\documentclass[12pt]{iopart}
\usepackage{graphicx,bm,url,color}
\bibliographystyle{iopart-num}
\begin{document}

\title[Quadratic instead of exponential growth]{Quadratic instead of exponential growth}

\author{Axel Brandenburg$^{1,2,3,4}$}

\address{
$^1$Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes Alfv\'ens v\"ag 12, SE-10691 Stockholm, Sweden \\
$^2$The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden\\
$^3$McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA\\
$^4$School of Natural Sciences and Medicine, Ilia State University, 3-5 Cholokashvili Avenue, 0194 Tbilisi, Georgia
}
\ead{brandenb@nordita.org}
\vspace{10pt}
\begin{indented}
\item[]June 2022
\end{indented}

\begin{abstract}
The existence of an exponential growth phase during early stages of a
pandemic is often taken for granted.
For the 2019 novel coronavirus epidemic, the early exponential phase
lasted only for five days, while the quadratic growth prevailed for 40
days until it spread to other countries and continued again quadratically,
but with a larger coefficient.
We show here that this rapid phase is followed by a subsequent slow-down
where the coefficient is reduced to almost the original value at the
outbreak.
This can be explained by the merging of previously disconnected sites
that emerged after the disease jumped (nonlocally) to a relatively small
number of separated sites.
\end{abstract}

\vspace{2pc}
\noindent{\it Keywords}: quadratic growth, SIR model, front propagration

\submitto{\jpa}

% Uncomment if a separate title page is required
%\maketitle
% 
% For two-column output uncomment the next line and choose [10pt] rather than [12pt] in the \documentclass declaration
%\ioptwocol
%

\section{Introduction}

Soon after the news about the 2019 novel coronavirus epidemic emerged,
people in Europe followed the increasing case numbers with concern.
The first deaths occurred on January 20, and, for a long time, the
ratio of the numbers of deaths to that of cases was around 0.02.

It became soon clear, that the number of cases increased subexponentially
\cite{Bra20, Maier+Brockmann20, Bodova+Kollar20, Radicchi+Bianconi20,
Blanco+21, Triambak+21}.
This was called peripheral growth, which means that the rate of increase
of the number of cases or deaths is proportion to the length $\ell$
of the periphery of a patch on a map containing the population with the
disease \cite{Bra20}.
If it is just a circular patch of radius $r$, the circumference is $\ell=2\pi r$
and the number of cases is $N_i=n_i\pi r^2$, where $n_i$ is the density of cases ($i={\rm C}$) or deaths ($i={\rm D}$)
per unit area.
The rate of increase is then
\begin{equation}
\frac{\dd N_i}{\dd t}=\alpha\ell=2\alpha (\pi N_i/n_i)^{1/2},
\quad i={\rm C}, {\rm D}.
\label{rate}
\end{equation}
with the solution
\begin{equation}
N_i^{1/2}=N_i_0^{1/2}+\alpha (\pi/n_i)^{1/2}(t-t_0)
\label{soltn}
\end{equation}
with $N_0=N(t_0)$ being the initial condition at $t=t_0$.
Thus, we expect a quadratic growth where $N_i^{1/2}$ versus
$t$ increases linearly with $t$.
This was clearly seen in the original study in Ref.~\cite{Bra20}.
Subsequent work confirmed the existence of algebraic growth, although
the exponent was sometimes found to deviate from 2.
The work by Ref.~\cite{Ziff+Ziff20} found that the exponent was between
to 2.28 and 3.09.

Of particular importance is the fact that, at some point, the exponential
growth sped up.
This was possible to model through the emergence of multiple nucleation
sites from where the disease spreads.
This means that the coefficient $\alpha$ should be replaced by
$\alpha\to\alpha_1+\alpha_2+\alpha_3+...$, depending on the number
of sites, the

iFart increases that by the slope of N is increased.

It was already anticipated in the appendix of B 20 that the subsequent decrease of the growth of the number of cases could be associated with the merging of independence sites from which separated France continue to expand after an initial prefer period of quadratic increase in the number of cases.
The details of this process we are not, however, explored in detail.
This is the purpose of the present paper.

\section{The spatially extended essay are model of epidemics}

In epidemics, the S I am either and it’s extensions are an important corner stone in the theory of epidemics.
The extension to including advection and diffusion operators, i.e., you know blah and Cavanagh blah tool important, when complete mixing among the population can no longer be assumed.
The inclusion has dramatic consequences for the evolution of the total number of cases or death.
In the following, the disease was consequences in detail.
We begin by outlining the essence of the essay and model

Mathematical formulation of the model model.
It’s original form, the SAR model assumes perfect mixing and therefore space sure how much unity.
Therefore, space sure gradients are absent. When SAR oh.
In essence, the model describes the increase of cases based on the current number of susceptible one.
Once this number begins to be depleted, it can only decrease, although it can still increase in neighboring locations, where the number of cases may still be smaller.
This leads to especially spreading of the disease and thereby tour and ultimately quadratic increase in the total number of cases.
Thus, the SAI model with spatial extend included is fully capable of describing the increase of cases and death.
It remains of cost unclear whether the space should increase corresponds to a completely or only partial space filling increase in the number of cases over the surface of the earth.
Given that the current number of cases reaching the total population size of people on earth, it no longer seems absurd to assume that every simple single individual on earth is in divorce suspect it to the disease.

What is the original as I am what it is not taking into account is the concept of reinfection, i.e., the fact that infected people can come out after certain period of time, big come effect it again.
This means that we must account for a term that describes the decrease of reinfected individuals, which leads to a source in the number of people that can be infected.
The purpose of the following work is to explore in more detailed the effect of the sustainment of cases by the phenomenon of reinfection.

\section{Model with reinfection}

Let us know study models where are the effect of reinfection is included, i.e., Gamma zero.
In Fair X, we consider models with different the rates of,.
It turns out that the continued decline in the cases in the number of cases can be explained when camera is large enough. Neck, i.e., where minors can I is, is
Stephen or unstable equilibrium

\section{Conclusions}

The present work has shown that the number of infected people will not increase exponentially, as expected for a very mixed model without space should extend, but that it can increase instead both still one, if the space a number of infected people it’s already exhausted, and faster, if the number of susceptible people can still increase by enabling locations.

Obviously, the total number of cases cannot increase indefinitely, but it can increase somewhat going to the fact that the people in neighboring locations can be reinfected and infected people can even be reinfected. Next line in the present world, we have studied in more detail the effect of reinfections, which is especially important in cases when most of the population has been affected.
It that’s it turns out that he a decrease in the total number of infections Road weight can be explained by the merging of originally separated spreading centers.
In the case of the task of two, this happens happened at more or less the same time for the different spreading centers on the earth.
Subsequent variations in the number of cases and deaths can be explained by variations in the reinfection rate.
In addition, there are

\section*{Acknowledgments}

This work was supported in part through the Swedish Research Council,
grant 2019-04234.
Nordita is sponsored by Nordforsk.
We acknowledge the allocation of computing resources provided by the
Swedish National Infrastructure for Computing (SNIC)
at the PDC Center for High Performance Computing Stockholm
and the National Supercomputer Centre (NSC) at Link\"oping.
\vspace{2mm}

\noindent
{\em Code and data availability.} The source code used for the
simulations of this study, the {\sc Pencil Code} \cite{JOSS},
is freely available on \url{https://github.com/pencil-code/}.
The DOI of the code is https://doi.org/10.5281/zenodo.2315093.
The simulation setup and the corresponding data are freely
available on \url{https://doi.org/10.5281/zenodo.5760126}; see also
\url{https://www.nordita.org/~brandenb/projects/SelfGrav} for easier
access to the same material as on the Zenodo site.

\begin{equation}
\label{cases}
\end{equation}


%\appendix

\bibliography{ref}
\end{document}
