An interacting particle system for the front of an epidemic advancing through a susceptible population Eliana Fausti and Andreas Sjmark

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An interacting particle system for the front of an epidemic

advancing through a susceptible population

Eliana Fausti and Andreas Søjmark

Abstract

We propose an interacting particle system with a moving boundary to model the spread of

an epidemic. Each individual in a given population starts from some level of shielding, given by

a non-negative real number, and this level then evolves over time according to diﬀusive dynamics

driven by independent Brownian motions. If the level of shielding gets too low, individuals will

ﬁnd themselves in ‘at-risk’ situations, as captured by proximity to a lower moving boundary,

which we call the ‘advancing front’ of the epidemic. Speciﬁcally, local time is accumulated along

this boundary and, while individuals are initially reﬂected, infection may eventually occur with

a likelihood depending on the accumulated local time as well as the intrinsic infectiousness of the

disease and the current contagiousness within the population. Our main technical contribution

is to give a rigorous formulation of this model in the form of a well-posed interacting particle

system. Moreover, we exploit the precise construction of the system to establish an important

martingale property of the infected proportion and present a result on its limiting behaviour if

the size of the population is sent to inﬁnity.

1 Introduction

Consider a given population of size nand let Itdenote the infected proportion accumulated up to

time t, let Ctdenote the currently infectious proportion at time t, and Stdenote the susceptible

population at time t, with St+It= 1.

In the classical SIR model, the evolution of these quantities is deterministic and governed by

the so-called basic reproduction number R0. This number is deﬁned as the (average) total number

of new cases caused by a single infectious individual in a population where everyone is susceptible,

and it is decomposed as

R0:= β¯

d=θ¯c¯

where θis the intrinsic transmissibility of the disease, ¯cis the rate of contact in the population, and

dis the duration of infectiousness. The SIR model then amounts to saying that Ievolves as

dtIt=βStCt,d

dtCt=d

dtIt−¯

d−1Ct,

where the term ¯

d−1Ctaccounts for recovery from infection at rate ¯

d−1. Over the duration of an

infection, we see that the SIR model produces a total amount of new infections

n(It+¯

d−It)≈nβStCt¯

d=R0St·nCt,(1)

where Rt:= R0Stis called the eﬀective reproduction number at time t.

arXiv:2210.09286v1 [math.PR] 17 Oct 2022

Taking away some of the focus from the basic reproduction number in (1), it can be instructive

to instead decompose the amount of new infections as

n(It+¯

d−It)≈nβStCt¯

d=nSt

|{z}

number of susceptibles

·θCt

|{z}

eﬀective rate of infection given contact

·¯c¯

|{z}

total contact

(2)

Here we stress that ‘contact’ just refers to contact with another individual, not necessarily contact

with a currently infectious individual, which is why Ctenters in the eﬀective rate of infection given

contact. In our model, we wish to change perspective from thinking about a general constant rate

of contact, ¯c, across the population, to instead thinking about a changing rate at which susceptibles

bring themselves in at-risk situations (being in places and behaving in ways that make infection

possible). Being at-risk at a given instant means that infection is a real possibility at that time,

but it may or may not happen. Replacing Iand Cwith their stochastic counterparts Inand Cn

from our model, the analogous version of the decomposition (2) then takes the form

nE[In

t+¯

d−In

t|¯

t]≈Ehn

i=1

|{z}

sum over suceptibles: t<τi

γ(t, Cn

| {z }

eﬀective rate of infection given at-risk

·(`i

(t+¯

d)∧τi−`i

t∧τi)

| {z }

total (at-)riskyness

|¯

ti(3)

where τiis the infection time of the ith individual and ( ¯

t)t≥0is a suitable ﬁltration to which all

the processes are adapted. For any h > 0, the increment `i

(t+h)∧τi−`i

t∧τicaptures the magnitude of

the at-risk situations that the ith susceptible individual has found him or herself in during a given

time interval [t, t +h]. The details of what we mean by this are presented in Section 1.1 below:

in short, `iis the local time of the ith individual along what we term the ‘advancing front’ of the

epidemic, An(deﬁned in (6) below).

Before turning to the ﬁner details of our model, we wish to brieﬂy discuss how one can think

of the eﬀective reproduction number for our model vis-a-vis the SIR model. In the SIR model,

when accounting for new infections in terms of the decomposition (2), we see that the eﬀective

reproduction number Rt=R0Stcorresponds to

n(It+¯

d−It) keeping ﬁxed C·+t≡1

n≈nSt·θ1

n·¯c¯

d=Rt(4)

In view of (3), the analogous expression for the eﬀective reproduction number at time tin our model

nE[In

t+¯

d−In

t|¯

t] keeping ﬁxed Cn

·+t≡1

n≈Ehn

i=1

γt, 1

n(`i

(t+¯

d)∧τi−`i

t∧τi)|¯

ti,(5)

where the advancing front An

·+t, along which the local times accumulate, is also taken to move only

in accordance with Cn

·+t≡1

n. To be precise, what we mean by keeping Cn

·+tﬁxed at 1/n is that all

further infections (generated by one initial infection) are not allowed to impact the eﬀective rate

of infection γand the advancing front An. As our model is stochastic, the conditional expectation

is needed, but we can refer to the same interpretations as in (3) to see that each term within the

expectation can be compared with the deterministic and constant quantities on the right-hand side

of (4).

1.1 The key mechanisms of the particle system

The starting point of our analysis is to represent each individual in the population by an initial

level of shielding Xi,n

0. This level of shielding is a summary measure of all relevant characteristics

such as inherent susceptibility to the disease, lifestyle, precautionary measures, and spatial distance

from areas where infection is possible. Over time, the level of shielding Xi,n

tevolves as a stochastic

diﬀerential equation driven by Brownian motion.

As the epidemic progresses, it is the individuals with lower levels of shielding that will become

infected ﬁrst. We model this by a moving boundary An

twhich represents the level of shielding above

which one is presently safe from being at-risk of infection. Since the level-of-shielding is a stochastic

process, there is randomness involved in whether an individual is about to reach this level. We refer

to An

tas the advancing front of the epidemic and take it to be of the form

t=a0+αZt

t−¯

%(t−s)In

sds(6)

for given constants a0, α, ¯

d≥0 and a given infection-to-recovery kernel %≥0 with supp(%) = [0,¯

and k%kL1= 1. Here In

tis the accumulated infected proportion over the period [0, t], deﬁned by

t=1

i=1

[0,t](τi),

where τiis the infection time of the ith individual. We will soon return to discuss the mechanism

by which infection may or may not occur given that an individual is at-risk, but right now we only

wish to highlight the following: in the deﬁnition of the advancing front (6), the infection-to-recovery

kernel %models how the impact of an infected individual starts increasing some time after the actual

infection happens, then it slows down, and ﬁnally stops aﬀecting the system altogether after ¯

dunits

of time.

Considering the recent COVID-19 pandemic, an example of a sensible infection-to-recovery

kernel could be a Weibull distribution with negligible mass after ¯

d= 15 days, cut oﬀ at that point

and normalised. This would be in line with what is typically used to model incubation time and

the severity of contagiousness during the infected period for COVID-19 infections in the literature.

As the epidemic progresses, we assume that higher levels of shielding are required to keep

individuals away from risk, because the disease is increasing its reach. This needs not just be

geographical, at a country-wide scale or similar, but it is also intended to capture the increased

reach within cities and communities where the disease continues to linger. Of course, one could also

allow for an entirely diﬀerent notion of spatial reach, if one wishes to model epidemic spread within

non-human species. The increasing reach of the disease is captured by the front Anbeing non-

decreasing in time. For example, once the disease has spread out to certain areas of a country, we

assume that it always remains possible to become infected when visiting these areas, whereas before

the epidemic had spread its reach to these areas the probability of infection was null. Naturally,

the disease may cease to increase its reach any further for some period, eﬀectively staying dormant,

but this does not mean that it has retracted its reach. Rather, it means that the current level of

contagiousness has decreased to a low level: this feature is part of what we discuss next.

Since the advancing front is part of our mechanism for determining infections, it is clear that (6)

presents a ﬁrst nonlinearity of the system. Moreover, the infection times themselves will feed into

the mechanism for infection through a separate channel, namely the eﬀective rate of infection given

that an individual is at-risk. This will depend critically on the current level of contagiousness within

the population, in the sense of how large a proportion of the population is currently infected and

what stage of the infection each individual is currently at (as modelled by the infection-to-recovery

kernel). Whenever a given path of Xi,n

treaches the moving boundary An

t, then we say that the

individual iis at-risk of infection, and we measure the magnitude of this risk in a given period of

time by the corresponding increment of the local time of Xi,n

talong the boundary. An individual

who is at-risk (meaning that Xi,n

tis at the boundary) may or may not be infected. Informally, we

model this by asking that, conditionally on a realisation of the dynamics of each individual, the

probability of infection in (t, t +h], given non-infection up time to t, is

γ(t, Cn

t)·(`i,n

t+h−`i,n

t) + o(`i,n

t+h−`i,n

t),(7)

where we have used little-O notation as h↓0. Here Cn

tis the current level of contagiousness in the

population captured by

t=Zt

t−¯

%(t−s)(In

s−In

s−¯

d) ds. (8)

As we discussed in relation to (3), the value γ(t, Cn

t) is the eﬀective rate of infection given that an

individual is presently at-risk, while the local-time increment `i,n

t+h−`i,n

tquantiﬁes the magnitude

of at-risk situations for the ith individual in the interval of time [t, t +h]. Concerning Cn

t, we stress

that this captures both how large a proportion of the population is currently infected and how

contagious these individuals are (in terms of how far into the course of the disease they are).

In Section 2, we conﬁrm that one can indeed formulate a well-posed particle system which evolves

according to the above mechanisms, up to some technical adjustments. The precise statement is

given in Theorem 2.3. We note here that it is the property (12) in Theorem 2.3 that clariﬁes the

precise sense in which (7) holds, and we note that it is Theorem 2.5 which justiﬁes (3) and (5).

We conclude this section by discussing some interesting extensions. Firstly, as it is presented

here, our model is intended for the short or medium term, meaning a period for which it is acceptable

to perform predictions without worrying about re-infection of previously infected individuals. If one

is interested in a longer term model, then this could be addressed by allowing for re-insertion of

infected individuals at some later point in time when their immunity has waned. Secondly, it could

be interesting to study interventions in the system. It may be possible to study this dynamically, by

formulating suitable stochastic control problems where individuals and or a government may control

the drift bin the dynamics for the level of shielding or the size of the eﬀective rate of infection γ.

In this way, it may be possible to capture eﬀects of diﬀerent forms of interventions.

Figure 1: A ﬁrst wave of infections amongst individuals with low levels of shielding, followed by a

dormant period turning into a gradual uptick in infections amongst individuals with higher levels of

shielding. The vertical axis represents individual levels of shielding. The horizontal axis is time. The

red curve shows the advancing front. The green curve shows the (accumulated) infected proportion.

Figure 2: A ﬁrst wave followed by a long dormant period and then a steep second wave of great

severity. The vertical axis represents individual levels of shielding. The horizontal axis is time. The

red curve shows the advancing front. The green curve shows the (accumulated) infected proportion.

1.2 Related literature

In the existing literature, the interacting particle system that is closest to the one we study here

is the following system considered in [Bar20]: it consists of nindependent Brownian motions (in

one dimesion) reﬂected oﬀ a moving boundary, which moves at a speed proportional to the sum

of the local times of each Brownian motion along this boundary. Another interesting work in this

direction is [BBF18a], which studies a (one-dimensional) SDE reﬂected oﬀ a moving boundary

which is a function of the local time of the SDE along the boundary itself (see also [BBF18b] for a

ﬁnancial application of this). These works, however, consider particles that are globally reﬂected,

thus making the analysis quite diﬀerent from what is needed to handle the notion of infection in our

system, which is what drives the moving boundary and determines the eﬀective rate of infection.

Still, one can make a preicse connection between a version of our system and a suitable extension

of the particle system from [Bar20], as we brieﬂy discuss in Remark 1 in the next section.

A rather diﬀerent, but still closely related line of work is the analysis of particle systems

approximating the one-dimensional Stefan problem for the freezing of a super-cooled liquid, see

e.g. [HLS19, LS21, NS19, NS20, CRSF]. Unlike our model, these works consider particle systems

with immedaite absorption upon ﬁrst reaching the boundary. While not concerned with particle

systems, [BS22], and later also [HM22], have recently studied a version of the aforementioned Stefan

problem with kinetic under-cooling, using that it can be analysed in terms of the law of a single

Brownian motion reﬂecting oﬀ a moving boundary proportional to the loss of mass. The loss of mass

comes from the Brownian motion being absorbed after a scalar multiple of its local time surpasses

an independent exponential random variable, a mechanism known as elastic killing.

Finally, we wish to stress that there are many interesting works in the PDE literature aimed

at modelling the spread of an epidemic, see for example [Had16], [DL15], and references therein.

We mention the two fomer works in particular because they represent well the PDE perspective on

approaches similar to ours, where the front of an epidemic is modelled through the free boundary

of a one-dimensional PDE capturing the density of a susceptible population. However, we note

that both works impose Stefan conditions at the free boundary to model the evolution of the front.

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AninteractingparticlesystemforthefrontofanepidemicadvancingthroughasusceptiblepopulationElianaFaustiandAndreasSjmarkAbstractWeproposeaninteractingparticlesystemwithamovingboundarytomodelthespreadofanepidemic.Eachindividualinagivenpopulationstartsfromsomelevelofshielding,givenbyanon-negativerealnumb...

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An interacting particle system for the front of an epidemic advancing through a susceptible population Eliana Fausti and Andreas Sjmark

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