#  Information Theory Unification
of
Epidemiological and Population Dynamics

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

We reformulate models in epidemiology and population dynamics in terms of
probability distributions. This allows us to construct the Fisher information,
which we interpret as the metric of a one-dimensional differentiable manifold.
For systems that can be effectively described by a single degree of freedom,
we show that their time evolution is fully captured by this metric. In this
way, we discover universal features across seemingly very different models.
This further motivates a reorganisation of the dynamics around zeroes of the
Fisher metric, corresponding to extrema of the probability distribution.
Concretely, we propose a simple form of the metric for which we can
analytically solve the dynamics of the system that well approximates the time
evolution of various established models in epidemiology and population
dynamics, thus providing a unifying framework.

${}^{\,}$11footnotemark: 122footnotemark: 2 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

33footnotemark: 3 INFN sezione di Napoli, via Cintia, 80126 Napoli, Italy

33footnotemark: 3 Quantum Theory Center ($\hbar$QTC) & D-IAS, Southern Denmark
Univ., Campusvej 55, 5230 Odense M, Denmark

###### Contents

  1. 1 Introduction
  2. 2 Review of Information Theory
    1. 2.1 Information Geometry
    2. 2.2 Time Evolution and Flow Equations
  3. 3 Information Theory in Simple Theoretical Models
    1. 3.1 Lotka-Volterra Model
    2. 3.2 SIR(S) Model
    3. 3.3 S$\text{I}^{n}$ Model
    4. 3.4 S$\text{I}^{n}$R(S) Model
  4. 4 General Metric
    1. 4.1 Monotonic Evolution
    2. 4.2 Full Time Evolution
      1. 4.2.1 Flow Between Real Zeroes of the Metric
      2. 4.2.2 Flow Between Imaginary Zeroes of the Metric
  5. 5 Conclusions

##  1 Introduction

Describing the probabilities for a discrete set of events to happen, is an
important question, both in mathematics and physics. In most cases, these
probabilities depend on a number of discrete or continuous parameters, related
to the system under consideration. Capturing these dependences in an efficient
manner is for example the basis for optimisation strategies and therefore the
foundation for many practical applications. The field of _information
geometry_ , as part of _information theory_ , proposes to describe probability
distributions in a geometric fashion. Indeed, in the pioneering works [1, 2,
3, 4], the question how to quantify the difference of two sets of
probabilities as a function of external parameters, has been approached using
geometric tools: concretely, a Riemannian manifold has been introduced, which
is equipped with a metric (called the _Fisher information metric_) and a
geodesic distance (called the _Fisher-Rao distance_). These works have further
been generalised in [5, 6, 7, 8, 9, 10] by adding additional structure to the
manifold, such as conjugate pairs of connections and higher order tensors,
leading to a more refined geometric realisation and description of probability
distributions.

In this paper, we propose to apply the ideas of information theory to (at
least partially) describe and in fact capture the time evolution of simple
systems that are relevant for population dynamics and epidemiology. The models
under consideration group a given population (of potentially variable size)
into different classes, _e.g._ ’predator’ and ’prey’ (in the case of the
Lotka-Volterra model [11, 12]) or ’susceptible’ and ’infectious’ (in the case
of the epidemiological SI-model [13]). The dynamics of each system is captured
by describing the size of these classes as a function of time
$t\in\mathbb{R}$, which is encoded in a set of coupled, non-linear, first-
order differential equations, supplemented by suitable initial conditions.
Within this framework, we can naturally444In fact, in many examples, we can
study different probability distributions, which exhibit different aspects of
the system under consideration. define a probability distribution
$p=(p_{1},\ldots,p_{N})$ (for $N\in\mathbb{N}$) as a set of normalised (_i.e._
such that $\sum_{i=1}^{N}p_{i}=1$) fractions of these classes within the
population. Following [1, 2, 3], we can construct a 1-dimensional
differentiable manifold that represents the time-dependence of these
probabilities. This manifold is concretely given through the Fisher
information metric $g_{tt}(t)$, which is a non-negative, differentiable
function of time.555Following the original definition in [1], $g_{tt}$ is
often simply called _Fisher information_. Although here it is the metric for
’only’ a 1-dimensional manifold, in the following we shall continue to refer
to it as ’Fisher (information) metric’ (or simply ’metric’ for short) to
highlight its property to provide a notion of ’distance’ between probability
distributions at different times, which is its essential property for our
discussion.

Notice that the Fisher metric is a priori calculated _from_ the time-
dependence of the probability distribution, which itself is determined by
other means. In this paper we are interested in the inverse problem, _i.e._ to
fix the time evolution of the probability distribution from the metric
$g_{tt}$, which serves as input for the dynamics. We first argue that under
certain conditions, the time dependence of the system can in fact be fully
described in terms of the Fisher metric: indeed, in cases where only two of
the probabilities vary strongly with time, while the remaining ones remain
approximately static (such that the full time evolution can be encoded in a
single, bounded function $x:\,\mathbb{R}\to[0,\alpha]$ for $0<\alpha<1$), we
can re-write the equation of motion for the dynamical probabilities completely
in terms of the Fisher metric $g_{tt}$. Such a scenario is trivially realised
in models with $N=2$, which is the case of the Lotka-Volterra or SI-model. We
also provide further, more sophisticated, examples in which such a scenario
can be realised, namely compartmental models in epidemiology (see [14, 15, 16,
17, 18, 19, 20] for reviews), particularly with multiple competing variants of
a pathogen (see _e.g._ [21]).

An important step for the metric to capture the full dynamics of the system is
to re-write $g_{tt}$ as a function of $x$ rather than a(n explicit) function
of $t$. For the examples mentioned above, this requires partially solving the
original equations of motion and in particular to incorporate the initial
conditions of the problem. In the case of the Lotka-Volterra model, the SIR
model and a simple compartmental model with competing variants, we manage to
perform this step analytically and write a closed form expression of the
metric as a function of $x$. For further examples, we provide numerical
solutions that allow to study the re-organisations of the time evolution in
terms of the metric. Comparing the specific form of $g_{tt}$ obtained in this
way, we find strong similarities among a priori very different systems, which
can be summarised as follows:

  1. _(i)_

real zeroes of the metric correspond to extrema of the probability $x(t)$

  2. _(ii)_

the metric can be described piecewise between two consecutive extrema of the
probability ($x_{1,2}$, such that $0\leq x_{1}<x_{2}\leq\alpha$),_i.e._ in
regimes where $x(t)$ is a monotonic function

     * •

in the simplest case, the metric is a convex function and can be very well
approximated in a simple form $g_{tt}=a(x-x_{1})^{b}(x_{2}-x)^{c}$, where
$a,b,c\in\mathbb{R}_{+}$ are suitable fitting coefficients (with the powers
$b,c$ of order $1$)

     * •

the metric can have local minima, in which case it can be well approximated to
include factors with complex zeroes, _e.g._
$g_{tt}=a(x-x_{1})^{b}(x_{2}-x)^{c}((x-x_{3})(x-\overline{x_{3}}))^{d}$ with
$x_{3}\in\mathbb{C}$ (and $x_{1}<\text{Re}(x_{3})<x_{2}$) and
$d\in\mathbb{R}_{+}$

  3. _(iii)_

by gluing together branches of the metric along their (real) zeroes, non-
monotonic behaviour of the probability $x(t)$ can be described, notably
periodic and oscillating solutions as in the case of the Lotka-Volterra model
and the SIR-model respectively.

Motivated by these common properties of different types of systems, we propose
a universal description of the dynamics of the probability distribution, which
uses the metric $g_{tt}$ as external input (supplemented by suitable initial
conditions): the monotonic evolution of $x$ can be described by a metric that
is proportional to (products of) monomials of its zeroes. For the simplest
choice $g_{tt}(x)\sim(x-x_{1})(x_{2}-x)$ as the product of two simple, real
zeroes at $x_{1,2}$, we provide an analytic, closed form solution for the
monotonic time evolution of $x(t)$ from its initial value to one of its
extrema at $x_{1,2}$. By gluing metrics of this form together, we can describe
explicitly more general solutions, notably periodic or oscillating types,
which are capable of approximating the dynamics of more complicated systems.
Due to the universality of our approach, it can be used to model various
different phenomena in a common form thereby unifying them within the
information theory framework.

This paper is organised as follows: in Section 2 we review important aspects
of Information Theory, defining notably the Fisher information metric. In
Section 3 we study simple theoretical models (_i.e._ the Lotka-Volterra model
and the SIR model as well as generalisations thereof that allow for multiple
variants of a disease to spread among the population): we calculate their
Fisher information metric and re-write it in terms of the probability
distribution, thereby uncovering its universal features. In Section 4 we
provide an analytic solution of the time evolution of a monotonic probability
distribution provided that a simple metric is given as input. Based on this,
we can describe more complicated behaviours by gluing together solutions of
this form, thereby providing a simple approximation to the systems studied in
Section 3. Finally, Section 5 contains our conclusions.

##  2 Review of Information Theory

In this Section we introduce and review a number of mathematical tools that
are aimed at quantifying the ’unevenness’ [22] in a probability distribution
and its characterisation in geometric terms. We follow, in particular, the
Riemannian approach advocated in [1, 2, 3, 4] and further generalised in [5,
6].

###  2.1 Information Geometry

Let $\mathbb{V}$ be a discrete set and define a _probability distribution_ as
a map [23, 6]

$\displaystyle p:\,\mathbb{V}\longrightarrow[0,1]\,,$ such that
$\displaystyle\sum_{X\in\mathbb{V}}p(X)=1\,.$ (2.1)

Given two such distributions $p$ and $q$ on the same set $\mathbb{V}$, their
’difference’ can be captured by various _divergences_ [24, 25] (see also [7,
6, 22, 26]), _e.g._ the _Kullback-Leibler divergence_ [27]666In information
theoretical works, the base of the logarithm is usually chosen to be 2. In
this paper we shall use the natural logarithm with base $e$.

$\displaystyle
D(p||q):=\sum_{X\in\mathbb{V}}p(X)\,\log\left(p(X)/q(X)\right)\,,$ (2.2)

which is non-negative ($D(p||q)\geq 0$) and vanishes only if the probability
distributions are identical. However, it is not symmetric (_i.e._ in general
$D(p||q)\neq D(q||p)$) and also does not satisfy the triangle inequality and
therefore does not constitute a distance function, which makes a geometric
interpretation difficult.

To remedy this, we shall follow the initial work of [1, 2, 3, 4, 28] to
provide a Riemann geometric interpretation of (families of) probability
distributions, which was further extended in [5, 6, 7] (see also [8, 9, 10]).
To this end, we first generalise the definition (2.1) by allowing $p$ to
depend on a number of (continuous) parameters. Let
$\xi^{i}=(\xi^{1},\ldots,\xi^{d})\in\Xi$ (which we assume to be an open subset
of $\mathbb{R}^{d}$) with $d\in\mathbb{N}$ (and $i,j=1,\ldots,d$) and consider
the probability distributions

$\displaystyle p:\,\mathbb{V}\times\Xi\longrightarrow[0,1]\,,$ with
$\displaystyle\sum_{X\in\mathbb{V}}p(X,\xi)=1\hskip
5.69046pt\forall\xi\in\Xi\,,$ (2.3)

which we assume to be (infinitely continuously) differentiable with respect to
$\xi$.777We shall also implicitly assume that the derivatives
$\partial_{i}:=\partial_{\xi^{i}}=\frac{\partial}{\partial\xi^{i}}$ $\forall
i=1,\ldots,d$ commute with integration. Following [6], we furthermore call a
family of probability distributions

$\displaystyle\mathcal{S}=\\{p(X,\xi)|X\in\mathbb{V}\text{ and
}\xi\in\Xi\\}\,,$ (2.4)

such that $\xi\to p(X,\xi)$ is injective $\forall X\in\mathbb{V}$, a
_statistical model_. It was shown in [6] that $\mathcal{S}$ can be endowed
with the structure of a differentiable manifold (also called _statistical
manifold_) and equipped with a Riemannian metric, called the _Fisher
information metric_ 888When simply viewed as a $d\times d$ matrix, this
quantity is also called the _Fisher information matrix_ or in the case $d=1$
simply the _Fisher information_. [3, 4, 28, 6, 26]

$\displaystyle g_{ij}(\xi):$
$\displaystyle=\sum_{X\in\mathbb{V}}\left(\partial_{i}\log
p(X,\xi)\right)\left(\partial_{j}\log p(X,\xi)\right)\,p(X,\xi)\hskip
62.59596pt\forall i,j\in\\{1,\ldots,d\\}\,,$
$\displaystyle=-\sum_{X\in\mathbb{V}}\left(\partial_{i}\partial_{j}\,\log
p(X,\xi)\right)\,p(X,\xi)=4\sum_{X\in\mathbb{V}}\left(\partial_{i}\sqrt{p(X,\xi)}\right)\left(\partial_{j}\sqrt{p(X,\xi)}\right)\,.$
(2.5)

For a given $\mathcal{S}$, this metric can be used to define a distance
between two probability distributions [3, 4] (see also [26, 29]): let
$\xi_{c}:\,[0,1]\to\Xi$ be a curve in $\Xi$ (parametrised by $\tau\in[0,1]$).
Such a curve is a geodesic if it satisfies

$\displaystyle\ddot{\xi}_{c}^{k}+{\Gamma^{k}}_{ij}\,\dot{\xi}_{c}^{i}\,\dot{\xi}_{c}^{j}=0\,,$
with
$\displaystyle\begin{array}[]{l}\dot{\xi}^{i}_{c}=\frac{d\xi_{c}^{i}}{d\tau}\,,\\\
{\Gamma^{k}}_{ij}=\frac{1}{2}\,g^{kl}\left(\partial_{i}g_{jl}+\partial_{j}g_{li}-\partial_{l}g_{ij}\right)\,,\end{array}$
(2.8)

where $g^{ij}$ is the inverse of the metric (2.5). We can then define the
_Fisher-Rao distance_ of the probability distributions $p(X,\xi_{1})$ and
$p(X,\xi_{2})$ for $\xi_{1,2}\in\Xi$ as

$\displaystyle
d(\xi_{1},\xi_{2})=\int_{0}^{1}d\tau\,\sqrt{\dot{\xi}^{i}_{c}\,\dot{\xi}^{j}_{c}\,g_{ij}(\xi_{c})}\,,$
(2.9)

where $\xi_{c}$ is the geodesic such that $\xi_{c}(\tau=0)=\xi_{1}$ and
$\xi_{c}(\tau=1)=\xi_{2}$.

###  2.2 Time Evolution and Flow Equations

In this paper, we shall exclusively be interested in the one-dimensional case
$\Xi\subset\mathbb{R}$, where we shall interpret $t\in\Xi$ as a time variable.
In this case, we use the following notation for the one-dimensional metric
(following from the general definition (2.5)).

$\displaystyle d\mathfrak{p}^{2}(t)=g_{tt}(t)\,dt^{2}\,,$ with $\displaystyle
g_{tt}(t)=\sum_{X\in\mathbb{V}}\left(\frac{\partial\log p(X,t)}{\partial
t}\right)^{2}\,p(X,t)\,,$ (2.10)

with $dt$ an infinitesimal time element.999As remarked before, $g_{tt}$ is the
Fisher information, originally introduced in [1]. In the following, to
highlight its property to keep track of changes or differences of $p(X,t)$, we
shall refer to it as ’Fisher (information) metric’ (or simply ’metric’ for
short). We first remark that the metric element $d\mathfrak{p}^{2}(t)$ can in
fact be related to the Kullback-Leibler divergence (2.2) of $p(X,t)$ and
$p(X,t+dt)$ (see [22]))

$\displaystyle
d\mathfrak{p}^{2}(t)=2\,D\left(p(\cdot,t)||p(\cdot,t+dt),t\right)\,.$ (2.11)

Furthermore, the Fisher-Rao distance (2.9) becomes the (absolute value) of the
proper time

$\displaystyle
d(t_{1},t_{2})=\left|\int_{t_{1}}^{t_{2}}\sqrt{g_{tt}(t^{\prime})}\,dt^{\prime}\right|\,.$
(2.12)

Moreover, we shall argue that (2.10) can be used to describe the time
evolution of the probability distribution $p(X,t)$, at least in certain cases.
To this end, we focus on a finite set $\mathbb{V}=\\{X_{1},\ldots,X_{N+1}\\}$
of cardinality $N+1$ and define the probability distribution

$\displaystyle p:\,\mathbb{V}\times\mathbb{R}$
$\displaystyle\longrightarrow[0,1]\,,$ with
$\displaystyle\sum_{i=1}^{N+1}p_{i}(t)=1\,,$ $\displaystyle(X_{i},t)$
$\displaystyle\longmapsto p(X_{i},t)=p_{i}(t)\,.$ (2.13)

We resolve the normalisation condition by defining $(x_{1},\ldots,x_{N})$
(with $x_{i}(t)\in[0,1]$), such that

$\displaystyle p_{i}(t)=x_{i}(t)\hskip 14.22636pt\forall
i\in\\{1,\ldots,N\\}\,,$ and $\displaystyle
p_{N+1}(t)=1-\sum_{i=1}^{N}x_{i}(t)\,,$ (2.14)

The Fisher information metric (2.10) can then be written as (with
$\dot{x}_{i}=\frac{dx_{i}}{dt}$)

$\displaystyle g_{tt}=\sum_{i=1}^{N+1}p_{i}(t)\,\left(\frac{d}{dt}\,\log
p_{i}(t)\right)^{2}=\sum_{i=1}^{N}\frac{\left(1-\sum_{j\neq
i}^{N}x_{j}\right)\,\dot{x}_{i}^{2}}{x_{i}\left(1-\sum_{j=1}^{N}x_{j}\right)}+2\sum_{1\leq
i<j\leq N}\frac{\dot{x}_{i}\dot{x}_{j}}{1-\sum_{k=1}^{N}x_{k}}\,.$ (2.15)

Rather than calculating the metric for a (given) probability distribution, we
shall in the following try to extract information about $\dot{x}_{i}$ in terms
of the metric $g_{tt}$, taking the latter as an input. In general, it is not
possible to extract the time evolution for all $x_{i}$ in this manner,
however, by making further assumptions on the form of the dynamics, we can
derive certain insights. As an example, consider the case in which one of the
probabilities is very close to zero, _e.g._ without loss of generality
$x_{1}\ll 1$. In this case, the metric (2.15) is dominated by a single term

$\displaystyle x_{1}\ll 1:$ $\displaystyle
g_{tt}=\frac{\dot{x}_{1}^{2}}{x_{1}}+\mathcal{O}(x_{1}^{0})\,,$ such that
$\displaystyle\frac{dx_{1}}{dt}\sim\pm\sqrt{x_{1}\,g_{tt}}\,.$ (2.16)

The sign in the last equation cannot be uniquely fixed due to time inversion
symmetry and is determined whether the probability $x_{1}$ is growing or
diminishing. Notice, that (2.16) is an effective description of the time
evolution of $x_{1}$, where the entire dynamics of the rest of the system
(_i.e._ $x_{2,\ldots,N}$) is encoded in the metric $g_{tt}$.

In the remainder of this paper, we shall study in detail a different scenario
than (2.16): consider the case where the time derivative of one probability is
much larger than the others101010From the perspective of the $p_{i}$, the
condition (2.17) implies that $\dot{x}_{1}$ grows (diminishes) at the expense
of $p_{N+1}$, while the remaining probabilities remain essentially constant.
This scenario therefore corresponds to a competition between two
probabilities, which can both take values in the interval $[0,\alpha]$., thus,
without loss of generality

$\displaystyle|\dot{x}_{1}|\gg|\dot{x}_{i}|\sim 0\hskip 14.22636pt\forall
i\in\\{2,\ldots,N\\}\,,$ such that $\displaystyle\sum_{i=2}^{N}x_{i}(t)\sim
1-\alpha=\text{const}.\,.$ (2.17)

In this case, the metric can be approximated by

$\displaystyle
g_{tt}\sim\frac{\left(1-\sum_{j=2}^{N}x_{j}\right)}{x_{1}\left(1-\sum_{j=1}^{N}x_{j}\right)}\,\dot{x}_{1}^{2}+\mathcal{O}(\dot{x}_{1})=\frac{\alpha\,\dot{x}_{1}^{2}}{x_{1}(\alpha-
x_{1})}+\mathcal{O}(\dot{x}_{1})\,.$ (2.18)

This equation can be re-written in the following fashion

$\displaystyle\frac{dx_{1}}{dt}\sim\pm\sqrt{g_{tt}\,x_{1}\,\left(1-\frac{x_{1}}{\alpha}\right)}\,,$
with $\displaystyle x_{1}\in[0,\alpha]\,,$ (2.19)

where the sign cannot be fixed and is determined by whether the probability is
growing or diminishing. If $g_{tt}$ is considered as an (external) input,
equation (2.19) can be understood as an effective description of the dynamics
for $x_{1}$. In particular, in the case that $g_{tt}$ is given as a function
of $x_{1}$ (thereby encoding the influence of the remainder of the system on
$x_{1}$), (2.19) uniquely fixes the time evolution of $x_{1}$. In this case,
(2.19) plays a similar role to the flow equations in epidemiology discussed in
[21]. Notice in this respect that $x_{1}=0$ and $x_{1}=\alpha$ denote the
extremal values $x_{1}$ can take.

We furthermore remark, that in the particular case (2.17) the dynamics of
$x_{1}$ can in fact be solved in terms of the proper time, which is the
Fisher-Rao distance (2.9) in its form (2.12). To this end, we rewrite the flow
equation (2.19)

$\displaystyle\frac{dx_{1}}{\sqrt{x_{1}\left(1-\frac{x_{1}}{\alpha}\right)}}=\pm\sqrt{g_{tt}}\,dt\,.$
(2.20)

Integrating this equation, with initial condition $x_{1}(t_{0})=x_{0}$ (with
$t>t_{0}$), we obtain

$\displaystyle d(t,t_{0})=\pm
2\sqrt{\alpha}\,\left[\arcsin\left(\sqrt{{x_{1}}/\alpha}\right)-\arcsin\left(\sqrt{x_{0}/\alpha}\right)\right]$
(2.21)

which can be inverted to give

$\displaystyle
x_{1}(t)=\alpha\sin^{2}\left[\arcsin\left(\sqrt{\tfrac{x_{0}}{\alpha}}\right)\pm\tfrac{d(t,t_{0})}{2\sqrt{\alpha}}\right]\,,$
(2.22)

where the sign determines whether $x_{1}$ is growing or decreasing as a
function of $d$.

In the following Sections 3 we shall consider simple models in epidemiology
and population dynamics and show that their dynamics can indeed be re-written
in the form (2.19). In particular, we shall provide an explicit form of the
metric $g_{tt}$ in terms of $x_{1}$ (either analytically or, in more
complicated cases, numerically). Based on these examples, we shall propose in
Section 4 a unified description of the various examples discussed in Sections
3, which in its simplest form allows a completely analytic solution.

##  3 Information Theory in Simple Theoretical Models

In this Section we first discuss as simple examples the Lotka-Volterra model
[11, 12] in Subsection 3.1 and the SIR-model [13] in Subsection 3.2. The
latter shall be generalised in Subsections 3.3 and 3.4 to include multiple
variants of a pathogen.

###  3.1 Lotka-Volterra Model

A simple model for population dynamics are the Lotka-Volterra equations. They
describe the time evolution of two species, usually referred to as ’prey’ and
’predator’, whose population growth depends on each other: denoting the
respective size of their population (as a function of time
$t\in\mathbb{R}_{+}$) by $\xi_{1,2}:\,\mathbb{R}_{+}\to\mathbb{R}_{+}$, the
dynamics is described by

$\displaystyle\frac{d\xi_{1}}{dt}=a_{1}\,\xi_{1}-b_{1}\,\xi_{1}\,\xi_{2}\,,$
$\displaystyle\frac{d\xi_{2}}{dt}=-a_{2}\,\xi_{2}+b_{2}\,\xi_{1}\,\xi_{2}\,,$
with $\displaystyle a_{1,2}\,,b_{1,2}\in\mathbb{R}_{+}\,.$ (3.1)

In short, the population of prey animals ($\xi_{1}$) grows exponentially on
its own, but is limited by the presence of the predators ($\xi_{2}$). The
latter in turn decline exponentially on their own and only grow depending on
the available prey. An exemplary numerical solution for $(\xi_{1},\xi_{2})$ is
shown in Figure 1: the solutions never reach one of the equilibrium points at
$(\xi_{1},\xi_{2})=(0,0)$ or
$(\xi_{1},\xi_{2})=\left(\tfrac{a_{2}}{b_{2}},\tfrac{a_{1}}{b_{1}}\right)$ but
are in fact periodic, as can be seen from the trajectory of the system in the
$(\xi_{1},\xi_{2})$-plane in the right panel of Figure 1.

Figure 1: Left panel: numerical solution of (3.1) for $a_{1}=0.2$,
$a_{2}=0.1$, $b_{1}=0.1$, $b_{2}=0.2$ and initial conditions $\xi_{1}(0)=0.3$,
$\xi_{2}(0)=1.5$. Right panel: trajectory of the system (red and blue points)
in the $(\xi_{1},\xi_{2})$-plane. The arrows indicate the size and direction
of the derivatives of $\xi_{1,2}$ (following from (3.1)) and the black dots
denote the fixed points of the system at $(\xi_{1},\xi_{2})=(0,0)$ and
$(\xi_{1},\xi_{2})=\left(\tfrac{a_{2}}{b_{2}},\tfrac{a_{1}}{b_{1}}\right)$. As
explained in the text, in both panels the evolution of the system is separated
into regions with monotonically growing (red) or decreasing (blue) $x$ (see
also left panel of Figure 2).

To facilitate the further discussion, we have divided the trajectory in Figure
1 into two parts (coloured red and blue respectively): to explain this, we
first define the probability distribution
$p:\,\\{1,2\\}\times\mathbb{R}\to[0,1]$ with

$\displaystyle p_{1}=\frac{\xi_{1}}{\xi_{1}+\xi_{2}}=:x\,,$ $\displaystyle
p_{2}=\frac{\xi_{2}}{\xi_{1}+\xi_{2}}=1-x\,.$ (3.2)

The probability $p_{1}(t)=x(t)$ (which depends on time $t$) takes values only
in a finite interval and has extrema for

$\displaystyle a_{1}+a_{2}=b_{1}\xi_{2}+b_{2}\xi_{1}\,.$ (3.3)

On the blue portion of the trajectory, $x$ is monotonically decreasing, while
on the red portion of the trajectory, $x$ is monotonically increasing, as can
be seen in the left panel of Figure 2. The respective regimes in the time-
evolution of the system are also indicated in red and blue respectively in the
left panel of Figure 1 (and all subsequent Figures).

Figure 2: Left panel: Numerical plot of the probability $x$ as a function of
time, which is bounded in the intervall $x\in[0.102,0.304]$. Right panel: the
Fisher information metric (3.4). Both plots use the same numerical values as
in Figure 1 and follow the same colouring scheme: the extrema of $x$
correspond to zeroes of the metric.

Using the probabilities (3.2), we can calculate the Fisher information metric
(2.10) (which is plotted as a function of time in the right panel of Figure
2):

$\displaystyle
g_{tt}=(\partial_{t}\log(p_{1}))^{2}\,p_{1}+(\partial_{t}\log(p_{2}))^{2}\,p_{2}=\frac{\xi_{1}\xi_{2}(a_{1}+a_{2}-b_{2}\xi_{1}-b_{1}\xi_{2})^{2}}{(\xi_{1}+\xi_{2})^{2}}\,.$
(3.4)

Following (2.19), extrema of the probability $x$ correspond to zeroes of the
Fisher metric. Instead of writing the latter as a function of time $t$, we
shall now re-write it as a function of $x$. To this end, as a first step, we
can express $\xi_{2}$ as a function of $x$ and $\xi_{1}$ to obtain

$\displaystyle
g_{tt}=\frac{1-x}{x}\,\left(a_{1}x+a_{2}x-\left(b_{1}(1-x)+b_{2}x\right)\xi_{1}\right)^{2}\,.$
(3.5)

However, to further express $\xi_{1}$ in terms of $x$, we need to solve the
dynamics of the system by integrating the quotient of the two equations in
(3.1)

$\displaystyle\left(b_{2}+b_{1}\frac{1-x}{x}\right)\xi_{1}-(a_{1}+a_{2})\,\log\xi_{1}-a_{1}\log\left(\frac{1-x}{x}\right)=L\,,$
(3.6)

where
$L=b_{2}\xi_{1}(0)-a_{2}\log\xi_{1}(0)+b_{1}\xi_{2}(0)-a_{1}\log\xi_{2}(0)$ is
a constant that is determined by the initial conditions. Equation (3.6) can be
solved to express $\xi_{1}$ as a function of $x$

$\displaystyle\xi_{1}=-\frac{(a_{1}+a_{2})x}{b_{1}(1-x)+b_{2}x}\,W\left(-\frac{e^{-\frac{L}{a_{1}+a_{2}}}\left(\frac{1-x}{x}\right)^{-\frac{a_{1}}{a_{1}+a_{2}}}(b_{1}(1-x)+b_{2}x)}{x(a_{1}+a_{2})}\right)\,,$
(3.7)

where $W$ is the Lambert function. Notice that (3.7) are in fact two
solutions, corresponding to the two branches of the Lambert function, as is
schematically shown in the left panel of Figure 3. These two branches are
again related to the sign of $\frac{dx}{dt}$ (as indicated by the colouring of
the plot in Figure 3). The Fisher information metric (3.5) then takes the form

$\displaystyle g_{tt}$
$\displaystyle=(a_{1}+a_{2})^{2}\,x\,(1-x)\left[1+W\left(-\frac{e^{-\frac{L}{a_{1}+a_{2}}}\left(\frac{1-x}{x}\right)^{-\frac{a_{1}}{a_{1}+a_{2}}}(b_{1}(1-x)+b_{2}x)}{x(a_{1}+a_{2})}\right)\right]^{2}\,,$
(3.8)

which (as (3.7)) has two branches, as can be seen from the right panel of
Figure 3. The zeroes of the metric are the solutions of the equation

$\displaystyle
b_{1}\left(\frac{1-x}{x}\right)^{\frac{a_{2}}{a_{1}+a_{2}}}+b_{2}\,\left(\frac{1-x}{x}\right)^{-\frac{a_{1}}{a_{1}+a_{2}}}=(a_{1}+a_{2})\,e^{\frac{L}{a_{1}+a_{2}}-1}\,,$
(3.9)

which cannot be solved analytically for generic values of $a_{1,2}$. We note,
however, that the form of the metric (3.8) can be very well approximated by a
function of the form

$\displaystyle g_{tt}(x)=a\,(x-x_{1})^{b}\,(x_{2}-x)^{c}\,,$
$\displaystyle\forall x\in[x_{1},x_{2}]\,.$ (3.10)

where $x_{1,2}\in[0,1]$ (with $x_{1}<x_{2}$) are two real solutions of (3.9)
(which are independent of time $t$) and $a,b,c$ suitable fitting coefficients
of order 1. Indeed, an approximation of this form is shown in Figure 4, along
with the flow equation

$\displaystyle\frac{dx}{dt}=\pm\sqrt{g_{tt}\,x\,\left(1-x\right)}\,,$ (3.11)

following from (2.19) (with $\alpha=1$).

Figure 3: Left panel: solution $\xi_{1}$ as a function of $x$ in (3.7). Right
panel: metric (3.8) as a function of $x$. In both cases, the red and blue
curve represent the two branches of the Lambert function, while the black dots
represent a numerical solution. The parameters were chosen to be $a_{1}=0.2$,
$a_{2}=0.1$, $b_{1}=0.1$, $b_{2}=0.2$ and initial conditions $\xi_{1}(0)=0.3$,
$\xi_{2}(0)=1.5$ (such that $L\sim 0.249$).

Figure 4: Left panel: Approximation of the metric $g_{tt}$ in (3.8) as a
function of $x$, using the same numerical values as in Figure 3. The black
dashed lines are interpolations using the functions (3.10), with $x_{1}\sim
0.1$, $x_{2}\sim 0.34$ and $(a,b,c)\sim(0.31,1.19,1.13)$ for the blue curve
and $(a,b,c)\sim(0.09,1.05,0.93)$ for the red curve. Right panel: time
derivative of $x$, following the flow equation (2.19) for $x$ computed from
the metric $g_{tt}$.

To make this approximation more tangible, we shall consider a concrete
example, namely111111By defining $\xi_{1}=\frac{a_{2}}{b_{2}}\,\Xi_{1}$,
$\xi_{2}=\frac{a_{2}}{b_{1}}\,\Xi_{2}$ and $t=\tau/a_{2}$, the equations (3.1)
can always be brought into the form
$\displaystyle\frac{d\Xi_{1}}{d\tau}=\alpha\,\Xi_{1}-\Xi_{1}\,\Xi_{2}\,,$ and
$\displaystyle\frac{d\Xi_{2}}{d\tau}=-\Xi_{1}+\Xi_{1}\,\Xi_{2}\,.$ (3.12) with
$\alpha=a_{1}/a_{2}$. (3.13) is therefore the particular choice $\alpha=1$.

$\displaystyle a_{1}=a_{2}=b_{1}=b_{2}=1\,.$ (3.13)

In this case, relation (3.6) allows for real solutions $\xi_{1}$ in terms of
$x$ only for $L\geq 2$. Furthermore, (3.9) has two real solutions in the
interval $[0,1]$121212For $L=2$ we have $x_{1}=x_{2}=1/2$. Furthermore, the
only real solutions of (3.6) arise for $x=0$ or $x=1/2$, in which case, the
system is in one of the two equilibrium points. In the following, we shall
therefore implicitly assume $L>2$.

$\displaystyle x_{1}=\frac{1}{2}\,\left(1-\sqrt{1-e^{2-L}}\right)\,,$
$\displaystyle x_{2}=\frac{1}{2}\,\left(1+\sqrt{1-e^{2-L}}\right)\,,$
$\displaystyle\forall L>2\,.$ (3.14)

Furthermore, the metric (3.8) takes the simpler form

Figure 5: Best fit of the parameters $\kappa$ and $d$ in (3.16) as functions
of $c$. The top panels show the case $\frac{dx}{dt}>0$ and the top panels
$\frac{dx}{dt}>0$, corresponding to two different branches of the Lambert
function in (3.15).

$\displaystyle
g_{tt}=4\,x\,(1-x)\left[1+W\left(-\frac{e^{-L/2}}{2\sqrt{x(1-x)}}\right)\right]^{2}\,,$
(3.15)

which can be approximated by

$\displaystyle g_{tt}\sim
4\kappa\left(x-\frac{1}{2}\,\left(1-\sqrt{1-e^{2-L}}\right)\right)^{d}\,\left(\frac{1}{2}\,\left(1+\sqrt{1-e^{2-L}}\right)-x\right)^{d}\,,$
(3.16)

where $(\kappa,d)$ are fit parameters that depend on $L$: The best fits are
plotted schematically in Figure 5, while explicit plots of the metric (for
$L=3$) are shown in Figure 6.

Figure 6: The two branches of the metric (3.15) for $L=3$ (left: blue solid
line, right: red solid line). The dashed black lines indicate the best
interpolation with $(\kappa,d)=(0.765,0.997)$ (left panel) and
$(\kappa,d)=(6.119,1.403)$. The dashed straight lines show the tangents at the
two zeroes of the metric.

###  3.2 SIR(S) Model

We next consider a simple compartmental model in epidemiology that describes
the spread of a pathogen through a population. Here, we only consider the
spread of a single pathogen, while in the following Subsections, we shall
consider multiple different variants at the same time.

We assume a(n isolated) population of constant size (normalised to 1), which
we group into different compartments, regarding to their state of affliction
with the pathogen: we denote with $S(t)$ the number of susceptible
individuals, with $I(t)$ the number of infectious individuals and with $R(t)$
the number of removed individuals at time $t\in\mathbb{R}$.131313Here we
define as a susceptible an individual, who is currently not infectious (_i.e._
not inflicted by the pathogen), but may become so if in contact with an
infectious individual. In contrast, a removed individual is also not currently
infectious but may also not become infectious upon contact with an infectious
one. We assume these numbers to be (differentiable) functions of time
$S,I,R:\,\mathbb{R}\to[0,1]$, whose evolution is governed by [13]

$\displaystyle\frac{dS}{dt}=-\gamma\,S\,I+\zeta\,R\,,$
$\displaystyle\frac{dI}{dt}=\gamma\,S\,I-\sigma I\,,$
$\displaystyle\frac{dR}{dt}=\sigma\,I-\zeta R\,,$ (3.17)

and we choose initial conditions such that $(S+I+R)(t=0)=1$. Furthermore,
$\gamma$, $\sigma$, $\zeta\in\mathbb{R}_{+}$ are constant parameters, which
denote the rate at which susceptible individuals are infected, the rate at
which infectious individuals become removed and the rate at which removed
individuals become susceptible again141414The model with $\zeta=0$ is usually
referred to as the SIR model [13], while the model with $\zeta\neq 0$, _i.e._
with the possibility of re-infection, is sometimes called the SIRS model (see
_e.g._ [20] for a review)., respectively. A numerical solution of (3.17)
(distinguishing $\zeta\neq 0$ and $\zeta$=0) is shown in Figure 7.

Figure 7: Numerical solution for the SIR-model (3.17) with and without
reinfection: Left-panel: $\gamma=1.7$, $\sigma=0.6$, $\zeta=0.2$ and initial
conditions $I(t=0)=0.00075$ (and $R(t=0)=0$). The times $t_{1,2}^{(2)}$
indicate extrema of $I+R$, while $t_{1,2,3}^{(3)}$ indicate extrema of $I$.
Right panel: $\gamma=1.7$, $\sigma=0.4$, $\zeta=0$ and initial conditions
$I(t=0)=0.00075$ (and $R(t=0)=0$). The time $t_{1}^{(3)}$ denotes the maximum
of $I$.

Given the initial condition (and the fact that (3.17) preserves the size of
the population), such that $(S+I+R)(t)=1$ $\forall t\geq 0$, we can define
different probability distributions that satisfy the normalisation condition
(2.1). While a natural choice is to define (using the definition (2.13))

$\displaystyle p^{(1)}:\,\\{1,2,3\\}\times\mathbb{R}\to[0,1]$ with
$\displaystyle p_{1}^{(1)}(t)=S(t)\,,$ $\displaystyle p_{2}^{(1)}(t)=I(t)\,,$
$\displaystyle p_{3}^{(1)}(t)=R(t)\,,$ (3.18)

in order to make contact with the formalism described in Section 2.2, we shall
be more interested in probability distributions as maps
$\\{1,2\\}\times\mathbb{R}\to[0,1]$. Concretely, we shall consider two
particular choices

$\displaystyle p^{(2)}:\,\\{1,2\\}\times\mathbb{R}\longrightarrow[0,1]$ with
$\displaystyle p_{1}^{(2)}(t)=S(t)\,,$ $\displaystyle
p_{2}^{(2)}(t)=(I+R)(t)\,,$ $\displaystyle
p^{(3)}:\,\\{1,2\\}\times\mathbb{R}\longrightarrow[0,1]$ with $\displaystyle
p_{1}^{(3)}(t)=(S+R)(t)\,,$ $\displaystyle p_{2}^{(3)}(t)=I(t)\,.$ (3.19)

Notice, in the case $\zeta=0$, $p_{2}^{(2)}$ describes the cumulative number
of infected individuals, which has been the object of study in the eRG
approach (see _e.g._ [30, 31, 32, 33, 34, 35, 36, 37, 21, 38, 39, 40]). For
each of the probability distributions, we also introduce the Fisher
information metrics (2.10)

Figure 8: Fisher metric associated with the probability distributions (3.19)
for a SIR model with (left) and without (right) reinfection. The plots use the
same numerical values as in Figure 7 and the vertical lines indicate extrema
of $I+R$ and $I$ respectively.

$\displaystyle g_{tt}^{(1)}(t)$
$\displaystyle=\sum_{i=1}^{3}p_{i}^{(1)}\,(\partial_{t}\log
p_{i}^{(1)})^{2}=\frac{(\gamma SI-\zeta R)^{2}}{S}+I(\gamma
S-\sigma)^{2}+\frac{(\sigma I-\zeta R)^{2}}{R}\,,$ $\displaystyle
g_{tt}^{(2)}(t)$ $\displaystyle=\sum_{i=1}^{2}p_{i}^{(2)}\,(\partial_{t}\log
p_{i}^{(2)})^{2}=\frac{(\gamma SI-\zeta R)^{2}}{(I+R)S}\,,$ $\displaystyle
g_{tt}^{(3)}(t)$ $\displaystyle=\sum_{i=1}^{2}p_{i}^{(3)}\,(\partial_{t}\log
p_{i}^{(3)})^{2}=\frac{I(\gamma S-\sigma)^{2}}{R+S}\,,$ (3.20)

which are plotted as functions of time in Figure 8.151515Notice that
$g_{tt}^{(1)}$ (blue curve in Figure 8) is always large than $g_{tt}^{(2)}$ or
$g_{tt}^{(3)}$ (orange and green curve). This follows on general grounds,
since $p^{(2,3)}$ can be obtained as sums of $p^{(1)}$: indeed, following [6,
41, 42] let $\mathbb{W}$, $\mathbb{V}$ be discrete sets and
$F:\,\mathbb{V}\rightarrow\mathbb{W}$ a map, which allows to associate with
each probability distribution $p(X,\xi)$ on $\mathbb{V}$ a probability
distribution $q(F(X),\xi)$ on $\mathbb{W}$. If we denote by $g_{ij}^{(F)}$ the
induced Fisher metric associated with $q(F(X),\xi)$, then
$g_{ij}(\xi)-g_{ij}^{(F)}(\xi)$ is positive semidefinite. Since $p^{(2)}$ and
$p^{(3)}$ are probability distributions defined on a set $\mathbb{V}$ of
cardinality $2$, the flow equation (2.19) (with $\alpha=1$) can be applied
without any additional approximations (since (2.17) is automatically
satisfied). Indeed, we can write

$\displaystyle\frac{dS}{dt}=-\frac{d(I+R)}{dt}=\pm\sqrt{g_{tt}^{(2)}\,(R+I)S}\,,$
(3.21)
$\displaystyle\frac{d(S+R)}{dt}=-\frac{dI}{dt}=\pm\sqrt{g_{tt}^{(3)}\,(R+S)I}\,,$
(3.22)

where the sign can change during the different phases of the spread of the
disease. In order for these equations to fully encode (part of) the dynamics
of the system, the metrics $g_{tt}^{(2,3)}$ need to be expressed in terms of
the probabilities themselves. This is shown for a numerical solution in Figure
9 for a model with reinfection and Figure 10 for a model without reinfection.
We remark, although the model is not periodic, the schematic form of the
metrics is similar to the shape found in Figure 4 in the Lotka Volterra model:
indeed the dashed black lines give interpolations of all cuves with functions
of the form (3.10). At least for the blue curves (which describe the initial
phase of the epidemic, in which $I$ or $I+R$ are monotonically growing), the
powers $b$ and $c$ are close to $1$.

Figure 9: Fisher metrics $g_{tt}^{(2)}$ as function of $I+R$ (left) and
$g_{tt}^{(3)}$ as function of $I$ (right) in a model with reinfection. Both
plots use the same parameters as in the left panel of Figure 7. In the left
panel, the blue curve shows the metric $g_{tt}^{(2)}$ from $t=0$ to
$t=t_{1}^{(2)}$, where $d(I+R)/dt$ is positive (and thus the minus sign is
applicable in (3.21)), while the orange curve shows the metric between
$t=t_{1}^{(2)}$ and $t=t_{2}^{(2)}$, where $d(I+R)/dt$ is negative (and thus
the plus sign is applicable in (3.21)). The dashed black lines represent
interpolations of the form (3.10) with
$(x_{1},x_{2},a,b,c)\sim(0,0.75,1.81,1.01,1.28)$ and
$(x_{1},x_{2},a,b,c)\sim(0.63,0.75,1.11,1.41,0.74)$ respectively. In the right
panel the blue curve shows the metric $g_{tt}^{(3)}$ from $t=0$ to
$t=t_{1}^{(3)}$, where $dI/dt$ is positive (and thus the minus sign is
applicable in (3.22)), while the orange curve shows the metric between
$t=t_{1}^{(3)}$ and $t=t_{2}^{(3)}$, where $dI/dt$ is negative (and thus the
plus sign is applicable in (3.22)). The dashed black lines represent
interpolations of the form (3.10) with
$(x_{1},x_{2},a,b,c)\sim(0,0.31,4.88,1.01,1.16)$ and
$(x_{1},x_{2},a,b,c)\sim(0.14,0.31,2.99,1.56,0.79)$.

Figure 10: Fisher metrics $g_{tt}^{(2)}$ as function of $I+R$ (left) and
$g_{tt}^{(3)}$ as function of $I$ (right) in a model without reinfection. Both
plots use the same parameters as in the right panel of Figure 7. In the left
panel, the dashed black line represents an interpolation of the form (3.10)
with $(x_{1},x_{2},a,b,c)\sim(0,0.98,1.99,1.06,1.45)$. In the right panel the
blue curve shows the metric $g_{tt}^{(3)}$ from $t=0$ to $t=t_{1}^{(3)}$,
where $dI/dt$ is positive (and thus the minus sign is applicable in (3.22)),
while the orange curve shows the metric between $t=t_{1}^{(3)}$ and
$t=t_{2}^{(3)}$, where $dI/dt$ is negative (and thus the plus sign is
applicable in (3.22)). The dashed black lines represent interpolations of the
form (3.10) with $(x_{1},x_{2},a,b,c)\sim(0,0.42,4.68,1.02,1.11)$ and
$(x_{1},x_{2},a,b,c)\sim(0,0.42,0.85,1.56,0.68)$ respectively.

For both $\zeta=0$ and $\zeta\neq 0$, the exact form of the metrics in Figures
9 and 10 depends not only on the parameters $\gamma$, $\sigma$ and $\zeta$,
but also on the exact initial conditions. In the case of $\zeta=0$ (_i.e._ a
model without re-infection) we can make this concrete by solving the dynamics
analytically. To this end, we consider the two probability distributions
$p^{(2)}$ and $p^{(3)}$ separately:

  * •

metric $g_{tt}^{(2)}$: We start with the discussion of the metric
$g_{tt}^{(2)}$ in the second line of (3.20). Let
$x(t)=p_{2}^{(2)}(t)=(I+R)(t)$, such that for $\zeta=0$

$\displaystyle g_{tt}^{(2)}=\frac{\gamma^{2}}{x}\,(1-x)\,I^{2}\,,$ (3.23)

which is not only a function of $x$ due to the presence of the factor $I^{2}$:
in fact, precisely by replacing this factor by a function of $x$ we encode the
non-trivial dynamics and turn (2.19) into a non-trivial flow equation (rather
than a mere definition of $g_{tt}$). To obtain $I(x)$, we divide the second
and third equations in (3.17)

$\displaystyle\frac{dR}{dI}=\frac{d(x-I)}{dI}=\frac{\sigma}{\gamma(1-x)-\sigma}\,,$
such that $\displaystyle\frac{dx}{dI}=\frac{\sigma}{\gamma(1-x)-\sigma}+1\,.$
(3.24)

This differential equation can be integrated by taking into account the
initial conditions $x_{0}=x(t=0)=I(t=0)$ (since we are assuming $R(t=0)=0$)

$\displaystyle
I=x_{0}+\int_{x_{0}}^{x}dx^{\prime}\left(\frac{\sigma}{\gamma(1-x^{\prime})-\sigma}+1\right)=x+\frac{\sigma}{\gamma}\,\log\left(\frac{1-x}{1-x_{0}}\right)\,.$
(3.25)

Inserting this into (3.23) we therefore find the metric $g_{tt}^{(2)}$ as a
function of $x$ as well as $\gamma,\sigma$ and the initial conditions $x_{0}$

$\displaystyle
g_{tt}^{(2)}=\frac{1-x}{x}\,\left(x\gamma+\sigma\,\,\log\left(\frac{1-x}{1-x_{0}}\right)\right)^{2}\,.$
(3.26)

We have verified that the dynamics (3.25) is indeed compatible with the
numerics discussed above and in particular that the form of the metric (3.26)
indeed matches the numerical form plotted in the left panel of Figure 10.
Notice in particular, since $x=I+R$ is monotonically growing for the entire
dynamics, the metric has only a single branch. Furthermore, the metric has
zeroes at

$\displaystyle
x=1+\frac{\sigma}{\gamma}\,W\left(-\frac{\gamma}{\sigma}\,(1-x_{0})\,e^{-\gamma/\sigma}\right)\,,$
(3.27)

which are in fact two values due to the two branches of the Lambert function.

  * •

metric $g_{tt}^{(3)}$: We can repeat the above analysis for the metric
$g_{tt}^{(3)}$ in the third line of (3.20). Let this time
$x(t)=p_{1}^{(3)}(t)=(S+R)(t)$, such that for $\zeta=0$

$\displaystyle g_{tt}^{(3)}=\frac{(1-x)}{x}\,(\gamma S-\sigma)^{2}\,.$ (3.28)

In order to express $g_{tt}^{(3)}$ entirely in terms of $x$, we need to
resolve the dynamics by expressing $S$ as a function of $x$. To this end, we
divide the first and third equation in (3.17)

$\displaystyle\frac{dR}{dS}=\frac{d(x-S)}{dS}=-\frac{\sigma}{\gamma\,S}$ such
that $\displaystyle\frac{dx}{dS}=1-\frac{\sigma}{\gamma S}\,.$ (3.29)

This differential equation can be integrated, where we consider the initial
conditions $S(t=0)=x_{0}=1-I(t=0)$

$\displaystyle
x=x_{0}+\int_{x_{0}}^{S}dS^{\prime}\,\left(1-\frac{\sigma}{\gamma
S^{\prime}}\right)=S-\frac{\sigma}{\gamma}\,\log(S/x_{0})\,.$ (3.30)

This relation can be inverted to express $S$ as a function of $x$

$\displaystyle
S=-\frac{\sigma}{\gamma}\,W\left(-\frac{x_{0}\gamma}{\sigma}\,e^{-\frac{x\gamma}{\sigma}}\right)\,,$
(3.31)

where $W$ is the Lambert function. Notice, however, that this solution is not
unique, but has two branches, reflecting the fact that $S+R$ is not a
monotonic function throughout the entire time evolution. Indeed, the metric
takes the form

$\displaystyle
g_{tt}^{(3)}(x)=\frac{1-x}{x}\,\sigma^{2}\,\left(1+W\left(-\frac{x_{0}\gamma}{\sigma}\,e^{-\frac{x\gamma}{\sigma}}\right)\right)^{2}\,.$
(3.32)

and also has two different branches. This is indeed, what we find from the
numerical solution of (3.17), which is plotted in the right panel of Figure
10.

We only briefly remark that generalising the discussion for the case
$\zeta\neq 0$ (for example for the metric $g_{tt}^{(2)}$) is more difficult,
due to the fact that the differential equation (3.24) becomes

$\displaystyle\frac{dx}{dI}=1+\frac{\zeta
x-(\zeta+\sigma)I}{I(\sigma-\gamma(1-x))}\,,$ (3.33)

which is no longer directly integrable. However, a numerical solutions (see
Figure 9) can nevertheless be obtained, which again reveals a metric that is
piecewise of the form (3.10).

###  3.3 S$\text{I}^{n}$ Model

Combining (and generalising) the ideas of both the Lotka-Volterra model in
Section 3.1 and the SIR model in Section 3.2, we next consider a simple
compartmental model with multiple variants of the pathogen. Concretely, we
denote $S(t)$ the number of susceptible individuals at time $t\in\mathbb{R}$
and $I_{i}(t)$ the number of infectious individuals who are infected with the
$i$th variant, with $i\in\\{1,\ldots,n\\}$ (for $n\in\mathbb{N}$ with $n\geq
2$). We assume that each infectious carries exactly one variant and cannot be
infected with any of the other variants. Here, we also exclude the possibility
of recovery (we shall discuss the generalisation in the next Subsection), such
that the dynamics of the epidemiological process is described by the following
$n+1$ coupled differential equations

$\displaystyle\frac{dS}{dt}(t)=-S(t)\,\sum_{i=1}^{n}\gamma_{i}\,I_{i}(t)\,,$
and $\displaystyle\frac{dI_{i}}{dt}(t)=\gamma_{i}\,S(t)\,I_{i}(t)\,,\hskip
5.69046pt\forall i=1,\ldots,n\,.$ (3.34)

Here $\gamma_{1,\ldots,n}\in\mathbb{R}_{+}$ are constants that describe the
infection rate of the $i$th variant. We furthermore consider initial
conditions at $t=0$ such that $S(0)+\sum_{i=1}^{n}I_{i}(0)=1$ and therefore
(since $\frac{d}{dt}\left(S(t)+\sum_{i=1}^{n}I_{i}(t)\right)=0$), the size of
the population remains constant. For the system (3.34) we can define various
different probability distributions. Here we shall be interested in two in
particular

$\displaystyle
p^{(1)}:\,\\{1,\ldots,n\\}\times\mathbb{R}\longrightarrow[0,1]\,,$ with
$\displaystyle p_{i}^{(1)}(t):=\frac{I_{i}(t)}{\sum_{j=1}^{n}I_{j}(t)}\,,$
$\displaystyle
p^{(2)}:\,\\{1,\ldots,n\\}\times\mathbb{R}\longrightarrow[0,1]\,,$ with
$\displaystyle
p_{i}^{(2)}(t):=\frac{\gamma_{i}\,I_{i}(t)}{\sum_{j=1}^{n}\gamma_{j}I_{j}(t)}\,,$
(3.35)

corresponding to the (normalised) distribution of (all) infectious individuals
or individuals newly becoming infectious at time $t$ (’newly infected’) among
all variants, respectively. The metrics (2.10) associated with these
probability distributions can be expressed explicitly in terms of $(S,I_{i})$
as follows

$\displaystyle
g_{tt}^{(1)}=\frac{S^{2}}{\left(\sum_{j=1}^{n}I_{j}\right)^{2}}\sum_{1\leq
i<j\leq n}I_{i}I_{j}(\gamma_{i}-\gamma_{j})^{2}\,,$ and $\displaystyle
g_{tt}^{(2)}=\frac{S^{2}}{\left(\sum_{j=1}^{n}\gamma_{j}I_{j}\right)^{2}}\sum_{1\leq
i<j\leq n}I_{i}I_{j}\gamma_{i}\gamma_{j}(\gamma_{i}-\gamma_{j})^{2}\,,$ (3.36)

Figure 11: Left panel: Time evolution of $(S,I_{1},I_{2})$ (with $n=2$) as a
function of time $t$. The parameters of the model were chosen to be
$\gamma_{1}=0.4$, $\gamma_{2}=0.1$, $I_{1}(t=0)=0.001$ and $I_{2}(t)=0.15$.
The plot shows a numerical Runge-Kutta solution of (3.34) with a time-step of
$\tau=0.0001$. Right panel: Fisher metrics (3.36) as functions of time. The
vertical dashed lines indicate the maxima $t_{1,2}$ of $g_{tt}^{(1,2)}$.

A typical time evolution of a solution of (3.34) for the simplest example
$n=2$ (of multiple variants) is schematically shown in the left panel of
Figure 11 and the metrics (3.36) are plotted in the right panel: here variant
2 was chosen to be more infectious than variant 1, but has a much lower
initial condition. Such a setup models the emergence of a new variant (the
more infectious variant 2) in a population where variant 1 is already
established. One can generalise such a scenario to simulate the emergence of a
new variant in a population where already several others are circulating. To
this end we shall label the number of infectious $I_{i}$ in (3.34) in a
slightly different fashion: let $\alpha,\beta\in\\{2,\ldots,n\\}$ with $n\geq
2$ and let

$\displaystyle\gamma_{1}\gg\gamma_{\alpha}\hskip 5.69046pt\text{and}\hskip
5.69046pt|\gamma_{\alpha}-\gamma_{\beta}|\ll\,,1\,,$ $\displaystyle
I_{1}(t=0)<I_{\alpha}(t=0)\,,$
$\displaystyle\forall\alpha,\beta\in\\{2,\ldots,n\\}\,,$ (3.37)

which simulates the emergence of a more infectious variant (called variant 1)
in the presence of an already established class of very similar variants. In
this case, the summations in (3.36) can be approximated by

$\displaystyle
g_{tt}^{(1)}\sim\frac{S^{2}I_{1}\bar{I}\,(\gamma_{1}-\bar{\gamma})^{2}}{\left(I_{1}+\bar{I}\right)^{2}}\,,$
and $\displaystyle
g_{tt}^{(2)}\sim\frac{S^{2}\,I_{1}\bar{I}\,\gamma_{1}\bar{\gamma}(\gamma_{1}-\bar{\gamma})^{2}}{\left(\gamma_{1}I_{1}+\bar{\gamma}\bar{I}\right)^{2}}\,,$
(3.38)

where we have defined

$\displaystyle\bar{\gamma}=\frac{1}{n-1}\sum_{\alpha=2}^{n}\gamma_{\alpha}\,,$
and $\displaystyle\bar{I}=\sum_{\alpha=2}^{n}I_{\alpha}\,,$ (3.39)

which are respectively the average infection rate of the variants
$\\{I_{\alpha}\\}_{\alpha\in\\{2,\ldots,n\\}}$ and their combined number of
infectious individuals. The metrics (3.38), however, are exactly the Fisher
metrics for two variants $(I_{1},\bar{I})$. Thus, from the perspective of the
information in the system, the variants $I_{\alpha}$ combined behave
essentially as a single one (which justifies the use of clustering tools as
used for example in [40]).

We next model the dynamics of the emergence of a new variant using the flow
equation (2.19) by expressing the metrics $g_{tt}^{(1,2)}$ in(3.39) as a
function of the probability distributions. As explained above, we shall
consider the case $n=2$ in (3.34) as approximation for a new variant (variant
1) emerging among already established ones, which collectively behave as
variant 2. We can treat both probability distributions in (3.35) in parallel
by defining

$\displaystyle\left\\{\begin{array}[]{lclr}x=\frac{I_{1}}{I_{1}+I_{2}}\,,&\text{such
that}&I_{2}=\frac{1-x}{x}\,I_{1}\,,&\text{for }p^{(1)}\,,\\\\[8.0pt]
y=\frac{\gamma_{1}I_{1}}{\gamma_{1}I_{1}+\gamma_{2}I_{2}}\,,&\text{such
that}&I_{2}=\frac{1-y}{y}\,\frac{\gamma_{1}}{\gamma_{2}}\,I_{1}\,,&\text{for
}p^{(2)}\,,\end{array}\right.$ (3.42)

and therefore the metric becomes

$\displaystyle
g_{tt}^{(1)}=\frac{1-x}{x}\,(\gamma_{1}-\gamma_{2})^{2}\,(x-I_{1})^{2}\,,$
$\displaystyle
g_{tt}^{(2)}=\frac{1-y}{y}\,\frac{(\gamma_{1}-\gamma_{2})^{2}}{\gamma_{2}^{2}}\,\left(y\gamma_{2}(1-I_{1})-\gamma_{1}I_{1}(1-y)\right)^{2}\,.$
(3.43)

To write these metrics entirely as functions of $x$ or $y$, we need to resolve
the dynamics of the system to express $I_{1}$ as a function of $x$ or $y$
respectively. To this end, we divide the two equations (3.34)

$\displaystyle\left\\{\begin{array}[]{lcr}\frac{dI_{2}}{dI_{1}}=\frac{1-x}{x}-\frac{I_{1}}{x^{2}}\,\frac{dx}{dI_{1}}=\frac{1-x}{x}\,\frac{\gamma_{2}}{\gamma_{1}}&\text{for}&p^{(1)}\,,\\\\[8.0pt]
\frac{dI_{2}}{dI_{1}}=\frac{\gamma_{1}}{\gamma_{2}}\left(\frac{1-y}{y}-\frac{I_{1}}{y^{2}}\,\frac{dy}{dI_{1}}\right)=\frac{1-y}{y}&\text{for}&p^{(2)}\,,\end{array}\right.$
(3.46)

such that

$\displaystyle\frac{du}{dI_{1}}=\frac{\gamma_{1}-\gamma_{2}}{\gamma_{1}}\,\frac{u(1-u)}{I_{1}}\,,$
with $\displaystyle u=\left\\{\begin{array}[]{lcl}x&\text{for}&p^{(1)}\,,\\\
y&\text{for}&p^{(2)}\,.\end{array}\right.$ (3.49)

We can integrate this equation with the initial conditions

$\displaystyle I_{1}(t=0)=I_{1}^{(0)}\,,$ and $\displaystyle
u(t=0)=u_{0}=\left\\{\begin{array}[]{lcl}\frac{I_{1}}{I_{1}+I_{2}}\big{|}_{t=0}&\text{for}&p^{(1)}\,,\\\\[8.0pt]
\frac{\gamma_{1}I_{1}}{\gamma_{1}I_{1}+\gamma_{2}I_{2}}\big{|}_{t=0}&\text{for}&p^{(2)}\,.\end{array}\right.$
(3.52)

to find

$\displaystyle\log(I_{1}/I_{1}^{(0)})=\int_{u_{0}}^{u}\frac{\gamma_{1}dz}{(1-z)z(\gamma_{1}-\gamma_{2})}=\frac{\gamma_{1}}{\gamma_{1}-\gamma_{2}}\,\log\left(\frac{u(1-u_{0})}{u_{0}(1-u)}\right)\,.$
(3.53)

which leads to

$\displaystyle
I_{1}=I_{1}^{(0)}\,\left(\frac{u(1-u_{0})}{u_{0}(1-u)}\right)^{\frac{\gamma_{1}}{\gamma_{1}-\gamma_{2}}}\,.$
(3.54)

Figure 12: Left panel: comparison of the numerical solutions for $I_{1}$
(coloured lines) with the form (3.54). Right panel: comparison of the
numerical solutions for the metrics $g_{tt}^{(1,2)}$ (coloured lines) with the
form (3.55). Both plots use the same numerical values and initial conditions
as in Figure 11.

The form of this solution is compared with the numerical solutions discussed
above in the left panel of Figure 12. Furthermore, the metrics become

$\displaystyle g_{tt}^{(1)}$
$\displaystyle=\frac{1-x}{x}\,(\gamma_{1}-\gamma_{2})^{2}\,\left(x-I_{1}^{(0)}\,\left(\frac{x(1-x_{0})}{x_{0}(1-x)}\right)^{\frac{\gamma_{1}}{\gamma_{1}-\gamma_{2}}}\right)^{2}\,,$
$\displaystyle g_{tt}^{(2)}$
$\displaystyle=\frac{1-y}{y}\,\frac{(\gamma_{1}-\gamma_{2})^{2}}{\gamma_{2}^{2}}\,\left(\gamma_{2}y-I_{1}^{(0)}\,\left(\frac{y(1-y_{0})}{y_{0}(1-y)}\right)^{\frac{\gamma_{1}}{\gamma_{1}-\gamma_{2}}}(y\gamma_{2}+(1-y)\gamma_{1})\right)^{2}\,,$
(3.55)

which are compared to the numerical results in the right panel of Figure 12.
We note that the metrics (3.55) can be well approximated by a function of the
form (3.10) with

$\displaystyle(x_{1},x_{2},a,b,c)\sim\left\\{\begin{array}[]{lcl}(0,0.347,0.233,0.879,1.745)&\text{for}&g_{tt}^{(1)}\,,\\\
(0,0.676,0.087,0.907,1.559)&\text{for}&g_{tt}^{(2)}\,.\end{array}\right.$
(3.58)

###  3.4 S$\text{I}^{n}$R(S) Model

As a generalisation of the S$\text{I}^{n}$ model discussed in the previous
Subsection, we now consider an S$\text{I}^{n}$R(S) model, in which infectious
individuals may recover (and potentially may become re-infected at a later
time). To this end, in addition to $S$ and $I_{i}$ (with $i\in 1,\ldots,n$) we
introduce as further compartment the number of removed (recovered) individuals
$R(t)$ at time $t$. As before, we consider $n\in\mathbb{R}$ variants, with
(slightly) different properties regarding infection and removal:

$\displaystyle\frac{dS}{dt}=-S\,\sum_{i=1}^{n}\gamma_{i}\,I_{i}+\zeta\,R\,,$
$\displaystyle\frac{dR}{dt}=\sum_{i=1}^{n}\sigma_{i}\,I_{i}-\zeta\,R\,,$
$\displaystyle\frac{dI_{i}}{dt}=\gamma_{i}\,S\,I_{i}-\sigma_{i}\,I_{i}\,,\hskip
5.69046pt\forall i=1,\ldots,n\,,$ (3.59)

with $\sigma_{i},\gamma_{i}\in\mathbb{R}_{+}$ denoting the infection and
recovery rates of the $i$th variant and $\zeta\in\mathbb{R}_{+}$ the rate at
which removed individuals become susceptible again. We furthermore assume
initial conditions at time $t=0$ such that
$\left(S+\sum_{i=1}^{n}I_{i}+R\right)(t)=1$. For the model (3.59) we shall
consider the following various probability distributions

$\displaystyle
p^{(1)}:\,\\{1,\ldots,n\\}\times\mathbb{R}\longrightarrow[0,1]\,,$ with
$\displaystyle p_{i}^{(1)}(t):=\frac{I_{i}(t)}{\sum_{j=1}^{n}I_{j}(t)}\,,$
$\displaystyle
p^{(2)}:\,\\{1,\ldots,n\\}\times\mathbb{R}\longrightarrow[0,1]\,,$ with
$\displaystyle
p_{i}^{(2)}(t):=\frac{\gamma_{i}\,I_{i}(t)}{\sum_{j=1}^{n}\gamma_{j}I_{j}(t)}\,,$
$\displaystyle
p^{(3)}:\,\\{1,\ldots,n\\}\times\mathbb{R}\longrightarrow[0,1]\,,$ with
$\displaystyle
p_{i}^{(3)}(t):=\frac{\sigma_{i}\,I_{i}(t)}{\sum_{j=1}^{n}\sigma_{j}I_{j}(t)}\,,$
(3.60)

corresponding to the normalised distribution of infectious individuals,
individuals becoming infectious (’newly infected’) and individuals recovering,
respectively. The associated Fisher information metrics (2.10) take the form

$\displaystyle g_{tt}^{(1)}$ $\displaystyle=\frac{\sum_{1\leq i<j\leq
n}I_{i}I_{j}\left(S(\gamma_{i}-\gamma_{j})-(\sigma_{i}-\sigma_{j})\right)^{2}}{\left(\sum_{k=1}^{n}I_{k}\right)^{2}}\,,$
$\displaystyle g_{tt}^{(2)}$ $\displaystyle=\frac{\sum_{1\leq i<j\leq
n}I_{i}I_{j}\gamma_{i}\gamma_{j}\left(S(\gamma_{i}-\gamma_{j})-(\sigma_{i}-\sigma_{j})\right)^{2}}{\left(\sum_{k=1}^{n}\gamma_{k}I_{k}\right)^{2}}\,,$
$\displaystyle g_{tt}^{(3)}$ $\displaystyle=\frac{\sum_{1\leq i<j\leq
n}I_{i}I_{j}\sigma_{i}\sigma_{j}\left(S(\gamma_{i}-\gamma_{j})-(\sigma_{i}-\sigma_{j})\right)^{2}}{\left(\sum_{k=1}^{n}\sigma_{k}I_{k}\right)^{2}}\,.$
(3.61)

These metrics can be written in a unified form as a function of the
probabilities $p_{i=1,\ldots,n-1}^{(a)}$

$\displaystyle
g_{tt}^{(a)}=\sum_{i=1}^{n-1}p_{i}^{(a)}\tau_{in}\left(\tau_{in}-\sum_{j=1}^{n-1}p_{j}^{(a)}\,\tau_{jn}\right)\,,$
$\displaystyle\forall a=1,2,3\,,$ (3.62)

where we have used $p_{n}^{(a)}=1-\sum_{i=1}^{n-1}p_{i}^{(a)}$ for the
probabilities (3.60). The combinations

$\displaystyle\tau_{ij}=S(\gamma_{i}-\gamma_{j})-(\sigma_{i}-\sigma_{j})\,,$
$\displaystyle\forall i,j=1,\ldots,n\,,$ (3.63)

depend on $S$, such that (3.62) is not a function of the $p_{i}^{(a)}$ (and
thus the $I_{1,\ldots,n}$) alone. Following (2.15), the metric can also be
represented in the following factorised form

$\displaystyle
g_{tt}^{(a)}=\sum_{i,j=1}^{n-1}\left(1-\frac{\delta_{ij}}{p_{i}^{(a)}}\sum_{k\neq
i}p_{k}^{(a)}\right)\,\frac{dp^{(a)}_{i}}{dt}\,\frac{dp_{j}^{(a)}}{dt}\,,$
$\displaystyle\forall a=1,2,3\,.$ (3.64)

Comparing (3.64) with (3.62) indeed allows to express the dynamics of the
$p_{i}^{(a)}$ as161616Since (3.64) is bilinear in the derivatives of
$p_{i}^{(a)}$, there remains a sign ambiguity which is related to the
direction of time.

$\displaystyle\frac{dp_{i}^{(a)}}{dt}=p_{i}^{(a)}\left(\tau_{in}-\sum_{j=1}^{n-1}\tau_{jn}\,p_{j}^{(a)}\right)\,,$
$\displaystyle\begin{array}[]{l}\forall a=1,2,3\,,\\\ \forall
i=1,\ldots,n-1\,,\end{array}$ (3.67)

which still implicitly depends on $S(t)$. Notice also that the probabilities
$p_{i}^{(a)}$ are not necessarily monotonic functions. For $n=2$, this can be
seen by writing the time-derivatives in the following manner171717Here we
implicitly assume $\gamma_{1}\neq\gamma_{2}$.

$\displaystyle\left(S-\frac{\sigma_{1}-\sigma_{2}}{\gamma_{1}-\gamma_{2}}\right)\,\frac{dp_{1}^{(a)}}{dt}=-\left(S-\frac{\sigma_{1}-\sigma_{2}}{\gamma_{1}-\gamma_{2}}\right)\,\frac{dp_{2}^{(a)}}{dt}=\frac{g_{tt}^{(a)}}{\gamma_{1}-\gamma_{2}}\,,$
$\displaystyle\forall a\in\\{1,2,3\\}\,.$ (3.68)

Since $S-\frac{\sigma_{1}-\sigma_{2}}{\gamma_{1}-\gamma_{2}}$ is not positive
definite, but $g_{tt}^{(a)}\geq 0$, the sign of the time-derivatives of the
probabilities are not fixed.

As in the previously discussed examples, in order to write (3.67) as an
effective model for the dynamics of the system, we need to resolve the
dynamics. In the present case, we shall do so in the particular case $n=2$ and
provide numerical solutions of the metric as functions of $p_{1}^{(a)}$ in
Figure 13. The dashed black lines give approximations of the metrics following
(3.10). Specifically, for $\zeta=0$ (left panel) we find as best fits

$\displaystyle g_{tt}^{(1)}:$
$\displaystyle(x_{1},x_{2},a,b,c)\sim(0,1,0.019,0.552,1.269)\,,$
$\displaystyle g_{tt}^{(2)}:$
$\displaystyle(x_{1},x_{2},a,b,c)\sim(0,1,0.030,0.612,1.426)\,,$
$\displaystyle g_{tt}^{(3)}:$
$\displaystyle(x_{1},x_{2},a,b,c)\sim(0.011,0.522,1.011)\,,$ (3.69)

Figure 13: Numerical plots of the metrics $g_{tt}^{(1,2,3)}$ as functions of
$x=p_{1}^{(1,2,3)}$ respectively for $\zeta=0$ (left) and $\zeta=0.2$ (right).
The dotted black lines give interpolations of the metrics following (3.10), as
explained in the text.

while for $\zeta=0.2$ (right panel) we find the approximation

$\displaystyle g_{tt}^{(1)}:$
$\displaystyle(x_{1},x_{2},a,b,c)\sim\left\\{\begin{array}[]{lcl}(0.006,0.120,1.237,0.183,3.375)&\text{for}&\frac{I_{1}}{I_{1}+I_{2}}\in[0,0.044]\,,\\\
(0.029,1.023,0.002,0.651,1.324)&\text{for}&\frac{I_{1}}{I_{1}+I_{2}}\in(0.044,1]\,,\end{array}\right.$
(3.72) $\displaystyle g_{tt}^{(2)}:$
$\displaystyle(x_{1},x_{2},a,b,c)\sim\left\\{\begin{array}[]{lcl}(0.003,0.110,2.825,0.604,2.583)&\text{for}&\frac{\gamma_{1}I_{1}}{\gamma_{1}I_{1}+\gamma_{2}I_{2}}\in[0,0.080]\,,\\\
(0.037,1.021,0.002,0.751,1.439)&\text{for}&\frac{\gamma_{1}I_{1}}{\gamma_{1}I_{1}+\gamma_{2}I_{2}}\in(0.080,1]\,,\end{array}\right.$
(3.75) $\displaystyle g_{tt}^{(3)}:$
$\displaystyle(x_{1},x_{2},a,b,c)\sim\left\\{\begin{array}[]{lcl}(0.001,0.091,34.405,0.656,3.384)&\text{for}&\frac{\sigma_{1}I_{1}}{\sigma_{1}I_{1}+\sigma_{2}I_{2}}\in[0,0.055]\,,\\\
(0.033,1.023,0.002,0.682,1.373)&\text{for}&\frac{\sigma_{1}I_{1}}{\sigma_{1}I_{1}+\sigma_{2}I_{2}}\in(0.055,1]\,.\end{array}\right.$
(3.78)

We remark that another way to fit these metrics is of the form

$\displaystyle g_{tt}^{(a)}\sim
a(x-x_{1})^{b}\,(x_{2}-x)^{c}\,\left((x-x_{3})(x-\overline{x_{3}})\right)^{d/2}\,,$
(3.79)

with $x_{1,2}$ real- and $x_{3}\in\mathbb{C}$ a complex zero (and $a,b,c,d$
suitable fitting parameters). Concretely, for the right panel of Figure 13 we
find

$\displaystyle g_{tt}^{(1)}:$
$\displaystyle(a,b,c,d,x_{1},x_{2},x_{3})\sim(0.001,0.472,0.825,0.049,0.989,0.908+0.052i)\,,$
$\displaystyle g_{tt}^{(2)}:$
$\displaystyle(a,b,c,d,x_{1},x_{2},x_{3})\sim(0.002,0.1,1.292,0.099,1.011,0.277+0.034i)\,,$
$\displaystyle g_{tt}^{(3)}:$
$\displaystyle(a,b,c,d,x_{1},x_{2},x_{3})\sim(0.002,0.321,1.517,0.063,1.031,-0.466+0.00001i)\,.$
(3.80)

Finally, before closing this Section we remark that for $n=2$,181818Similarly
we may assume as in (3.37) that several variants behave very similarly and can
thus be modelled collectively. we can find exact expressions of $g_{tt}^{(a)}$
as functions of $p_{1}^{(a)}$ respectively in particular cases: for example we
may impose that $S$ and $R$ remain (approximately) constant

$\displaystyle\frac{dS}{dt}=0=-S(\gamma_{1}I_{1}+\gamma_{2}I_{2})+\zeta R\,,$
and $\displaystyle\frac{dR}{dt}=0=\sigma_{1}I_{1}+\sigma_{2}I_{2}-\zeta R\,.$
(3.81)

In this case the overall number of infectious individuals remains constant and
only the distribution between the two variants changes. The conditions (3.81)
are solved by

$\displaystyle
S=\frac{\sigma_{1}I_{1}+\sigma_{2}I_{2}}{\gamma_{1}I_{1}+\gamma_{2}I_{2}}\,,$
and $\displaystyle
R=\frac{\sigma_{1}}{\zeta}\,I_{1}+\frac{\sigma_{2}}{\zeta}\,I_{2}\,,$ (3.82)

which allows us to simplify the expressions for the metrics:

$\displaystyle g_{tt}^{(1)}(x)$
$\displaystyle=\frac{x(1-x)\left(\gamma_{1}\sigma_{2}-\gamma_{2}\sigma_{1}\right)^{2}}{\left(x(\gamma_{1}-\gamma_{2})+\gamma_{2}\right)^{2}}\,,$
with $\displaystyle x=\frac{I_{1}}{I_{1}+I_{2}}\,,$ $\displaystyle
g_{tt}^{(2)}(x)$
$\displaystyle=\frac{x(1-x)\left(\gamma_{1}(1-x)+x\gamma_{2}\right)^{2}\left(\gamma_{1}\sigma_{2}-\gamma_{2}\sigma_{1}\right)^{2}}{\gamma_{1}^{2}\gamma_{2}^{2}}\,,$
with $\displaystyle
x=\frac{\gamma_{1}I_{1}}{\gamma_{1}I_{1}+\gamma_{2}I_{2}}\,,$ $\displaystyle
g_{tt}^{(3)}(x)$
$\displaystyle=\frac{x(1-x)\left(\sigma_{1}(1-x)+x\sigma_{2}\right)^{2}\left(\gamma_{1}\sigma_{2}-\gamma_{2}\sigma_{1}\right)^{2}}{\left((1-x)\gamma_{2}\sigma_{1}+x\gamma_{1}\sigma_{2}\right)^{2}}\,,$
with $\displaystyle
x=\frac{\sigma_{1}I_{1}}{\sigma_{1}I_{1}+\sigma_{2}I_{2}}\,.$ (3.83)

In all three cases, the solution of (2.19) (with $\alpha=1$) yields the
following transcendental equations for $x$ as a function of time

$\displaystyle p_{1}^{(1)}:$ $\displaystyle
t(\gamma_{2}\sigma_{1}-\gamma_{1}\sigma_{2})=\gamma_{1}\,\log(1-x)-\gamma_{2}\log
x\,,$ $\displaystyle p_{1}^{(2)}:$ $\displaystyle
t(\gamma_{2}\sigma_{1}-\gamma_{1}\sigma_{2})=\gamma_{1}\,\log(1-x)-\gamma_{2}\log
x-(\gamma_{1}-\gamma_{2})\log\left(\gamma_{1}(1-x)+\gamma_{2}x\right)\,,$
$\displaystyle p_{1}^{(3)}:$ $\displaystyle
t(\gamma_{2}\sigma_{1}-\gamma_{1}\sigma_{2})=\gamma_{1}\,\log(1-x)-\gamma_{2}\log
x-(\gamma_{1}-\gamma_{2})\log\left(\sigma_{1}(1-x)+\sigma_{2}x\right)\,.$
(3.84)

##  4 General Metric

In the previous Section, we have re-organised the dynamics of a number of
simple examples in the form of the flow equation (2.19). To this end, we have
defined probability distributions for this models of the form
$p:\,\\{1,2\\}\times\mathbb{R}\to[0,1]$ with $p=(x(t),1-x(t))$ and have
expressed the metric $g_{tt}$ as a function of $x$. In all cases, we have seen
that the latter can (at least piecewise) be very well approximated by a
function of the form (3.10) with $b,c$ constants of order $1$. In this Section
we consider the inverse problem, _i.e._ we start with the simplest form of the
metric (3.10) and develop the corresponding dynamics in an analytic fashion.

###  4.1 Monotonic Evolution

We start by considering the following simple form of the metric

$\displaystyle g_{tt}(x)=a\,(x-x_{1})(x_{2}-x)\,,$ for
$\displaystyle\begin{array}[]{l}a\in\mathbb{R}_{+}\,,\\\ 0\leq x_{1}\leq x\leq
x_{2}\leq 1\,,\end{array}$ (4.3)

_i.e._ (3.10) for the simplest case $b=c=1$. We assume this metric as the
defining input for the dynamics, which fixes $x(t)$ according to the flow
equation (2.19) for $\alpha=1$. Assuming the initial condition
$x(t=t_{0})=x_{0}\in[x_{1},x_{2}]$191919The case $x_{1}=0$ or $x_{2}=1$ needs
particular care, as we shall discuss below., the flow equation can be
integrated

$\displaystyle
t-t_{0}=\kappa\,\int_{x_{0}}^{x}\frac{dz}{\sqrt{az(1-z)(z-x_{1})(x_{2}-z)}}=\frac{2i\kappa\left(F(u(x),m)-F(u(x_{0}),m)\right)}{\sqrt{a(1-x_{1})x_{2}}}\,,$
(4.4)

where we have assumed that the sign $\kappa\in\\{\pm 1\\}$ of the first
derivative of $x$ remains constant for all times in the interval $[t_{0},t]$
and $x_{1}\leq x\leq x_{2}$. Furthermore, $F$ in (4.4) is the incomplete
elliptic integral of the first kind

$\displaystyle
F(\phi,k^{2})=\int_{0}^{\phi}\frac{dz}{\sqrt{1-k^{2}\sin^{2}z}}\,,$ (4.5)

with the arguments

$\displaystyle m=\frac{x_{1}(1-x_{2})}{x_{2}(1-x_{1})}\,,$ and $\displaystyle
u(x)=\arcsin\left(\sqrt{\frac{x(1-x_{1})}{x_{1}(1-x)}}\right)\,.$ (4.6)

The relation (4.4) can be solved to express $x$ as a function of time $t$

$\displaystyle
x(t)=\frac{x_{1}\,\text{sn}\left(\tfrac{i\kappa(t-t_{0})}{2}\,\sqrt{a(1-x_{1})x_{2}}-F(u(x_{0}),m),m\right)^{2}}{1-x_{1}+x_{1}\text{sn}\left(\tfrac{i\kappa(t-t_{0})}{2}\,\sqrt{a(1-x_{1})x_{2}}-F(u(x_{0}),m),m\right)^{2}}\,,$
(4.7)

where sn is a Jacobi elliptic function, which can be expressed in terms of
Jacobi theta-functions

$\displaystyle\text{sn}(u,k^{2})=\frac{\theta_{3}(0,\tau)}{\theta_{2}(0,\tau)}\,\frac{\theta_{1}(u/\theta_{3}^{2}(0,\tau),\tau)}{\theta_{4}(u/\theta_{3}^{2}(0,\tau),\tau)}\,,$
with $\displaystyle
k=\frac{\theta_{2}^{2}(0,\tau)}{\theta_{3}^{2}(0,\tau)}\,.$ (4.8)

The solution (4.7) is periodic in $t$ with periodicity

$\displaystyle T=\frac{4K(1-m)}{\sqrt{a(1-x_{1})x_{2}}}\,,$ (4.9)

where $K$ is the complete elliptic integral of the first kind

$\displaystyle
K(m)=F(\pi/2,m)=\frac{\pi}{2}\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\,m^{n}\,.$
(4.10)

In the case of $x_{1}=0$ and $x_{2}=1$ care needs to be taken that only
$x\in(x_{1},x_{2})$ is allowed (_i.e._ notably $x_{0}\notin\\{0,1\\}$). In
this case, the integral in (4.4) simplifies

$\displaystyle
t-t_{0}=\frac{\kappa}{\sqrt{a}}\,\log\left(\frac{x(1-x_{0})}{x_{0}(1-x)}\right)\,,$
(4.11)

such that

$\displaystyle
x(t)=\frac{x_{0}\,e^{\sqrt{a}\,\kappa(t-t_{0})}}{1-x_{0}+x_{0}\,e^{\sqrt{a}\kappa(t-t_{0})}}\,,$
(4.12)

which corresponds to a logistic function and is plotted schematically in
Figure 14. Notice that

$\displaystyle\lim_{t\to\infty}x(t)=\left\\{\begin{array}[]{lcl}1&\text{if}&\kappa=1\,,\\\
0&\text{if}&\kappa=-1\,,\end{array}\right.$ (4.15)

such that the function $x(t)$ is monotonic for the entire time evolution.

Figure 14: Metric (4.3) (left) and $x(t)$ (4.12) (right) for $a=1$,
$\kappa=1$, $t_{0}=0$ and $x_{0}=\tfrac{1}{100}$.

###  4.2 Full Time Evolution

Using the result of the previous Subsection, we can construct more complicated
time evolutions, by defining the metric $g_{tt}$ to be piecewise of the form
(4.3).

####  4.2.1 Flow Between Real Zeroes of the Metric

Notice that the integral (4.4) is only defined for $x_{1}<x<x_{2}$ (since
outside of this interval the integrand becomes imaginary) and for fixed sign
of the time derivative $\dot{x}$. In the examples of Section 3, we have seen
that the zeroes of the metric $x(t)=x_{1}$ or $x_{2}$, correspond to points,
where the sign of $\dot{x}$ changes, which also leads to a change in the
metric. Thus, in this case, we can describe the time evolution by defining the
metric piecewise, for fixed sign of $\dot{x}$. In the following we discuss two
examples in more detail: the first one inspired by the Lotka-Volterra equation
in Section (3.1) and the second one inspired by the SIR model in Section 3.2:

  1. 1.

_Periodic Time Evolution:_ We first shall consider a metric of the form

$\displaystyle
g_{tt}=\left\\{\begin{array}[]{lcl}a_{1}\,(x-x_{1})(x_{2}-x)&\text{for}&\dot{x}>0\,,\\\
a_{2}\,(x-x_{1})(x_{2}-x)&\text{for}&\dot{x}<0\end{array}\right.$
$\displaystyle\begin{array}[]{l}a_{1},a_{2}\in\mathbb{R}_{+}\,,\\\ 0\leq
x_{1}\leq x\leq x_{2}\leq 1\,,\end{array}$ (4.20)

where both zeroes of the two branches of the metric coincide (see left panel
of Figure 15).

Figure 15: Left panel: Metric (4.20) for $x_{1}=1/8$, $x_{2}=7/8$, $a_{1}=1$
(blue curve) and $a_{2}=1/4$ (red curve). Right panel: corresponding dynamics
$x(t)$ (4.12) for $t_{0}=0$ and $x_{0}=\tfrac{1}{8}$. The dashed lines
indicate extrema of $x$, corresponding to zeroes of the metric (_i.e._ points
where the branch of the metric changes). The colours of the dynamics $x(t)$ in
the right panel match the colours of the branch of the metric in the left
panel.

In this case, the motion of $x(t)$ is periodic and piecewise described by
(4.7). The periodicity is given by

$\displaystyle
T=\frac{2(\sqrt{a_{1}}+\sqrt{a_{2}})\,K\left(\tfrac{x_{2}-x_{1}}{(1-x_{1})x_{2}}\right)}{\sqrt{a_{1}a_{2}(1-x_{1})x_{2}}}\,.$
(4.21)

as is schematically shown in the right panel of Figure 15.

  2. 2.

_Oscillating Time Evolution:_ Let $\\{x_{1}(n)\\}_{n\in\mathbb{N}^{*}}$ and
$\\{x_{2}(n)\\}_{n\in\mathbb{N}^{*}}$ be two (convergent) series of real
integers, such that $0<x_{1}(n)<x_{1}(n+1)\leq x_{2}(n+1)<x_{2}(n)<1$ $\forall
n\in\mathbb{N}^{*}$ and
$\lim_{n\to\infty}x_{1}(n)=\lim_{n\to\infty}x_{2}(n)=x_{f}\in[0,1]$. Then
consider the series of metrics

$\displaystyle
g_{tt}(n)=\left\\{\begin{array}[]{lcl}a_{1}\,(x-x_{1}(n))(x_{2}(n)-x)&\text{for}&\dot{x}>0\,,\\\
a_{2}\,(x-x_{1}(n))(x_{2}(n)-x)&\text{for}&\dot{x}<0\end{array}\right.$
$\displaystyle\begin{array}[]{l}a_{1},a_{2}\in\mathbb{R}_{+}\,,\\\ 0\leq
x_{1}(n)\leq x\leq x_{2}(n)\leq 1\,,\end{array}$ (4.26)

This series of metrics then describes a time evolution, where $x$ oscillates
around $x_{f}$ and is piecewise described by (4.7). An example for the series

$\displaystyle
x_{1}(n)=\frac{1}{8}+\frac{9}{4\pi^{2}}\sum_{m=1}^{n}\frac{1}{m^{2}}\,,$
$\displaystyle
x_{2}(n)=\frac{7}{8}-\frac{9}{4\pi^{2}}\sum_{m=1}^{n}\frac{1}{m^{2}}\,,$
$\displaystyle x_{f}=\frac{1}{2}\,,$ (4.27)

(and $a_{1}=a_{2}=1$) is shown in Figure 16, leading to a solution oscillating
around $x_{f}$.

Figure 16: Metric (4.26) (left) and evolution of $x$ (right) using the series
(4.27) and initial conditions $t_{0}=0$ and $x_{0}=\tfrac{1}{8}$. The black
vertical lines indicate extrema of $x$, corresponding to zeroes of the metric:
solid lines mark the increase $n\to n+1$, while the dashed lines mark the
switch of the branch of the metric in (4.26). The colours of the dynamics
$x(t)$ in the right panel match the colours of the branch of the metric in the
left panel.

####  4.2.2 Flow Between Imaginary Zeroes of the Metric

Before closing this Section, we want to remark on another possibility for
defining metrics that are piecewise of the form (4.3): in the previous
examples, we have glued metrics at their (real) zeros, corresponding to
extrema of $x$. In principle, we can also glue metrics along points with
$g_{tt}\neq 0$, an example of which we have encountered in Section 3.4 (see
right panel of Figure 13) in the time evolution of the S$\text{I}^{n}$R model
with re-infection. A simpler example is given by the dashed black curve in the
left panel of Figure 17 (black dashed curve). This metric is approximated by a
function of the form

$\displaystyle
g_{tt}=a\,(x-x_{1})^{b}\,(x_{2}-x)^{c}\,\left((x-x_{3})(x-\overline{x_{3}})\right)^{d/2}\,,$
with $\displaystyle\begin{array}[]{l}x_{1}\,,x_{2}\in[0,1]\text{ with
}x_{1}<x_{2}\,,\\\ x_{3}\in\mathbb{C}\text{ with
}x_{1}<\text{Re}(x_{3})<x_{2}\,,\end{array}$ (4.30)

Figure 17: Left panel: example of a metric glued along complex zeroes. The
dashed black curves are given by $g_{tt}=(x-1/8)(5/8-x)$ for $1/8<x<0.6075$
and $g_{tt}=(x-23/40)(7/8-x)$ for $0.6075<x<1$. Both curves are joined along
$x_{3}=0.6075$ (for which $g_{tt}\neq 0$). The solid blue line represents an
approximation of this metric of the form (4.30) with $a=1.72$, $b=1.16$,
$c=1.24$, $d=0.6$ and $x_{1}=1/8$, $x_{2}=7/8$ and $x_{3}=0.62+0.009i$. Right
panel: corresponding time evolution $x(t)$: the dashed line represents a
piecewise solution of the form (4.7), which are joined at $t_{3}=6.76$. The
blue curve is a numerical solution using the approximate metric of the from
(4.30).

where $a,b,c,d$ are suitable fitting parameters and $x_{1,2}$ real zeroes of
$g_{tt}$, while $x_{3}$ (and its conjugate $\overline{x_{3}}$) are complex
zeroes. From this perspective, the metric is joined along a complex zero and
is not convex on the entire interval $[x_{1},x_{2}]$. The solution of (2.19)
for the piecewise defined $g_{tt}$ is shown in the right panel of Figure 17
(black dashed curve), along with a numerical solution using the approximated
metric (4.30). Indeed, the sign of $dx/dt$ remains positive for the entire
time evolution, consistent with the fact that the metric never vanishes. Such
a behaviour was used in [34] to model periods between two waves of a SARS-
CoV-2 epidemic. This behaviour was re-interpreted in [21] as a system coming
close to a(n unstable) fixed point in the context of multiple variants of
SARS-CoV-2.

##  5 Conclusions

In this paper we have studied a reformulation of simple epidemiological- and
population dynamical processes in terms of the Fisher information (metric). To
this end, we have first re-organised the degrees of freedom of the system into
(normalised) probability distributions (in the sense of (2.1)) that are
functions of time. This is a natural step in epidemiology or population
dynamics, since in many approaches a given population is grouped into distinct
classes (see _e.g._ compartmental models [13]) and probabilities can be
defined as normalised ratios of their sizes. We have associated the Fisher
information [1] with a chosen probability distribution, which (following [2,
3, 4, 5, 6]) we have interpreted as a metric $g_{tt}$ of a one-dimensional
Riemannian manifold, to highlight its property to capture the distance between
the state of the system at different times. In fact, for configurations where
the entire probability distribution can be described by a single bounded
function $x:\,\mathbb{R}\to[0,\alpha]$ (with $0<\alpha<1$), we have shown that
the dynamics can be re-cast in the flow equation (2.19), which is entirely
determined by the metric. Partially resolving the dynamics of various
different models, namely the Lotka-Volterra model (Section 3.1), the SIR(S)
model (Section 3.2) and compartmental models describing the evolution of two
(or more) variants of a disease (Sections 3.3 and 3.4), such that $g_{tt}$ can
be expressed as a function of $x$ alone, has revealed strong similarities
between the different models: the time evolution of $x$ can be described
piecewise for regimes in which $x$ is a monotonic function, _i.e._ between two
of its local extrema, which correspond to real zeroes of $g_{tt}$. Moreover,
in any such regime the metric $g_{tt}$ can be very well approximated simply
through powers of its zeroes: if $g_{tt}$ is a convex function it can be
approximated through the form (3.10), which can be generalised by including
imaginary zeroes (see (4.30)) if $g_{tt}$ has one or more local minima. The
complete dynamics of the system can be described by gluing such metrics
together at their zeroes. In fact, using such a metric as a full input for the
flow equation (2.19) and focusing on the simplest type of metric (_i.e._
(4.3)) we have provided an analytic, closed form solution of the monotonic
time evolution for $x$ in (4.7). By gluing this solution, we have constructed
more general periodic and oscillating solutions in Section 4.2.

Our approach highlights that the Fisher information metric not only describes
families of probabilities, but in fact is capable of capturing their full time
evolution provided that enough information about the dynamics of the system
can be encoded in it. In our current setting, where the probability
distribution is described as a single function $x$, this is tantamount to re-
writing $g_{tt}$ as a function of $x$ alone, rather than an explicit function
of $t$. The emerging dynamics is quite similar among very different looking
models, since details of the system have been resolved, thus revealing
universal properties of the underlying time evolution. Specifically, our
analysis reveals the importance of zeroes of the metric, both real and
imaginary: real zeroes correspond to extrema of the probability distribution
$x$ and specifying the metric between two zeroes fixes a monotonic time
evolution of $x$. Complex zeroes (or local minima) of $g_{tt}$ provide
additional structure (resembling saddle points) to this monotonic time
evolution, as can be seen in the right panel of Figure 17.

The importance of (quasi) fixed points in the time evolution was previously
pointed out in [34, 21] in a concrete epidemiological system: inspired by
ideas of renormalisation group methods [43, 44, 45, 46, 47, 48] the (M)eRG
(Mutation epidemiological Renormalisation Group) model (based on an original
proposal in [30, 31]) conjectured a (set of) first order differential
equations (called $\beta$-functions or flow equations) that interpolate the
cumulative number of individuals infected with (a variant of) a disease
between two (disease free) fixed points. This simple model was very successful
in describing individual waves of Covid-19 in various regions of the world
[32, 33, 36, 37, 39] (see also the review [38]), while the multi-wave
structure was attributed to unstable fixed points of the system [35, 34] which
were linked to the appearance of new variants of SARS-CoV-2 replacing previous
ones [21]. Our current analysis is not only in line with this interpretation,
but in fact gives a clearer picture of how to describe the notion of ’fixed
point’ in a more general class of systems: by reducing the dynamics of quite
different models (with conceptually different degrees of freedom) to a common
form (2.19), we can translate the notion of a ’fixed point’ (in the language
of [34, 21]) to (complex) zeroes of the Fisher information, for which moreover
a geometric interpretation exists. This allows to re-organise the dynamics of
a much larger class of quite different models around these ’fixed points’, as
is evidenced by our simple approximations in (3.10). Moreover, this re-
formulation leads to a potentially simpler description (focused on properties
of the system around highly symmetric points), as is showcased by the analytic
solutions we have found for the simplest approximation in Section 4.

For the future, we anticipate further applications and generalisations of our
work. First of all it will be interesting to extend the description to include
cases where the (effective) probability distribution is described by more than
one function $x$. Similarly, the inclusion of additional parameters beyond the
time-variable will lead to more refined Fisher metrics, capable of describing
more sophisticated models and their fixed points.

## Acknowledgements

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.

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