reports concerning the
incubation period of COVID-19 who_novel_2020-1 ; cdc_2019_2020 . The remaining
fit parameters are shown in Tab. 1. For these values, the effective basic
reproduction number is found to range between $1.7\leq R_{0,\mathrm{eff}}\leq
3.3$ for the discussed provinces, consistent with estimates found in previous
early assessment studies cdc_2019_2020 ; kucharski_early_2020 ;
read_novel_2020 ; zhao_preliminary_2020 .

A detailed analysis of the obtained values for quarantine probability $Q$ and
public containment leverage $P$ indicates that a wide range of these
parameters can account for similar shapes of the respective case counts.
Consequently, the model is structurally stable with respect to these
parameters and the numerical value is of less importance than the quality of
the mechanism they control (see App. A).

For the remaining 20 provinces, we find growth behaviors similar to the ones
discussed before. Some follow a functional form comparable to the case counts
in Hubei, others display a stronger agreement with pure algebraic growth, as
can be seen in Fig. 3. Generally, a stronger agreement with algebraic growth
can be associated with stronger public containment leverages.

In order to stress the general importance of public containment measures and
to clarify to what extent quarantine is connected to case confirmation we also
analyzed a model variant where (i) the general public and infecteds are
removed from the transmission process in the same way and (ii) case counting
is decoupled from the quarantine process (see App. C). We find that these
model variants describe the real case count data reasonably well for the
majority of provinces, further evidence of the importance of containment
policies that target the susceptible population. In App. B we present further
analytical evidence for this conclusion.

Note that the described saturation behavior of confirmed cases requires that
eventually all susceptibles will effectively be removed from the transmission
process. In reality, not every susceptible person can be isolated for such an
extended period of time as the model suggests. One might expect instead that
the number of infecteds will decay more slowly and saturate to a small, yet
non-zero level. At this point it would be crucial to identify unquarantined
infecteds more efficiently in order to completely shut down the transmission
process. Due to potential difficulties in upholding the containment policies
for such an extended period of time, we expect that our predictions will
underestimate the final total amount of confirmed cases.

##  V Discussion and Conclusion

In summary, we find that one of the key features of the dynamics of the
COVID-19 epidemic in Hubei Province but also in all other provinces is the
robust sub-exponential rise in the number of confirmed cases according to the
scaling law $t^{\mu}$ during the first episode of the epidemic. This general,
almost universal behavior is evidence that fundamental principles are at work
associated with this particular epidemic that are dominated by the interplay
of the contagion process with endogeneous behavioral changes in the
susceptible population and external mitigation policies. We find that the
generic scaling of case counts with time is independent of many other factors
that often shape the time-course of an epidemic.

The model defined by Eqs. (1)-(3) and discussed here indicates that this type
of behavior can generally be expected if the supply of susceptible individuals
is systematically decreased by means of implemented containment strategies or
behavioral changes in response to information about the ongoing epidemic.
Unlike contagion processes that develop without external interference at all
or processes that merely lead to parametric changes in the dynamics, our
analysis suggests that non-exponential growth is expected when the supply of
susceptibles is depleted on a timescale comparable to the infectious period
assiociated with the infection. The effective depletion of the susceptible
populations on a timescale $\kappa_{0}^{-1}$ in the model does not imply that
the pool of susceptibles must be depleted in a physical sense. Rather, this
depletion could be achieved indirectly.

Containment that targets the susceptible population is a requirement for the
observed algebraic scaling of case counts with time, unlike the reduction of
transmission rates, or the reduction of the duration infected individuals
remain infectious. Both of the latter measures address infected individuals
only, which leads to a reduction of the exponential growth rate of new
infections, or an exponential decay if the corresponding measures are
sufficiently effective, but do not alter the functional growth behavior.

The fact that scaling laws were observed in Hubei after an initial exponential
phase and that an exponential phase was not observed in the other
substantially affected provinces is consistent with the model predictions if
one assumes that the response in the population and the effect of policies
lagged the onset of the epidemic in Hubei province and was immediate in all
other provinces that were in an alert status already when cases were
increasing substantially in Hubei province.

The model reproduces the empirical case counts in all provinces well for
plausible parameter values. The quality of the reproduction of the case counts
in all 29 affected provinces can be used to estimate the peak time of the
number of asymptomatic or oligo-symptomatic infected individuals in the
population, which is the key quantity for estimating the time when this
outbreak will wane. The current analysis indicates that this peak time was
reached around Feb. 7th for Hubei and within the first days of February in the
remaining affected provinces.

The model further suggests that the public response to the epidemic and the
containment measures put in place by the Chinese administration were effective
despite the increase in confirmed cases. That this behavior was universally
observed in all provinces also indicates that mitigation strategies were
universally effective. Based on our analysis, we argue that the implemented
containment strategies should stay in effect for a longer time than the
incubation period after the saturation in confirmed cases sets in for this
particular outbreak.

Our analysis further implies and provides evidence that mitigation strategies
that target the susceptible population and induce behavioral changes at this
“end” of the transmission process are most effective to contain an
epidemic—especially in situations when asymptomatic or mildly symptomatic
infectious periods are long or their duration unknown. This may be of
importance for developing containment strategies in future scenarios or if the
current COVID-19 epidemic were to trigger large scale outbreaks in other
regions of the world by exports and subsequent proliferation.

###### Acknowledgements.

We want to express our gratitude to L. H. Wieler and L. Schaade for helpful
comments regarding the manuscript. B. F. M. is financially supported as an
_Add-on Fellow for Interdisciplinary Life Science_ by the Joachim Herz
Stiftung.

##  Appendix A Fitting routine and analysis

Fits were performed using the Levenberg-Marquardt method of least squares. We
fixed the epidemiological parameters to duration of infection
$T_{I}=8\,\mathrm{d}$ and basic reproduction number
$R_{0,\mathrm{free}}=\alpha T_{I}=6.2$. The population size $N$ of each of the
affected Chinese provinces was obtained from the Geonames project
geonames_geonames_2019  and is listed in Tab. 1. For each confirmed cases data
set, we set the initial conditions $X(t_{0})=C(t_{0})/N$,
$I(t_{0})=(I_{0}/X_{0})X(t_{0})$, and $S(t_{0})=1-I(t_{0})-X(t_{0})$ where
$t_{i},C(t_{i})$ is the $i$-th pair of a province’s time and aggregated
confirmed case number. Since the number of unidentified infectious is unknown
per definition, the prefactor $I_{0}/X_{0}\geq 1$ was chosen as a fit
parameter. The remaining fit parameters were quarantine rate $\kappa>0$ and
containment rate $\kappa_{0}>0$. For the fit procedure, Eqs. (1)-(3) were
integrated using the Dormand–Prince method which implements a fourth-order
Runge–Kutta method with step-size control, yielding $I(t)$ and $X(t)$ for
every parameter configuration and every data set. The residuals were computed
as $\mathcal{R}(t_{i})=NX(t_{i})-C(t_{i}).$

We find that the model accurately reflects both the sub-exponential growth as
well as the saturating behavior observed in the data discussed in the main
text, with the obtained fit parameters displayed in Tab. 1. For the 9 majorly
affected provinces and the aggregated data over all provinces except Hubei,
the quarantine probability is found to have similar values of $Q=0.6\pm 0.1$.
The public containment leverage is fluctuating more strongly with values
ranging between $P=0.07$ and $P=1$, where generally, lower values appear
concurrently with a higher quarantine probability, which suggests that
stronger quarantine implementation requires less public isolation. Yet, we
advise not to overinterpret the values of these fit parameters, as many
different parameter values generate similar developments of confirmed cases.

As we fixed the population size to be equal to the total population of the
respective provinces, one might also ask how system size changes the
results—for instance, if the outbreak is contained in a small region of a
province, the effective population available to the transmission process will
be substantially smaller. Therefore, we repeated the fit procedure with $N$ as
a free fit parameter to find that the form of $X(t)$ is not altered
substantially for different values of $N$, while $I(t)$ can vary more strongly
as reflected by a larger variation in $I_{0}$. This result suggests that the
estimation of the number of infecteds is associated with a larger uncertainty.
The general shape of $I(t)$, however, remains stable such that an inference of
the peak time of unidentified infecteds is reasonable. Furthermore, the model
favours larger values of the containment rate $\kappa_{0}$ for larger
population sizes in order to reproduce emprical data. This is a reasonable
because the number of available transmission pathways grows quadratically with
the population size, making it easier for the disease to spread faster. In
order to still observe sub-exponential growth, a large part of the susceptible
population has to be removed quickly. Consistently, we find that a decay of
susceptibles is necessary to reliably obtain the observed growth behavior
(i.e. $P>0$, $\kappa_{0}>0$).

Concerning variation in the fixed model parameters, larger values of up to
$R_{0,\mathrm{free}}=12$ yield results similar to the ones described above
when adjusting the infectious period $T_{I}$ to larger values, as well. For
lower values of $R_{0,\mathrm{free}}<6$, the model fails to reproduce the
scaling laws observed in the data. Similarly, the fit results are reasonably
robust against variations of the duration of infection in a range of
$T_{I}\in[6\,\mathrm{d},20\,\mathrm{d}]$ with concurrent adjustment of
$R_{0,\mathrm{free}}$.

Similar effects are found for the remaining 20 affected provinces for which
model fits are displayed in Fig. 3 with parameters given in Tab. 2. In
general, provinces with larger values of public containment leverage $P$
exhibit a stronger agreement with the hypothesis of algebraic growth.

All data and the analysis material is available online
maier_benmaiercovid19casenumbermodel_2020 .

Province | $N/10^{6}$ | $Q$ | $P$ | $I_{0}/X_{0}$ | $R_{0,\mathrm{eff}}$
---|---|---|---|---|---
Shandong | 94.2 | 0.79 | 0.04 | 18.86 | 1.33
Sichuan | 87.2 | 0.75 | 0.12 | 9.74 | 1.58
Heilongjiang | 38.2 | 0.49 | 0.70 | 6.13 | 3.19
Beijing | 14.9 | 0.77 | 0.07 | 1.50 | 1.42
Shanghai | 22.3 | 0.58 | 0.56 | 1.99 | 2.58
Fujian | 36.9 | 0.76 | 0.11 | 16.10 | 1.47
Hebei | 69.9 | 0.79 | 0.03 | 9.41 | 1.27
Guangxi | 48.2 | 0.81 | 0.03 | 6.83 | 1.15
Shaanxi | 37.6 | 0.75 | 0.13 | 4.33 | 1.55
Yunnan | 45.4 | 0.51 | 1.00 | 14.99 | 3.01
Hainan | 9.3 | 0.82 | 0.01 | 1.77 | 1.12
Guizhou | 37.9 | 0.71 | 0.11 | 2.23 | 1.81
Shanxi | 34.1 | 0.72 | 0.14 | 4.93 | 1.71
Liaoning | 43.1 | 0.79 | 0.06 | 3.51 | 1.28
Tianjin | 14.0 | 0.78 | 0.05 | 1.96 | 1.37
Gansu | 26.3 | 0.79 | 0.06 | 2.50 | 1.30
Jilin | 27.3 | 0.48 | 0.81 | 3.79 | 3.22
Xinjiang | 21.3 | 0.80 | 0.02 | 1.13 | 1.22
Inner Mongolia | 24.3 | 0.82 | 0.01 | 4.11 | 1.11
Ningxia | 6.2 | 0.49 | 1.00 | 3.99 | 3.16
Table 2: Fit parameters as described in Sec. III, fixed population size $N$,
and resulting effective basic reproduction number $R_{0,\mathrm{eff}}$ for the
remaining affected provinces, decreasingly ordered by largest case number. The
infectious period $T_{I}$ and the basic reproduction number
$R_{0,\mathrm{free}}=\alpha T_{I}$ were fixed to values $T_{I}=8\,\mathrm{d}$
and $R_{0,\mathrm{free}}=6.2$, respectively. Figure 3: Case numbers of all
affected provinces reproduced by the SIR-X model defined by Eqs. (1)-(3) and
the fraction of unidentified infecteds $I(t)$ as obtained from least-squares
fits. All fits were performed with fixed values of $R_{0,\mathrm{free}}=6.2$
and $T_{I}=8\mathrm{d}$. For more details concerning the fitting procedure,
cf. App. A.

##  Appendix B Case-number development in early phase of the epidemic

In a typical outbreak scenario only few people are infected initially, such
that $S(0)=1-\varepsilon$ and $I(0)=\epsilon$ with $\varepsilon\ll 1$, which
implies that the depletion of susceptibles available to the transmission
process will be dominated by shutdown policies. This assumption effectively
linearizes Eq. (1), yielding the solution

$S(t)=\exp(-\kappa_{0}t).$

Further integration of the linearized Eq. (2) yields

$I(t)=I_{0}\exp\left[-(\beta+\kappa+\kappa_{0})t\right]\times\exp\left[-\frac{\alpha}{\kappa_{0}}\left(\mathrm{e}^{-\kappa_{0}t}-1\right)\right].$

For small values of $t$ we can expand the exponent to obtain the approximate
growth function

$\displaystyle I(t)/\tilde{I}_{0}$
$\displaystyle\approx\exp\left\\{\left(\alpha-\beta-\kappa-\kappa_{0}\right)t\right\\}\times\exp\left\\{-\frac{1}{2}\alpha\kappa_{0}t^{2}\right\\}$

Here, the first factor implies that quarantining infecteds merely decreases
the rate with which their number will grow exponentially, while the second
term suppresses the whole transmission process well enough to alter the growth
behavior. This implies that public shutdown policies facilitate epidemic
containment in a more effective way than quarantine measures.

##  Appendix C Case confirmation without quarantine

In an alternative approach to the one discussed in the main text one may
assume that isolation affects all citizens equally, i.e. $\kappa=0$, but that
the confirmation and counting of a new case is decoupled from whether the
respective person is quarantined or not—rather, an infected is discovered and
counted at rate $\tilde{\kappa}$. This implies an ODE system of

$\displaystyle\partial_{t}S$ $\displaystyle=-\alpha SI-\kappa_{0}S$ (4)
$\displaystyle\partial_{t}I$ $\displaystyle=\alpha SI-\beta I-\kappa_{0}I$ (5)
$\displaystyle\partial_{t}X$ $\displaystyle=\tilde{\kappa}I,$ (6)

where as before, $X$ will be equal to the number of confirmed cases $C$.

A numerical investigation of this alternative model shows that confirmed case
data can be reasonably fit for all provinces, as well, which confirms the
argumentation of App. B that removal of susceptibles from the transmission
process is more effective than quarantine in mitigating epidemic spread.

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