#  Renormalisation Group Methods
for
Effective Epidemiological Models

Stefan<EMAIL_ADDRESS>, and Francesco
<EMAIL_ADDRESS>
###### Abstract

Epidemiological models describe the spread of an infectious disease within a
population. They capture microscopic details on how the disease is passed on
among individuals in various different ways, while making predictions about
the state of the entirety of the population. However, the type and structure
of the specific model considered typically depend on the size of the
population under consideration. To analyse this effect, we study a family of
effective epidemiological models in space and time that are related to each
other through scaling transformations. Inspired by a similar treatment of
diffusion processes, we interpret the latter as renormalisation group
transformations, both at the level of the underlying differential equations
and their solutions. We show that in the large scale limit, the microscopic
details of the infection process become irrelevant, safe for a simple real
number, which plays the role of the infection rate in a basic compartmental
model.

11footnotemark: 1 Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I
Lyon, UMR 5822, F-69622, Villeurbanne, France

${}^{}\,{}^{}$22footnotemark: 233footnotemark: 3 Dept. of Physics E. Pancini,
Università di Napoli Federico II, via Cintia, 80126 Napoli, Italy

22footnotemark: 2 Scuola Superiore Meridionale, Largo S. Marcellino, 10, 80138
Napoli NA, Italy

22footnotemark: 2 INFN sezione di Napoli, via Cintia, 80126 Napoli, Italy

22footnotemark: 2 Quantum Theory Center ($\hbar$QTC) & D-IAS, Southern Denmark
Univ., Campusvej 55, 5230 Odense M, Denmark

###### Contents

  1. 1 Introduction
  2. 2 Diffusion: Random Walks and Heat Equation
    1. 2.1 Stochastic Processes and Difference Equations
    2. 2.2 Deterministic Model and Heat Equation
      1. 2.2.1 Coarse Graining: Continuous Space-Time and Probability Densities
      2. 2.2.2 Continuum Limit
      3. 2.2.3 Heat Equation
    3. 2.3 Perturbation around the Continuum Limit
      1. 2.3.1 Perturbation of the Probability Density
      2. 2.3.2 Perturbation of the Heat Equation
  3. 3 Epidemiology: Simple Epidemic- and the SI-Model
    1. 3.1 Stochastic Approach
    2. 3.2 Simple Epidemic Model
      1. 3.2.1 Deterministic Approximation: Continuous Time
      2. 3.2.2 Coarse Graining: Continuous Spatial Variables
    3. 3.3 Family of Effective Models
    4. 3.4 Length Scales at the Level of Solutions
      1. 3.4.1 Travelling Waves and Similarity Transformation in $d=1$
      2. 3.4.2 Linearisation Around a Position Independent Background
  4. 4 Conclusions and Outlook
  5. A Notation and Conventions
    1. A.1 Approximation of the Identity
    2. A.2 Review of Intermediate Asymptotics and Renormalisation Group
  6. B Mathematical Expansion

##  1 Introduction

The mathematical modelling of the spread of an infectious diseases through a
population has a longstanding history. Indeed, numerous conceptually different
approaches have been discussed in the literature (see _e.g._ [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15] for reviews). Classifying them in an
exhaustive fashion is rather difficult, however, a common distinction sets
apart333A complete separation is indeed difficult, since models can contain
elements of both. Furthermore, more recent data- and computer-driven
approaches are difficult to attribute to either of the two.

  * •

stochastic approaches: the basic processes that govern the spread of the
disease (_e.g._ the infection or recovery of individuals) are of a
probabilistic nature. Models using this approach are usually discretised
(_i.e._ the population consists of discretely many individuals and time is a
discrete variable) and are characterised by a set of probabilities (_e.g._ the
probability for an infected individual to infect another one in close
proximity).

  * •

deterministic approaches: the spread of the disease is understood as a
predictable process in time. The number of individuals that are affected (in
various different ways) by the disease are described by differential equations
in space and time. Models using this approach are characterised by a set of
rates (_e.g._ the rate at which infectious individuals infect others).

The choice of approach largely depends on the concrete question under
consideration: on the one hand side, since a stochastic modelling of the
(microscopic) processes that lead to the spread of a disease is usually closer
to the (biological) reality, deterministic models are typically considered as
approximations (see _e.g._ [16, 17]). Stochastic models are well adapted for
(short term) simulations of diseases, but their potentially higher
computational cost makes long-term predictions difficult. On the other hand,
deterministic models try to capture properties of the disease that arise from
the basic microscopic processes. If done correctly, due to the intrinsic
nature of these models, they allow predictions of general trends and (long-
term) developments of the disease.

Among other things, a crucial aspect for the choice of the approach is the
scale at which the time-evolution of the disease is described, both in terms
of the size of the population and the time-frame. For example, a model that is
well adapted to describe the initial outbreak and spread of Covid-19 in a
small town might be much less successful in predicting the cumulative number
of infected individuals in a large country at the end of an entire wave.
Indeed, even though the basic (biological) microscopic processes that govern
the transmission of the disease (as well as the recovery) on an individual
level are the same, the dynamics in the two cases can be influenced very
differently by other aspects, such as initial conditions (_i.e._ the number
and geographic distribution of infected individuals in the beginning), the
impact of so-called super-spreaders (_i.e._ individuals with a significantly
higher chance of infecting other individuals) and super-spreader events
(_i.e._ events during which the disease is transmitted much more than usual),
the geographic mobility of the population, _etc._. Depending on the disease in
question, this list can further be extended to include socio-geographic,
economical and political factors.

A similar scale dependence occurs in many physical problems (see _e.g._ [18,
19]), for example phase transitions in thermodynamical systems, effective
descriptions of quantum field theory and particle physics or even quantum
aspects of black holes (see _e.g._ [20, 21, 22, 23]). Despite the large number
of vastly different examples, a general framework to approach them exists in
the form of _renormalisation group_ methods [24, 25, 26, 27], which for
example in [28] have been characterised in the following fashion: ’The essence
of the renormalisation group method is to extract structurally stable features
of a system which are insensitive to details.’ The current paper is dedicated
to applying this idea to epidemiology.

Indeed, we consider as a template model the well studied relation between
diffusion processes and random walks: for sufficiently large times, the
probability (density) to find a random walker in a finite region of a
$d$-dimensional (regular) lattice can be approximated by a function of
continuous space- and time variables. Furthermore, in a specific continuum
limit, this probability density becomes the causal Green’s function of the
heat equation, which vanishes at spatial infinity. We establish a similar line
of reasoning in the case of a simplistic model of epidemiology: the starting
point is a stochastic model on a regular lattice, where the lattice sites
represent individuals, who may be infected with a communicable disease and the
probability for a healthy individual to become infected is modelled as a
function of the infectious individuals at neighbouring lattice sites. We
approximate this stochastic model by a compartmental model [29] (see also [1,
2, 3, 5, 10, 11, 14] for reviews), where the number of infectious individuals
is given as a function of continuous space- and time-variables that satisfy a
first order non-linear differential equation in time [17, 30]. The details of
how individuals infect one another depending on their spatial position is
encoded in a single integrable function $f\in L^{1}$. We argue that a
continuous rescaling of the spatial variable leads to a family of effective
models, that describe the average number of infected individuals in a variable
region of space. In the limit of the latter becoming infinite, we recover the
well known SI-model [29]: in this case, the entire information of the function
$f$ is scaled away, except for its $L^{1}$-norm, which is interpreted as the
infection rate. Since this rescaling procedure falls into the definition given
in [28] cited above, we propose this mechanism to describe a renormalisation
group method in epidemiology, as advocated previously in [31, 32, 33, 34, 35,
36, 37, 38, 39, 15, 40, 41] based on phenomenological arguments. Indeed, the
SI-model found in the limiting case of our family of epidemic models coincides
with the simplest version of the so-called _epidemiological renormalisation
group_ (eRG) equation proposed in [31].

This work focuses on the mathematical and conceptual ideals, thus sacrificing
epidemiological realistic features for computational simplicity: for example,
our family of models (following [17]) does not allow recovery of infectious
individuals and infections are instantaneous in time (_i.e._ we do not allow
for an incubation time of the disease). Furthermore, countermeasures of the
population (vaccinations, non-pharmaceutical interventions, _etc._) are
neglected as are details of the mobility or the structure of the population.
We plan to discuss more sophisticated models in the future.

Concretely, this work is organised as follows: In Section 2 we first consider
the simple example of diffusion processes, which allows us to introduce
various concepts related to scaling, namely coarse graining of degrees of
freedom which leads to families of self-similar models. In Section 3 we shall
apply these concepts to a simple model of epidemiology [17, 30], leading to
the definition of a family of related models, whose fixed point is the SI-
model [29]. Finally, Section 4 contains our conclusions and an outlook for
future work. This paper is supplemented by two appendices, which contain
additional mathematical and conceptual definitions as well as computational
details that were deemed to lengthy for the main body of the text.

##  2 Diffusion: Random Walks and Heat Equation

Before discussing epidemiological models that describe the spread of a disease
throughout a population, we discuss a simple system in which the concept of
scaling can be made evident, namely a simple diffusion process in
$d$-dimensions. This examples showcases a number of features that we shall
implement into the description of epidemiological processes (concretely into
the ’simple’ epidemic model described in [30]) in the subsequent Section.

We first present a stochastic approach to the phenomenon of diffusion in the
form of random walks on a (regular) lattice. This description can be
approximated by deterministic equations, which in the limit of very small
lattice spacing become the $d$-dimensional heat equations. While many of the
computations and conceptual ideas are well-known in the literature (see _e.g._
the textbook [42] for a pedagogical presentation that we shall follow in this
Section), this example allows us to showcase the deterministic approximation
of a stochastic process in a controllable setting: we shall discuss the
approximation both at the level of the probabilities and the equation
governing the dynamics.

###  2.1 Stochastic Processes and Difference Equations

Let $\Gamma$ be a $d$-dimensional infinite, hypercubic lattice in Euclidean
space

$\displaystyle\Gamma=\\{\vec{X}\in\mathbb{R}^{d}|\vec{X}=\sum_{i=1}^{d}n_{i}\,\vec{E}_{i}\text{
with }n_{i}\in\mathbb{Z}\\}\,,$ (2.1)

that is generated by $d$ orthogonal vectors $(\vec{E}_{1},\ldots,\vec{E}_{d})$

$\displaystyle\vec{E}_{i}\cdot\vec{E}_{j}=a^{2}\,\delta_{ij}\,,$
$\displaystyle\forall i,j=1,\ldots,d\,.$ (2.2)

Every lattice site has $2d$ neighbours (with lattice spacing
$a\in\mathbb{R}_{+}$) such that the coordination number is $q=2d$. We consider
a walker, who at regular time intervals $\tau\in\mathbb{R}_{+}$ takes a step
in a random direction by moving to one of the neighbouring lattice sites. We
assume that the probability to move to each of the $2d$ neighbouring sites is
equal and independent of previous movements or the current position. We denote
by $P(\vec{X}_{1}-\vec{X}_{0},T_{1}-T_{0})$ the probability for the walker to
be at the time $T_{1}$ at the lattice site $\vec{X}_{1}$ when they were at
lattice site $\vec{X}_{0}$ at time $T_{0}$. Due to translational invariance,
$P$ only depends on the differences $\vec{X}=\vec{X}_{1}-\vec{X}_{0}$ and
$T=T_{1}-T_{0}$. Furthermore, the probability satisfies the following
recursion relation in time

$\displaystyle
P(\vec{X},T+\tau)=\frac{1}{2d}\sum_{i=1}^{d}\left[P(\vec{X}+\vec{E}_{i},T)+P(\vec{X}-\vec{E}_{i},T)\right]\,,$
(2.3)

and is normalised to satisfy

$\displaystyle P(\vec{X},0)=\delta_{\vec{X},0}\,,$ and
$\displaystyle\sum_{\vec{X}\in\Gamma}P(\vec{X},T)=1\,.$ (2.4)

Upon defining the Fourier transform $\widetilde{P}(\vec{K},T)$ of
$P(\vec{X},T)$ with respect to the spatial coordinates

$\displaystyle
P(\vec{X},T)=\frac{1}{(2\pi)^{d}}\int_{[-\pi,\pi]^{d}}d^{d}K\,e^{\frac{i\vec{K}\cdot\vec{X}}{a^{2}}}\,\widetilde{P}(\vec{K},T)\,,$
with
$\displaystyle\int_{[-\pi,\pi]^{d}}d^{d}K=\prod_{i=1}^{d}\int_{-\pi}^{\pi}dK_{i}\,,$
(2.5)

the recursion relation (2.3) and the initial conditions (2.4) take the form

$\displaystyle\widetilde{P}(\vec{K},T+\tau)=\frac{1}{d}\,\widetilde{P}(\vec{K},T)\sum_{j=1}^{d}\cos\left(\frac{\vec{K}\cdot\vec{E}_{j}}{a^{2}}\right)\,,$
and $\displaystyle\widetilde{P}(\vec{K},0)=1\,.$ (2.6)

They can be solved to give the following integral representation of the
probability

$\displaystyle P(\vec{X},T)$
$\displaystyle=\frac{a^{d}}{(2\pi)^{d}}\int_{\left[-\frac{\pi}{a},\frac{\pi}{a}\right]^{d}}d^{d}k\,e^{\frac{i\vec{k}\cdot\vec{X}}{a}}\,\left(\frac{1}{d}\sum_{j=1}^{d}\cos\left(\vec{k}\cdot\vec{e}_{j}\right)\right)^{\frac{T}{\tau}}\,,$
(2.7)

where we have introduced the following orthonormal set of vectors in
$\mathbb{R}^{d}$

$\displaystyle\vec{e}_{i}=\frac{1}{a}\,\vec{E}_{i}\in\mathbb{R}^{d}\,,$ with
$\displaystyle\vec{e}_{i}\cdot\vec{e}_{j}=\delta_{ij}\,,$ (2.8)

and the new integration variables $k_{i}=\frac{K_{i}}{a}$ with

$\displaystyle\vec{k}=\sum_{i=1}^{d}k_{i}\,\vec{E}_{i}=a\,\sum_{i=1}^{d}k_{i}\,\vec{e}_{i}=\frac{\vec{K}}{a}\,.$
(2.9)

A plot of the probabilities (2.7) for different times $T$ in one dimension
($d=1$) is shown in Figure 1.

Figure 1: Random walks in one dimension ($d=1$): The left panel shows the
probability (2.7) for different times $T$. The dots represent the values of
the integral (2.7) (for fixed $a$), while the solid lines are interpolations
in the form of Gaussian functions. The right panel shows the width of the
later, which is interpolated by $\Delta P(T)=\sqrt{T/\tau}$.

###  2.2 Deterministic Model and Heat Equation

As a next step we develop a deterministic approximation of the stochastic
process of random walks: to this end, we shall replace the probabilities
$P(\vec{X},T)$ by coarse grained quantities that are determined by a
differential equation which replaces the recursion relation (2.3).

####  2.2.1 Coarse Graining: Continuous Space-Time and Probability Densities

Instead of the probability of finding the walker at the fixed lattice position
$\vec{X}_{1}$ at time $T_{1}$ (assuming that they started at
$\vec{X}_{0}=\vec{X}_{1}-\vec{X}$ at time $T_{0}=T_{1}-T$), we consider the
probability to find the walker during a time interval $\Delta T$ in a region
$\Delta X$ (with $\vec{X}_{1}\in\Delta X$). As schematically shown in Figure
2, we consider $\Delta T$ to be labelled by a continuous time variable
$t\in\mathbb{R}_{+}$ and $\Delta X$ by (Euclidean) coordinates444For
convenience, we choose a parametrisation, such that $\vec{x}=0$ parametrises a
$\Delta X$ that includes $\vec{X}_{0}$.
$\vec{x}=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}$ such that the length of the
time interval $|\Delta T|$ is fixed and independent of $t$ and the volume of
$\Delta X$ is independent of $\vec{x}$. Notice, while in general555Since here
we are ultimately interested in the limit $a\to 0$ we can allow $\Delta X$ to
be labelled by a coordinate that does not necessarily represent a lattice site
itself. $\vec{x}\notin\Gamma$ it can still be expanded in a series expansion
in the basis vectors $\vec{E}_{i}$ (or equivalently $\vec{e}_{i}$)

$\displaystyle\vec{x}=\sum_{i=1}^{d}x_{i}\,\vec{e}_{i}=a\,\sum_{i=1}^{d}\,X_{i}\,\vec{e}_{i}=\sum_{i=1}^{d}\,X_{i}\,E_{i}\,,$
with $\displaystyle x_{i}=a\,X_{i}\in\mathbb{R}\,$ $\displaystyle\forall
i=1,\ldots,d\,.$ (2.10)

Furthermore, in the following we shall assume that $\Delta T$ and $\Delta X$
have the following properties

$a$$\vec{X}_{0}$$\vec{x}$$\Delta
X$$\vec{X}_{1}$$\vec{E}_{1}$$\vec{E}_{2}$$\tau$$T_{0}$$T_{1}$$t$$\Delta T$

Figure 2: Coarse Graining of the time and space-variables.

  * _(i)_

the length $|\Delta T|$ of the interval $\Delta T$ is large compared to $\tau$
and the volume of $\Delta X$ (measured in $\mathbb{R}^{d}$) is large compared
to the volume of the unit cell of $\Gamma$

$\displaystyle|\Delta T|\gg\tau\,,$ and $\displaystyle|\Delta
X|=\text{vol}_{\mathbb{R}^{d}}(\Delta X)\gg a^{d}\,.$ (2.11)

  * _(ii)_

the probability to find the walker does not vary too much over $\Delta T$ and
$\Delta X$ (which is satisfied in general if
$\frac{T}{\tau}=\frac{T_{1}-T_{0}}{\tau}\gg 1$)

Both assumptions imply that we coarse grain the system at a time at which the
walker has performed sufficiently many steps such that the probabilities
$P(\vec{X},T)$ are slowly varying functions on $\Gamma$ which can be
considered constant over the scales given by the volumes $|\Delta T|$ and
$|\Delta X|$. These are in fact exactly the requirements for the system to be
in a state that was called _intermediate asymptotics_ in [18] (see Apendix
A.2), _i.e._ the system being in a state which no longer (strongly) depends on
the precise details of the initial conditions but has not yet settled into a
static configuration. Furthermore, the assumptions _(i)_ and _(ii)_ allow us
to introduce the following probability density

$\displaystyle p_{a,\tau}(\vec{x},t)|\Delta X|=\sum_{\vec{X}\in\Delta
X}P(\vec{X},T)\sim\frac{|\Delta X|}{a^{d}}P(\vec{X},T)\,,$ (2.12)

which is implicitly a function of $a$ and $\tau$.

####  2.2.2 Continuum Limit

We are interested in the limit where both the lattice spacing $a$ and the time
interval $\tau$ tend to zero, thus leading to a continuum theory. We are thus
lead to consider

$\displaystyle p(\vec{x},t):=\lim_{a,\tau\to
0}\frac{1}{(2\pi)^{d}}\,\int_{\left[-\frac{\pi}{a},\frac{\pi}{a}\right]^{d}}d^{d}k\,e^{i\sum_{i=1}^{d}k_{i}x_{i}}\,\left(\frac{1}{d}\sum_{j=1}^{d}\cos(a\,k_{j})\right)^{\frac{t}{\tau}}\,.$
(2.13)

This expression, however, strongly depends on the way the limit $a,\tau\to 0$
is taken. To understand this, we consider (for $\alpha\in\mathbb{R}$ and
$\beta\in\mathbb{R}_{+}$)

$\displaystyle\lim_{u\to
0^{+}}\left(\cos(u\alpha)\right)^{\frac{\beta}{u^{p}}}=\lim_{u\to
0^{+}}\left(1-\frac{u^{2}\alpha^{2}}{2}+\mathcal{O}(u^{4})\right)^{\frac{\beta}{u^{p}}}=\left\\{\begin{array}[]{lcl}1&\text{if}&0<p<2\,,\\\
e^{-\frac{\beta\alpha^{2}}{2}}&\text{if}&p=2\,,\\\
0&\text{if}&p>2\,.\end{array}\right.$ (2.17)

Therefore, we define the limit in (2.13) as

$\displaystyle\tau\rightarrow 0\,,$ and $\displaystyle a\rightarrow 0\,,$ such
that $\displaystyle\frac{\tau}{a^{2}}\rightarrow\frac{1}{2d\kappa}\hskip
8.5359pt(\text{finite})\,,$ (2.18)

where $\kappa\in\mathbb{R}\setminus\\{0\\}$ is an arbitrary constant, such
that

$\displaystyle\lim_{a,\tau\to
0}\left(\frac{1}{d}\sum_{j=1}^{d}\cos(a\,k_{j})\right)^{\frac{t}{\tau}}\longrightarrow\,\lim_{u\to
0}\left(\frac{1}{d}\sum_{j=1}^{d}\cos\left(u\,k_{j}\right)\right)^{\frac{2d\kappa
t}{u^{2}}}=\text{exp}\left(-\kappa t\sum_{j=1}^{d}k_{j}^{2}\right)\,.$ (2.19)

We therefore find for the probability density (2.13) the following Gaussian
integral

$\displaystyle
p(\vec{x},t)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}d^{d}k\,e^{-\sum_{j=1}^{d}(\kappa
tk_{j}^{2}-ik_{j}x_{j})}=\frac{e^{-\frac{|\vec{x}|^{2}}{4\kappa
t}}}{\left(4\pi\kappa t\right)^{d/2}}\,,$ $\displaystyle\forall t>0\,.$ (2.20)

We remark that this probability density is normalised such that $\forall t>0$

$\displaystyle\int_{\mathbb{R}^{d}}d^{d}x\,p(\vec{x},t)=1\,,$ (2.21)

and it furthermore satisfies the following Markov property

$\displaystyle\int_{\mathbb{R}^{d}}d^{d}x\,p(\vec{x}_{2}-\vec{x},t_{2}-t_{1})\,p(\vec{x}-\vec{x}_{0},t_{1}-t_{0})=\,p(\vec{x}_{2}-\vec{x}_{0},t_{2}-t_{0})\,,$
$\displaystyle\forall t_{2}>t_{1}>t_{0}\,.$ (2.22)

To get a better physical understanding of the result (2.20) it is instructive
to calculate $\langle|\vec{x}|\rangle$ (_i.e._ the expectation value of the
distance from the origin that the walker has moved) as a function of time

$\displaystyle\langle|\vec{x}|\rangle(t)$
$\displaystyle=\int_{\mathbb{R}^{d}}d^{d}x^{\prime}\,|\vec{x}\,^{\prime}|\,p(\vec{x}\,^{\prime},t)=2\frac{\Gamma\left(\frac{1+d}{2}\right)}{\Gamma\left(\frac{d}{2}\right)}\,\sqrt{\kappa
t}\,.$ (2.23)

The fact that this expectation value behaves as $t^{1/2}$ is the reason why
only $p=2$ in (2.17) leads to a non-trivial limit (2.18) for the probability
density. We can see from Figure 1 that this behaviour is already encoded in
(2.7) and is not a consequence of the limit (2.18): in the case $d=1$, the
left panel shows the probabilities (2.7) for different times $T$ which are
fitted with Gaussian functions, whose width in turn is fitted by $\Delta
P=\sqrt{\frac{T}{\tau}}$ in the right part of the Figure.

####  2.2.3 Heat Equation

In the previous Subsubsection we have taken the continuum limit at the level
of the probability density (2.13). We can equally discuss this limit at the
level of the recursion relation (2.3) which implies for the probability
density

$\displaystyle
p_{a,\tau}(\vec{x},t+\tau)-p_{a,\tau}(\vec{x},t)=\frac{1}{2d}\,\sum_{i=1}^{d}\left[p_{a,\tau}(\vec{x}+a\,\vec{e}_{i},t)+p_{a,\tau}(\vec{x}-a\,\vec{e}_{i},t)-2p_{a,\tau}(\vec{x},t)\right]\,.$
(2.24)

Expanding the left hand side as a series in $\tau$ and the right hand side as
a series in $a$ we find

$\displaystyle\tau\,\frac{\partial p_{a,\tau}}{\partial
t}(\vec{x},t)+\mathcal{O}(\tau^{2})$
$\displaystyle=\frac{a^{2}}{2d}\sum_{i,j,k=1}^{d}(\vec{e}_{i})_{j}\,(\vec{e}_{i})_{k}\frac{\partial^{2}p_{a,\tau}}{\partial
x_{j}\partial x_{k}}(\vec{x},t)+\mathcal{O}(a^{3})\,.$ (2.25)

Since $\\{\vec{e}_{1},\ldots,\vec{e}_{d}\\}$ is a complete (orthonormalised)
basis of $\mathbb{R}^{d}$, we have
$\sum_{i=1}^{d}(\vec{e}_{i})_{j}\,(\vec{e}_{i})_{k}=\delta_{jk}$. Furthermore,
we divide (2.25) by $\tau$ and implement the limit (2.18) by first identifying
$\tau=\frac{a^{2}}{2d\kappa}$ and by defining

$\displaystyle
p_{a}(\vec{x},t):=p_{a,\tau=\frac{a^{2}}{2d\kappa}}(\vec{x},t)=\frac{1}{(2\pi)^{d}}\,\prod_{i=1}^{d}\int_{\left[-\frac{\pi}{a},\frac{\pi}{a}\right]}dk_{i}\,e^{ik_{i}x_{i}}\,\left(\frac{1}{d}\sum_{j=1}^{d}\cos(a\,k_{j})\right)^{\frac{2d\kappa
t}{a^{2}}}\,.$ (2.26)

The limit $a\to 0$ then becomes

$\displaystyle\lim_{a\to 0}\,\frac{\partial p_{a}}{\partial
t}(\vec{x},t)=\lim_{a\to
0}\left[\kappa\,\sum_{j=1}^{d}\,\frac{\partial^{2}p_{a}}{\partial
x_{j}^{2}}(\vec{x},t)+\mathcal{O}(a)\right]\,.$ (2.27)

Upon introducing the Laplace operator
$\Delta_{\vec{x}}=\sum_{i=1}^{d}\frac{\partial^{2}}{\partial x_{i}^{2}}$ on
$\mathbb{R}^{d}$ and using (2.13), the limit in (2.27) becomes (for $t>0$)

$\displaystyle(\partial_{t}-\kappa\,\Delta_{\vec{x}})\,p(\vec{x},t)=0\,,$
(2.28)

which is the heat equation with (thermal) diffusivity $\kappa$. This result is
indeed consistent with the fact that (2.20) is the causal Green’s function of
the differential operator $(\partial_{t}-\kappa\,\Delta_{\vec{x}})$ on
$\mathbb{R}^{d}\times\mathbb{R}$ and with the boundary condition

$\displaystyle\lim_{|\vec{x}|\rightarrow\infty}\,p(\vec{x},t)=0\,,$
$\displaystyle\forall t>0\,.$ (2.29)

###  2.3 Perturbation around the Continuum Limit

While we have seen in the previous Subsection that the limit (2.18) leads to a
well-defined and non-trivial probability density $p(\vec{x},t)$, in this
Subsection we are interested in perturbations around this limit. To this end,
we treat $a$ (and thus also $\tau=\frac{a^{2}}{2d\kappa}$) as a small
parameter and study the leading perturbations of (2.20) and (2.28).

####  2.3.1 Perturbation of the Probability Density

To obtain a first intuitive picture for the probabilities for small $a$, we
return to the stochastic model described by the probability (2.7) and consider
the expectation value

$\displaystyle\langle|\vec{X}|\rangle(T)=\sum_{\vec{X}\,^{\prime}\in\Gamma}|\vec{X}\,^{\prime}|\,P(\vec{X}\,^{\prime},T)\big{|}_{\tau=\frac{a^{2}}{2d\kappa}}\,,$
(2.30)

For $d=1$, this function of $T$ (quantised in units of
$\tau=\frac{a^{2}}{2d\kappa}$) is shown for different values of $a$ in Figure
3. This plot suggests that the corrections to the expectation value (2.23) are
small and decrease as $T$ grows larger.

Figure 3: Expectation value (2.30) to leading order in $a$: left panel
$\langle|\vec{X}|\rangle$ for $d=1$, $\kappa=1$ and different values of $a$.
The thin dashed line represents the limit $a\to 0$ in (2.23). Right panel: the
difference $\langle|\vec{X}|\rangle-\langle|\vec{x}|\rangle$ representing the
correction to the limit (2.23). The values for $a$ are the same as in the left
panel.

To obtain an analytic expression, we consider the deterministic description
and (based on the definition (2.12)) study the family of probability densities
(2.26) for finite (but small) values of $a$. The leading correction to $p_{a}$
is calculated in Appendix B (see (B.5)), and can be written in the form

$\displaystyle p_{a}(\vec{x},t)$
$\displaystyle=p(\vec{x},t)\,\left[1-\frac{a^{2}}{192d(\kappa
t)^{3}}\left(3|\vec{x}|^{4}-d\sum_{j=1}^{d}x_{j}^{4}-24\kappa
t|\vec{x}|^{2}+24d\kappa^{2}t^{2}\right)\right]+\mathcal{O}(a^{4})\,.$ (2.31)

While the limit (2.20) is spherically symmetric, the leading correction breaks
this symmetry. Indeed, in Figure 4 the subleading contribution is plotted for
$d=2$ showing a breaking of the spherical symmetry. Finally, the subleading
corrections to the expectation value (2.23) are given by

$\displaystyle\langle|\vec{x}|\rangle=2\,\frac{\Gamma\left(\frac{1+d}{2}\right)}{\Gamma\left(\frac{d}{2}\right)}\,\sqrt{\kappa
t}\left[1+\frac{a^{2}}{8d(d+2)\kappa t}+\mathcal{O}(a^{4})\right]\,,$
$\displaystyle\forall t>\frac{a^{2}}{2d\kappa}\,.$ (2.32)

In general, perturbations of the expectation value (2.23) of the continuum
theory are suppressed by powers of $\frac{a^{2}}{\kappa t}$. This means, even
for finite values of $a$, the continuum theory provides a good approximation
for times $t\gg\frac{a^{2}}{\kappa}$, in which case (2.28) can be seen as an
effective description.

Figure 4: Loss of spherical symmetry of the probability density (2.31) for
$d=2$: the left panel shows $p_{a}/p$ as a function of $\vec{x}=(x_{1},x_{2})$
for $\kappa t=10$ and $a=1/2$. The coloured lines highlight the curves
parametrised by $(r\cos\theta,r\sin\theta)$ for
$\theta\in\\{0,\pi/3,\pi/4\\}$, which are re-plotted in the right panel for
better comparison.

####  2.3.2 Perturbation of the Heat Equation

Before closing this Section, we remark that we can compute sub-leading
contributions not only directly to the probability densities as in (2.31), but
also to the differential equation that describes them (_i.e._ (2.28)). To this
end, we formally expand the probability density as an even power series in $a$

$\displaystyle
p_{a}(\vec{x},t)=p(\vec{x},t)+a^{2}\,p^{(2)}(\vec{x},t)+\mathcal{O}(a^{4})\,.$
(2.33)

Inserting this expansion into the difference equation (2.24) (for
$\tau=\frac{a^{2}}{2d\kappa}$) we can expand it order by order in $a$ to find
the following equations

$\displaystyle\text{order }\mathcal{O}(a^{2}):$
$\displaystyle\frac{1}{2d\kappa}\left(\partial_{t}\,p(x,t)-\kappa\,\Delta\,p(x,t)\right)=0\,,$
(2.34) $\displaystyle\text{order }\mathcal{O}(a^{4}):$
$\displaystyle\frac{1}{2d\kappa}\left(\partial_{t}\,p^{(2)}(x,t)-\kappa\,\Delta\,p^{(2)}(x,t)\right)=\frac{1}{4!d}\sum_{i=1}^{d}\frac{\partial^{4}p(\vec{x},t)}{\partial
x_{i}^{4}}-\frac{1}{8d^{2}\kappa^{2}}\,\partial_{t}^{2}p(\vec{x},t)\,.$ (2.35)

Here the equation (2.34) at order $\mathcal{O}(a^{2})$ is the result obtained
in (2.25) and (2.28) namely the heat equation. The solution of this equation
$p(\vec{x},t)$ acts as inhomogeneity in the heat equation (2.35) for the
correction $p^{(2)}$. While both (2.34) and (2.35) a priori allow for more
solutions, we have verified that for $p(x,t)$ as in (2.20), the subleading
contribution in (2.31) is indeed a solution of (2.35).

##  3 Epidemiology: Simple Epidemic- and the SI-Model

After discussing diffusion processes in the previous Section, we now consider
a system that describes in a simplistic fashion the spread of a disease within
a population. We shall not allow for the possibility of recovery. We first
discuss a purely stochastic approach, which can be approximated by a
deterministic model similar to the one considered in [43].

###  3.1 Stochastic Approach

We consider a(n infinite) population whose inhabitants are represented as the
lattice sites on the $d$-dimensional infinite, hypercubic lattice $\Gamma$
defined in eq. (2.1). At any given time $T$, we separate the population into
two compartments, namely each individual is either _infectious_ (_i.e._ they
carry a certain communicable disease) or _susceptible_ (see Figure 5 for a
sample configuration in $d=2$).666As is customary in epidemiology, a
susceptible individual is currently not infected with the disease, however,
upon contact with an infectious individual may contract the disease and become
infectious themselves (_i.e._ capable of infecting other susceptible). At this
stage, we do not consider the possibility of recovery, _i.e._ an infectious
individual will remain so forever. We plan to come back to this point in later
work. We assume that in fixed time-intervals $\tau$ infectious individuals
can, with a certain probability, infect nearby susceptible individuals (such
that they turn into infectious individuals themselves). To this end, we
introduce the function

$F:\hskip 14.22636pt\Gamma\longrightarrow[0,1/\tau]$ with $\displaystyle
0\leq\gamma^{\prime}:=\sum_{\vec{X}\in\Gamma}F(\vec{X})<\infty\,,$ (3.1)

which defines the rate at which an individual at $\vec{X}$ infects susceptible
individuals at $\vec{X}^{\prime}$.777For simplicity, we assume in this work
that this rate is translational invariant, _i.e._ it is only a function of the
difference $\vec{X}-\vec{X}\,^{\prime}$. Furthermore, let $P(\vec{X},T)$
denote the probability for the individual at the lattice site $\vec{X}$ to be
infectious at time $T\in\mathbb{R}$, then we have the following recursion
relation

$\vec{E}_{1}$$\vec{E}_{2}$$a$

Figure 5: Configuration of infectious (red) and susceptible (green)
individuals as sites on the lattice $\Gamma$ (for $d=2$) generated by the
basis vectors $(\vec{E}_{1},\vec{E}_{2})$.

$\displaystyle
P(\vec{X},T+\tau)=P(\vec{X},T)+\tau\,\left(1-P(\vec{X},T)\right)\,\sum_{\vec{X}\,^{\prime}\in\Gamma}F(\vec{X}-\vec{X}^{\prime})\,P(\vec{X}\,^{\prime},T)\,.$
(3.2)

Starting from an initial lattice configuration of susceptible and infectious
individuals at some fixed $T_{0}$, this relation allows us to iteratively
compute the probability for an individual at any $\vec{X}\in\Gamma$ to be
infectious at a time $T_{0}+n\,\tau$ for any $n\in\mathbb{N}$. Notice, if
$P(\vec{X},T)=1$ it will remain $1$ for all $T^{\prime}\geq T$.

For later use and in order to gain some intuition for this model, we shall
discuss some examples for the particular case $d=1$ numerically. In
particular, we shall consider different initial conditions and choices of the
function $F$, that governs the way in which individuals infect one
another:888For simplicity, and to simplify the notation in $d=1$, we shall
denote the positions simply by $X$ (instead of $\vec{X}$).

Figure 6: Probabilities $P(X,T)$ in $d=1$ for the initial conditions (3.3).
The red dots show the solution (3.2) for $P_{0}=5\cdot 10^{-6}$, while the
black line gives an interpolation by a logistic function.

  * •

homogeneous distribution: the simplest initial condition is to assume that at
some time $T_{0}$ all individuals have the same probability to be
infectious999Since we use $d=1$, we denote the spatial variable simply with
$X\in a\mathbb{Z}$.

$\displaystyle P(X,T=T_{0})=P_{0}\hskip 28.45274pt\forall
X\in\Gamma\cong\mathbb{Z}\,.$ (3.3)

In this case, the time evolution shows a (trivial) self-similarity in the
spatial distribution: the probabilities are the same at each lattice site,
such that they become independent of $X$. This behaviour is still
(approximately) observed if the initial probabilities $P_{0}$ are perturbed by
a small $X$-dependent function. Schematically, the iterative solution of (3.2)
is plotted in Figure 6 for a 1-dimensional configuration. While the plot was
computed using a specific $F$ (the one in (3.15)), the result is independent
of the details of this choice of $F$ but only depends on $\gamma^{\prime}$ in
(3.1). Indeed, the solid black line in Figure 6 is an interpolation by a
logistic function $P(0,T)=\frac{1}{1+e^{-\lambda(T-T_{0})}}$ with
$\lambda\sim\gamma^{\prime}\sim 0.6927$. As is showcased in Figure 6, the time
evolution in each point can be well approximated with this logistic growth.

  * •

travelling waves: we consider a starting configuration consisting of a number
of infectious individuals placed (symmetrically) around the origin, _i.e._ for
$T_{0}=0$ we have

$\displaystyle P(X,T_{0})=\left\\{\begin{array}[]{lcl}1&\text{if}&|X|\leq
R_{0}\,,\\\ 0&\text{if}&|X|>R_{0}\end{array}\right.$ (3.6)

Figure 7: Numerical evaluation of the probabilities following (3.2) with the
initial conditions (3.6) for $d=1$ and with $F(X,X^{\prime})$ given in (3.9).
The left panel shows the dependence of the evolution on the value of $R_{0}$
(for early times), while the right panel shows the evolution in the form of a
traveling wave at different times for fixed $R_{0}=0$. Both plots use $\tau=1$
and $\gamma=0.15$.

Figure 7 shows the numerical evaluation of the probability for an individual
at $X$ to be infected at time $T$ for $d=1$ and for $F=F_{\text{adj}}$:

$\displaystyle F_{\text{adj}}(X)=\left\\{\begin{array}[]{ll}\kappa&\text{if
}X=\pm a\,,\\\ 0&\text{else}\end{array}\right.$ (3.9)

with $\kappa=\in[0,\frac{1}{\tau}]$ a constant. This means, in this model only
nearest neighbours can infect one another. Both plots in 7 indicate that also
in this case the probability exhibits a self-similarity behaviour for large
enough $T$, namely in the form of a travelling wave. Indeed, the solid lines
in Figure 7 represent interpolations of the form

$\displaystyle
P(X,T)=\frac{1}{1+e^{-\lambda(T)(X+X_{0}(T))}}-\frac{1}{1+e^{-\lambda(T)(X-X_{0}(T))}}\,.$
(3.10)

Since (due to the choice of initial conditions (3.6)), the time evolution is
symmetric with respect to $X\to-X$, it is sufficient to focus on $X>0$, in
which case, one can equivalently (for sufficiently large $T$) approximate the
solution by the function

$\displaystyle P(X,T)=1-\frac{1}{1+e^{-\lambda(T)(X-X_{0}(T))}}\,,$
$\displaystyle\forall X>0\,.$ (3.11)

Here and in (3.10), $\lambda,X_{0}:\mathbb{R}\to\mathbb{R}_{+}$ are functions
of time, which are plotted in Figure 8. These suggest that for sufficiently
large $T$ (_i.e._ once the system has settled into a state of intermediate
asymptotics), $\gamma$ approaches a constant (that only depends on $\lambda$),
while $X_{0}$ tends to a linear function in time. This means that the
travelling wave moves approximately with a constant velocity, which is in
contrast to the behaviour of the expectation value of the position of a random
walker found in (2.23).

Figure 8: Numerical plots of the functions $\lambda$ (left panel) and $X_{0}$
(right panel) appearing in the interpolating function (3.10) as functions of
time.

  * •

periodic initial conditions: Another class of initial conditions are
periodically fluctuating ones of the form

$\displaystyle P(X,T_{0})=P_{0}\,(1+\cos(LX))\,,$ (3.12)

which are schamtically plotted in the left panel of Figure 9. Such conditions
could be used to mimic outbreak centers that are geographically spaced out
with a characteristic length scale $L$.

Figure 9: Numerical evaluation of the probabilities following (3.2) with
periodic initial conditions (3.12) for $P_{0}=5\cdot 10^{-6}$ and
$L=\frac{\pi}{30}$. The left panel shows the initial conditions, while the
right panel shows the evolution of $P(X,T)$ as a function of $T$ for different
fixed $X$ (for $F$ as in (3.15)). The solid lines give interpolations with
logistic functions.

Here we choose the $F(X)$ function to be

$\displaystyle
F(X)=\left\\{\begin{array}[]{lcl}\frac{e^{-X^{2}/20}}{10}&\text{if}&X\neq
0\,,\\\ 0&\text{if}&X=0\,.\end{array}\right.$ (3.15)

The iterative solution of equation (3.2) is shown in the right panel of Figure
9 for different values of $X$. Each of these solutions can be very well
approximated by a logistic function, however, with parameters that depend on
$X$

$\displaystyle P(X,T)\sim\frac{1}{1+e^{-\lambda(X)(T-T_{0}(X))}}\,$ (3.16)

Schematic plots of $\lambda(X)$ and $T_{0}(X)$ are shown in Figure 10.

Figure 10: Parameters $\lambda$ (left panel) and $T_{0}$ (right panel) of the
approximations in (3.16) for the probabilities computed from (3.2) with
periodic boundary conditions (3.12). Both plots use the same remaining
parameters as in Figure 9.

###  3.2 Simple Epidemic Model

Rather than working with the stochastic model (3.2), we shall ultimately work
with a deterministic ’simple epidemic model’ introduced in [17] (see also
[16]). To this end, we shall approximate the former by replacing the time- and
spatial variables by continuous ones.

####  3.2.1 Deterministic Approximation: Continuous Time

To motivate the relation to the stochastic approach discussed above, we first
write the difference equation (3.2) in the following form

$\displaystyle\frac{P(\vec{X},T+\tau)-P(\vec{X},T)}{\tau}=\left(1-P(\vec{X},T)\right)\,\sum_{\vec{X}\,^{\prime}\in\Gamma}F(\vec{X}-\vec{X}^{\prime})\,P(\vec{X}\,^{\prime},T)$
(3.17)

and take the limit $\tau\to 0$

$\displaystyle\frac{\partial P(\vec{X},t)}{\partial
t}=\left(1-P(\vec{X},t)\right)\,\sum_{\vec{X}\,^{\prime}\in\Gamma}F(\vec{X}-\vec{X}^{\prime})\,P(\vec{X}\,^{\prime},t)\,,$
$\displaystyle\forall\vec{X}\in\Gamma\text{ and }t\in\mathbb{R}\,.$ (3.18)

Here we have furthermore replaced $T\to t\in\mathbb{R}$ and we consider the
latter as a continuous variable (that is not quantised in units of $\tau$).
This equation, however, can also be interpreted to come from a compartmental
SI-model defined on the lattice $\Gamma$, similar to [43]: to see this, we
define

$\displaystyle\mathfrak{S}:\hskip
14.22636pt\Gamma\times\mathbb{R}_{+}\longrightarrow[0,1]\,,$
$\displaystyle\mathfrak{I}:\hskip
14.22636pt\Gamma\times\mathbb{R}_{+}\longrightarrow[0,1]\,,$
$\displaystyle\hskip
38.41139pt(\vec{X},t)\longmapsto\mathfrak{S}(\vec{X},t)\,,$
$\displaystyle\hskip
38.41139pt(\vec{X},t)\longmapsto\mathfrak{I}(\vec{X},t)\,,$ (3.19)

where $\mathfrak{S}$ and $\mathfrak{I}$ are interpreted as the (normalised)
number of susceptible and infectious individuals respectively at the lattice
site $\vec{X}$ and at the time $t$ (which is considered to be
continuous).101010For simplicity, we place the start of the epidemic at $t=0$.
In order to model the contact between susceptible and infectious individuals
(and therefore also the infection dynamics), we define a function
$F:\,\Gamma\longrightarrow\mathbb{R}_{+}$ (similar to (3.1)) with the property

$\displaystyle\sum_{\vec{X}\in\Gamma}F(\vec{X})=\gamma^{\prime}\,,$ with
$\displaystyle 0<\gamma^{\prime}<\infty\,,$ (3.20)

and model the spread of the disease by the following differential equations in
time

$\displaystyle\frac{\partial\mathfrak{S}}{\partial t}(\vec{X},t)$
$\displaystyle=-\mathfrak{S}(\vec{X},t)\,\sum_{\vec{X}\,^{\prime}\in\Gamma}\,F(\vec{X}-\vec{X}\,^{\prime})\,\mathfrak{I}(\vec{X}\,^{\prime},t)\,,$
$\displaystyle\frac{\partial\mathfrak{I}}{\partial t}(\vec{X},t)$
$\displaystyle=\mathfrak{S}(\vec{X},t)\,\sum_{\vec{X}\,^{\prime}\in\Gamma}\,F(\vec{X}-\vec{X}\,^{\prime})\,\mathfrak{I}(\vec{X}\,^{\prime},t)\,,$
(3.21)

together with the initial conditions

$\displaystyle\mathfrak{S}(\vec{X},0)=\mathfrak{S}_{\text{init}}(\vec{X})\,,$
$\displaystyle\mathfrak{I}(\vec{X},0)=\mathfrak{I}_{\text{init}}(\vec{X})\,,$
$\displaystyle\forall\vec{X}\in\Gamma\,,$ (3.22)

where
$\mathfrak{S}_{\text{init}}\,,\mathfrak{I}_{\text{init}}:\,\Gamma\longrightarrow[0,1]$
are functions that satisfy
$\mathfrak{S}_{\text{init}}(\vec{X})+\mathfrak{I}_{\text{init}}(\vec{X})=1$
$\forall\vec{X}\in\Gamma$. Since (3.24) implies
$\frac{d}{dt}\left(\mathfrak{S}+\mathfrak{I}\right)(\vec{X},t)=0$ we have

$\displaystyle(\mathfrak{S}+\mathfrak{I})(\vec{X},t)=1\,,$
$\displaystyle\forall\,\vec{X}\in\Gamma\,,\hskip 5.69046pt\forall
t\in\mathbb{R}\,.$ (3.23)

This allows us to write the coupled system of differential equations (3.21) in
the form of a single non-linear differential equation

$\displaystyle\frac{\partial\mathfrak{I}}{\partial
t}(\vec{X},t)=\left(1-\mathfrak{I}(\vec{X},t)\right)\,\sum_{\vec{X}\,^{\prime}\in\Gamma}\,F(\vec{X}-\vec{X}\,^{\prime})\,\mathfrak{I}(\vec{X}\,^{\prime},t)\,.$
(3.24)

This is the same equation as (3.18), upon identifying directly the
probabilities $P(\vec{X},t)$ for a lattice site $\vec{X}$ to be infected with
the number of infectious individuals $\mathfrak{I}$ at $\vec{X}$. Identifying
(3.24) with (3.18) therefore marks the transition from a stochastic approach
(characterised by probabilities) to a deterministic approach (with the number
of infectious individuals as the observable).

The model (3.24) is motivated by the mass action law, which underlies many
compartmental models, notably the SIR model [29]: the rate at which
susceptible individuals (at $\vec{X}$) are turned into infectious ones depends
on the number of susceptible individuals at $\vec{X}$ and the number of
infectious ones they can get into contact with. In (3.24) we are allowing for
the possibilities that infectious from anywhere on $\Gamma$ may infect a
susceptible at $\vec{X}$, however, the rate at which this is happening is
weighted by the function $F$. Since the latter is bounded from below and
$\mathfrak{I}(\vec{X},t)\in[0,1]$ $\forall t\in\mathbb{R}$, the condition
(3.20) is a necessary condition for the equations (3.24) to be well-defined
since

$\displaystyle
0\leq\sum_{\vec{X}\,^{\prime}\in\Gamma}\,F(\vec{X}-\vec{X}\,^{\prime})\,\mathfrak{I}(\vec{X}\,^{\prime},t)\leq\sum_{\vec{X}\,^{\prime}\in\Gamma}\,F(\vec{X}-\vec{X}\,^{\prime})=\gamma^{\prime}<\infty\,.$
(3.25)

$F(X-X^{\prime})$$X^{\prime}$$X$

Figure 11: Examples for the schematic forms of the function $F$ in $d=1$.

Apart from (3.20), we pose no further restrictions on $F$. Intuitively,
however, to model an epidemic, we expect $F(\vec{X}-\vec{X}^{\prime})$ to be
peaked around $\vec{X}$, different examples of which are schematically shown
in Figure 11 for the case $d=1$.111111Notice that we consider in general
$F(0)\neq 0$: indeed, $\mathfrak{I}$ models the number of infectious
individuals at $\vec{X}$ which are also capable of infecting susceptible
individuals at the same lattice site. This is in contrast to the functions $F$
that we had used in the examples in Section 3.1 to model the probabilities for
an individual at $\vec{X}$ to become infected by their neighbours. While each
one of these functions leads to a different spread of the disease across the
lattice (_i.e._ a qualitatively and quantitatively different solution of
(3.24)), we shall define a family of effective (_i.e._ coarse grained)
theories, in which the detailed shape of this function becomes parametrically
less and less important.

####  3.2.2 Coarse Graining: Continuous Spatial Variables

Following the example of the diffusion process in Section 2.2.1, we shall now
pass to continuous space-variables and approximate (3.24) by a partial
differential equation for continuous variables $\vec{x}\in\mathbb{R}$ instead
of discrete $\vec{X}\in\Gamma$. To this end, instead of the number of
infectious individuals at lattice site $\vec{X}\in\Gamma$ we introduce the
coarse grained quantity
$I:\,\mathbb{R}^{d}\times\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ as the
average of the number of infectious individuals over a compact subset $\Delta
X\subset\Gamma$

$\displaystyle\mathfrak{I}(\vec{X},t)\longrightarrow
I(\vec{x},t)=\frac{1}{|\Delta X|}\sum_{\vec{X}\,^{\prime}\in\Delta
X}\mathfrak{I}(\vec{X}\,^{\prime},t)\sim\mathfrak{I}(\vec{X},t)\,,$ with
$\displaystyle\vec{x}\in\mathbb{R}^{d}\,,$ (3.26)

where $|\Delta X|$ is the number of lattice points contained in $\Delta
X$.121212We note that one could introduce a constant factor in eq. (3.26),
which reflects the liberty of rescaling of the (normalised) $I$. For the sake
of simplicity, we refrain from doing so here. Similarly, we note that we could
also define $|\Delta X|=\text{vol}_{\mathbb{R}^{d}}(\Delta X)/a^{d}$, which is
equivalent up to a rescaling of $I(\vec{x},t)$. We shall comment on the effect
of the latter below. The region $\Delta X$ is labeled by a continuous variable
$\vec{x}\in\mathbb{R}^{d}$ (where potentially $\vec{x}\notin\Gamma$), as is
schematically shown in Figure 12. In the last step in (3.26) we have assumed
that the number of infectious individuals does not vary significantly over
$\Delta X$, such that the value of $I(\vec{x},t)$ can be identified with
$\mathfrak{I}(\vec{X},t)$. We furthermore assume that $I(\vec{x},t)$ is a
continuous function, such that in (3.24) we can replace

$a$$\vec{x}$$\Delta X$$\vec{X}$$\vec{E}_{1}$$\vec{E}_{2}$

Figure 12: Coarse graining in the case of epidemical model (3.24).

$\displaystyle\sum_{\vec{X}\,^{\prime}\in\Gamma}\,F(\vec{X}-\vec{X}\,^{\prime})\,\mathfrak{I}(\vec{X}\,^{\prime},t)\longrightarrow\int_{\mathbb{R}^{d}}d^{d}x^{\prime}\,f(\vec{x}-\vec{x}\,^{\prime})\,I(\vec{x}\,^{\prime},t)=(f*I)(\vec{x},t)\,,$
(3.27)

where $*$ denotes the convolution (with respect to the spatial variable
$\vec{x}$) and $f:\,\mathbb{R}^{d}\longrightarrow\mathbb{R}_{+}$ is a
continuous function which satisfies

$\displaystyle\int_{\mathbb{R}^{d}}d^{d}x\,f(\vec{x})=\gamma\,,$ and
$\displaystyle f(\vec{X})=F(\vec{X})\,,\hskip
14.22636pt\forall\vec{X}\in\Gamma\cap\mathbb{R}^{d}\,,$ (3.28)

for some real constant $\gamma$ with $0<\gamma<\infty$.131313The relation
between $\gamma$ and $\gamma^{\prime}$ in (3.20) depends on the details how
$F$ is continued from a function defined on $\Gamma$ to a function defined on
all $\mathbb{R}^{d}$, which shall not be important in the following. Notice,
since $f$ is assumed to be non-negative for all $\vec{x}\in\mathbb{R}^{d}$,
the first condition implies that $f\in L^{1}(\mathbb{R}^{d})$. We therefore
write the following coarse grained approximation of eq.(3.24)

$\displaystyle\frac{\partial I}{\partial
t}(\vec{x},t)=\left(1-I(\vec{x},t)\right)\,\left((f*I)(\vec{x},t)\right)\,.$
(3.29)

This model has been studied in [17] for $d=1$, in particular searching for
travelling waves and relating it to stochastic approaches [30].

###  3.3 Family of Effective Models

The equation (3.29) provides an effective description of a (microscopic) model
based on a lattice $\Gamma$ with lattice spacing $a$ through coarse graining.
In the following, we shall consider a family of such effective theories, where
we rescale $a$ by a dimensionless parameter $\mu$

$\displaystyle a\longrightarrow\mu\,a\,,$ with
$\displaystyle\mu\in\mathbb{R}_{+}\,.$ (3.30)

We shall, however, keep $\Delta X$ as a subset of $\mathbb{R}^{d}$ fixed, as
is schematically shown in Figure 13 for $0<\mu<1$.141414In the following we
shall be mostly interested in the limit $\mu\to 0$, such that implicitly
(3.30) corresponds to considering more fine-grained lattices. A possible way
to interpret this from an epidemiological perspective, is to associate $\Delta
X$ (labelled by the fixed $\vec{x}\in\mathbb{R}^{d}$) with a (fixed)
geographical region, while the lattice points represent (finer and finer)
subdivisions of the population (_e.g._ provinces, cities, households, _etc._).
Concretely one may think of $\Delta X$ as a country with the lattice points
representing its different provinces. After rescaling with a certain $\mu$, we
obtain a new model in which the spread of the disease within the same country
$\Delta X$ is now described at the level of cities (represented by the finer
grained lattice).

$a$$\Delta X$$0$$\vec{x}$$a\longrightarrow\mu\,a$$\mu\,a$$0$$\vec{x}$

Figure 13: Rescaling of $\Gamma$ for $0<\mu<1$: both $\Delta
X\subset\mathbb{R}^{d}$ and $\vec{x}\in\mathbb{R}^{d}$ remain fixed.

We remark that there is an alternative way to interpret this rescaling from
the perspective of the variables $I$ defined in (3.26): instead of a change of
the lattice spacing (3.30), we can think of keeping $a$ fixed, but instead
rescaling $\Delta X$ (and the variable $\vec{x}$ labelling it) to include more
lattice sites. This is schematically shown in Figure 14. From an
epidemiological perspective, a smaller value of $\mu$ implies that we are
averaging the number of infectious individuals over a larger geographical
region: _e.g._ instead of counting the number of infected individuals in a
single province (made up from several cities represented by the lattice
points), $I$ after rescaling counts the (averaged) number of infected in an
entire country (with the lattice points still representing single cities).

$a$$\Delta X$$0$$\vec{x}$$a$$\Delta X_{\mu}$$0$$\vec{x}/\mu$

Figure 14: Equivalent interpretation of the rescaling for $0<\mu<1$: instead
of changing the lattice spacing $a$ of $\Gamma$ the coarse graining is
performed over a larger subset $\Delta X_{\mu}$ of $\Gamma$ and $\vec{x}$ is
replaced by $\frac{\vec{x}}{\mu}$.

Implementing the rescaling (3.30) at the level of the equation (3.29), we
first realise that, from the perspective of the lattice variables it requires
to replace $\mathfrak{I}(\vec{X},t)$ by $\mathfrak{I}(\vec{X}/\mu,t)$ (see
Figure 14), which therefore suggests to rescale

$\displaystyle I(\vec{x},t)$ $\displaystyle\longrightarrow
I_{\mu}(\vec{x},t)=I(\vec{x}/\mu,t)\,,$ (3.31)

where $I_{\mu}:\,\mathbb{R}^{d}\longrightarrow\mathbb{R}_{+}$ is a regular
function. This in turn suggests for the convolution in (3.27)

$\displaystyle(f*I)(\vec{x},t)$
$\displaystyle\longrightarrow\int_{\mathbb{R}^{d}}d^{d}x^{\prime}f\left(\frac{\vec{x}}{\mu}-\vec{x}\,^{\prime}\right)I(\vec{x}\,^{\prime},t)$
$\displaystyle\hskip
45.52458pt=\int_{\mathbb{R}^{d}}d^{d}x^{\prime}\frac{1}{\mu^{d}}\,f\left(\frac{\vec{x}-\vec{x}\,^{\prime}}{\mu}\right)\,I(\vec{x}\,^{\prime}/\mu,t)=:(f_{\mu}*I_{\mu})(\vec{x},t)\,,$
(3.32)

where we have introduced the function

$\displaystyle f_{\mu}:\,\mathbb{R}^{d}$
$\displaystyle\longrightarrow\mathbb{R}_{+}$ $\displaystyle\vec{x}$
$\displaystyle\longmapsto
f_{\mu}(\vec{x})=\frac{1}{\mu^{d}}\,f(\vec{x}/\mu)\,.$ (3.33)

$f(x-x^{\prime})$$x^{\prime}$$x$$f_{\mu}(x-x^{\prime})$$x^{\prime}$$x$

Figure 15: Schematic examples of the rescaled functions $f_{\mu}$ for
$0<\mu<1$ in $d=1$.

Schematic examples of the rescaled functions are shown in Figure 15: for
$0<\mu<1$, the functions become more peaked around the origin, however, in
such a way that the total area below the graph remains invariant. Indeed, for
$f\in L^{1}(\mathbb{R}^{d})$ (satisfying the first equation in (3.28) and
$f(\vec{x})\geq 0$ $\forall\vec{x}\in\mathbb{R}^{d}$) and any $\mu>0$, the
function $f_{\mu}$ has the following properties

  1. _(i)_

$f_{\mu}\in L^{1}(\mathbb{R}^{d})$, since

$\displaystyle\int_{\mathbb{R}^{d}}d^{d}x\,|f_{\mu}(\vec{x})|=\frac{1}{\mu^{d}}\int_{\mathbb{R}^{d}}d^{d}x\,|f(\vec{x}/\mu)|=\int_{\mathbb{R}^{d}}d^{d}x\,|f(\vec{x})|<\infty\,,$
(3.34)

where the last step follows due to $f\in L^{1}(\mathbb{R}^{d})$.

  2. _(ii)_

Due to the first equation in (3.28)

$\displaystyle\int_{\mathbb{R}^{d}}d^{d}x\,f_{\mu}(\vec{x})=\frac{1}{\mu^{d}}\int_{\mathbb{R}^{d}}d^{d}x\,f(\vec{x}/\mu)=\int_{\mathbb{R}^{d}}d^{d}x\,f(\vec{x})=\gamma\,.$
(3.35)

  3. _(iii)_

For any $\epsilon>0$

$\displaystyle\lim_{\mu\to
0}\int_{|\vec{x}|\geq\epsilon}\,|f_{\mu}(\vec{x})|=0\,,$ (3.36)

which follows since

$\displaystyle\lim_{\mu\to
0}\int_{|\vec{x}|\geq\epsilon}d^{d}x\,|f_{\mu}(\vec{x})|=\lim_{\mu\to
0}\int_{|\vec{x}|\geq\epsilon}d^{d}x\,\frac{1}{\mu^{d}}\,|f(\vec{x}/\mu)|=\lim_{\mu\to
0}\int_{|\vec{x}|\geq\frac{\epsilon}{\mu}}d^{d}x\,f(\vec{x})\longrightarrow
0\,.$ (3.37)

As explained in Appendix A.1, the family of functions $\\{f_{\mu}\\}_{\mu>0}$
is therefore an _approximation to the identity_.

The replacement (3.32) leads to the following family of differential equations

$\displaystyle\frac{\partial I_{\mu}}{\partial
t}(\vec{x},t)=\left(1-I_{\mu}(\vec{x},t)\right)\,\left((f_{\mu}*I_{\mu})(\vec{x},t)\right)\,,$
$\displaystyle\forall\mu>0\,.$ (3.38)

This is the same type of differential equation as in (3.29), except that the
’interaction’ between infectious and susceptible individuals is now governed
by a different function $f_{\mu}$. From an epidemiological perspective, we are
interested in the continuum limit $\mu\to 0$. Using the properties of the
family of functions $\\{f_{\mu}\\}_{\mu>0}$ and denoting
$I_{0}(\vec{x},t)=\lim_{\mu\to\infty}I_{\mu}(\vec{x},t)$, we therefore find in
the limit

$\displaystyle\lim_{\mu\to
0}(f_{\mu}*I_{\mu})(\vec{x},t)=\gamma\,I_{0}(\vec{x},t)\,,$ (3.39)

such that we find for the limit $\lim_{\mu\to 0}$ of the differential equation
(3.38)

$\displaystyle\frac{\partial I_{0}}{\partial t}(\vec{x},t)$
$\displaystyle=\gamma\,\left(1-I_{0}(\vec{x},t)\right)\,I_{0}(\vec{x},t)\,.$
(3.40)

We note here that despite the dependence of $I_{0}(\vec{x},t)$, the dynamics
in the limit is entirely local at $\vec{x}$: the temporal evolution of the
number of infectious individuals at $\vec{x}$ is independent of the number of
infectious individuals at $\vec{x}\,^{\prime}\neq\vec{x}$. From the
perspective of the lattice, $\Delta X$ is now averaging the number of
infectious over an infinite subset of the lattice $\Gamma$. From an
epidemiological perspective, we are now averaging over a continuous function
of infectious individuals in the geographic region $\Delta X$. If the latter
represents _e.g._ a country (labelled by $\vec{x}$), the temporal evolution of
the disease no longer takes into account any geographic spread within the
country. Furthermore, this evolution is completely independent of the
evolution in neighbouring countries (_i.e._ for
$\vec{x}\,^{\prime}\neq\vec{x}$) and describes the independent spread of a
disease in different ’countries’. For given fixed $\vec{x}$ the equation
(3.40) is equivalent to the logistic equation stemming from the SI-model
(without recovery) [29], which is among the simplest compartmental models
(with only two compartments).

Going beyond the limit $\mu\to 0$ and following the example of the heat
equation in Section 2.2.3, we can consider an expansion of (3.38) for small
$\mu$. To this end, we define

$\displaystyle
I_{\mu}(\vec{x},t)=I^{(0)}(\vec{x},t)+\mu\,I^{(1)}(\vec{x},t)+\mathcal{O}(\mu^{2})\,.$
(3.41)

The main difficulty in expanding (3.38) is the convolution of the function
$f_{\mu}$ with $I_{\mu}$ of the following form

$\displaystyle(f_{\mu}*I_{\mu})(\vec{x},t)=\int_{\mathbb{R}^{d}}\frac{d^{d}x^{\prime}}{\mu^{d}}\,f\left(\frac{\vec{x}-\vec{x}\,^{\prime}}{\mu}\right)\left(I^{(0)}(\vec{x}\,^{\prime},t)+\mu\,I^{(1)}(\vec{x}\,^{\prime},t)+\mathcal{O}(\mu^{2})\right)$
(3.42)

Upon changing variables $\vec{y}=\frac{\vec{x}-\vec{x}\,^{\prime}}{\mu}$ we
can re-write this convolution in the form

$\displaystyle(f_{\mu}*I_{\mu})(\vec{x},t)$
$\displaystyle=\int_{\mathbb{R}^{d}}d^{d}y\,f(\vec{y})\left(I^{(0)}(\vec{x}-\mu\,\vec{y},t)+\mu\,I^{(1)}(\vec{x}-\mu\,\vec{y},t)+\mathcal{O}(\mu^{2})\right)$
$\displaystyle=\gamma\,I^{(0)}(\vec{x},t)+\mu\left[\gamma\,I^{(1)}(\vec{x},t)-\int_{\mathbb{R}^{d}}d^{d}y\,f(\vec{y})(\vec{y}\cdot\vec{\nabla}_{x})I^{(0)}(\vec{x},t)\right]+\mathcal{O}(\mu^{2})\,.$
(3.43)

Equation (3.38) can then be expanded to first order to yield

$\displaystyle\text{order }\mathcal{O}(\mu^{0}):$ $\displaystyle\frac{\partial
I^{(0)}}{\partial
t}(\vec{x},t)=\gamma\,(1-I^{(0)}(\vec{x},t))\,I^{(0)}(\vec{x},t)\,,$ (3.44)
$\displaystyle\text{order }\mathcal{O}(\mu^{1}):$ $\displaystyle\frac{\partial
I^{(1)}}{\partial
t}(\vec{x},t)=\gamma\,(1-2I^{(0)}(\vec{x},t))\,I^{(1)}(\vec{x},t)$
$\displaystyle\hskip
68.28644pt-(1-I^{(0)}(\vec{x},t))\int_{\mathbb{R}^{d}}d^{d}y\,f(\vec{y})\,(\vec{y}\cdot\vec{\nabla}_{x})\,I^{(0)}(\vec{x},t)\,.$
(3.45)

The solution to (3.44) is a logistic function (see (3.68) below), which may
acquire an $\vec{x}$ dependence through the initial conditions. The solution
$I^{(0)}$ enters into (3.45) as the coefficient of the term linear in
$I^{(1)}$ as well as an inhomogeneity (which encodes additional information of
the function $f$ beyond the simple constant $\gamma$). Notice, however, that
the latter vanishes if initial conditions for $I^{(0)}$ are chosen that are
independent of $\vec{x}$.

###  3.4 Length Scales at the Level of Solutions

Formally, the family of equations (3.38) established in the previous
Subsection can be thought of as being obtained from the simple epidemic model
(3.29) through the transformation

$\displaystyle I\longrightarrow I_{\mu}\,,$ and $\displaystyle
f\longrightarrow f_{\mu}\,.$ (3.46)

From a physics perspective this transformation leads to a family of effective
models in which the ’interaction’ of infectious and susceptible individuals is
(through the function $f_{\mu}$) governed by the dimensionless scaling
parameter $\mu$. In this Subsection we study the behaviour of particular
classes of (approximate) solutions to the equations (3.38) and consider their
behaviour under rescaling or the inclusion of a characteristic length scale.

####  3.4.1 Travelling Waves and Similarity Transformation in $d=1$

As a first type of solutions, we shall consider travelling waves for $d=1$.
Indeed, in one dimension and for the particular choice
$f(\vec{x})=\frac{1}{2}\beta\,e^{-\beta|x|}$ (with $\beta\in\mathbb{R}_{+}$)
travelling wave solutions of (3.29) have been found and discussed in [30] (see
also [44]) and it has been speculated that such solutions also exist for other
classes of functions. A travelling wave is a solution
$I_{w}:\,\mathbb{R}\rightarrow\mathbb{R}_{+}$ of (3.29) that depends on
$(x,t)$ only through the particular combination $z:=x-vt$ for some constant
velocity $v$. Inserting $I(x,t)=u(x-vt)$ into (3.29)

$\displaystyle-v\,u^{\prime}(z)=(1-u(z))\int_{-\infty}^{\infty}dx^{\prime}\,f(x-x^{\prime})\,u(x^{\prime}-vt)\,,$
(3.47)

and changing variables to $x^{\prime}-vt=\xi$, travelling wave solutions are
characterised by the integro-differential equation

$\displaystyle
u^{\prime}(z)=\frac{u(z)-1}{v}\int_{-\infty}^{\infty}d\xi\,f\left(z-\xi\right)\,u(\xi)=\frac{u(z)-1}{v}\,(f*u)(z)\,.$
(3.48)

At least approximately we can demonstrate the existence of such waves by
numerically solving the discretised equations (3.24). For
simplicity151515Other functions that are sufficiently peaked around $x=0$
yield very similar results., we consider (for $d=1$) $F$ to be a Gaussian
function $F(X)=\frac{e^{-X^{2}/16}}{10}$. Numerical solutions for the initial
condition $I(X,t=0)=\delta_{X,0}/100$ at different times $t$ (and for $a=1$
are shown in Figure 16). Since $F$ is a symmetric function, we also have
$I(X,t)=I(-X,t)$. After a certain initial time (as shown in the left panel of
Figure 16), the system reaches a state of the intermediate asymptotics in the
sense of [18] (see Appendix A.2), the solution corresponds to a steady
travelling wave moving from $X=0$ to the left and right: indeed, focusing on
$X>0$, $I(X,t)$ can be well approximated by a sigmoid function of the form

Figure 16: Numerical solutions of (3.24) for $d=1$ and the initial conditions
$I(X,t=0)=\delta_{X,0}/4$ for $F(X)=\frac{e^{-X^{2}/16}}{10}$ and $a=1$. Left
panel: full solution shortly after the initial conditions. Right panel: the
wave travelling to the right after sufficiently large time for the system to
reach a state of intermediate asymptotics. The solid lines are fits of the
numerical solution with a function of the form (3.54).

$\displaystyle I(X,t)=1-\frac{1}{1+e^{-\lambda(t)(X-X_{0}(t))}}\,,$
$\displaystyle\forall X>0\,.$ (3.49)

This form is the same as (3.11), which was used to approximate the stochastic
model (3.2), thus justifying that (3.24) is a good (deterministic)
approximation of the stochastic model (3.2). The quantities $\lambda(t)$ and
$X_{0}(t)$ in (3.49) are functions of time, which are plotted in Figure 17.
These suggest that for large $t$ to good approximation161616We note that here
and in the following, the error estimates are only with regards to the fitting
of the numerical solution.

Figure 17: Numerical plots of the functions $\lambda(t)$ and $X_{0}(t)$ as
functions of $t$, along with the approximations in eq. (3.53)

$\displaystyle\lambda(t)\sim\lambda_{0}\,,$ $\displaystyle
X_{0}(t)\sim\kappa+vt\,,$ with
$\displaystyle\begin{array}[]{l}\lambda_{0}=0.3900\pm 0.0008\,,\\\
\kappa=-12.154\pm 0.35\,,\\\ v=2.001\pm 0.002\,.\end{array}$ (3.53)

Therefore, the solution $I(X,t)$ can (approximately) be written in the form

$\displaystyle
I(X,t)=1-\frac{1}{1+e^{-\lambda_{0}(X-(\kappa+vt))}}=1-\frac{1}{1+B\,e^{-\lambda_{0}(X-vt)}}\,,$
with $\displaystyle B=e^{\lambda_{0}\kappa}\,,$ (3.54)

which is indeed only a function of the combination $X-vt$, where $v=$ const.
and is therefore of the form of a steady travelling wave [45] of velocity $v$.

Next, we consider a rescaled model, in which we replace the function $F(X)$ by
$F_{\mu}(X)$ following (15)

$\displaystyle
F_{\mu}(X)=\frac{1}{\mu}\,F(X/\mu)=\frac{1}{10\,\mu}\,e^{-\frac{X^{2}}{16\mu^{2}}}\,,$
$\displaystyle\forall\mu>0\,.$ (3.55)

Figure 18: Numerical plots of $\mu$ dependence of $(\lambda_{0},v,\kappa)$
(red dots) and their approximation (solid black lines) using eq.(3.62)

A numerical analysis suggest that the corresponding solutions $I_{\mu}(X,t)$
of (3.24) still correspond to traveling waves, which can be described by
(3.54) except for coefficients $(\lambda_{0},v,\kappa)$ that depend on $\mu$.
Numerically, the functional dependence is plotted in Figure 18, which can be
fitted by

$\displaystyle\begin{array}[]{l}\lambda_{0}\sim\frac{\gamma_{0}}{\mu}\,,\\\
v\sim\mu\,v_{0}\,,\\\ \kappa\sim\mu\,\kappa_{0}\,,\end{array}$ with
$\displaystyle\begin{array}[]{l}\gamma_{0}=0.3941\pm 0.0002\,,\\\
v_{0}=2.0581\pm 0.0003\,,\\\ \kappa_{0}=-11.93\pm 0.01\,,\end{array}$ (3.62)

such that we find for the solution

$\displaystyle
I_{\mu}(X,t)=1-\frac{1}{1+e^{-\frac{\gamma_{0}}{\mu}(X-\mu(\kappa_{0}+v_{0}t))}}=1-\frac{1}{1+B_{\mu}\,e^{-\gamma_{0}(X/\mu-
v_{0}t)}}\,,$ with $\displaystyle B_{\mu}=e^{\gamma_{0}\kappa_{0}}\,.$ (3.63)

This is indeed approximately $I(X/\mu,t)$ with the function $I$ given in
(3.54).171717We note that a different choice of initial conditions also
requires a shift $X\to X+c$ for some $c\in[-1,1]$. We furthermore note that
our computation includes further sources of numerical errors (_e.g._ the fact
that the computations were performed on a finite array of $201$ points.) This
not only shows the consistency of the replacements (3.31) and (3.32), it also
shows that these latter can be seen as similarity transformations which have
the structure of a group: performing two consecutive rescalings with
parameters $\mu_{1}$ and $\mu_{2}$ is equivalent to performing a single
rescaling with $\mu=\mu_{1}\mu_{2}$. Therefore (3.31) and (3.32) bear the
hallmark of a renormalisation group transformation.

Before continuing to other solutions we remark that the existence of
travelling wave solutions of (3.38) for $d>1$ is a more delicate question.
Indeed, substitution of the ansatz $I_{\mu}(\vec{x},t)=u(z)$ with
$z=|\vec{x}|-vt$ leads to

$\displaystyle-v\,u^{\prime}(z)=(1-u)\int_{\mathbb{R}^{d}}d^{d}x^{\prime}f_{\mu}(\vec{x}-\vec{x}\,^{\prime})\,u(|\vec{x}\,^{\prime}|-vt)\,.$
(3.64)

For general $L^{1}(\mathbb{R}^{d})$-functions, the latter, however, is not
consistent, since the right hand side is not necessarily a function of $z$
alone.

####  3.4.2 Linearisation Around a Position Independent Background

As another type of solution of (3.29) we consider a linearisation around a
position independent solution

$\displaystyle I(\vec{x},t)=I_{0}(t)+h(\vec{x},t)\,,$ with
$\displaystyle\begin{array}[]{l}I_{0}:\,\mathbb{R}\longrightarrow\mathbb{R}_{+}\,,\\\
h(\vec{x},t):\,\mathbb{R}^{d}\times\mathbb{R}\longrightarrow\mathbb{R}\,.\end{array}$
(3.67)

where we consider $h$ to be a small fluctuation. Expanding (3.29) for small
$h$ we find to leading order (3.40) _i.e._

$\displaystyle\frac{\partial I_{0}}{\partial
t}(t)=\gamma\,(1-I_{0}(t))\,I_{0}(t)$ with solution $\displaystyle
I_{0}(t)=\frac{1}{1+e^{-\gamma(t-\tau_{0})}}\,,$ (3.68)

with $\gamma$ given in (3.28) and $\tau_{0}\in\mathbb{R}_{+}$ is an
integration constant. To first order in $h$, (3.29) becomes

$\displaystyle\frac{\partial h}{\partial
t}(\vec{x},t)=-\gamma\,h(\vec{x},t)\,I_{0}(t)+(1-I_{0}(t))\,(f*h)(\vec{x},t)\,.$
(3.69)

In (3.43) we have expanded the convolution $(f*h)$ for small scaling parameter
$\mu$. Here we shall take a slightly different route and assume that for both
$f$ and $h$ their Fourier transforms $\widehat{f}$ and $\widehat{h}$ exist and
that the inversion can be written in the form

$\displaystyle
f(\vec{x})=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}d^{d}k\,e^{i\vec{k}\cdot\vec{x}}\,\widehat{f}(\vec{k})\,,$
$\displaystyle
h(\vec{x},t)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}d^{d}k\,e^{i\vec{k}\cdot\vec{x}}\,\widehat{h}(\vec{k},t)\,.$
(3.70)

The convolution of the two functions is then written as

$\displaystyle(f*h)(\vec{x},t)=\int_{\mathbb{R}}d^{d}k\,e^{i\vec{k}\cdot\vec{x}}\,\widehat{f}(\vec{k})\,\widehat{h}(\vec{k},t)\,,$
(3.71)

such that (3.69) becomes

$\displaystyle\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}d^{d}k\,e^{i\vec{k}\cdot\vec{x}}\,\frac{\partial\widehat{h}}{\partial
t}(\vec{k},t)=-\frac{\gamma
I_{0}(t)}{(2\pi)^{d/2}}\,\int_{\mathbb{R}^{d}}d^{d}k\,e^{i\vec{k}\cdot\vec{x}}\,\widehat{h}(\vec{k},t)+(1-I_{0}(t))\int_{\mathbb{R}}d^{d}k\,e^{i\vec{k}\cdot\vec{x}}\,\widehat{f}(\vec{k})\,\widehat{h}(\vec{k},t)\,.$

We thus obtain the following differential equation for the Fourier
transformation of $h$

$\displaystyle\frac{\partial\widehat{h}}{\partial
t}(\vec{k},t)=-\gamma\,I_{0}(t)\,\widehat{h}(\vec{k},t)+(2\pi)^{d/2}\,(1-I_{0}(t))\,\widehat{f}(\vec{k})\,\widehat{h}(\vec{k},t)\,,$
(3.72)

which can be integrated to yield

$\displaystyle\widehat{h}(\vec{k},t)=\widehat{h}(\vec{k},t_{0})\,\text{exp}\left[\int_{t_{0}}^{t}dt^{\prime}\left((2\pi)^{d/2}\,(1-I_{0}(t^{\prime}))\,\widehat{f}(\vec{k})-\gamma\,I_{0}(t^{\prime})\right)\right]\,.$
(3.73)

To leading order we therefore find for $h(\vec{x},t)$

$\displaystyle
h(\vec{x},t)=\frac{e^{-\gamma\int_{t_{0}}^{t}dt^{\prime}I_{0}(t^{\prime})}}{(2\pi)^{d/2}}\,\int_{\mathbb{R}^{d}}d^{d}k\,\widehat{h}(\vec{k},t_{0})\,\text{exp}\left[(2\pi)^{d/2}\int_{t_{0}}^{t}dt^{\prime}(1-I_{0}(t^{\prime}))\,\widehat{f}(\vec{k})\right]\,e^{i\vec{k}\cdot\vec{x}}\,.$
(3.74)

To continue further, we next consider averaging the solution (3.67) over an
interval $[-L,L]^{d}$ (with $L\in\mathbb{R}_{+}$), _i.e._ we define

$\displaystyle\overline{I}_{L}(\vec{x},t)=\frac{1}{(2L)^{d}}\left(\prod_{i=1}^{d}\int_{-L+x_{i}}^{L+x_{i}}dx_{i}^{\prime}\right)I(\vec{x}\,^{\prime},t)=I_{0}(t)+\overline{h}_{L}(\vec{x},t)\,.$
(3.75)

Assuming (absolute) convergence of the integral in (3.74), such that the order
of integration of $\vec{k}$ and $\vec{x}\,^{\prime}$ in (3.75) can be
exchanged, we can write explicitly (with $\vec{k}=(k_{1},\ldots,k_{d})$)

$\displaystyle\overline{h}_{L}(\vec{x},t)=\frac{e^{-\gamma\int_{t_{0}}^{t}dt^{\prime}I_{0}(t^{\prime})}}{(2\pi)^{d/2}}\,\int_{\mathbb{R}^{d}}d^{d}k\,\widehat{h}(\vec{k},t_{0})\,\left(\prod_{i=1}^{d}\frac{\sin(k_{i}L)}{k_{i}L}\right)\,\text{exp}\left[(2\pi)^{d/2}\int_{t_{0}}^{t}dt^{\prime}(1-I_{0}(t^{\prime}))\,\widehat{f}(\vec{k})\right]\,e^{i\vec{k}\cdot\vec{x}}\,.$
(3.76)

The main difference of this expression compared to (3.74) is that individual
Fourier modes (characterised by $\vec{k}$) are weighted by the $L$-dependent
factor $\prod_{i=1}^{d}\frac{\sin(k_{i}L)}{k_{i}L}$. This can lead to
resonance effects: indeed, if for example the initial conditions are
characterised by a dominant Fourier mode, specific choices of $L$ can lead to
a strong suppression of the fluctuation $\overline{h}_{L}$, such that the
averaged number of infected individuals is very well approximated by the
(homogeneous) $I_{0}(t)$, which is the solution of the limiting equation
(3.40). The existence of a dominant Fourier mode in the initial conditions
$h(\vec{x},t_{0})$ indicates the presence of a certain length scale in the
distribution of initial infectious individuals, which can for example be
caused by the characteristic length distance between population centers and
agglomerations. Such information is very useful to extract critical length
scales for effective theories.

To further illustrate this idea, we shall discuss a concrete examples for
$d=1$, with $f$ a Gaussian function of the form

$\displaystyle f(x)=f_{0}\,e^{-\beta x^{2}}\,,$ with $\displaystyle
f_{0},\beta\in\mathbb{R}_{+}\,,$ (3.77)

such that we have

$\displaystyle\gamma=f_{0}\,\sqrt{\tfrac{\pi}{\beta}}\,,$ and
$\displaystyle\widehat{f}(k)=\frac{f_{0}}{\sqrt{2\beta}}\,e^{-\frac{k^{2}}{4\beta}}\,.$
(3.78)

For the initial conditions at $t=0$ we choose $I(x,t=0)=A\,(\cos(\alpha x)+1)$
(with $A,\alpha\in\mathbb{R}_{+}$). Decomposing these initial conditions
according to (3.67) we have

$\displaystyle\begin{array}[]{l}I_{0}(0)=A\,,\\\ h(x,0)=A\,\cos(\alpha
x)\,,\end{array}$ with
$\displaystyle\widehat{h}(k,0)=\frac{A}{2}\,\sqrt{2\pi}\,\left(\delta(k-\alpha)+\delta(k+\alpha)\right)\,,$
(3.81)

which leads to the background solution (3.68)
$I_{0}(t)=\frac{A}{A+(1-A)\,e^{-\gamma t}}$ and the fluctuation (3.74)

$\displaystyle h(x,t)$
$\displaystyle=\frac{A\,\text{exp}\left(\gamma\,t\,e^{-\frac{\alpha^{2}}{4\beta}}\right)}{1+A(e^{t\gamma}-1)}\,\cos(\alpha
x)\,\left(1+A(e^{t\gamma}-1)\right)^{-\text{exp}\left(-\frac{\alpha^{2}}{4\beta}\right)}\,.$
(3.82)

Given this explicit form of the fluctuation, we can verify whether it is
indeed a small fluctuation compared to the homogeneous background solution
(3.68). The function $h(x,t)$ satisfies

$\displaystyle h(x,t=0)=A\,,$ and $\displaystyle\lim_{t\to\infty}h(x,t)=0\,,$
(3.83)

and has an extremum at

$\displaystyle
t_{\text{ext}}=\frac{8\beta\text{arctanh}(1-2A)-\alpha^{2}}{4\beta\gamma}\,,$
(3.84)

for which $I_{0}(t_{\text{ext}})=\frac{1}{1+e^{\alpha^{2}/(4\beta)}}$ which is
independent of $A$, while $\lim_{A\to 0}h(x,t_{\text{ext}})=0$. Thus the
fluctuation can be made arbitrarily small compared to the background solution
if the initial fluctuation is sufficiently small.

The averaged fluctuation (3.76) then takes the form

$\displaystyle\overline{h}_{L}(x,t)=\cos(\alpha
x)\,\overline{\mathfrak{h}}_{L}(t)\,,$ with
$\displaystyle\overline{\mathfrak{h}}_{L}(t)=\frac{A\,\text{exp}\left(\gamma\,t\,e^{-\frac{\alpha^{2}}{4\beta}}\right)}{1+A(e^{t\gamma}-1)}\,\frac{\sin(L\alpha)}{L\alpha}\,\left(1+A(e^{t\gamma}-1)\right)^{-\text{exp}\left(-\frac{\alpha^{2}}{4\beta}\right)}\,.$
(3.85)

Graphical plots of $\overline{\mathfrak{h}}_{L}$ for different values of $L$,
as well as a comparison to the numerical solution of the full non-linear
equation (3.29) are shown in Figure 19. As is evident from (3.85), the
fluctuation $\overline{\mathfrak{h}}_{L}$ is suppressed by an inverse power of
$L$. Moreover, if $L\alpha$ is an integer multiple of $\pi$, the fluctuation
is vanishing altogether and the number of infected individuals is very well
approximated by $I_{0}$, which is a solution of the limiting equation (3.40).

Figure 19: Left panel: plots of $\overline{\mathfrak{h}}_{L}$ in (3.85) for
different values of $L$. Right panel: comparison between
$\overline{\mathfrak{h}}_{L}$ in (3.85) (solid black line) and a numerical
solution of (3.29) (red) for $L=4$. Both plots use $A=1/200000$,
$\alpha=\pi/8$, $f_{0}=1/10$ and $\beta=1/20$.

This fact can also be visualised in a different fashion: averaged solutions of
(3.29) can be roughly approximated by a logistic function

$\displaystyle\overline{I}_{L}(t)=\frac{1}{2L}\int_{-L}^{L}I(x,t)\simeq\frac{1}{1+e^{-\kappa(t-t_{0})}}\,.$
(3.86)

Figure 20: Numerical fits of $\kappa$ (left) and $t_{0}$ (right) in (3.86).
Both plots use $A=1/200000$, $\alpha=\pi/8$, $f_{0}=1/10$ and $\beta=1/20$.
The dashed horizontal line in the left plot represents the value of $\gamma$
in (3.78). The enveloping curves in black are compatible with a $1/L$
behaviour.

Numerical fits for the effective values of $\kappa$ and $t_{0}$ as functions
of $L$ are plotted in Figure 20. The effective description oscillates around
the solution of the limiting equation (3.40), with an amplitude that is
roughly decaying as $1/L$.

##  4 Conclusions and Outlook

In this paper we have studied effective models and renormalisation group
methods for simple epidemiological theories, following the example of
diffusion processes, as is schematically shown in the following diagram:

diffusionepidemiological modelstochastic model random walks
probability (2.7) stochastic lattice model
probability (3.2) approximationcoarse graining deterministic effective model
deformed heat eq. (2.35)
deformed probability density (2.26) ’simple epidemic model’ eq.(3.38)
continuumlimit limiting model heat equation (2.28)
Green’s function (2.20) SI-model (3.40)

The starting point is a stochastic approach, which models the microscopic
dynamics (_i.e._ the infection among individuals) in a probabilistic manner.
Such models can be approximated by deterministic models, which also include a
step of coarse graining, _i.e._ discrete time- and space-variables are
replaced by constant ones, by averaging over suitable regions of the
coordinate space. At this point we assume that the variables do not vary too
much over the averaged region, which corresponds to a status of _intermediate
asymptotic_ in the language of [19, 18] (see Appendix A.2). Rescaling the
spatial variable in this model leads to a family of effective models, which
are distinguished by the size of the population, over which an averaging has
taken place. In this way, the initial microscopic details of the
epidemiological process become less and less important: indeed, while the
latter in the case of the ’simple epidemic model’ [17, 30] are encoded in an
$L^{1}$-function $f$, in the continuum limit only its $L^{1}$-norm remains,
which is interpreted as the infection rate of the resulting SI-model.
Mathematically, this is due to the fact that the series of functions (called
$f_{\mu}$ in (3.33)) that govern the infection in an arbitrary member of the
family of effective models, form an approximation to the identity (see
Appendix A.1). In Section 3, we have studied the transition of the effective
models towards the continuum limit, both at the level of the defining
differential equation and its solutions. Since the transition from one
effective model to another takes the form of a self-similarity transformation,
we interpret it as a renormalisation group transformation, in the sense of
[28] .

While the type of models we discuss in this work are simplistic from an
epidemiological perspective, they still allow to draw some conceptual
conclusions: in order to describe the epidemiological evolution on a small
population (_i.e._ for $\mu\gg 0$), the detailed structure of the function
$f_{\mu}$ in (3.33) is crucial. In other words, to describe the spread of the
disease in a small group of people, the (microscopic) details of how the
disease is passed on are important to drive the dynamics. For larger groups,
these details become less important, while in the continuum limit only a
single number (the infection rate) remains relevant. This explains, why simple
epidemiological descriptions, such as the eRG [31, 32, 33, 34, 35, 36, 37, 38,
39, 15, 40, 41] are capable of successfully describing large scale
developments (_e.g._ the time evolution at the level of countries). We have
furthermore seen indications in Section 3.4.2 that the crucial length scale
above which such effective models can be used, may also depend on the initial
conditions of the problem and require a more careful analysis of the
solutions.

In the the future, we plan to extend our results to include more realistic
features into the models, such as the possibility of recovery, non-
instantaneous infection among individuals, the mobility of the population or
the possibility of the spread of competing variants of a single disease.

## Acknowledgements

We thank E. Battista, G. Cacciapaglia, C. Cot, D. Iacobacci and S.Vatani for
enlightening discussions and exchanges. This study is supported by Ministero
dell’Università e della Ricerca (Italy), Piano Nazionale di Ripresa e
Resilienza, and EU within the Extended Partnership initiative on Emerging
Infectious Diseases project number PE00000007 (One Health Basic and
Translational Actions Addressing Unmet Needs on Emerging Infectious Diseases).
The work of F.S. is partially supported by the Carlsberg Foundation, semper
ardens grant CF22-0922.

##  Appendix A Notation and Conventions

###  A.1 Approximation of the Identity

Def: (see _e.g._ [46, 47, 48, 49]) An approximation of the identity (also
called _approximate identity_) is a family of functions
$\\{g_{\mu}\\}_{\mu>0}$ with $g_{\mu}\in L^{1}(\mathbb{R}^{d})$ such that

  1. 1.

$\\{g_{\mu}\\}$ is bounded in the $L^{1}(\mathbb{R}^{d})$ norm, _i.e._
$\exists\,c>0$ such that $||g_{\mu}||\leq c$ $\forall\mu>0$

  2. 2.

$\int_{\mathbb{R}^{d}}g_{\mu}(x)\,dx=1$ $\forall\mu>0$

  3. 3.

For any $\epsilon>0$: $\lim_{\mu\to
0}\int_{|\vec{x}|\geq\epsilon}|g_{\mu}(x)|dx=0$.

Let $\\{g_{\mu}\\}_{\mu>0}$ be an approximation to the identity, we then have

  1. 1.

If $f\in L^{p}(\mathbb{R}^{d})$ (with $1\leq p<\infty$), then $f*g_{\mu}\in
L^{p}(\mathbb{R}^{d})$ and $f*g_{\mu}\longrightarrow f$ in the
$L^{p}(\mathbb{R}^{d})$ norm, as $\mu\to 0$.

  2. 2.

If $f$ is uniformly continuous and bounded on $\mathbb{R}^{d}$, then
$f*g_{\mu}$ is also uniformly continuous and bounded on $\mathbb{R}^{d}$.
Furthermore, $f*g_{\mu}\longrightarrow f$ uniformly, _i.e._ in the
$L^{\infty}(\mathbb{R}^{d})$ norm, as $\mu\to 0$.

  3. 3.

If $f\in L^{\infty}(\mathbb{R}^{d})$ is continuous on an open subset
$U\subset\mathbb{R}^{d}$, then $f*g_{\mu}\in L^{\infty}(\mathbb{R}^{d})$, is
uniformly continuous on $\mathbb{R}^{d}$ and $f*g_{\mu}\longrightarrow f$
uniformly on compact subsets $K\subset U$, as $\mu\to 0$.

###  A.2 Review of Intermediate Asymptotics and Renormalisation Group

Our main reference for this appendix is [19], building on the earlier work
[18]: consider a dimensionless quantity $Z$ that depends on $n+1$
dimensionless variables $\\{\mu,z_{1},\ldots,z_{n}\\}$

$\displaystyle Z=f(\mu,z_{1},\ldots,z_{n})\,.$ (A.1)

Following [18], the system is said to be in an _intermediate asymptotic_
state, if its dynamics is independent on the exact initial and/or boundary
conditions, but has not reached a final (_i.e._ static) state. If $\mu\to 0$
corresponds to an intermediate asymptotic limit, there are three different
possibilities [18] (see also [19])

  * •

_self-similarity of the first kind:_ the limit
$\lim_{\mu}f(\mu,z_{1},\ldots,z_{n})$ exists

  * •

_self-similarity of the second kind:_ the limit
$\lim_{\mu}f(\mu,z_{1},\ldots,z_{n})$ is not well defined, but instead there
exist $(\alpha,\alpha_{1},\ldots,\alpha_{n})\in\mathbb{R}^{n+1}$ such that the
limit

$\displaystyle\lim_{\mu\to 0}\frac{Z}{\mu^{\alpha}}=\lim_{\mu\to
0}\frac{1}{\mu^{\alpha}}\,f\left(\mu,\frac{z_{1}}{\mu^{\alpha_{1}}}\,,\ldots\,,\frac{z_{n}}{\mu^{\alpha_{n}}}\right)\,,$
(A.2)

exists. The parameters
$(\alpha,\alpha_{1},\ldots,\alpha_{n})\in\mathbb{R}^{n+1}$ cannot be obtained
by dimensional analysis alone, but they following in principle from the
differential equation that describes the dynamics of $Z$. As remarked in [19],
from the perspective of the RG, the parameters
$(\alpha,\alpha_{1},\ldots,\alpha_{n})\in\mathbb{R}^{n+1}$ are the anomalous
dimensions.

  * •

None of the above, in which case a different rescaling than (A.2) might be
required.

##  Appendix B Mathematical Expansion

In this appendix we calculate $p_{a}$ in (2.26) up to order
$\mathcal{O}(a^{2})$. To this end, we begin by writing

$\displaystyle p_{a}(\vec{x},t)$
$\displaystyle=\frac{1}{(2\pi)^{d}}\prod_{i=1}^{d}\int_{\left[-\frac{\pi}{a},\frac{\pi}{a}\right]}dk_{i}\,e^{ik_{i}x_{i}}\left[1-\frac{a^{2}}{2d}\sum_{j=1}^{d}k_{j}^{2}+\frac{a^{4}}{24d}\sum_{j=1}^{d}k_{j}^{4}+\mathcal{O}(a^{6})\right]^{\frac{2d\kappa
t}{a^{2}}}$
$\displaystyle=\frac{1}{(2\pi)^{d}}\prod_{i=1}^{d}\int_{\left[-\frac{\pi}{a},\frac{\pi}{a}\right]}dk_{i}\,e^{ik_{i}x_{i}}\,e^{-\kappa
t\sum_{j=1}^{d}k_{j}^{2}}\left[1-\frac{\kappa
ta^{2}}{12d}\left(3\left(\sum_{j=1}^{d}k_{j}^{d}\right)^{2}-d\sum_{j=1}^{d}k_{j}^{4}\right)\right]+\mathcal{O}(a^{4})\,.$
(B.1)

The first term in the square bracket yields the limiting contribution (2.20)
up to exponentially suppressed terms (stemming from the dependence of the
integration boundaries on $a$), which we are not interested in. Similarly, for
the second term in the square bracket in (B.1), the dependence of the integral
boundaries on $a$ only produce exponentially suppressed contributions, which
we neglect. To order $\mathcal{O}(a^{2})$ we can therefore write

$\displaystyle p_{a}(\vec{x},t)=p(\vec{x},t)-\frac{a^{2}\kappa
t}{12d(2\pi)^{2}}\int_{\mathbb{R}^{d}}d^{d}k\,e^{-\sum_{j=1}^{d}(\kappa
tk_{j}^{2}-ik_{j}x_{j})}\,\left[(3-d)\sum_{j=1}^{d}k_{j}^{4}+6\sum_{1\leq
i<j\leq d}k_{i}^{2}k_{j}^{2}\right]+\mathcal{O}(a^{4})\,.$ (B.2)

The remaining integrals are of Gauss type, for which we can use the results

$\displaystyle\int_{\mathbb{R}}dk_{j}\,e^{ik_{j}x_{j}-\kappa tk_{j}^{2}}$
$\displaystyle=\sqrt{\frac{\pi}{\kappa t}}\,e^{-\frac{x_{j}^{2}}{4\kappa
t}}\,,$
$\displaystyle\int_{\mathbb{R}}dk_{j}\,k_{j}^{2}\,e^{ik_{j}x_{j}-\kappa
tk_{j}^{2}}$ $\displaystyle=\frac{(2\kappa t-x_{j}^{2})\sqrt{\pi}}{4(\kappa
t)^{5/2}}\,e^{-\frac{x_{j}^{2}}{4\kappa t}}\,,$
$\displaystyle\int_{\mathbb{R}}dk_{j}\,k_{j}^{4}\,e^{ik_{j}x_{j}-\kappa
tk_{j}^{2}}$ $\displaystyle=\frac{(12\kappa^{2}t^{2}-12\kappa
tx_{j}^{2}+x_{j}^{4}-x_{j}^{2})\sqrt{\pi}}{16(\kappa
t)^{9/2}}\,e^{-\frac{x_{j}^{2}}{4\kappa t}}\,,$ (B.3)

such that we find

$\displaystyle p_{a}(\vec{x},t)$
$\displaystyle=p(\vec{x},t)-\frac{a^{2}\pi^{d/2}e^{-\frac{|\vec{x}|^{2}}{4\kappa
t}}}{192d(2\pi)^{d}(\kappa
t)^{d/2+3}}\,\bigg{[}(3-d)\sum_{j=1}^{d}\left(x_{j}^{4}-12\kappa
tx_{j}^{2}+12\kappa^{2}t^{2}\right)$ $\displaystyle\hskip
128.0374pt+6\sum_{1\leq i<j\leq d}\left(x_{i}^{2}x_{j}^{2}-2\kappa
t(x_{i}^{2}+x_{j}^{2})+4\kappa^{2}t^{2}\right)\bigg{]}+\mathcal{O}(a^{4})\,.$
(B.4)

Using the form of (2.20), this result can be simplified to give

$\displaystyle p_{a}(\vec{x},t)$
$\displaystyle=\frac{e^{-\frac{|\vec{x}|^{2}}{4\kappa t}}}{(4\pi\kappa
t)^{d/2}}\left[1-\frac{a^{2}}{192d(\kappa
t)^{3}}\left(3|\vec{x}|^{4}-d\sum_{j=1}^{d}x_{j}^{4}-24\kappa
t|\vec{x}|^{2}+24d\kappa^{2}t^{2}\right)\right]+\mathcal{O}(a^{4})\,.$ (B.5)

## References

  * [1] N. Bailey, The Mathematical Theory of Infectious Diseases, 2nd ed. Hafner, New York, 1975.
  * [2] N. Becker, “The use of epidemic models,” Biometrics, vol. 35, pp. 295–305, 1978.
  * [3] K. Dietz and D. Schenzle, “Mathematical models for infectious disease statistics, in: A. Atkinson (Ed.), A Celebration of Statistics,” Springer, pp. 167–204, 1985.
  * [4] J. Frauenthal, Mathematical Modelling in Epidemiology.  Springer-Verlag Universitext Berlin, 1980.
  * [5] e. C. Castillo-Chavez, “Mathematical and Statistical Approaches to AIDS Epidemiology, Lecture Notes in Biomath,” Springer-Verlag, Berlin, vol. 83, 1989.
  * [6] V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomath. 97.  Springer-Verlag, Berlin, 1993.
  * [7] D. Daley and J. Gani, Epidemic Modelling: An Introduction.  Cambridge University Press, Cambridge, UK, 1999.
  * [8] O. Diekmann and J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases.  Wiley, New York, 2000.
  * [9] H. Lauwerier, Mathematical Models of Epidemics.  Mathematisch Centrum, Amsterdam, 1981.
  * [10] K. Dietz, “Epidemics and rumours: A survey,” J. Roy. Statist. Soc. Ser. A, vol. 130, pp. 505–528, 1967.
  * [11] K. Dietz, “Density dependence in parasite transmission dynamics,” Parasit. Today, vol. 4, pp. 91–97, 1988.
  * [12] R. Anderson and E. R.M. May, Population Biology of Infectious Diseases.  Springer Verlag, Berlin, Heidelberg, New York, 1982.
  * [13] E. R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control.  Oxford University Press, UK, 1991.
  * [14] H. Hethcote, “A thousand and one epidemic models, in Frontiers in Theoretical Biology, S.A. Levin, ed., Lecture Notes in Biomath.,” Springer-Verlag, Berlin, vol. 100, pp. 504–515, 1994.
  * [15] G. Cacciapaglia, C. Cot, M. Della Morte, S. Hohenegger, F. Sannino, and S. Vatani, “The field theoretical ABC of epidemic dynamics,” 1 2021.
  * [16] D. Kendall, “Mathematical models of the spread of infection,” Mathematics and Computer Science in Biology and Medicine, Medical Research Council, pp. 213–225, 1965.
  * [17] D. Mollison, “Spatial contact models for ecological and epidemic spread,” Journal of the Royal Statistical Society. Series B (Methodological), vol. 39, no. 3, pp. 283–326, 1977.
  * [18] G. Barenblatt, “Self-similarity: Similarity and intermediate asymptotic form,” Radiophysics and Quantum Electronics, vol. 19, pp. 643–664, 1976\.
  * [19] N. Goldenfeld, O. Martin, and Y. Oono, “Intermediate asymptotics and renormalization group theory,” J. Sci. Comput., vol. 4, pp. 355–372, 12 1989.
  * [20] A. Bonanno and M. Reuter, “Renormalization group improved black hole space-times,” Phys. Rev. D, vol. 62, p. 043008, 2000.
  * [21] E. Binetti, M. Del Piano, S. Hohenegger, F. Pezzella, and F. Sannino, “Effective theory of quantum black holes,” Phys. Rev. D, vol. 106, no. 4, p. 046006, 2022.
  * [22] A. D’Alise, G. Fabiano, D. Frattulillo, S. Hohenegger, D. Iacobacci, F. Pezzella, and F. Sannino, “Positivity conditions for generalized Schwarzschild space-times,” Phys. Rev. D, vol. 108, no. 8, p. 084042, 2023\.
  * [23] M. Del Piano, S. Hohenegger, and F. Sannino, “Quantum black hole physics from the event horizon,” Phys. Rev. D, vol. 109, no. 2, p. 024045, 2024.
  * [24] Y. Oono, “In advances in chemical physics; prigogine, i., rice, s. a., eds,” 1985\.
  * [25] N. Goldenfeld, Lectures On Phase Transitions And The Renormalization Group.  CRC Press, 2018.
  * [26] G. C. Paquette, L.-Y. Chen, N. Goldenfeld, and Y. Oono, “Structural stability and renormalization group for propagating fronts,” Phys. Rev. Lett., vol. 72, pp. 76–79, Jan 1994.
  * [27] L.-Y. Chen, N. Goldenfeld, and Y. Oono, “Renormalization group theory and variational calculations for propagating fronts,” Phys. Rev. E, vol. 49, pp. 4502–4511, May 1994.
  * [28] L.-Y. Chen, N. Goldenfeld, and Y. Oono, “Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory,” Physical Review E, vol. 54, p. 376–394, Jul 1996.
  * [29] W. O. Kermack, A. McKendrick, and G. T. Walker, “A contribution to the mathematical theory of epidemics,” Proceedings of the Royal Society A, vol. 115, pp. 700–721, 1927.
  * [30] D. Mollison, “Possible velocities for a simple epidemic,” Advances in Applied Probability, vol. 4, no. 2, pp. 233–257, 1972.
  * [31] M. Della Morte, D. Orlando, and F. Sannino, “Renormalization Group Approach to Pandemics: The COVID-19 Case,” Front. in Phys., vol. 8, p. 144, 2020\.
  * [32] M. Della Morte and F. Sannino, “Renormalisation Group approach to pandemics as a time-dependent SIR model,” Front. in Phys., vol. 8, p. 583, 2021\.
  * [33] G. Cacciapaglia, C. Cot, and F. Sannino, “Second wave covid-19 pandemics in europe: A temporal playbook,” Sci Rep, vol. 10, p. 15514, 2020.
  * [34] G. Cacciapaglia, C. Cot, and F. Sannino, “Mining google and apple mobility data: Temporal anatomy for covid-19 social distancing,” Scientific Reports, vol. 11, p. 4150, 2021.
  * [35] G. Cacciapaglia and F. Sannino, “Evidence for complex fixed points in pandemic data,” Front. Appl. Math. Stat., vol. 7, p. 659580, 2021.
  * [36] G. Cacciapaglia, C. Cot, and F. Sannino, “Multiwave pandemic dynamics explained: How to tame the next wave of infectious diseases,” Scientific Reports, vol. 11, p. 6638, 2021.
  * [37] G. Cacciapaglia and F. Sannino, “Interplay of social distancing and border restrictions for pandemics (COVID-19) via the epidemic Renormalisation Group framework,” Sci Rep, vol. 10, p. 15828, 5 2020.
  * [38] G. Cacciapaglia, C. Cot, A. S. Islind, M. Óskarsdóttir, and F. Sannino, “Impact of us vaccination strategy on covid-19 wave dynamics,” Scientific Reports, vol. 11(1), pp. 1–11, 2021.
  * [39] G. Cacciapaglia, C. Cot, A. de Hoffer, S. Hohenegger, F. Sannino, and S. Vatani, “Epidemiological theory of virus variants,” Physica A, vol. 596, p. 127071, 2022.
  * [40] S. Hohenegger, G. Cacciapaglia, and F. Sannino, “Effective mathematical modelling of health passes during a pandemic,” Scientific Reports, vol. 12, p. 6989, 2022.
  * [41] A. de Hoffer, S. Vatani, C. Cot, G. Cacciapaglia, M. L. Chiusano, A. Cimarelli, F. Conventi, A. Giannini, S. Hohenegger, and F. Sannino, “Variant-driven multi-wave pattern of covid-19 via a machine learning analysis of spike protein mutations,” Sci Rep, vol. 12, p. 9275, 2022.
  * [42] C. Itzykson and J. M. Drouffe, Statistical Field Theory. Vol. 1: From Brownian motion to Renormalization and Lattice Gauge Theory.  Cambridge Monographs on Mathematical Physics, CUP, 1989.
  * [43] P. Grassberger, “On the critical behavior of the general epidemic process and dynamical percolation,” Mathematical Biosciences, vol. 63, no. 2, pp. 157 – 172, 1983.
  * [44] A. Kolmogorov, I. Petrovskii, and N. Piscunov, “A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem,” Byul. Moskovskogo Gos. Univ., vol. 1, no. 6, pp. 1–25, 1937.
  * [45] G. I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics.  Cambridge Texts in Applied Mathematics, Cambridge University Press, 1996\.
  * [46] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces.  Princeton University Press, 1971.
  * [47] W. Rudin, Functional Analysis (2nd ed.).  McGraw-Hill, 1991.
  * [48] G. B. Folland, Real Analysis, Modern Techniques and Applications, 2nd Edition.  John Wiley & Sons, 1999.
  * [49] L. Grafakos, Classical Fourier Analysis, 3rd Edition.  Springer, Graduate Texts in Mathematics 249, 2014.