#  Noise can lead to exponential epidemic spreading despite $R_{0}$ below one

Johannes Pausch  Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 0WA, United Kingdom  St. Catharine’s
College, Cambridge CB2 1RL, United Kingdom  Rosalba Garcia-Millan  Department
of Applied Mathematics and Theoretical Physics, University of Cambridge,
Cambridge CB3 0WA, United Kingdom  Gunnar Pruessner  Department of
Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

###### Abstract

Branching processes are widely used to model evolutionary and population
dynamics as well as the spread of infectious diseases. To characterize the
dynamics of their growth or spread, the basic reproduction number $R_{0}$ has
received considerable attention. In the context of infectious diseases, it is
usually defined as the expected number of secondary cases produced by an
infectious case in a completely susceptible population. Typically $R_{0}>1$
indicates that an outbreak is expected to continue and to grow exponentially,
while $R_{0}<1$ usually indicates that an outbreak is expected to terminate
after some time.

In this work, we show that fluctuations of the dynamics in time can lead to a
continuation of outbreaks even when the expected number of secondary cases
from a single case is below $1$. Such fluctuations are usually neglected in
modelling of infectious diseases by a set of ordinary differential equations,
such as the classic SIR model. We showcase three examples: 1) extinction
following an Ornstein-Uhlenbeck process, 2) extinction switching randomly
between two values and 3) mixing of two populations with different $R_{0}$
values. We corroborate our analytical findings with computer simulations.

##  I Introduction

A branching process Harris (1963); Athreya and Ney (1972) is a stochastic
process in which individuals randomly create copies of themselves or become
extinct. This model has a wide range of applications and individuals might
represent offspring Watson and Galton (1875); Nowak (2006); Bacaër (2011),
particles Pázsit and Pál (2007); Williams (2013); Garcia-Millan  _et al._
(2018), active neurons Wilting and Priesemann (2018); Zierenberg  _et al._
(2018); Pausch  _et al._ (2020); Zierenberg  _et al._ (2020); Pausch (2021),
or infected individuals Farrington and Grant (1999); Farrington  _et al._
(2003); Wilting and Priesemann (2018); Corral (2021) among others Kimmel and
Axelrod (2002). At the centre of many investigations of branching processes is
the statistics of spells of activity, which are called avalanches Pázsit and
Pál (2007); Williams (2013); Garcia-Millan  _et al._ (2018); Wilting and
Priesemann (2018); Zierenberg  _et al._ (2018); Pausch  _et al._ (2020);
Zierenberg  _et al._ (2020); Pausch (2021) in most applications or outbreaks
in the context of infectious diseases. Avalanches, or outbreaks, are defined
as the activity in the branching process which is initiated by a single
individual and lasts until all subsequent individuals have become extinct. In
the context of particles, an avalanche starts with a single particle and ends
when the system is empty. In the context of infectious diseases, an outbreak
starts with a single infected individual and ends when no infected individual
is left (here we use the words _infected_ and _infectious_ without
distinction).

A branching process is subcritical if the expected size of an outbreak
_decays_ exponentially. It is called supercritical if the expected size of the
outbreak _grows_ exponentially. Otherwise, if the size approaches a constant
value in time, the branching process is said to be critical. We will use this
criterion, asymptotically constant expected outbreak size, as the definition
of the critical point throughout this work. The critical point generally
divides a parameter region resulting in asymptotically exponential growth from
one resulting in asymptotically exponential decline. For particle avalanches
as well as for outbreaks of infectious diseases, it is of great interest to
identify simple parameters that indicate whether the process is supercritical
or not. One of those parameters is the basic reproduction number $R_{0}$.

The basic reproduction number $R_{0}$ is defined as the expected number of
secondary individuals that are created from a single individual Dietz (1993);
Heesterbeek and Dietz (1996); Heesterbeek (2002); Li  _et al._ (2011);
Delamater  _et al._ (2019). More explicitly in a branching process, an
existing individual waits until a branching event occurs. In many models, the
waiting time between branching events is fixed Harris (1963); Corral  _et al._
(2018, 2016), however in the present work, we consider only exponentially
distributed waiting times Bordeu  _et al._ (2019); Garcia-Millan  _et al._
(2018); Pausch  _et al._ (2020). At a branching event, an individual is
replaced by its offspring, which is a random number $K\in\mathbb{N}_{0}$ of
individuals. The case $K=0$ corresponds to the extinction of the parent
individual. In general, the offspring distribution can be defined by
specifying the branching probabilities $p_{0}$, $p_{1}$, $p_{2}$, …$\in[0,1]$
such that

$\displaystyle
P(K=k)=p_{k}\qquad\text{with}\qquad\sum\limits_{k=0}^{\infty}p_{k}=1,$ (1)

i.e. the probability $P(K=k)$ that an individual has $K=k$ offspring equals
$p_{k}$. Although our analytical results hold for any distribution of $K$, in
our simulations we use only the binary offspring distribution with
$K\in\\{0,2\\}$. All individuals are independent and there is no bound to the
number of individuals in the system, which can be interpreted as an unbounded
population of susceptible individuals. In this setup, the basic reproduction
number $R_{0}$ is defined as the expected number of offspring,

$\displaystyle R_{0}=\sum_{k=0}^{\infty}kp_{k}=\mathbb{E}\left[K\right].$ (2)

In particular, $R_{0}$ is dimensionless and does not indicate how quickly or
slowly an avalanche/outbreak evolves.

There are many difficulties in deriving $R_{0}$ from data Li  _et al._ (2011);
Diekmann  _et al._ (1990); van den Bosch  _et al._ (2008) and researchers have
defined several similar quantities related to $R_{0}$ Anderson (1992); Cao
_et al._ (2020). In addition, more detailed models of infectious diseases take
other characteristics such as age, immunity, behaviour or the evolution of the
disease itself into account, which make the definition of $R_{0}$ more
difficult Diekmann  _et al._ (1990); Anderson (1992); van den Bosch  _et al._
(2008); Li  _et al._ (2011); Ridenhour  _et al._ (2014). Rather than including
more detailed aspects into our models, we restrict ourselves to the basic
model outlined above and keep the discussion at the level of stochastic
processes. For example, we assume that infected individuals are also
infectious.

In many real-world occurrences of branching processes, the environment and the
process itself are imperfect in the sense that they fluctuate in time, for
example because individuals and the environment change the conditions for
disease transmission Ariel and Louzoun (2021). Such fluctuations will be
affecting the branching process over time and are not easily dealt with
analytically. Of particular importance are fluctuations that affect the
population as a whole. As our calculations show, basic approximations of
branching processes with noise can be misleading by predicting subcritical
dynamics where a more detailed calculation reveals supercritical behaviour. It
is the main aim of this article to highlight such, often counter-intuitive,
phenomena, which are easily missed by traditional modelling of epidemics,
which draw on a coarse-grained set of equations, such as the classic SIR model
Vasiliauskaite  _et al._ (2021).

The article is organized as follows: In Sec. II, the branching model without
noise is presented as a Master Equation. It forms the basis of the models with
noise that follow. In Sec. III, we introduce a branching process coupled to an
Ornstein-Uhlenbeck process. Although a mean-field approximation predicts a
critical point at $R_{0}=1$, our detailed analysis supported by simulations
reveal a shift of the critical $R_{0}$ to values smaller than $1$. In Sec. IV,
the branching process is coupled to a stochastic process called telegraphic
noise, which implements a random switch between two different extinction rates
of the branching process. In this system, it is much less obvious what a
suitable definition of $R_{0}$ would be. Neither of the two $R_{0}$ values
associated with the two extinction rates, nor a simple weighted average of
them predict $R_{0}=1$ to be the critical point. An exact calculation, reveals
that the critical $R_{0}$ is smaller than $1$. In Sec. V, two branching
processes with two different $R_{0}$ are coupled. We show that neither of the
two $R_{0}$ nor a linear combination of them correctly predict the critical
point of the system. We conclude in Sec. VI. The detailed analytical results
are based on field-theoretic approaches which are presented in the appendix.

Figure 1: Trajectory of a branching processes (top) whose extinction rate is
coupled to an Ornstein-Uhlenbeck process (bottom), see Sec. III. Parameters
for this simulation are $s=1$, $R_{0}=1$, $\lambda=0.18$, $\beta=3.5$ and
$D=2$. Figure 2: Trajectory of a branching process (top) whose extinction rate
is coupled to a Telegraphic noise (bottom), see Sec. IV. Parameters for this
simulation are $s=1$, $R_{0}=1.2$, $\mu_{\text{on}}=\mu_{\text{off}}=1$,
$\epsilon_{\text{on}}=0.444$ and $\epsilon_{\text{off}}=0.000$. Figure 3:
Trajectories of two coupled branching process with $s=1$, basic reproduction
numbers $R_{0A}=1.2$, $R_{0B}=0.75$ and transmutation rates
$\mu_{A}=\mu_{B}=1$, see Sec. V.

##  II Basic Branching Process

At the center of our models is the Master Equation for a branching process
Garcia-Millan  _et al._ (2018). In order to use a consistent language, we
refer to individuals being created or becoming infected in branching events.
They are then present in the population. We will say that they disappear from
the population when an extinction occurs. We hope that the reader can
translate this language to their application by replacing individual with
particle, signal, … and population with system, neural network, … according to
their requirements.

Let the population size $N(t)$ denote the number of individuals present at
time $t$, with initial condition $N(0)=1$, and $P(N(t)=n)$ the probability
that there are $n$ individuals present at time $t$. Then, the probabilistic
dynamics of the basic branching process are described by the master equation
Garcia-Millan  _et al._ (2018)

$\displaystyle{\partial_{t}}P(N;t)=$
$\displaystyle\,s\sum\limits_{k=0}^{\infty}p_{k}(N-k+1)P(N-k+1;t)$
$\displaystyle\,-sNP(N;t),$ (3)

where $p_{k}$ is the offspring distribution and $s$ is the overall event rate.
In particular, there are two probabilistic components: 1) $s$ is the parameter
for an exponentially distributed waiting time until a branching event occurs,
and 2) at a branching event of an individual, the offspring distribution
$p_{k}$ determines by how many individuals it is replaced, i.e. the original
individual does not continue to exist alongside the $k$ new individuals.
Alternatively, one could say that $k-1$ new individuals are created while the
original individual continues to be present.

The independence of the processes implies that the product $sp_{k}$ equals the
event rate for an individual to be replaced by $k$ other individuals. In the
case $k=0$, we denote by $\epsilon:=sp_{0}$ the extinction rate, i.e. the rate
of the exponential distribution that determines the time when an individual
disappears from the population without producing any offspring. In the present
work, we consider branching dynamics for which $\epsilon$ fluctuates in time.
To clarify the rôle of the extinction rate $\epsilon$, we rewrite Eq. (II) as

$\displaystyle{\partial_{t}}P(N;t)$ $\displaystyle=$
$\displaystyle\,s\sum\limits_{k=2}^{\infty}p_{k}\Big{\\{}(N-k+1)P(N-k+1;t)-NP(N;t)\Big{\\}}$
$\displaystyle\,+\epsilon\Big{\\{}(N+1)P(N+1;t)-NP(N;t)\Big{\\}}\,.$ (4)

In our approach, there is no modelling of a healthy population and there is no
saturation of a population with infected individuals. In particular, there is
no upper bound to the number of infected individuals. This is because we want
to identify the effect of the noise on the extinction rate without getting
lost in the too many details and model parameters.

The time-homogeneous branching process described by the Master Equation (II)
has been studied before Harris (1963); Garcia-Millan  _et al._ (2018). The
temporal evolution of the expected number of infected individuals,

$\mathbb{E}\left[N(t)\right]=\sum_{N=0}^{\infty}NP(N;t)\,,$ (5)

follows from (II),

$\displaystyle{\partial_{t}}\mathbb{E}\left[N(t)\right]=s(R_{0}-1)\mathbb{E}\left[N(t)\right],$
(6)

with $R_{0}$ as defined in Eq. (2). The solution
$\mathbb{E}\left[N(t)\right]=\mathchoice{e^{s(R_{0}-1)t}}{\operatorname{exp}\\!\left(s(R_{0}-1)t\right)}{\operatorname{exp}\\!\left(s(R_{0}-1)t\right)}{\operatorname{exp}\\!\left(s(R_{0}-1)t\right)}$
illustrates the important role of the basic reproduction number $R_{0}$: if
$R_{0}<1$, the expected number of individuals decreases over time (subcritical
case), while it increases exponentially if $R_{0}>1$ (supercritical case). The
case $R_{0}=1$ is called the critical point as the expected number of
individuals stays constant in time.

The model described by Eq. (II) does not rely on the law of large numbers,
i.e. it is valid for small numbers and even for one individual. The equation
also allows deriving non-trivial dynamics for higher moments. For example, the
equation that governs the variance $\mathbb{V}[N(t)]$ of the number of
infected individuals is

$\displaystyle{\partial_{t}}\mathbb{V}[N(t)]=2s(R_{0}-1)\mathbb{V}[N(t)]+s\mathbb{E}\left[(K-1)^{2}\right]\mathbb{E}\left[N(t)\right].$
(7)

Here the appearance of $\mathbb{E}\left[(K-1)^{2}\right]$ implies that the
second moment of the offspring distribution affects the dynamics too – not
just $R_{0}$.

The model in Eq. (II), and therefore also Eqs. (II), (6) and (7), assumes a
static setup, i.e. branching always happens with the same rates and the same
offspring distribution. However, this assumption may be unrealistic for many
applications. What if event rates and offspring distributions fluctuate in
time? This is the topic of the next two sections, after which we also consider
the case where two populations with different basic reproduction numbers
$R_{0}$ interact.

##  III Branching coupled to an Ornstein-Uhlenbeck process

As a first example of noisy branching processes, we couple the extinction rate
$\epsilon$ in Eq. (II) to an Ornstein-Uhlenbeck (OU) process

$\displaystyle\epsilon(t)=sp_{0}+\lambda y(t),$ (8)

where the rate $y(t)$ is governed by

$\displaystyle{\partial_{t}}y(t)=-\beta y(t)+\eta(t).$ (9)

Here, $\eta(t)$ is a Gaussian white noise with mean
$\mathbb{E}\left[\eta(t)\right]=0$ and correlator
$\mathbb{E}\left[\eta(t)\eta(t^{\prime})\right]=2D\delta(t-t^{\prime})$. The
dimensionless parameter $\lambda$ is the coupling strength and $\beta$ is the
return rate. The persistence time $\beta^{-1}$ induces temporal correlations,
or a memory, in the noise $y(t)$ in a similar way to active fluctuations in
the motion of active Ornstein-Uhlenbeck particles Dabelow  _et al._ (2019);
Walter  _et al._ (2021), Eq. (18). In principle, $\epsilon(t)$ may become
negative, which would render the process ill-defined. In numerical
simulations, we can guard against that by monitoring the value of $\epsilon$
and replacing it by $0$ whenever it becomes negative. For the parameters
considered below, this is exceedingly rare, affecting a single realisation in
well over $10^{10}$. An example trajectory is shown in Fig. 1.

Eq. (9) implies that the steady state distribution of $y$ is the Gaussian
distribution

$\lim_{t\to\infty}P(y;t)=\sqrt{\frac{\beta}{2\pi
D}}\mathchoice{e^{-\frac{\beta
y^{2}}{2D}}}{\operatorname{exp}\\!\left(-\frac{\beta
y^{2}}{2D}\right)}{\operatorname{exp}\\!\left(-\frac{\beta
y^{2}}{2D}\right)}{\operatorname{exp}\\!\left(-\frac{\beta
y^{2}}{2D}\right)}\,$ (10)

and therefore the time average equals $\mathbb{E}_{\tau}\left[y\right]=0$. The
_noisy branching process_ Eq. (II) with Eq. (8) is described by the following
master equation:

$\displaystyle{\partial_{t}}P(N;t)=$
$\displaystyle\,s\sum\limits_{k=0}^{\infty}p_{k}(N-k+1)P(N-k+1;t)$
$\displaystyle+\lambda y(N+1)P(N+1;t)$ $\displaystyle-(s+\lambda y)NP(N;t)\,.$
(11)

The additional contribution $y(t)$ to the extinction rate $\epsilon(t)$ is
affecting all individuals equally, so that rather than being reduced by the
law of large numbers, the effect of $y(t)$ grows linearly with the population
size.

What is the effect of this noise on the dynamics of the branching process? A
mean-field approximation predicts that this perturbation does not have any
impact, because in the mean-field approach all occurrences of $y(t)$ are
replaced by its mean $\mathbb{E}_{\tau}\left[y\right]=0$ and therefore the
mean-field expected number of secondary cases equals $R_{0}$, Eq. (2). In
other words, mean-field theory predicts that $R_{0}=\sum kp_{k}=1$ results in
a critical process.

However, closer inspection reveals that with OU noise, the offspring
distribution $P(K=k)=p_{k}$, Eq. (1), has effectively become time-dependent.
To see this, we regard the branching process as a collection of simultaneous,
independent, exponentially-distributed waiting processes — one process for
each $K\in\mathbb{N}_{0}$. Because they are independent, the waiting time
until the first of these events occurs is exponentially distributed with rate
$s+\lambda y(t)$. Which of the processes actually occurs first can be answered
probabilistically by calculating the ratio of the rate of that process divided
by the sum of the rates of all simultaneous processes:

$\displaystyle P(K=k)=$ $\displaystyle\,\frac{sp_{k}}{s+\lambda y}$
$\displaystyle\text{ for }k\neq 0,$ (12a) $\displaystyle P(K=0)=$
$\displaystyle\,\frac{sp_{0}+\lambda y}{s+\lambda y}$ $\displaystyle\text{ for
}k=0,$ (12b)

which satisfies normalisation, $\sum_{k\geq 0}P(K=k)=1$, and describes the
effective offspring distribution given a rate $y=y(t)$. Hence, the time-
dependent expected number of secondary cases equals

$\displaystyle\mathbb{E}\left[K(t)\right]=\frac{s}{s+\lambda y(t)}R_{0},$ (13)

with $R_{0}=\sum kp_{k}$, Eq. (2). The effect of $y(t)$ in Eq. (13) is not
symmetric about $0$, as can be seen by expanding $s/(s+\lambda y)=1-(\lambda
y/s)+(\lambda y/s)^{2}-\ldots$ Taking the expectation over $y$, suggests
$\mathbb{E}\left[K(t)\right]=R_{0}(1+(\lambda/s)^{2}\mathbb{E}\left[y^{2}\right]+\ldots)$.
As shown in Appendix B.2, the average
$\mathbb{E}_{y}\left[\mathbb{E}\left[K(t)\right]\right]$ of
$\mathbb{E}\left[K(t)\right]$ over the stationary distribution of $y$, Eq.
(10), effectively the time-average of $\mathbb{E}\left[K(t)\right]$, can be
calculated in closed form

$\displaystyle\mathbb{E}_{y}\left[\mathbb{E}\left[K(t)\right]\right]=\frac{s}{\lambda}\sqrt{\frac{2\beta}{D}}{{\mathcal{D}_{+}}}\\!\left(\frac{s}{\lambda}\sqrt{\frac{\beta}{2D}}\right)R_{0}\,,$
(14)

where $\mathcal{D}_{+}$ denotes the Dawson function, defined in (53).

Figure 4:  Expected number of infected individuals
$\mathbb{E}\left[N(t)\right]$ of a branching process driven by an Ornstein-
Uhlenbeck noise estimated from numerical simulations (symbols) based on
$10^{7}$ iterations. The parameters of the noise in each panel are as follows:
$\lambda=0.05$ (top row); $\lambda=0.1$ (bottom row); $\beta=0.5$, $D=0.125$
(left column); $\beta=1$, $D=1$ (middle column); $\beta=1$, $D=2$ (right
column). For each set of noise parameters, we can choose different values of
$R_{0}$ such that the population has supercritical (red symbols, black lines),
critical (orange symbols, grey lines), and subcritical dynamics (blue symbols,
pink lines). Our numerical results are in agreement with Eq. (16). In
particular, we find values $R_{0}<1$ such that the population displays
supercritical behaviour due to the external noise.

Demanding the expected offspring number
$\mathbb{E}_{y}\left[\mathbb{E}\left[K(t)\right]\right]$ to be unity at the
critical point, produces

$\displaystyle R_{0}=1-\frac{D\lambda^{2}}{\beta
s^{2}}-2\left(\frac{D\lambda^{2}}{\beta
s^{2}}\right)^{2}+\mathcal{O}\left(\left(\frac{D\lambda^{2}}{\beta
s^{2}}\right)^{3}\right)\ ,$ (15)

which generally is less than unity, as can also be gleaned from
$R_{0}=1-(\lambda/s)^{2}D/\beta+\ldots$ using the expansion discussed after
Eq. (13) and $\mathbb{E}\left[y^{2}\right]=D/\beta$ from Eq. (10).

However, (15) needs further scrutiny as it is based on the wrong assumption,
as we will explain now. The process is at the critical point when the average
number of offspring spawned per reproductive event is unity. However,
$\mathbb{E}_{y}\left[\mathbb{E}\left[K(t)\right]\right]$ in (15) is an average
over the stationary distribution of $y$, assuming that the same number of
reproductive events take place at any such value. Because of the temporal
correlations in $y(t)$, however, episodes of high extinction typically occur
when the population size is small anyway. As a result, fewer reproductive
events are affected by high extinction rates than by low extinction rates.

Using a Doi-Peliti field theory, which is derived and explained in Appendix
B.1, we can calculate the expected population size directly, producing the
final result

$\displaystyle\mathbb{E}\left[N(t)\right]$
$\displaystyle=\mathchoice{e^{-\left(s(1-R_{0})+\frac{\beta}{2}\right)t}}{\operatorname{exp}\\!\left(-\left(s(1-R_{0})+\frac{\beta}{2}\right)t\right)}{\operatorname{exp}\\!\left(-\left(s(1-R_{0})+\frac{\beta}{2}\right)t\right)}{\operatorname{exp}\\!\left(-\left(s(1-R_{0})+\frac{\beta}{2}\right)t\right)}\left(\cosh\left(\frac{\beta
t}{2}\sqrt{1+\frac{4D\lambda^{2}}{\beta^{3}}}\right)\right.$
$\displaystyle\left.+\frac{1}{\sqrt{1+\frac{4D\lambda^{2}}{\beta^{3}}}}\sinh\left(\frac{\beta
t}{2}\sqrt{1+\frac{4D\lambda^{2}}{\beta^{3}}}\right)\right)$
$\displaystyle+\mathcal{O}\left(\left(\frac{D\lambda^{2}}{\beta^{3}}\right)^{2}\right)\,.$
(16)

The basic reproduction number $R_{0}$ that results in asymptotically constant
expected population size,
$\infty>\lim_{t\to\infty}\mathbb{E}\left[N(t)\right]>0$, according to (16), is
the one that makes the exponent vanish for all $t$, namely
$s(1-R_{0})=\beta/2(\sqrt{1+4D\lambda^{2}/\beta^{3}}-1)$, or

$\displaystyle R_{0}$
$\displaystyle=1+\frac{\beta}{2s}\left({1-\sqrt{1+\frac{4D\lambda^{2}}{\beta^{3}}}}\right)+\mathcal{O}\left(\left(\frac{D\lambda^{2}}{\beta^{3}}\right)^{2}\right)$
(17a)
$\displaystyle=1-\frac{D\lambda^{2}}{\beta^{2}s}+\mathcal{O}\left(\left(\frac{D\lambda^{2}}{\beta^{3}}\right)^{2}\right).$
(17b)

Remarkably, in any non-trivial setup, this critical $R_{0}$ is less than
unity. The average number of secondary infections of an isolated individual
not subject to noise needed to sustain an outbreak, is thus less than unity.
The explanation for this counter-intuitive result is similar to the reason why
Eq. (15) is based on the wrong assumptions: Because the noise is correlated,
in general population sizes experiencing low extinction rates are larger than
those experiencing large extinction rates. The noise correlator of the the
Ornstein-Uhlenbeck process (8) is van Kampen (1992)

$\mathbb{E}\left[y(t)y(t^{\prime})\right]=\frac{D}{\beta}\mathchoice{e^{-\beta|t-t^{\prime}|}}{\operatorname{exp}\\!\left(-\beta|t-t^{\prime}|\right)}{\operatorname{exp}\\!\left(-\beta|t-t^{\prime}|\right)}{\operatorname{exp}\\!\left(-\beta|t-t^{\prime}|\right)}\,,$
(18)

whose characteristic time $\beta^{-1}$ is a measure of the persistence of
active fluctuations. Although the noise has vanishing mean, its effect is
biased towards larger population sizes. In addition, Eq. (14) does not
incorporate the change in frequency with which events take place overall — at
times of high extinction rates, more events take place than at times of low
extinction rates.

Our field theoretic result Eq. (17a) provides a systematic expansion of the
critical $R_{0}$ in orders of $\lambda$ and is subtly different from the _ad
hoc_ result (15), as the denominator of the leading order correction in (15)
is $\beta s^{2}$ rather than $\beta^{2}s$ in Eq. (17a).

To test Eq. (17a) numerically, we have performed Monte-Carlo simulations to
estimate the critical $R_{0}$ as shown in Fig. 5. Given the other parameter
values, a fairly small range of $\lambda$ is available, as otherwise
$\epsilon(t)$ might stray in to negative territory. The perturbative result
Eq. (17a) is in excellent agreement with the numerics. Fig. 4 shows some
examples of subcritical, near critical and supercritical trajectories of
$\mathbb{E}\left[N(t)\right]$.

Figure 5:  Phase diagram of branching process coupled to an Ornstein-Uhlenbeck
process for $s=1$ and different values of $D$ and $\beta$. We estimated
numerically the critical $R_{0}$ for some values of $\lambda$ (symbols), see
Fig. 4. Critical values separate subcritical (below) and supercritical regimes
(above). Solid lines (labelled as "theory") indicate the critical $R_{0}$ for
a given $\lambda$ as approximated by Eq. (17a). These theoretical curves are
first-order approximations in orders of $D\lambda^{2}/\beta^{3}$, which
explains the deviation of the numerical estimates from the theory. We expect
larger deviations between true values and our approximation (17a) for larger
ratios of $D/\beta^{3}$. The mean-field approximation in (15) (dashed line) is
common to the three sets of values since $D/\beta$ is the same in all three
cases.

##  IV Extinction Rate coupled to Telegraphic noise

As a second example of how noise can shift the critical point in unexpected
ways, we consider a branching process in which the extinction rate switches
spontaneously between two values. We call this random switching telegraphic
noise Horsthemke and Lefever (1989) and write,

$\epsilon(t)=sp_{0}+T(t)\,,$ (19)

where the binary random variable $T$ switches between the two values
$\epsilon_{\text{on}}>0$ and $\epsilon_{\text{off}}=0$. An example trajectory
is shown in Fig. 2. Analogously to Eq. (13), we can immediately deduce the
time-dependent expected number of secondary cases

$\displaystyle\mathbb{E}\left[K(t)\right]=\frac{s}{s+T(t)}R_{0}\,,$ (20)

with $R_{0}$ as defined by the distribution $p_{k}$, Eq. (2). The waiting
times between switching events are exponentially distributed with rates
$\mu_{\text{off}}$ to go from $\epsilon_{\text{on}}$ to
$\epsilon_{\text{off}}$, and $\mu_{\text{on}}$ to go from state
$\epsilon_{\text{off}}$ to $\epsilon_{\text{on}}$,

$\displaystyle\epsilon_{\text{on}}\xrightarrow[]{\mu_{\text{off}}}\epsilon_{\text{off}}\,,$
$\displaystyle\epsilon_{\text{off}}\xrightarrow[]{\mu_{\text{on}}}\epsilon_{\text{on}}\,.$
(21)

The switching rates $\mu_{\text{on}}$ and $\mu_{\text{off}}$ induce temporal
correlations in the noise $T(t)$ in the same way as the active fluctuations in
the motion of run-and-tumble particles Dhar  _et al._ (2019); Garcia-Millan
and Pruessner (2021), Eq. (27).

Figure 6:  Expected number of infected individuals
$\mathbb{E}\left[N(t)\right]$ of a branching process driven by a dichotomous
Telegraphic noise estimated from numerical simulations (symbols) based on
$10^{7}$ iterations. The parameters of the noise are as follows: equal
switching rates $\mu_{\text{on}}=\mu_{\text{off}}=1$ (top row), and unequal
switching rates $\mu_{\text{on}}=1$ and $\mu_{\text{off}}=2$ (bottom row). For
each set of noise parameters, we can choose different values of
$\epsilon_{\text{on}}$ such that the population has supercritical (red
symbols, black lines), critical (orange symbols, grey lines), and subcritical
dynamics (blue symbols, pink lines). In all instances,
$\epsilon_{\text{off}}=0$. Our numerical results are in agreement with Eq.
(71b).

The master equation describing the telegraphic noise only is

$\displaystyle{\partial_{t}}P(\epsilon_{\text{on}};t)=$
$\displaystyle\mu_{\text{on}}P(\epsilon_{\text{off}};t)-\mu_{\text{off}}P(\epsilon_{\text{on}};t),$
(22a) $\displaystyle{\partial_{t}}P(\epsilon_{\text{off}};t)=$
$\displaystyle\mu_{\text{off}}P(\epsilon_{\text{on}};t)-\mu_{\text{on}}P(\epsilon_{\text{off}};t)$
(22b)

where $P(T;t)$ is the probability distribution of $T$ at time $t$. From Eq.
(22) we derive the expected value of $T$ at time $t>0$:

$\displaystyle\mathbb{E}\left[T(t)\right]=\mathbb{E}\left[T(0)\right]e^{-(\mu_{\text{on}}+\mu_{\text{off}})t}+\frac{\epsilon_{\text{on}}\mu_{\text{on}}}{\mu_{\text{on}}+\mu_{\text{off}}}.$
(23)

In this set-up, the expected number of secondary cases $\mathbb{E}[K]$, Eq.
(20), switches between two values. If one of them predicts supercritical
behaviour and the other one predicts subcritical behaviour, we cannot
immediately deduce the criticality of the population that randomly switches
between them.

One way of determining the critical point is to demand that the rate with
which offspring are _produced_ in branching events equals that with which they
go extinct. In each branching into $k$ particles, $k-1$ offspring are
produced, so that the production rate of particles is

$s\left(\sum_{k=2}^{\infty}(k-1)p_{k}\right)=s(R_{0}+p_{0}-1)$ (24)

using Eq. (2). This _production_ rate is unaffected by the state of the
system. The extinction, on the other hand, depends on the state. As the
transitioning times are exponentially distributed, the system spends on
average $1/\mu_{\text{on}}$ amount of time in the $T=\epsilon_{\text{off}}=0$
state and $1/\mu_{\text{off}}$ amount of time in the $T=\epsilon_{\text{on}}$
state. This implies that at an arbitrary point in time, the system is in state
$T=\epsilon_{\text{off}}=0$ with probability
$\mu_{\text{off}}/(\mu_{\text{on}}+\mu_{\text{off}})$ and in state
$T=\epsilon_{\text{on}}$ with probability
$\mu_{\text{on}}/(\mu_{\text{on}}+\mu_{\text{off}})$. The effective extinction
rate is therefore

$sp_{0}\frac{\mu_{\text{off}}}{\mu_{\text{on}}+\mu_{\text{off}}}+(sp_{0}+\epsilon_{\text{on}})\frac{\mu_{\text{on}}}{\mu_{\text{on}}+\mu_{\text{off}}}=sp_{0}+\frac{\epsilon_{\text{on}}\mu_{\text{on}}}{\mu_{\text{on}}+\mu_{\text{off}}}\
.$ (25)

Equating this with Eq. (24) produces the criterion for the critical point,

$\displaystyle\frac{\epsilon_{\text{on}}}{s}=(R_{0}-1)\left(1+\frac{\mu_{\text{off}}}{\mu_{\text{on}}}\right)\
.$ (26)

However, Eq. (26) does not correctly predict the critical point as generally a
larger population is affected by small extinction rates than by large
extinction rates, because $\mu_{\text{on}}$ and $\mu_{\text{off}}$ are finite,
so that the system lingers in either state. The mean field theory is expected
to describe only the case of $\mu_{\text{on}},\mu_{\text{off}}\gg s$
correctly, when the telegraphic noise changes so quickly that population size
and state of the noise become uncorrelated. In its steady state, the
telegraphic noise has a Pearson correlation coefficient $\rho$ of

$\displaystyle\rho_{T(t)T(t^{\prime})}=$
$\displaystyle\,\frac{\mathbb{E}\left[T(t)T(t^{\prime})\right]-\mathbb{E}\left[T(t)\right]\mathbb{E}\left[T(t^{\prime})\right]}{\sqrt{\mathbb{V}[T(t)]\mathbb{V}[T(t^{\prime})]}}$
$\displaystyle=$
$\displaystyle\,e^{-(\mu_{\text{on}}+\mu_{\text{off}})|t-t^{\prime}|},$ (27)

which indicates that correlations become irrelevant as $\mu_{\text{on}}$ and
$\mu_{\text{off}}$ become large compared to $s$, the other event rate of the
system. The derivation of Eq. (27) is presented in Appendix C.1.

As in Sec. III, the expectation of the number of offspring averaged over all
branching events is not a simple average of Eq. (20), as it lacks a weighting
by population size.

Figure 7: Two phase diagrams of branching process coupled to telegraphic noise
(blue and pink). The critical $R_{0}$-$\epsilon_{\textit{on}}/s$ line
separating the subcritical (above left) from the supercritical regime (below
right). Solid line: exact result, Eq. (28); dashed line: mean-field result,
Eq. (26). The red dots indicate where the critical line was verified by Monte-
Carlo simulations, see Fig. 6.

In order to capture whether outbreaks are supercritical or not, we inspect
again the expected number of infected individuals over time
$\mathbb{E}\left[N(t)\right]$. For the calculation of
$\mathbb{E}\left[N(t)\right]$, we use a Doi-Peliti field theory, derived in
Appendix C, with initial condition $N(0)=1$ and $T(0)=\epsilon_{\text{on}}$.
We determine in which parameter region $\mathbb{E}\left[N(t)\right]$ grows
over time (supercritical phase) and which it decreases (subcritical phase).
The boundary between the two regions defines the critical hypersurface

$\frac{\epsilon_{\text{on}}}{s}=(R_{0}-1)\left(1+\frac{\mu_{\textit{off}}}{s(1-R_{0})+\mu_{\textit{on}}}\right)\,,$
(28)

which is shown in Fig. 7.

The direct comparison between the mean-field approach, Eq. (26), and the exact
result, Eq. (28), in Fig. 7 shows that the mean-field approximation predicts
subcritical behaviour in regions where the dynamics are actually
supercritical, i.e. in the regions between solid and dashed lines. As
expected, Eqs. (26) and (28) coincide when
$\mu_{\text{on}},\mu_{\text{off}}\gg s$, in which case they reduce to
$\epsilon_{\text{on}}/s=(R_{0}-1)(1+\mu_{\text{off}}/\mu_{\text{on}})$.

We verified the shifted critical line using Monte-Carlo simulations, shown in
Fig. 6.

##  V Coupled Branching Processes

While in the previous sections the branching process was coupled to different
noises via a dynamic change in the extinction rate, here we study a type of
noise introduced by the interaction between different populations. In the
context of infectious diseases, we can think of population subgroups with
different susceptibility, perhaps as a matter of lifestyle, behaviour or
underlying health condition. As before, we leave various interpretations and
applications to the reader and focus on analyzing the dynamics of an example
process.

We consider a branching process that is coupled to another branching process.
Individuals from two populations $A$ and $B$ with branching probabilities
$p_{kA}$ and $p_{kB}$, Eq. (1), respectively, change from one population to
the other with transmutation rates $\mu_{A}$ (from $A$ to $B$) and $\mu_{B}$
(from $B$ to $A$),

$\displaystyle A\xrightarrow[]{\mu_{A}}\,B\,,$ $\displaystyle
B\xrightarrow[]{\mu_{B}}\,A\,.$ (29)

An example trajectory is shown in Fig. 3. The master equation of this process
involves the joint probability $P(N_{A},N_{B};t)$, where $N_{A}$ and $N_{B}$
are the number of individuals of populations $A$ and $B$ respectively at time
$t$. The master equation is made of three blocks describing each subprocess:
two independent branching processes for sub-populations $A$ and $B$, modelled
in (II) and a coupling term that models the interaction (29) between the two
populations,

$\displaystyle{\partial_{t}}P(N_{A},N_{B};t)=$
$\displaystyle{\partial_{t}}P_{A}(N_{A},N_{B};t)+{\partial_{t}}P_{B}(N_{A},N_{B};t)$
$\displaystyle+{\partial_{t}}P_{AB}(N_{A},N_{B};t)\,.$ (30)

Here, the term ${\partial_{t}}P_{A}(N_{A},N_{B};t)$ is given by (II) replacing
$N$ by $N_{A}$, $p_{k}$ by $p_{kA}$ and $P(N;t)$ by $P(N_{A},N_{B};t)$; the
term ${\partial_{t}}P_{B}(N_{A},N_{B};t)$ is given by (II) replacing $N$ by
$N_{B}$, $p_{k}$ by $p_{kB}$ and $P(N;t)$ by $P(N_{A},N_{B};t)$; and the term
${\partial_{t}}P_{AB}(N_{A},N_{B};t)$ captures the transmutation of indviduals
in (29),

$\displaystyle{\partial_{t}}P_{AB}(N_{A},N_{B};t)$
$\displaystyle=\mu_{A}\bigl{(}(N_{A}+1)P(N_{A}+1,N_{B}-1;t)-N_{A}P(N_{A},N_{B};t)\bigr{)}$
$\displaystyle\,+\mu_{B}\bigl{(}(N_{B}+1)P(N_{A}-1,N_{B}+1;t)-N_{B}P(N_{A},N_{B};t)\bigr{)}\,.$
(31)

This coupling term describes how an individual of population $A$ joins
population $B$ with rate $\mu_{A}$ and how individuals from $B$ convert to $A$
with rate $\mu_{B}$.

Figure 8:  Expected number of individuals in two coupled branching processes
$\mathbb{E}\left[N_{A}(t)\right]$ (blue symbols, black lines) and
$\mathbb{E}\left[N_{B}(t)\right]$ (orange symbols, grey lines). The
transmutation rates are $\mu_{A}=\mu_{B}=1$ (cf. Fig. 9 for $\mu_{A}=1$ and
$\mu_{B}=2$). The basic reproduction number $R_{0A}$ is above $1$ and
increases from left to right panels. The basic reproduction number $R_{0B}$ is
adjusted in each panel to illustrate supercritical processes (top row),
critical processes (middle row), and subcritical processes (bottom row). The
supercritical cases shown in the top row are incorrectly predicted to be
subcritical by the mean-field theory. Numerical estimates (symbols), based on
$2\cdot 10^{5}$ trajectories, are in good agreement with exact predictions
(lines) in Eq. (85).  Figure 9:  Expected number of individuals in two coupled
branching processes $\mathbb{E}\left[N_{A}(t)\right]$ (blue symbols, black
lines) and $\mathbb{E}\left[N_{B}(t)\right]$ (orange symbols, grey lines). The
transmutation rates are $\mu_{A}=1$ and $\mu_{B}=2$ (cf. Fig. 8 for
$\mu_{A}=\mu_{B}=1$). The basic reproduction number $R_{0A}$ is above $1$ and
increases from left to right panels. The basic reproduction number $R_{0B}$ is
adjusted in each panel to illustrate supercritical processes (top row),
critical processes (middle row), and subcritical processes (bottom row). The
supercritical cases shown in the top row are incorrectly predicted to be
subcritical by the mean-field theory. Numerical estimates (symbols), based on
$2\cdot 10^{5}$ trajectories, are in good agreement with exact predictions
(lines) in Eq. (85).

To derive the dynamics of one of the two populations, say $A$, we marginalise
the joint probability $P(N_{A},N_{B};t)$ by summing over $M$, which gives the
probability that population $A$ has $N_{A}$ individuals at time $t$,

$P(N_{A};t)=\sum_{N_{B}\geq 0}P(N_{A},N_{B};t).$ (32)

By summing over $N_{B}$ in (30) we obtain

$\displaystyle{\partial_{t}}P(N_{A};t)=$ $\displaystyle
s\sum\limits_{k=0}^{\infty}p_{kA}(N_{A}-k+1)P(N_{A}-k+1;t)$ $\displaystyle-
sN_{A}P(N_{A};t)$
$\displaystyle+\mu_{A}\bigl{(}(N_{A}+1)P(N_{A}+1;t)-N_{A}P(N_{A};t)\bigr{)}$
$\displaystyle+\mu_{B}\bigl{(}\mathbb{E}\left[N_{B}(t)|N_{A}(t)-1\right]P(N_{A}-1;t)$
$\displaystyle-\mathbb{E}\left[N_{B}(t)|N_{A}(t)\right]P(N_{A};t)\bigr{)}\,,$
(33)

which shows that, from the perspective of sub-population $A$, its dynamics can
be cast into a branching process (II) with slightly adjusted extinction rate
and an additional influx, akin to spontaneous creation. The first two terms on
the right hand side of Eq. (V) are indeed identical to the branching process
in Eq. (II), the term parameterised by $\mu_{A}$ corresponds to a spontaneous
extinction and the last term, parameterised by $\mu_{B}$, is reminiscent of a
spontaneous creation. However, the rates of the gain and loss terms of this
creation differ and depend on $\mathbb{E}\left[N_{B}(t)|N_{A}(t)\right]$,
which is a deterministic function of the stochastically varying size $N_{A}$
of sub-population $A$. It is the only term that links the dynamics of the two
sub-populations. In particular, it encapsulates the conservation of
individuals by transmutation. The branching dynamics of sub-population $B$
disappears from the dynamics of sub-population $A$ otherwise.

If the branching processes of both populations are supercritical, we expect
the coupled populations to remain supercritical, irrespective of the
transmutation, as it conserves the total population size and cannot reduce it.
Similarly if both processes are subcritical, as the transmutation cannot
increase the population size either. However, the overall dynamics is not
straightforward if the two populations lie in different criticality regimes.
Without loss of generality, we assume in the following that basic reproduction
numbers are $R_{0A}>1$ and $R_{0B}<1$, both defined by Eq. (2) with $p_{k}$
replaced $p_{kA}$ and $p_{kB}$ respectively. Is the joint population of $A$
and $B$ super- or subcritical?

A naive approach would be to consider how much time an individual spends as
part of population $A$ before joining population $B$ and vice versa. As the
waiting time between transmutations is exponentially distributed with
parameters $\mu_{A}$ and $\mu_{B}$ respectively, an individual spends on
average $1/\mu_{A}$ time in population $A$ and $1/\mu_{B}$ time in population
$B$. Thus, a weighted average of the two $R_{0}$ values is given by

$\frac{R_{0A}/\mu_{A}+R_{0B}/\mu_{B}}{1/\mu_{A}+1/\mu_{B}}=\frac{R_{0A}\mu_{B}+R_{0B}\mu_{A}}{\mu_{A}+\mu_{B}}$
(34)

and demanding that this weighted average is unity at the critical point
determines the critical hypersurface as

$\displaystyle R_{0B}=1+\frac{(1-R_{0A})\mu_{B}}{\mu_{A}},$ (35)

which is shown in Fig. 10 as dashed lines. This _ad hoc_ approximation ignores
some important details of the interaction between the two sub-populations:
Firstly, many more individuals are initiated in sub-population $A$, a bias not
accounted for by the time-averaging taken above. Secondly, whenever a
individual resides in $A$, many more branching events, namely those of its
many offspring, will be characterised by $R_{0A}$. As in Sec. IV, only in the
limit of large $\mu_{A},\mu_{B}$ with constant $\mu_{A}/\mu_{B}$, can we
expect Eq. (35) to be correct.

In order to find the critical point where the average total population size
starts displaying exponential growth, we use a Doi-Peliti field theory,
Appendix D. Given transmutation rates $\mu_{A}$ and $\mu_{B}$, the border
between the subcritical and the supercritical regime is the critical line
defined by

$\displaystyle R_{0B}=$ $\displaystyle
1+\frac{(1-R_{0A})\mu_{B}}{s(1-R_{0A})+\mu_{A}}\,,$ (36)

shown in Fig. 10 with solid lines. Indeed, the mean-field result Eq. (35) is
recovered in the limit of $\mu_{A},\mu_{B}\to\infty$ at constant
$\mu_{A}/\mu_{B}$, when $R_{0B}=1+(1-R_{0A})\mu_{B}/\mu_{A}$.

Figure 10: Two phase diagrams of coupled branching processes (blue and pink).
The critical $R_{0A}$-$R_{0B}$ line between supercritical (top right) and
subcritical (bottom left) regimes for $\mu_{A}/s=\mu_{B}/s=1$ (pink) and
$\mu_{A}/s=1$, $\mu_{B}/s=2$ (blue). A mean-field approach would predict the
critical line to equal the dashed line. The red dots indicate where the
critical line was verified by Monte-Carlo simulations, see Figs. 8 and 9.

As in the previous sections (Figs. 5 and 7), Fig. 10 shows that the mean-field
approach, Eq. (35), may estimate subcritical behaviour where an exact
calculation reveals supercritical dynamics, i.e. in the regions between solid
and dashed lines. The shift of the critical line Eq. (36) is confirmed in
Monte-Carlo simulations, shown in Figs. 8 and 9.

##  VI Discussion and Conclusion

Branching processes are used to model a variety of avalanche-like dynamics
ranging from neuronal activity to infectious diseases. One of the key points
of interest is the prediction of the criticality of the dynamics, i.e. whether
activity might diverge and continue forever or whether it will die out
eventually. Here, we show in several examples that global noise in the
branching dynamics induce correlations in the entire population which
influence the growth or decline of the population over time.

The three examples are i) Sec. III, a branching process in which the
subprocess of extinction follows an Ornstein-Uhlenbeck process, ii) Sec. IV, a
branching process in which the subprocess of extinction displays telegraphic
noise, i.e. its extinction rate randomly switches between two values and iii)
Sec. V, two coupled branching processes where individuals can convert from one
branching process and its parameters to another branching process with a
different set of parameters.

In each case, we saw that mean-field arguments fail, even when they seem to
capture much of the dynamics. In each case, correlations and fluctuations are
important to be captured correctly, something that coarse-grained
descriptions, such as SIR models, routinely ignore. Instead, we made use of
more sophisticated, field-theoretical techniques to characterise the branching
processes either perturbatively, as in the case of Ornstein-Uhlenbeck noise
(Secs. III and B) or exactly (Secs. IV, V, C and D). This approach allowed us
to determine effective critical values of $R_{0}$ that, surprisingly, are less
than unity.

###  VI.1 Failure of Mean-Field Theories

In all three examples, the effect of the noise is that the critical point
dividing divergence from decline is shifted in unexpected ways. Mean-field
approaches predict subcritical behaviour in some parameter regions where in
fact supercritical dynamics occur. The reason for the failure of the mean-
field theory is that it is based on averages that do not take the population
size into account. When the extinction rate fluctuates, typically large
populations are exposed to low extinction, and typically small populations are
exposed to high extinction rates, unless the change of the extinction is fast
compared to the process. And yet, at the critical point, as gains and losses
are balanced, we expect the typical population sizes to be identical in the
low and the high extinction rate regimes. In other words, according to this
argument, the mean-field theory may fail to characterise the average number of
offspring produced, but it should still predict the critical point correctly.
However, critical or not, small, symmetric fluctuations in the time spent in a
state of either high or low extinction rate have a disproportionate effect on
the population size: Spending additional time $\Delta t$ in a state of low
extinction creates many more offspring, given the initially large population
size, than go extinct by spending the same additional time $\Delta t$ in the
state of high extinction, or that are failed to be created when the time spent
in the low extinction rate is reduced by $\Delta t$.

A similar bias is visible in the expectation of the exponential
$\mathbb{E}[\mathchoice{e^{z_{t}}}{\operatorname{exp}\\!\left(z_{t}\right)}{\operatorname{exp}\\!\left(z_{t}\right)}{\operatorname{exp}\\!\left(z_{t}\right)}]=1+\mathbb{E}[z_{t}^{2}]+\ldots>1$
of a symmetric random walk $z_{t}$ with $\mathbb{E}[z_{t}]=0$. Somewhat _ad
hoc_ , the population size of the branching process with Ornstein-Uhlenbeck
noise, Sec. III and Sec. B, may in fact be approximated by assuming that it is
the exponential of the time integral of the instantaneous effective mass $r$
Garcia-Millan  _et al._ (2018), and correspondingly averaged,

$\displaystyle\mathbb{E}[N(t)]\approx\mathbb{E}\left[\operatorname{exp}\\!\left(-\int_{0}^{t}(r+\lambda
y(t^{\prime}))\mathrm{d}t^{\prime}\right)\right]$ (37a)
$\displaystyle=\mathchoice{e^{-rt}}{\operatorname{exp}\\!\left(-rt\right)}{\operatorname{exp}\\!\left(-rt\right)}{\operatorname{exp}\\!\left(-rt\right)}\Big{\\{}1+\frac{\lambda^{2}}{2}\mathbb{E}\left[\left(\int_{0}^{t}y(t^{\prime})\mathrm{d}t^{\prime}\right)^{2}\right]+\ldots\Big{\\}}$
(37b)

as the total, effective, instantaneous mass is $r+\lambda y_{t}$ with
$r=s(1-R_{0})$, Eq. (39), and $\mathbb{E}[y(t)]=0$. Using
$\mathbb{E}[y(t)y(t^{\prime})]=(D/\beta)\mathchoice{e^{-\beta|t-t^{\prime}|}}{\operatorname{exp}\\!\left(-\beta|t-t^{\prime}|\right)}{\operatorname{exp}\\!\left(-\beta|t-t^{\prime}|\right)}{\operatorname{exp}\\!\left(-\beta|t-t^{\prime}|\right)}$
of the Ornstein-Uhlenbeck process van Kampen (1992), this integral produces
indeed the correct first order correction, Eq. (61),

$\displaystyle\mathbb{E}[N(t)]\approx\mathchoice{e^{-rt}}{\operatorname{exp}\\!\left(-rt\right)}{\operatorname{exp}\\!\left(-rt\right)}{\operatorname{exp}\\!\left(-rt\right)}\Big{\\{}1+\frac{D\lambda^{2}}{\beta^{3}}(\mathchoice{e^{-\beta
t}}{\operatorname{exp}\\!\left(-\beta
t\right)}{\operatorname{exp}\\!\left(-\beta
t\right)}{\operatorname{exp}\\!\left(-\beta t\right)}-1+\beta
t)+\ldots\Big{\\}}\ ,$ (38)

and, apparently, all higher order corrections of the population size, Eq.
(60). Given the _ad hoc_ nature of this expression, we cannot confirm its
validity to all orders, nor do we know whether correlation functions are
correctly captured (cf. Eqs. (6) and (32) in Pausch  _et al._ , 2020).

###  VI.2 Implications for infectious disease modelling

In this paper we consider a branching process with an external noise as a
basic model of infectious disease spreading. This model of unmitigated
epidemic assumes an infinite population of susceptible individuals and hence
it does not account for elements such as immunity or saturation. However, our
results show an important aspect in disease spreading that, to our knowledge,
has not been accounted for before Andreasen (2011): _fluctuations present in
the transmission of the infection can shift the critical value of $R_{0}$
below $1$_. Therefore, we may observe an epidemic outbreak despite $R_{0}$
being smaller than $1$. This needs to be taken into account when designing
interventions aimed at containing an epidemic outbreak.

Moreover, we expose the failure of mean-field theories, which are widely used
in epidemic modelling, to predict the critical point, see discussion above.
The main reason for this failure is that mean-field theories and deterministic
models, such as the classic SIR model, are not designed for capturing
fluctuations and noise. These may provide useful approximations in the mist of
an ongoing epidemic crisis Gog and Hollingsworth (2021); Gog  _et al._ (2021);
Kucharski  _et al._ (2020); Abbott  _et al._ (2020); Riccardo  _et al._ (2020)
but lack the capacity to account for randomness. Therefore, incorporating the
stochastic nature of disease spreading in epidemiologic models will provide a
better understanding and prediction of the evolution of an outbreak Ariel and
Louzoun (2021). We leave for future work the study of the effect of
immunisation and the study of interventions that are able to contain the
outbreak as well as the calculations of observables such as the peak
prevalence (maximum value of infected individuals over time) and the final
size of the outbreak (proportion of population infected at any time during the
epidemic).

###  VI.3 Conclusion and Outlook

In practice, the failure of the mean-field theory means that a faithfully
defined critical $R_{0}$ value which is obtained by ignoring fluctuations and
correlations is unreliable. Even basic, handwaving arguments fail — we had
initially expected the noise to push the critical value of $R_{0}$ above
unity, because additional fluctuations might terminate a branching process by
wiping out the last individual of a small, highly volatile population of an
otherwise near-critical branching process. Instead, as long as the noise has a
finite correlation time, so that the system _lingers_ in a state of higher or
lower extinction rate, there is an intrinsic bias towards larger populations.
The most striking consequence of this bias is the critical basic reproduction
number $R_{0}$, in our naive definition Eq. (2), becoming less than unity.
There is no unique definition of $R_{0}$, yet in the case of unbiased
Ornstein-Uhlenbeck noise, Sec. III, we cannot think of a redefinition of
$R_{0}$ that renders its critical value unity.

Future research will focus on finding a more suitable observable or set of
observables which constitute a sufficient predictor for the criticality of the
noisy branching dynamics.

## Acknowledgements

We thank Andy Thomas for his brilliant technical support, Nanxin Wei and
Guillaume Salbreux for fruitful discussions. J.P. was supported by an EPSRC
Doctoral Prize Fellowship. R.G.M.’s work was funded by the European Research
Council under the EU’s Horizon 2020 Programme, grant number 740269.

## Author’s Contributions

All authors were involved in all aspects of the conceptualization and
investigation as well as the writing and editing of the manuscript. JP
acquired partial funding for this project through an EPSRC Doctoral Prize
Fellowship. Data visualization and other graphics were created by RGM and JP.
RGM and GP acquired computing resources through Imperial College London.

##  Appendix A Branching process field theory

We recall from Garcia-Millan  _et al._ (2018) the Doi-Peliti field theory for
the continuous-time branching process with arbitrary time-independent
offspring distribution, where $K\in\\{0,1,\dots\\}$ is the number of offspring
produced at a branching event with probability $P(K=k)=p_{k}$, Eq. (1). The
waiting times between two branching events is exponentially distributed with
rate $s$. The action of this field theory is

$\displaystyle\mathcal{A}_{\text{BP}}[\phi,\widetilde{\phi}]=$
$\displaystyle\int\mathchoice{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}\left\\{-\widetilde{\phi}\left(\partial_{t}+r\right)\phi+\sum\limits_{j=2}^{\infty}q_{j}\widetilde{\phi}^{j}\phi\right\\},$
(39a) $\displaystyle\text{with }r=$ $\displaystyle s\left(1-R_{0}\right)\text{
and }q_{j}=s\sum\limits_{k=0}^{\infty}{k\choose j}p_{k},$ (39b)

where the fields $\phi$ and $\widetilde{\phi}$ are functions of time $t$ and
the binomial coefficient is zero if $j>k$. We assume that the system is
initialised with one individual at time $t_{0}=0$, $N(t_{0})=1$, throughout.
To calculate an observable ${\mathcal{O}}$ in the dynamics of the branching
process, we perform the path integral

$\left\langle\mathcal{O}\right\rangle_{\text{BP}}=\int\mathcal{D}\phi\mathcal{D}\widetilde{\phi}\,\,\mathcal{O}\,\,\mathchoice{e^{\mathcal{A}_{\text{BP}}[\phi,\widetilde{\phi}]}}{\operatorname{exp}\\!\left(\mathcal{A}_{\text{BP}}[\phi,\widetilde{\phi}]\right)}{\operatorname{exp}\\!\left(\mathcal{A}_{\text{BP}}[\phi,\widetilde{\phi}]\right)}{\operatorname{exp}\\!\left(\mathcal{A}_{\text{BP}}[\phi,\widetilde{\phi}]\right)}\,,$
(40)

which generally involves the bilinear part in (39a) and the nonlinear
couplings $q_{j}$. However, the observables that we are concerned with in this
paper, do not involve the couplings $q_{j}$, so we do not consider them beyond
this point. The Gaussian model of the branching process follows from the
bilinear part of (39a),
$\mathcal{A}_{0}[\phi,\widetilde{\phi}]=-\int\mathchoice{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}\widetilde{\phi}\left(\partial_{t}+r\right)\phi$,
which gives the bare propagator from the Gaussian model Garcia-Millan  _et
al._ (2018),

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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\hat{=}\left\langle\phi(\omega)\widetilde{\phi}(\omega^{\prime})\right\rangle_{0}=\frac{\delta\mkern-8.0mu\mathchar
22\relax(\omega+\omega^{\prime})}{-\mathring{\imath}\omega+r}\,,$ (41)

where $\delta\mkern-8.0mu\mathchar
22\relax(\omega+\omega^{\prime})=2\pi\delta(\omega+\omega^{\prime})$ and

$\displaystyle\left\langle\bullet\right\rangle_{0}=\int\mathcal{D}\phi\mathcal{D}\widetilde{\phi}\dots\bullet
e^{\mathcal{A}_{0}[\phi,\widetilde{\phi},\dots]}.$ (42)

In (41), the frequencies $\omega$ and $\omega^{\prime}$ are reciprocal to
times $t$ and $t_{0}$ under the Fourier transform convention,

$\displaystyle\phi(\omega)=\int\mathchoice{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}\mathchoice{e^{\mathring{\imath}\omega
t}}{\operatorname{exp}\\!\left(\mathring{\imath}\omega
t\right)}{\operatorname{exp}\\!\left(\mathring{\imath}\omega
t\right)}{\operatorname{exp}\\!\left(\mathring{\imath}\omega
t\right)}\phi(t)\,,$
$\displaystyle\phi(t)=\int\mathchoice{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}\mathchoice{e^{-\mathring{\imath}\omega
t}}{\operatorname{exp}\\!\left(-\mathring{\imath}\omega
t\right)}{\operatorname{exp}\\!\left(-\mathring{\imath}\omega
t\right)}{\operatorname{exp}\\!\left(-\mathring{\imath}\omega
t\right)}\phi(\omega)\,,$ (43)

and similarly for $\widetilde{\phi}$, with $\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega=\mathrm{d}\omega/(2\pi)$.

Our general approach in the three examples illustrated in Appendices B, C and
D, where the extinction rate $sp_{0}$ is modulated by an external noise, is
the following. We first derive the action that governs the dynamics of the
external noise $\mathcal{A}_{\text{Y}}$ with Y either an Ornstein-Uhlenbeck
process (Y$=$OU), a telegraphic noise (Y$=$T) or a branching process (Y$=$BP).
Then, we derive the action $\mathcal{A}_{\text{int}}$ that describes the
interaction between the noise Y and the branching process. Merging the three
parts, the action that encapsulates all concurring processes is

$\mathcal{A}=\mathcal{A}_{\text{BP}}+\mathcal{A}_{\text{Y}}+\mathcal{A}_{\text{int}}\,.$
(44)

In each of the actions of the three subprocesses there are, or may be,
bilinear terms. We include those bilinear terms in the Gaussian model
$\mathcal{A}_{0}$ and group the rest in the perturbation $\mathcal{A}_{1}$, so
that the overall action $\mathcal{A}$ is written as

$\mathcal{A}=\mathcal{A}_{0}+\mathcal{A}_{1}\,.$ (45)

To calculate an observable, such as the expected number of infected
individuals
$\mathbb{E}\left[N(t)\right]=\left\langle\phi(t)\widetilde{\phi}(t_{0})\right\rangle$,
we then perform a perturbative expansion about the Gaussian model,

$\left\langle\mathcal{O}\right\rangle=\left\langle\int\mathcal{D}\phi\mathcal{D}\widetilde{\phi}\mathcal{D}\psi\mathcal{D}\widetilde{\psi}\ldots\,\,\mathcal{O}\,\,\mathchoice{e^{\mathcal{A}_{1}[\phi,\widetilde{\phi},\psi,\widetilde{\psi},\ldots]}}{\operatorname{exp}\\!\left(\mathcal{A}_{1}[\phi,\widetilde{\phi},\psi,\widetilde{\psi},\ldots]\right)}{\operatorname{exp}\\!\left(\mathcal{A}_{1}[\phi,\widetilde{\phi},\psi,\widetilde{\psi},\ldots]\right)}{\operatorname{exp}\\!\left(\mathcal{A}_{1}[\phi,\widetilde{\phi},\psi,\widetilde{\psi},\ldots]\right)}\right\rangle_{0}\,,$
(46)

where the fields $\psi,\widetilde{\psi}$ represent the external noise.

##  Appendix B Ornstein-Uhlenbeck Process

###  B.1 Field theory

While some noisy processes can be described by a Doi-Peliti field theory (Sec.
IV and V), where fields capture the time-dependent density of a degree of
freedom, others are easier described using a Langevin equation and the
response field formalism Martin  _et al._ (1973); Täuber (2014), where fields
represent the degree of freedom itself. The Langevin equation of the Ornstein-
Uhlenbeck process is

$\displaystyle{\partial_{t}}y=-\beta y+\eta(t)$ (47)

where $y(t)$ is the degree of freedom of the Ornstein-Uhlenbeck process,
$\beta$ is the inverse persistence time, and $\eta(t)$ is a Gaussian white
noise with mean $\mathbb{E}\left[\eta\right]=0$, and correlator
$\mathbb{E}\left[\eta(t)\eta(t^{\prime})\right]=2D\delta(t-t^{\prime})$.

Using the Janssen-DeDominicis-Martin-Siggia-Rose (JDMSR) response field
formalism Martin  _et al._ (1973); de Dominicis (1976); Janssen (1976); Täuber
(2014), the Langevin equation of the Ornstein-Uhlenbeck process, Eq. (47) can
be transformed into the action Täuber (2014)

$\displaystyle\mathcal{A}_{\text{OU}}[\psi,\widetilde{\psi}]=\int\mathchoice{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}\left\\{-\widetilde{\psi}\left({\partial_{t}}+\beta\right)\psi+D\widetilde{\psi}^{2}\right\\},$
(48)

which defines the field theory for the OU process. The field $\psi$ represents
the values of the random variable in the OU process. The auxiliary field
$\widetilde{\psi}$ is _not_ related to the $\psi$ field. It is not to be
confused with a Doi-shifted creator field. It is introduced in the JDMSR
response field formalism purely to enforce the system to obey the OU Langevin
equation, (47), Täuber (2014).

The bare propagator of the external noise is

$\leavevmode\hbox to35.34pt{\vbox to1.2pt{\pgfpicture\makeatletter\hbox{\hskip
17.67169pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{
{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-17.0717pt}{0.0pt}\pgfsys@lineto{17.0717pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\hat{=}\left\langle\psi(\omega)\widetilde{\psi}(\omega^{\prime})\right\rangle_{0}=\frac{\delta\mkern-8.0mu\mathchar
22\relax(\omega+\omega^{\prime})}{-\mathring{\imath}\omega+\beta}\,,$ (49)

and the coupling introduced by the constant $D$ in (48) is represented by the
source diagram

$\displaystyle\leavevmode\hbox to41.33pt{\vbox
to20.54pt{\pgfpicture\makeatletter\hbox{\hskip 16.6421pt\lower-6.43892pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{
{}{{}}{} {}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{17.0717pt}{0.0pt}\pgfsys@lineto{-16.04211pt}{-5.83893pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{12.79323pt}{3.93301pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$D$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{17.0717pt}{0.0pt}\pgfsys@lineto{-16.04211pt}{5.83893pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (50)

###  B.2 Mean-field approximation

The average $\mathbb{E}_{y}\left[\mathbb{E}\left[K(t)\right]\right]$ of
$\mathbb{E}\left[K(t)\right]$, Eq. (13), in the steady state of the Ornstein-
Uhlenbeck process for $y$, Eq. (10), requires us to determine the following
integral

$\displaystyle\mathbb{E}_{y}\left[\mathbb{E}\left[K(t)\right]\right]=$
$\displaystyle
sR_{0}\int_{-\infty}^{\infty}\mathchoice{\\!\mathrm{d}y\,}{\\!\mathrm{d}y\,}{\\!\mathrm{d}y\,}{\\!\mathrm{d}y\,}\frac{1}{s+\lambda
y}\sqrt{\frac{\beta}{2\pi D}}e^{-\frac{\beta y^{2}}{2D}}$ $\displaystyle=$
$\displaystyle\frac{s}{\lambda}\sqrt{\frac{\beta}{2\pi
D}}R_{0}\int_{-\infty}^{\infty}\mathchoice{\\!\mathrm{d}x\,}{\\!\mathrm{d}x\,}{\\!\mathrm{d}x\,}{\\!\mathrm{d}x\,}\frac{e^{-x^{2}}}{\frac{s}{\lambda}\sqrt{\frac{\beta}{2D}}-x}\,,$
(51)

which contains the Hilbert transform of a Gaussian. The Hilbert transform of a
function $f$ is defined as Hilbert (1912)

$\displaystyle
H[f(x)](z)=\frac{1}{\pi}\text{p.v.}\\!\\!\int_{-\infty}^{\infty}\mathchoice{\\!\mathrm{d}x\,}{\\!\mathrm{d}x\,}{\\!\mathrm{d}x\,}{\\!\mathrm{d}x\,}\frac{f(x)}{z-x}\,,$
(52)

where p.v. denotes Cauchy’s principal value. Using that the Dawson function
Dawson (1897)

$\displaystyle\mathcal{D}_{+}(z)=\mathchoice{e^{-z^{2}}}{\operatorname{exp}\\!\left(-z^{2}\right)}{\operatorname{exp}\\!\left(-z^{2}\right)}{\operatorname{exp}\\!\left(-z^{2}\right)}\int_{0}^{z}\mathchoice{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}\mathchoice{e^{t^{2}}}{\operatorname{exp}\\!\left(t^{2}\right)}{\operatorname{exp}\\!\left(t^{2}\right)}{\operatorname{exp}\\!\left(t^{2}\right)}.$
(53)

is the Hilbert transform of a Gaussian,

$H\left[\mathchoice{e^{-x^{2}}}{\operatorname{exp}\\!\left(-x^{2}\right)}{\operatorname{exp}\\!\left(-x^{2}\right)}{\operatorname{exp}\\!\left(-x^{2}\right)}\right]\left(z\right)=\frac{2}{\sqrt{\pi}}{\mathcal{D}_{+}}\\!\left(z\right)\,,$
(54)

we obtain the result in (14).

###  B.3 Branching process coupled to an Ornstein-Uhlenbeck process

We couple the Ornstein-Uhlenbeck process in (48) to the extinction rate of the
branching process in (39a), as described by the master equation, Eq. (III).
From (III), we derive the interaction term

$\displaystyle\mathcal{A}_{\text{int}}[\phi,\widetilde{\phi},\psi]=-\lambda\int\mathchoice{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}\widetilde{\phi}\phi\psi\,,$
(55)

which produces the nonlinear coupling

$\leavevmode\hbox to46.72pt{\vbox
to26.19pt{\pgfpicture\makeatletter\hbox{\hskip
23.36227pt\lower-11.98091pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{19.71236pt}{-11.38092pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.69443pt}{3.93301pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$-\lambda$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-22.76228pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (56)

The overall action is then

$\displaystyle\mathcal{A}=\mathcal{A}_{\text{BP}}[\phi,\widetilde{\phi}]+\mathcal{A}_{\text{OU}}[\psi,\widetilde{\psi}]+\mathcal{A}_{\text{int}}[\phi,\widetilde{\phi},\psi]\,,$
(57)

which combines the Doi-Peliti and JDMSR formalisms, drawing on Eqs. (39), (48)
and (55). The subprocess governed by the action
$\mathcal{A}_{\text{BP}}+\mathcal{A}_{\text{int}}$ describes a branching
processes with a factor $\psi$ that modulates the extinction process. The
expectation of any observable $\mathcal{O}$ of this subprocess is represented
by a $n$-point correlation function
$\left\langle\mathcal{O}(\psi)\right\rangle_{\text{BP}}$ that is a function of
$\psi$,

$\displaystyle\left\langle\mathcal{O}(\psi)\right\rangle_{\text{BP}}=\int\mathcal{D}[\phi,\widetilde{\phi}]\mathcal{O}e^{\mathcal{A}_{\text{BP}}[\phi,\widetilde{\phi}]+\mathcal{A}_{\text{int}}[\phi,\widetilde{\phi},\psi]}.$
(58)

The OU process renders $\psi$ a random variable with a probability
distribution given by (10). In order to take this distribution into account,
the expectation $\left\langle\mathcal{O}(\psi)\right\rangle_{\text{BP}}$ needs
to be considered as an observable for the OU process,

$\displaystyle\left\langle\mathcal{O}(\psi)\right\rangle=$
$\displaystyle\left\langle\left\langle\mathcal{O}\right\rangle_{\text{BP}}\right\rangle_{\text{OU}}$
(59a) $\displaystyle=$
$\displaystyle\int\mathcal{D}[\psi,\widetilde{\psi}]\left\langle\mathcal{O}(\psi)\right\rangle_{\text{BP}}e^{\mathcal{A}_{\text{OU}}[\psi,\widetilde{\psi}]}$
(59b) $\displaystyle=$
$\displaystyle\int\mathcal{D}[\phi,\widetilde{\phi},\psi,\widetilde{\psi}]\mathcal{O}e^{\mathcal{A}[\phi,\widetilde{\phi},\psi,\widetilde{\psi}]}.$
(59c)

In order to determine whether the dynamics are super- or subcritical, we
calculate the expected number of infected individuals $\left\langle
N(t)\right\rangle=\left\langle\phi(t)\widetilde{\phi}(0)\right\rangle$ and
look for its exponential growth and decay respectively. The extended action
(57) introduces the following loop corrections to this propagator,

$\displaystyle\left\langle\phi(t)\widetilde{\phi}(0)\right\rangle\hat{=}\,$
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to46.72pt{\vbox to15.43pt{\pgfpicture\makeatletter\hbox{\hskip
23.36227pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
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}{}\pgfsys@moveto{11.38092pt}{14.22638pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{11.38092pt}{14.22638pt}\pgfsys@lineto{2.84544pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (60a)
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-22.76228pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to49.57pt{\vbox to15.43pt{\pgfpicture\makeatletter\hbox{\hskip
23.36227pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{2.84544pt}{14.22638pt}\pgfsys@lineto{-17.07182pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{2.84544pt}{14.22638pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{25.60756pt}{14.22638pt}\pgfsys@lineto{5.69073pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{25.60756pt}{14.22638pt}\pgfsys@lineto{14.22621pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22646pt}{14.22638pt}\pgfsys@lineto{-5.69037pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22646pt}{14.22638pt}\pgfsys@lineto{2.84511pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-22.76228pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\dots$ (60b)
$\displaystyle+\,\leavevmode\hbox to46.72pt{\vbox
to29.65pt{\pgfpicture\makeatletter\hbox{\hskip
23.36227pt\lower-14.82637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{11.38092pt}{14.22638pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{11.38092pt}{14.22638pt}\pgfsys@lineto{2.84544pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope; {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{17.07182pt}{-14.22638pt}\pgfsys@lineto{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{17.07182pt}{-14.22638pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-22.76228pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\,\leavevmode\hbox
to46.72pt{\vbox to29.65pt{\pgfpicture\makeatletter\hbox{\hskip
23.36227pt\lower-14.82637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{8.536pt}{14.22638pt}\pgfsys@lineto{-11.38127pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{8.536pt}{14.22638pt}\pgfsys@lineto{0.00009pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22646pt}{-14.22638pt}\pgfsys@lineto{-5.69037pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22646pt}{-14.22638pt}\pgfsys@lineto{5.69055pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-22.76228pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\dots$ (60c)

The one-loop correction in (60a) is

$\displaystyle\leavevmode\hbox to35.34pt{\vbox
to15.43pt{\pgfpicture\makeatletter\hbox{\hskip 17.67181pt\lower-0.59999pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{11.38092pt}{14.22638pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{11.38092pt}{14.22638pt}\pgfsys@lineto{2.84544pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope; {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-17.07182pt}{0.0pt}\pgfsys@lineto{17.07182pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\hat{=}$
$\displaystyle\int\mathchoice{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}\mathchoice{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega^{\prime}\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega^{\prime}\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega^{\prime}\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega^{\prime}\,}\frac{2e^{-i\omega
t}D\lambda^{2}}{(-i\omega+r)^{2}(-i(\omega+\omega^{\prime})+r)(\omega^{\prime
2}+\beta^{2})}$ $\displaystyle=$
$\displaystyle\int\mathchoice{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}{\\!\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,}\frac{e^{-i\omega
t}D\lambda^{2}}{(-i\omega+r)^{2}(-i\omega+r+\beta)\beta}$ (61a)
$\displaystyle=$
$\displaystyle\frac{D\lambda^{2}}{\beta^{3}}e^{-rt}\Bigl{(}e^{-\beta
t}-1+\beta t\Bigr{)}\,,$ (61b)

so that the propagator is

$\displaystyle\left\langle\phi(t)\widetilde{\phi}(0)\right\rangle=$
$\displaystyle\mathchoice{e^{-rt}}{\operatorname{exp}\\!\left(-rt\right)}{\operatorname{exp}\\!\left(-rt\right)}{\operatorname{exp}\\!\left(-rt\right)}\left(1+\frac{D\lambda^{2}}{\beta^{3}}e^{-rt}\Bigl{(}e^{-\beta
t}-1+\beta t\Bigr{)}\right)$
$\displaystyle+\mathcal{O}\left(\left(\frac{D\lambda^{2}}{\beta^{3}}\right)^{2}\right)\,.$
(62)

The pre-factor of $e^{-rt}$ indicates that, in this approximation, the
critical point remains at $0=r=s(1-\mathbb{E}[K])$. Calculating only this one-
loop approximation is indeed insufficient to see a shift in the critical
point.

Calculating, on the other hand, all loop corrections in (60) is doable in
principle, but calculating a closed form expression for each of them and doing
appropriate bookkeeping is an arduous task, in particular for entangled loops
such as the ones in (60c). Instead, we find that the Dyson sum, which includes
all loop corrections of the form (60a)-(60b), and excludes entangled loops,
known as the non-approximation, provides a better approximation to the
expected number of infected individuals $\mathbb{E}\left[N(t)\right]$. This
is,

$\displaystyle\mathbb{E}\left[N(t)\right]_{\text{Dy}}\hat{=}\,$
$\displaystyle\leavevmode\hbox to35.34pt{\vbox
to1.2pt{\pgfpicture\makeatletter\hbox{\hskip 17.67169pt\lower-0.59999pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{
{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-17.0717pt}{0.0pt}\pgfsys@lineto{17.0717pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to35.34pt{\vbox to11.87pt{\pgfpicture\makeatletter\hbox{\hskip
17.67169pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{
{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{8.53569pt}{10.66978pt}\pgfsys@lineto{-6.40192pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{8.53569pt}{10.66978pt}\pgfsys@lineto{2.13408pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope; {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-17.0717pt}{0.0pt}\pgfsys@lineto{17.0717pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\,\leavevmode\hbox
to35.34pt{\vbox to11.87pt{\pgfpicture\makeatletter\hbox{\hskip
17.67169pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{
{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{2.13408pt}{10.66978pt}\pgfsys@lineto{-12.80386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
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}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{2.13408pt}{10.66978pt}\pgfsys@lineto{-4.26784pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{17.0717pt}{10.66978pt}\pgfsys@lineto{2.1341pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{17.0717pt}{10.66978pt}\pgfsys@lineto{10.66978pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-17.0717pt}{0.0pt}\pgfsys@lineto{17.0717pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\dots$ (63a)
$\displaystyle\hat{=}$ $\displaystyle{\displaystyle\int}\frac{e^{-i\omega
t}}{-i\omega+r}\sum\limits_{k=0}^{\infty}\left(\frac{D\lambda^{2}\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega}{\beta(-i\omega+r)(-i\omega+r+\beta)}\right)^{k}$ (63b)
$\displaystyle=$ $\displaystyle{\displaystyle\int}\frac{e^{-i\omega
t}(-i\omega+r+\beta)\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega}{(-i\omega+r+\beta)(-i\omega+r)-\frac{D\lambda^{2}}{\beta}}$
(63c) $\displaystyle=$
$\displaystyle\,\Theta(t)e^{-(r+\frac{\beta}{2})t}\left(\cosh\left(\frac{\beta
t}{2}\sqrt{1+\frac{4D\lambda^{2}}{\beta^{3}}}\right)\right.$
$\displaystyle\left.+\frac{1}{\sqrt{1+\frac{4D\lambda^{2}}{\beta^{3}}}}\sinh\left(\frac{\beta
t}{2}\sqrt{1+\frac{4D\lambda^{2}}{\beta^{3}}}\right)\right),$ (63d)

where from Eq. (63b) to Eq. (63c), we calculated a geometric sum of the one-
loop correction from Eq. (61). The exponential growth rate in Eq. (63d) gives
the critical point approximated by the Dyson sum,

$\displaystyle
r+\frac{\beta}{2}\left(1-\sqrt{1+\frac{4D\lambda^{2}}{\beta^{3}}}\right)=0,$
(64)

which forms a hypersurface in $r$-$\lambda$-$D$-$\beta$ space and which is
transformed into the critical hypersurface in Eq. (17a).

##  Appendix C Telegraphic noise

To cast the telegraphic noise in a field theory, we consider two species, "on"
and "off". The state of the telegraphic noise is given by the number of
particles $m$ and $n$ of species "on" and "off" respectively. Particles
transmute between the two species according to (21) such that the total number
of particles is conserved. In our case, the total number of particles is
$m+n=1$ since, initially, the telegraphic noise is
$T(t_{0})=\epsilon_{\text{on}}$.

Using a bra-ket notation, $\langle m,n|$ and $|m,n\rangle$, and ladder
operators $a$, $a^{\dagger}$, $b$, and $b^{\dagger}$ with commutators
$[a,a^{\dagger}]=1=[b,b^{\dagger}]$,
$[a,b]=[a,b^{\dagger}]=[a^{\dagger},b]=[a^{\dagger},b^{\dagger}]=0$ and
$a|m,n\rangle=m|m-1,n\rangle$, $a^{\dagger}|m,n\rangle=|m+1,n\rangle$,
$b|m,n\rangle=n|m,n-1\rangle$, and $b^{\dagger}|m,n\rangle=|m,n+1\rangle$, the
master equation (22) can be turned into an equation for the probability
generating function
$|\mathcal{M}(t)\rangle:=\sum_{\\{m,n\\}}P(m,n;t)|m,n\rangle$ Doi (1976),

$\displaystyle{\partial_{t}}|\mathcal{M}(t)\rangle=\left(\mu_{\textit{off}}\left(b^{\dagger}-a^{\dagger}\right)a+\mu_{\textit{on}}\left(a^{\dagger}-b^{\dagger}\right)b\right)|\mathcal{M}(t)\rangle$
(65)

Building on work by Peliti Peliti (1985), Eq. (65) can be turned into a field
theory for fields $\psi$, $\widetilde{\psi}$ for $m$ particles in state "on",
and $\chi$, $\widetilde{\chi}$ for $n$ particles in state "off" with action
$\mathcal{A}_{T}$,

$\displaystyle\mathcal{A}_{T}[\psi,\widetilde{\psi},\chi,\widetilde{\chi}]=\int\mathchoice{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}$
$\displaystyle\left\\{-\widetilde{\psi}(\partial_{t}+\mu_{\text{off}})\psi-\widetilde{\chi}(\partial_{t}+\mu_{\text{on}})\chi\right.$
$\displaystyle\left.+\mu_{\text{on}}\,\widetilde{\psi}\chi+\mu_{\text{off}}\,\widetilde{\chi}\psi\right\\}\,.$
(66)

The values $\epsilon_{\text{on}}$ and $\epsilon_{\text{off}}$, Eq. (19), can
be chosen arbitrarily. For simplicity, we set $\epsilon_{\text{off}}=0$.

To couple the telegraphic noise $T(t)$ to the branching process such that its
value is added to the death rate, we use the master equation from the previous
model (branching with OU noise), Eq. (III), where we replace $\lambda y$ by
$T(t)$. The value of the telegraphic noise $T(t)$ is represented by
$\epsilon_{\text{on}}\psi^{\dagger}\psi$ in the field theory. The creator
field is then Doi-shifted $\psi^{\dagger}=\widetilde{\psi}+1$, which leads to
the following term in the action

$\displaystyle\mathcal{A}_{\text{int}}[\phi,\widetilde{\phi},\psi,\widetilde{\psi}]=-\epsilon_{\text{on}}\int\mathchoice{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}{\\!\mathrm{d}t\,}\left\\{\psi\widetilde{\phi}\phi+\widetilde{\psi}\psi\widetilde{\phi}\phi\right\\}\,.$
(67)

As a result, the total extinction rate switches between values $sp_{0}$ and
$sp_{0}+\epsilon_{\text{on}}$ in intervals that are exponentially distributed
with rates $\mu_{\text{on}}$ and $\mu_{\text{off}}$, respectively. The overall
action of the process is then

$\displaystyle\begin{split}\mathcal{A}[\phi,\widetilde{\phi},\psi,\widetilde{\psi},\chi,\widetilde{\chi}]=&\\\
&\mathcal{A}_{\text{BP}}[\phi,\widetilde{\phi}]+\mathcal{A}_{T}[\psi,\widetilde{\psi},\chi,\widetilde{\chi}]+\mathcal{A}_{\text{int}}[\phi,\widetilde{\phi},\psi,\widetilde{\psi}]\,.\end{split}$
(68)

In this field theory we need the bare propagator of the driving noise,

$\displaystyle\leavevmode\hbox to35.34pt{\vbox
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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.13408pt}{0.0pt}\pgfsys@curveto{2.13408pt}{1.17862pt}{1.17862pt}{2.13408pt}{0.0pt}{2.13408pt}\pgfsys@curveto{-1.17862pt}{2.13408pt}{-2.13408pt}{1.17862pt}{-2.13408pt}{0.0pt}\pgfsys@curveto{-2.13408pt}{-1.17862pt}{-1.17862pt}{-2.13408pt}{0.0pt}{-2.13408pt}\pgfsys@curveto{1.17862pt}{-2.13408pt}{2.13408pt}{-1.17862pt}{2.13408pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-13.30182pt}{0.0pt}\pgfsys@lineto{-19.34807pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\ldots$ (69a)
$\displaystyle\hat{=}$
$\displaystyle\left\langle\psi(\omega)\widetilde{\psi}(\omega^{\prime})\right\rangle_{0}$
$\displaystyle=$ $\displaystyle\frac{\delta\mkern-8.0mu\mathchar
22\relax(\omega+\omega^{\prime})(-\mathring{\imath}\omega+\mu_{\text{off}})}{(-\mathring{\imath}\omega+\mu_{\text{on}})(-\mathring{\imath}\omega+\mu_{\text{off}})-\mu_{\text{on}}\mu_{\text{off}}}\,,$
(69b)

and the two couplings

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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.98508pt}{5.13855pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\epsilon_{\text{on}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-22.76228pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (70)

In order to identify the critical point, we calculate the expected number of
individuals,

$\displaystyle\mathbb{E}[N(t)]=$
$\displaystyle\langle\phi(t)\phi^{\dagger}(0)\psi^{\dagger}(0)\rangle$ (71a)
$\displaystyle=$
$\displaystyle\langle\phi(t)\widetilde{\phi}(0)\rangle+\langle\phi(t)\widetilde{\phi}(0)\widetilde{\psi}(0)\rangle\,.$
(71b)

While the first term in (71b) is simply

$\displaystyle\langle\phi(t)\widetilde{\phi}(0)\rangle=e^{-rt}\,,$ (72)

the second term in (71b) is more complicated and can be represented in Feynman
diagrams as a Dyson sum,

$\displaystyle\langle\phi(t)\widetilde{\phi}(0)\widetilde{\psi}(0)\rangle\hat{=}$
$\displaystyle\,\leavevmode\hbox to29.65pt{\vbox
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{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{32.00934pt}{-10.66977pt}\pgfsys@moveto{34.00124pt}{-10.66977pt}\pgfsys@curveto{34.00124pt}{-9.56967pt}{33.10944pt}{-8.67787pt}{32.00934pt}{-8.67787pt}\pgfsys@curveto{30.90924pt}{-8.67787pt}{30.01744pt}{-9.56967pt}{30.01744pt}{-10.66977pt}\pgfsys@curveto{30.01744pt}{-11.76987pt}{30.90924pt}{-12.66167pt}{32.00934pt}{-12.66167pt}\pgfsys@curveto{33.10944pt}{-12.66167pt}{34.00124pt}{-11.76987pt}{34.00124pt}{-10.66977pt}\pgfsys@closepath\pgfsys@moveto{32.00934pt}{-10.66977pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{10.66977pt}{-10.66977pt}\pgfsys@moveto{12.66167pt}{-10.66977pt}\pgfsys@curveto{12.66167pt}{-9.56967pt}{11.76987pt}{-8.67787pt}{10.66977pt}{-8.67787pt}\pgfsys@curveto{9.56967pt}{-8.67787pt}{8.67787pt}{-9.56967pt}{8.67787pt}{-10.66977pt}\pgfsys@curveto{8.67787pt}{-11.76987pt}{9.56967pt}{-12.66167pt}{10.66977pt}{-12.66167pt}\pgfsys@curveto{11.76987pt}{-12.66167pt}{12.66167pt}{-11.76987pt}{12.66167pt}{-10.66977pt}\pgfsys@closepath\pgfsys@moveto{10.66977pt}{-10.66977pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\cdots$ (73a)
$\displaystyle=$ $\displaystyle\int\frac{-\epsilon_{\text{on}}e^{-i\omega
t}(-i\omega_{1}+\mu_{\text{on}})}{(-i\omega+r)(-i(\omega-\omega_{1})+r)((-i\omega_{1}+\mu_{\text{off}})(-i\omega_{1}+\mu_{\text{on}})-\mu_{\text{off}}\mu_{\text{on}})}$
$\displaystyle\times\sum\limits_{k=0}^{\infty}\left(\int\frac{-\epsilon_{\text{on}}(-i\omega^{\prime}+\mu_{\text{on}})\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega^{\prime}}{(-i(\omega-\omega^{\prime})+r)((-i\omega^{\prime}+\mu_{\text{off}})(-i\omega^{\prime}+\mu_{\text{on}})-\mu_{\text{off}}\mu_{\text{on}})}\right)^{k}\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\mathrm{d}\mkern-6.0mu\mathchar 22\relax\omega_{1}$ (73b)
$\displaystyle=$ $\displaystyle\int\frac{-\epsilon_{\text{on}}e^{-i\omega
t}(-i\omega+r+\mu_{\text{on}})}{(-i\omega+r)\bigl{(}(-i\omega+r+\mu_{\text{off}})(-i\omega+r+\mu_{\text{on}})-\mu_{\text{off}}\mu_{\text{on}}+\epsilon_{\text{on}}(-i\omega+r+\mu_{\text{on}})\bigr{)}}\mathrm{d}\mkern-6.0mu\mathchar
22\relax\omega\,,$ (73c)

where from Eq. (73b) to Eq. (73c), a geometric sum over loop corrections is
calculated. Using the abbreviations,

$\displaystyle\epsilon=$
$\displaystyle\mu_{\text{on}}+\mu_{\text{off}}+\epsilon_{\text{on}}\,,$ (74a)
$\displaystyle\tau=$
$\displaystyle\sqrt{\epsilon^{2}-4\epsilon_{\text{on}}\mu_{\text{on}}}\,,$
(74b)

and based on Eqs. (72) and (73c), the expected number of infected individuals
can be calculated as

$\displaystyle\mathbb{E}[N(t)]=$
$\displaystyle\,\Theta(t)e^{-(r+\epsilon/2)t}\Bigl{(}\cosh(\tau t/2)$
$\displaystyle\,+\frac{\mu_{\text{on}}+\mu_{\text{off}}-\epsilon_{\text{on}}}{\tau}\sinh(\tau
t/2)\Bigr{)},$ (75)