Original: On Early Extinction and the Effect of Travelling in the SIR Model
Revised: On Early Extinction and the Effect of Travelling in the SIR Model

Original: Abstract
Revised: Abstract

Original: We consider a population protocol version of the
Revised: We consider a population protocol version of the

Original: SIR model.
Revised: SIR model.

Original: In every round, an individual is chosen uniformly at random.
Revised: In every round, an individual is chosen uniformly at random.

Original: If the individual is susceptible, then it becomes infected w.p. βI
Revised: If the individual is susceptible, then it becomes infected w.p. βI

Original: t /N , where
Revised: t /N , where

Original: I t is the number of infections at time t and N is the total number of individuals.
Revised: I t is the number of infections at time t and N is the total number of individuals.

Original: If the individual is infected, then it recovers w.p. γ , whereas, if the individual is already recovered, nothing happens.
Revised: If the individual is infected, then it recovers w.p. γ , whereas, if the individual is already recovered, nothing happens.

Original: We prove sharp bounds on the probability of the disease becoming pandemic vs extinguishing early (dying out quickly).
Revised: We prove sharp bounds on the probability of the disease becoming pandemic vs extinguishing early (dying out quickly).

Original: The probability of extinguishing early, Pr [ E ext ] , is typically neglected in prior work since most use (deterministic) differential equations.
Revised: The probability of extinguishing early, Pr [ E ext ] , is typically neglected in prior work since most use (deterministic) differential equations.

Original: Leveraging on this, using Pr [ E ext ] , we proceed by bounding the expected size of the population that contracts the disease E [ R ∞ ] .
Revised: Leveraging on this, using Pr [ E ext ] , we proceed by bounding the expected size of the population that contracts the disease E [ R ∞ ] .

Original: Prior work only calculated E (cid:2) R ∞ | E ext (cid:3) , or obtained non-closed form solutions.
Revised: Prior work only calculated E (cid:2) R ∞ | E ext (cid:3) , or obtained non-closed form solutions.

Original: We then study the two-country model also accounting for the role of Pr [ E ext ] .
Revised: We then study the two-country model also accounting for the role of Pr [ E ext ] .

Original: We assume that both countries have different infection rates β
Revised: We assume that both countries have different infection rates β

Original: ( i ) , but share the same recovery rate γ.
Revised: ( i ) , but share the same recovery rate γ .

Original: In this model, each round has two steps: First, an individual is chosen u.a.r. and travels w.p. p travel to the other country.
Revised: In this model, each round has two steps: First, an individual is chosen u.a.r. and travels w.p. p travel to the other country.

Original: Afterwards, the process continues as before with the respective infection rates.
Revised: Afterwards, the process continues as before with the respective infection rates.

Original: Finally, using simulations, we characterise the influence of p travel on the total number of infections.
Revised: Finally, using simulations, we characterise the influence of p travel on the total number of infections.

Original: Our simulations show that, depending on the β ( i ) , increasing p travel can decrease or increase the expected total number of infections E
Revised: Our simulations show that, depending on the β ( i ) , increasing p travel can decrease or increase the expected total number of infections E

Original: [ R ∞ ] .
Revised: [ R ∞ ] .

Original:  INTRODUCTION
Revised:  INTRODUCTION

Original: In this paper we consider the well-known SIR process which is used to study the spread of a contagious disease.
Revised: In this paper we consider the well-known SIR process which is used to study the spread of a contagious disease.

Original: The model was introduced in the early 20th century (Ross and
Revised: The model was introduced in the early 20th century (Ross and

Original: Hudson [1917], Kermack and McKendrick [1927]).
Revised: Hudson [1917], Kermack and McKendrick [1927]).

Original: The population is split into three compartments (or states): susceptible (S), infected (I) and recovered (R).
Revised: The population is split into three compartments (or states): susceptible (S), infected (I) and recovered (R).

Original: At the beginning of an epidemic, a small number of individuals are infected and the rest of the population is susceptible.
Revised: At the beginning of an epidemic, a small number of individuals are infected and the rest of the population is susceptible.

Original: Susceptible individuals can be infected and infected individuals can recover and become permanently immune to the disease.
Revised: Susceptible individuals can be infected and infected individuals can recover and become permanently immune to the disease.

Original: The model is often used to study the spread of diseases like
Revised: The model is often used to study the spread of diseases like

Original: COVID-19, measles, mumps and rubella.
Revised: COVID-19, measles, mumps and rubella.

Original: When an epidemic starts, the number of infected individuals increases rapidly.
Revised: When an epidemic starts, the number of infected individuals increases rapidly.

Original: At some point, the proportion of still susceptible and already recovered individuals will be such that the spread of the infection will begin to slow down to the point of complete stop.
Revised: At some point, the proportion of still susceptible and already recovered individuals will be such that the spread of the infection will begin to slow down to the point of complete stop.

Original: The model is characterised by transition rates between the (compartments/states) depending on the infection rate β and the recovery rate γ .
Revised: The model is characterised by transition rates between the (compartments/states) depending on the infection rate β and the recovery rate γ .

Original: The so-called reproduction number of an SIR process is defined as R 0 = β/γ .
Revised: The so-called reproduction number of an SIR process is defined as R 0 = β/γ .

Original: This number is equal to the expected number of secondary cases following the introduction of one infected individual into a fully susceptible population.
Revised: This number is equal to the expected number of secondary cases following the introduction of one infected individual into a fully susceptible population.

Original: This reproduction number determines how the process evolves over time and how many individuals will get infected until the infection dies out.
Revised: This reproduction number determines how the process evolves over time and how many individuals will get infected until the infection dies out.

Original: In general SIR processes behave as follows.
Revised: In general SIR processes behave as follows.

Original: If R 0 < 1 , the disease extinguishes early and the number of total infections follows an exponential distribution.
Revised: If R 0 < 1 , the disease extinguishes early and the number of total infections follows an exponential distribution.

Original: If R 0 > 1 , the number of infected individuals first grows exponentially (in expectation).
Revised: If R 0 > 1 , the number of infected individuals first grows exponentially (in expectation).

Original: If the number of infected and recovered individuals reaches a fraction of 1 − 1 / R 

Revised: If the number of infected and recovered individuals reaches a fraction of 1 − 1 / R

Original: the number of infected individuals decreases, in expectation, until it reaches zero.
Revised: the number of infected individuals decreases, in expectation, until it reaches zero.

Original: The fraction 1 − 1 / R 0 is also called herd immunity threshold .
Revised: The fraction 1 − 1 / R 0 is also called herd immunity threshold .

Original: Looking closer at the case R 0 > 1 , one can see that the total number of infections follows a bi-modal distribution.
Revised: Looking closer at the case R 0 > 1 , one can see that the total number of infections follows a bi-modal distribution.

Original: The first peak represents early extinction , meaning the process terminates early (see Section 2 for a precise definition).
Revised: The first peak represents early extinction , meaning the process terminates early (see Section 2 for a precise definition).

Original: If this unlikely event does not happen, then the process is likely to reach the herd immunity threshold and a much larger number of individuals gets infected.
Revised: If this unlikely event does not happen, then the process is likely to reach the herd immunity threshold and a much larger number of individuals gets infected.

Original: In this paper we study the SIR process as a population protocol.
Revised: In this paper we study the SIR process as a population protocol.

Original: We assume that the population has a fixed size
Revised: We assume that the population has a fixed size

Original: N and that the individuals are modelled as a finite state machine.
Revised: N and that the individuals are modelled as a finite state machine.

Original: The state space of our protocol is simple, each individual being in one of the three states S , I , or R .
Revised: The state space of our protocol is simple, each individual being in one of the three states S , I , or R .

Original: At the beginning of time t , we denote by S t , I t and R t the number of individuals susceptible to a disease, infected individuals and recovered individuals, respectively.
Revised: At the beginning of time t , we denote by S t , I t and R t the number of individuals susceptible to a disease, infected individuals and recovered individuals, respectively.

Original: At the beginning of the process t = 0 , one individual (or a small subset of them) is infected and all other individuals are susceptible.
Revised: At the beginning of the process t = 0 , one individual (or a small subset of them) is infected and all other individuals are susceptible.

Original: No individual is recovered yet.
Revised: No individual is recovered yet.

Original: The process is now defined as follows.
Revised: The process is now defined as follows.

Original: In each step a pair of individuals i, j is chosen uniformly at random and is allowed to change state.
Revised: In each step a pair of individuals i, j is chosen uniformly at random and is allowed to change state.

Original: If the individual i is currently infected, then it recovers with probability γ .
Revised: If the individual i is currently infected, then it recovers with probability γ .

Original: If i is susceptible and j is infected, then i becomes infected with probability β .
Revised: If i is susceptible and j is infected, then i becomes infected with probability β .

Original: As mentioned before, it may happen that the disease dies out quickly without ever evolving into a pandemic, i.e., without reaching the herd immunity threshold.
Revised: As mentioned before, it may happen that the disease dies out quickly without ever evolving into a pandemic, i.e., without reaching the herd immunity threshold.

Original: In this paper we calculate the probability of early extinction as a function of R 0 = β/γ
Revised: In this paper we calculate the probability of early extinction as a function of R 0 = β/γ

Original: and we present a fully rigorous analysis for such probability using coupling arguments.
Revised: and we present a fully rigorous analysis for such probability using coupling arguments.

Original: We show that, if we start with s infected individuals, the probability for early extinction
Revised: We show that, if we start with s infected individuals, the probability for early extinction

Original: Pr [ E ext ] is asymptotically (1 / R 0 ) s assuming β > γ + ε for some arbitrarily small constant ε .
Revised: Pr [ E ext ] is asymptotically (1 / R 0 ) s assuming β > γ + ε for some arbitrarily small constant ε .

Original: If β < γ − ε , then Pr [ E ext ] = 1 − o (1) .
Revised: If β < γ − ε , then Pr [ E ext ] = 1 − o (1) .

Original: We then calculate the expected size of the population that gets infected throughout the process E
Revised: We then calculate the expected size of the population that gets infected throughout the process E

Original: [ R ∞ ] .
Revised: [ R ∞ ] .

Original: Note that R ∞ denotes the number of recovered individuals at the end of the process.
Revised: Note that R ∞ denotes the number of recovered individuals at the end of the process.

Original: The recovered state is the only terminal state and equals the number of individuals that were infected throughout the process.
Revised: The recovered state is the only terminal state and equals the number of individuals that were infected throughout the process.

Original: We first show that E
Revised: We first show that E

Original: [ R ∞ | E ext ] is close to zero.
Revised: [ R ∞ | E ext ] is close to zero.

Original: Then we show, conditioning on E ext , that E (cid:2) r ∞ | E ext (cid:3) =
Revised: Then we show, conditioning on E ext , that E (cid:2) r ∞ | E ext (cid:3) =

Original: (cid:0) 1 + W (cid:0)
Revised: (cid:0) 1 + W (cid:0)

Original: [Equation]
Revised: [Equation]

Original: where r ∞ =
Revised: where r ∞ =

Original: R ∞ /N .
Revised: R ∞ /N .

Original: Combining our results, we show that, without any conditioning,
Revised: Combining our results, we show that, without any conditioning,

Original: [Equation]
Revised: [Equation]

Original: Many of the results found in the literature are based on firstorder methods (mean field approaches)
Revised: Many of the results found in the literature are based on firstorder methods (mean field approaches)

Original: (e.g., Kröger and
Revised: (e.g., Kröger and

Original: Schlickeiser [2020], Bicher and Popper [2013]) and they give an expression similar to our term for E (cid:2) r ∞ | E ext (cid:3) as an estimation of E [ r ∞ ] .
Revised: Schlickeiser [2020], Bicher and Popper [2013]) and they give an expression similar to our term for E (cid:2) r ∞ | E ext (cid:3) as an estimation of E [ r ∞ ] .

Original: This is due to the fact that, with a mean field approach, the process is regarded as deterministic, which means that early extinctions cannot happen for R 0 > 1 .
Revised: This is due to the fact that, with a mean field approach, the process is regarded as deterministic, which means that early extinctions cannot happen for R 0 > 1 .

Original: Furthermore, such approach neglects the variance of the process, possibly crucial in the analysis of random processes (see Berenbrink et al.
Revised: Furthermore, such approach neglects the variance of the process, possibly crucial in the analysis of random processes (see Berenbrink et al.

Original: [2017] for a majority type process where the expected change is the same, but the variance in the process determines the convergence time).
Revised: [2017] for a majority type process where the expected change is the same, but the variance in the process determines the convergence time).

Original: Although, it has been observed through simulations that the deterministic and stochastic processes differ in terms of the total number of infections (e.g., Fig.13 of Allen and Burgin [2000]).
Revised: Although, it has been observed through simulations that the deterministic and stochastic processes differ in terms of the total number of infections (e.g., Fig.13 of Allen and Burgin [2000]).

Original: In the second part of our paper we consider a two-country setting: we show through simulations how travelling affects the spread of the disease.
Revised: In the second part of our paper we consider a two-country setting: we show through simulations how travelling affects the spread of the disease.

Original: A possible ban, or restriction, on travelling has been considered and applied by governments in some cases, especially with respect to the current pandemic.
Revised: A possible ban, or restriction, on travelling has been considered and applied by governments in some cases, especially with respect to the current pandemic.

Original: Such measure falls under the label of nonpharmaceutical interventions (NPI).
Revised: Such measure falls under the label of nonpharmaceutical interventions (NPI).

Original: We introduce an extension of the model where each individual resides in one of two countries.
Revised: We introduce an extension of the model where each individual resides in one of two countries.

Original: We assume that both countries can have different infection rates, β (1)
Revised: We assume that both countries can have different infection rates, β (1)

Original: ̸
Revised: ̸

Original: = β (2) , and each country has population size N (1) , N (2) with N = N (1)
Revised: = β (2) , and each country has population size N (1) , N (2) with N = N (1)

Original: + N (2) .
Revised: + N (2) .

Original: The process is similar to our original process.
Revised: The process is similar to our original process.

Original: First, an individual i is chosen uniformly at random among the whole population of both countries (with probability 1 /N ).
Revised: First, an individual i is chosen uniformly at random among the whole population of both countries (with probability 1 /N ).

Original: The chosen individual then travels to the other country with a probability of p travel and remains in the current country w.p.
Revised: The chosen individual then travels to the other country with a probability of p travel and remains in the current country w.p.

Original: 1 − p travel .
Revised: 1 − p travel .

Original: After this, one of the two countries is chosen with a probability proportional to its population and one step of the single-country process is performed in that country.
Revised: After this, one of the two countries is chosen with a probability proportional to its population and one step of the single-country process is performed in that country.

Original: We study, via simulations, the connection between p travel and the total number of infections over time.
Revised: We study, via simulations, the connection between p travel and the total number of infections over time.

Original: We obtain interesting and surprising insights.
Revised: We obtain interesting and surprising insights.

Original: For example, one would think that increasing p travel increases the total number of infections.
Revised: For example, one would think that increasing p travel increases the total number of infections.

Original: While this is generally true for small values of p travel , (though for some values of β and γ , not even this is true)
Revised: While this is generally true for small values of p travel , (though for some values of β and γ , not even this is true)

Original: it turns out that this does not hold if one country has
Revised: it turns out that this does not hold if one country has

Original: [Equation]
Revised: [Equation]

Original: /γ > 1 and the other country has sufficiently small
Revised: /γ > 1 and the other country has sufficiently small

Original: [Equation]
Revised: [Equation]

Original: < 1 .
Revised: < 1 .

Original: In that case for large enough values of p travel , the total number of infections is decreased.
Revised: In that case for large enough values of p travel , the total number of infections is decreased.

Original: Our simulations suggest that early extinction affects the size of the epidemic, leading its average to be reduced and its variance to increase.
Revised: Our simulations suggest that early extinction affects the size of the epidemic, leading its average to be reduced and its variance to increase.

Original: 1.1 RELATED WORK
Revised: 1.1 RELATED WORK

Original: The SIR model is a very basic mathematical model for the spread of diseases (e.g. it does not model birth and death of individuals, the latter resulting from the illness or other reasons) and a very large number of generalizations were suggested in the literature.
Revised: The SIR model is a very basic mathematical model for the spread of diseases (e.g. it does not model birth and death of individuals, the latter resulting from the illness or other reasons) and a very large number of generalizations were suggested in the literature.

Original: Due to the vast literature, we will only discuss results in the SIR comparable to our results and we will concentrate on analytical results.
Revised: Due to the vast literature, we will only discuss results in the SIR comparable to our results and we will concentrate on analytical results.

Original: There are also many compartmental models similar to the SIR model.
Revised: There are also many compartmental models similar to the SIR model.

Original: See
Revised: See

Original: Li [2018], Daley [1999] for a nice overview about different mathematical models.
Revised: Li [2018], Daley [1999] for a nice overview about different mathematical models.

Original: The SIR model was introduced in 1927 by Kermack and
Revised: The SIR model was introduced in 1927 by Kermack and

Original: McKendrick [1927], building on work by Ross and Hudson [1917].
Revised: McKendrick [1927], building on work by Ross and Hudson [1917].

Original: The authors also introduced the well-known set of non-linear ordinary differential equations (which we introduce in this paper too) to describe the spread of the disease.
Revised: The authors also introduced the well-known set of non-linear ordinary differential equations (which we introduce in this paper too) to describe the spread of the disease.

Original: The authors showed the so-called threshold theorem, which predicts the critical fraction of susceptible individuals in the population that must be exceeded if an epidemic is to occur.
Revised: The authors showed the so-called threshold theorem, which predicts the critical fraction of susceptible individuals in the population that must be exceeded if an epidemic is to occur.

Original: The authors show that s ∞
Revised: The authors show that s ∞

Original: = 1 − r ∞ =
Revised: [Equation]

Original: s 

Revised: s

Original: e
Revised: e

Original: [Equation]
Revised: [Equation]

Original: R 0 ( r ∞ − r 0 ) , where s ∞ = | S ∞ | /N
Revised: R 0 ( r ∞ − r 0 ) , where s ∞ = | S ∞ | /N

Original: (fraction of susceptible individuals as time goes to infinity), r ∞ =
Revised: (fraction of susceptible individuals as time goes to infinity), r ∞ =

Original: | R ∞ | /N , s t =
Revised: | R ∞ | /N , s t =

Original: | S t | /N , and r t = | R t | /N .
Revised: | S t | /N , and r t = | R t | /N .

Original: This transcendental equation has a solution in terms of the Lambert W function:
Revised: This transcendental equation has a solution in terms of the Lambert W function:

Original: [Equation]
Revised: [Equation]

Original: Most of the publications investigating the SIR model use numerical methods, employing a wide number of different techniques (see Biazar [2006], Rafei et al.
Revised: Most of the publications investigating the SIR model use numerical methods, employing a wide number of different techniques (see Biazar [2006], Rafei et al.

Original: [2007a,b] and the references therein).
Revised: [2007a,b] and the references therein).

Original: There is a huge amount of literature studying very diverse effects in the SIR compartmental model.
Revised: There is a huge amount of literature studying very diverse effects in the SIR compartmental model.

Original: Much of it approximates the random process with a deterministic process defined via ordinary differential equations.
Revised: Much of it approximates the random process with a deterministic process defined via ordinary differential equations.

Original: Hence an early extinction of the process is not possible.
Revised: Hence an early extinction of the process is not possible.

Original: In Harko et al.
Revised: In Harko et al.

Original: [2014] the authors derived an exact analytical solution to the SIR model in parametric form.
Revised: [2014] the authors derived an exact analytical solution to the SIR model in parametric form.

Original: A similar result was shown by Miller [2012, 2017].
Revised: A similar result was shown by Miller [2012, 2017].

Original: In Shabbir et al.
Revised: In Shabbir et al.

Original: [2011] an exact analytical solution to the SIR and SIS models with constant population is obtained with the help of direct integration tools.
Revised: [2011] an exact analytical solution to the SIR and SIS models with constant population is obtained with the help of direct integration tools.

Original: Lefèvre and Simon [2020] propose a block-structured Markov process to describe the spread of epidemics of SIR type and they determine the distribution of the final state of the process.
Revised: Lefèvre and Simon [2020] propose a block-structured Markov process to describe the spread of epidemics of SIR type and they determine the distribution of the final state of the process.

Original: In Black and Ross [2015] the authors present a new method for the recursive computation of the epidemic size distributions.
Revised: In Black and Ross [2015] the authors present a new method for the recursive computation of the epidemic size distributions.

Original: The authors do not estimate the expected size of the epidemic nor do they give a closed form of its distribution.
Revised: The authors do not estimate the expected size of the epidemic nor do they give a closed form of its distribution.

Original: Pastor-Satorras et al.
Revised: Pastor-Satorras et al.

Original: [2015] give a great overview about epidemic processes in networks: Here the individuals are connected by a network modelling social contacts, such that the infection spreads from a node to its neighbours.
Revised: [2015] give a great overview about epidemic processes in networks: Here the individuals are connected by a network modelling social contacts, such that the infection spreads from a node to its neighbours.

Original: The authors of Janson et al.
Revised: The authors of Janson et al.

Original: [2014] consider random graphs with a given degree sequence and prove that there is a threshold as a function for γ , β , and given vertex degrees.
Revised: [2014] consider random graphs with a given degree sequence and prove that there is a threshold as a function for γ , β , and given vertex degrees.

Original: Below the threshold, only a small number of infections occurs, while above it most of the graph gets infected.
Revised: Below the threshold, only a small number of infections occurs, while above it most of the graph gets infected.

Original: Kempe et al.
Revised: Kempe et al.

Original: [2003] consider influence maximisation in the independent cascade model introduced in Goldenberg et al.
Revised: [2003] consider influence maximisation in the independent cascade model introduced in Goldenberg et al.

Original: [2001a,b].
Revised: [2001a,b].

Original: The process works in parallel rounds and it starts with a set of active (infected) nodes.
Revised: The process works in parallel rounds and it starts with a set of active (infected) nodes.

Original: Every active node infects every noninfected neighbour with a probability of p .
Revised: Every active node infects every noninfected neighbour with a probability of p .

Original: Then the active nodes become inactive and the newly infected neighbours become active.
Revised: Then the active nodes become inactive and the newly infected neighbours become active.

Original: The process runs until no more activations are possible.
Revised: The process runs until no more activations are possible.

Original: The optimization problem of selecting the most influential nodes to be activated in the beginning is NP-hard and the authors provide the first provable approximation guarantees for efficient algorithms.
Revised: The optimization problem of selecting the most influential nodes to be activated in the beginning is NP-hard and the authors provide the first provable approximation guarantees for efficient algorithms.

Original: For an overview about results in this model see Shakarian et al.
Revised: For an overview about results in this model see Shakarian et al.

Original: Bohman and Picollelli [2012] consider an SIR process of random graphs with a given degree sequence in an continuous time model.
Revised: Bohman and Picollelli [2012] consider an SIR process of random graphs with a given degree sequence in an continuous time model.

Original: Each infected node sends infections to each of its neighbors at times determined by independent exponential random variables with parameter λ .
Revised: Each infected node sends infections to each of its neighbors at times determined by independent exponential random variables with parameter λ .

Original: An infected node recovers at a random time given by an independent exponential random variable with parameter r ∞ .
Revised: An infected node recovers at a random time given by an independent exponential random variable with parameter r ∞ .

Original: The authors assume that the infection spreads from a single infected node and show that either the disease halts after infecting only a small number of nodes, or an epidemic spreads to infect a linear number of nodes.
Revised: The authors assume that the infection spreads from a single infected node and show that either the disease halts after infecting only a small number of nodes, or an epidemic spreads to infect a linear number of nodes.

Original: The authors also show that, conditioned on the event that more than a small number of nodes are infected, the epidemic is likely to follow a trajectory given by the solution of an associated system of ordinary differential equations.
Revised: The authors also show that, conditioned on the event that more than a small number of nodes are infected, the epidemic is likely to follow a trajectory given by the solution of an associated system of ordinary differential equations.

Original: Their approach gives bounds on the total number of infected nodes.
Revised: Their approach gives bounds on the total number of infected nodes.

Original: There is also related work that considers an agent-based modelling of the SIR model (e.g. Bicher and Popper [2013]).
Revised: There is also related work that considers an agent-based modelling of the SIR model (e.g. Bicher and Popper [2013]).

Original: These investigate some form of non-homogeneous populations (e.g. distances between agents given by a graph), and analyse results empirically, either using geographic information systems Perez and Dragicevic [2009], or social relationships Alzu’bi et al.
Revised: These investigate some form of non-homogeneous populations (e.g. distances between agents given by a graph), and analyse results empirically, either using geographic information systems Perez and Dragicevic [2009], or social relationships Alzu’bi et al.

Original: Many works model how different societal factors play a role in the evolution of an epidemic.
Revised: Many works model how different societal factors play a role in the evolution of an epidemic.

Original: In the case of the COVIDpandemic, many of them focus on accounting for different factors, like social distancing measures and testing regimens in order to build a reliable model.
Revised: In the case of the COVIDpandemic, many of them focus on accounting for different factors, like social distancing measures and testing regimens in order to build a reliable model.

Original: In Levesque et al.
Revised: In Levesque et al.

Original: [2021] the authors create a Crump-Mode-Jagers continuous time branching process modelling COVID-19 propagation in order to decide which mitigation strategies are more effective.
Revised: [2021] the authors create a Crump-Mode-Jagers continuous time branching process modelling COVID-19 propagation in order to decide which mitigation strategies are more effective.

Original: Similar works developed mathematical and data-driven models in order to establish the efficacy of such measures
Revised: Similar works developed mathematical and data-driven models in order to establish the efficacy of such measures

Original: Sun et al.
Revised: Sun et al.

Original: [2020], Choi and Shim [2021], Liu et al.
Revised: [2020], Choi and Shim [2021], Liu et al.

Original: However, there is still a lack of work studying the role travelling has on the epidemic.
Revised: However, there is still a lack of work studying the role travelling has on the epidemic.

Original: In Arino and Van Den Driessche [2003], the authors incorporate the concept of travel into the ordinary differential equations for a Susceptible-ExposedInfected-Removed-Susceptible (SEIRS) model.
Revised: In Arino and Van Den Driessche [2003], the authors incorporate the concept of travel into the ordinary differential equations for a Susceptible-ExposedInfected-Removed-Susceptible (SEIRS) model.

Original: They derive bounds on R 0 and show that a disease-free equilibrium is reached, though they hint at its uniqueness through simulations and no discussion is made about the final expected size of the population.
Revised: They derive bounds on R 0 and show that a disease-free equilibrium is reached, though they hint at its uniqueness through simulations and no discussion is made about the final expected size of the population.

Original: Their model is complex and detailed, with the central assumption that individuals come back to the origin country before leaving again for any other country.
Revised: Their model is complex and detailed, with the central assumption that individuals come back to the origin country before leaving again for any other country.

Original: This is different from our work, where we do not require individuals to follow travel patterns.
Revised: This is different from our work, where we do not require individuals to follow travel patterns.

Original: In Zakary et al.
Revised: In Zakary et al.

Original: [2017] a multi-regions discrete-time model describes the spatialtemporal spread of an epidemic.
Revised: [2017] a multi-regions discrete-time model describes the spatialtemporal spread of an epidemic.

Original: Starting from one region, this enters to regions connected with their neighbors by any mean favouring movement.
Revised: Starting from one region, this enters to regions connected with their neighbors by any mean favouring movement.

Original: Like in our case, the authors consider homogeneous SIR populations.
Revised: Like in our case, the authors consider homogeneous SIR populations.

Original:  MODELS AND RESULTS
Revised:  MODELS AND RESULTS

Original: In this section we introduce formally the two models considered in this paper.
Revised: In this section we introduce formally the two models considered in this paper.

Original: Both models work in sequential rounds and are population protocols Angluin et al.
Revised: Both models work in sequential rounds and are population protocols Angluin et al.

Original: At every round, two individuals i and j are picked uniformly at random.
Revised: At every round, two individuals i and j are picked uniformly at random.

Original: Individual i can either become infected, recover, or stay susceptible, depending on the current state of i and j .
Revised: Individual i can either become infected, recover, or stay susceptible, depending on the current state of i and j .

Original: In our or second model, individuals can also travel.
Revised: In our or second model, individuals can also travel.

Original: We study discrete-time models, where in each integer step, an action occurs.
Revised: We study discrete-time models, where in each integer step, an action occurs.

Original: One major advantage the discrete-time models have over continuous time models is that they allow to study the early extinction phenomenon.
Revised: One major advantage the discrete-time models have over continuous time models is that they allow to study the early extinction phenomenon.

Original: Note that continuous systems and, in particular, differential equations fail to capture early extinctions.
Revised: Note that continuous systems and, in particular, differential equations fail to capture early extinctions.

Original: Single Country Model.
Revised: Single Country Model.

Original: Let S t , I t , R t be the number of susceptible, infected and recovered individuals at time t , with S t
Revised: Let S t , I t , R t be the number of susceptible, infected and recovered individuals at time t , with S t

Original: + I t + R t
Revised: + I t + R t

Original: =
Revised: =

Original: N , where N is the total number of individuals.
Revised: N , where N is the total number of individuals.

Original: Our SIR process is defined as follows.
Revised: Our SIR process is defined as follows.

Original: In every round two individuals i, j are picked uniformly at random.
Revised: In every round two individuals i, j are picked uniformly at random.

Original: If individual i is infected, then it recovers w.p. γ . 2.
Revised: If individual i is infected, then it recovers w.p. γ . 2.

Original: If individual i is susceptible and j is infected, then i will becomes infected w.p. β .
Revised: If individual i is susceptible and j is infected, then i will becomes infected w.p. β .

Original: Note that β · I t /N is the probability that the selected susceptible individual i becomes infected.
Revised: Note that β · I t /N is the probability that the selected susceptible individual i becomes infected.

Original: We assume that β and γ are both constants.
Revised: We assume that β and γ are both constants.

Original: In expectation, the system evolves as follows.
Revised: In expectation, the system evolves as follows.

Original: [Equation]
Revised: [Equation]

Original: The reproduction number is defined as R 0 = β/γ .
Revised: The reproduction number is defined as R 0 = β/γ .

Original: This number is equal to the expected number of infections caused by an infected individual assuming that S = N .
Revised: This number is equal to the expected number of infections caused by an infected individual assuming that S = N .

Original: Over time, the number S t decreases such that every newly infected individual introduces less and less infections into the population.
Revised: Over time, the number S t decreases such that every newly infected individual introduces less and less infections into the population.

Original: The herd immunity threshold is defined as 1 − 1 / R 0 .
Revised: The herd immunity threshold is defined as 1 − 1 / R 0 .

Original: This is the value of S t /N
Revised: This is the value of S t /N

Original: such that R 0 · S t /N
Revised: such that R 0 · S t /N

Original: = 1 .
Revised: = 1 .

Original: In Last [2001], a pandemic is defined as “an epidemic occurring worldwide, or over a very wide area, crossing international boundaries and usually affecting a large number of individuals”.
Revised: In Last [2001], a pandemic is defined as “an epidemic occurring worldwide, or over a very wide area, crossing international boundaries and usually affecting a large number of individuals”.

Original: In this work we assume a simplified notion of pandemic, depending only on the total number of infected individuals.
Revised: In this work we assume a simplified notion of pandemic, depending only on the total number of infected individuals.

Original: We say a process results in a pandemic if at one point of time the number of active infections ( I t ) reaches √ N . 

Revised: We say a process results in a pandemic if at one point of time the number of active infections ( I t ) reaches √ N .

Original: Moreover, if the process does not result in a pandemic, then we say it extincts early .
Revised: Moreover, if the process does not result in a pandemic, then we say it extincts early .

Original: We denote the corresponding events by E pan and E ext .
Revised: We denote the corresponding events by E pan and E ext .

Original: Finally, we define
Revised: Theoretically, over time a constant fraction of the population might become infected, even though at any point of time the number of current infections stays below √ N . However, with the same techniques as we use in this paper, one can show that this event happens with an exponentially small probability—it is akin to a biased random walk on a line remaining in the interval (1 , √ N ) for a linear number of rounds.

Original:  R ∞ as the epidemic final size; since all individuals eventually recover this corresponds to R ∞ =
Revised: Finally, we define R ∞ as the epidemic final size; since all individuals eventually recover this corresponds to R ∞ =

Original: lim t →∞ R t .
Revised: lim t →∞ R t .

Original: We define r ∞ =
Revised: We define r ∞ =

Original: R ∞ /N .
Revised: R ∞ /N .

Original: Let W denote the Lambert W-function, which is the inverse of function f ( W ) =
Revised: Let W denote the Lambert Wfunction, which is the inverse of function f ( W ) =

Original: We W considered for W ∈ [ − 1 , ∞ ) (where function f increases monotonically from − 1 /e to infinity).
Revised: We W considered for W ∈ [ − 1 , ∞ ) (where function f increases monotonically from − 1 /e to infinity).

Original: Model for two countries.
Revised: Model for two countries.

Original: In this case we assume that the
Revised: In this case we assume that the

Original: N individuals are distributed over two countries.
Revised: N individuals are distributed over two countries.

Original: N (1) t and
Revised: N (1) t and

Original: N (2) t denotes the number of individuals staying in country 1 and 2 at time t .
Revised: N (2) t denotes the number of individuals staying in country 1 and 2 at time t .

Original: We assume N (1) t = N (1) t = N/ 2 .
Revised: We assume N (1) t = N (1) t = N/ 2 .

Original: The number of susceptible, infected and recovered individuals at time t in country i is denoted by S ( i ) t , I ( i ) t and R (1 i ) t for i ∈ { 1 , 2 } .
Revised: The number of susceptible, infected and recovered individuals at time t in country i is denoted by S ( i ) t , I ( i ) t and R (1 i ) t for i ∈ { 1 , 2 } .

Original: Initially we have I (1) t
Revised: Initially we have I (1) t

Original: =
Revised: =

Original: I (2) t =
Revised: I (2) t =

Original: s for s ≥ 1 .
Revised: s for s ≥ 1 .

Original: We assume that every country has its own infection rate: β (1) and β (2) .
Revised: We assume that every country has its own infection rate: β (1) and β (2) .

Original: The recovery rate in both countries is γ .
Revised: The recovery rate in both countries is γ .

Original: The process has two selection steps and step t works as follows.
Revised: The process has two selection steps and step t works as follows.

Original: Pick an individual i uniformly at random (with prob.
Revised: Pick an individual i uniformly at random (with prob.

Original: 1 /N ).
Revised: 1 /N ).

Original: With probability p travel individual i travels from its current country to the other country.
Revised: With probability p travel individual i travels from its current country to the other country.

Original: Adjust N (1) t and N (2) t accordingly.
Revised: Adjust N (1) t and N (2) t accordingly.

Original: Pick country ℓ as follows: ℓ = 1 w.p. N (1) t /N and otherwise.
Revised: Pick country ℓ as follows: ℓ = 1 w.p. N (1) t /N andotherwise.

Original: An SIR step as described above is applied on the chosen country.
Revised: An SIR step as described above is applied on the chosen country.

Original: (a) Pick a pair of individuals i and j uniformly at random from country ℓ .
Revised: (a) Pick a pair of individuals i and j uniformly at random from country ℓ .

Original: (b) If individual i is infected, then it recovers w.p. γ .
Revised: (b) If individual i is infected, then it recovers w.p. γ .

Original: (c) If individual i is susceptible and j is infected, then i will become infected w.p. β ( ℓ ) .
Revised: (c) If individual i is susceptible and j is infected, then i will become infected w.p. β ( ℓ ) .

Original: Compared to existing works, this model is simpler and posing minimal constraints to the movement of individuals.
Revised: Compared to existing works, this model is simpler and posing minimal constraints to the movement of individuals.

Original: This makes it easily adaptable, for example in the case of a change in the number of countries involved.
Revised: This makes it easily adaptable, for example in the case of a change in the number of countries involved.

Original: Further, a simpler model might facilitate future analytical results regarding the expected size of the epidemic.
Revised: Further, a simpler model might facilitate future analytical results regarding the expected size of the epidemic.

Original: 2.1 RESULTS
Revised: 2.1 RESULTS

Original: Our main result Theorem 4 shows a very tight characterisation of the early extinction probability.
Revised: Our main result Theorem 4 shows a very tight characterisation of the early extinction probability.

Original: Recall that a process extincts early if the number of infected dies out before ever reaching I t ≥ √ N , where N is the size of the population.
Revised: Recall that a process extincts early if the number of infected dies out before ever reaching I t ≥ √ N , where N is the size of the population.

Original: Recall that R 0 =
Revised: Recall that R 0 =

Original: β/γ is the reproduction number, R ∞ is the total number of infected individuals and r ∞ =
Revised: β/γ is the reproduction number, R ∞ is the total number of infected individuals and r ∞ =

Original: R ∞ /N .
Revised: R ∞ /N .

Original: First we present a simplified version of Theorem 4.
Revised: First we present a simplified version of Theorem 4.

Original: Theorem 1 (Simplified version of Theorem 4) .
Revised: Theorem 1 (Simplified version of Theorem 4) .

Original: Let β + γ ≤ 1 .
Revised: Let β + γ ≤ 1 .

Original: Consider the single country model starting with
Revised: Consider the single country model starting with

Original: I 

Revised: I

Original: = s ≤ [Equation] infections.
Revised: = s ≤ √ N 2log N loglog N infections.

Original: Let ε = 5 log N/ √ [Equation] and let Pr [ E ext ] be the probability of early extinction.
Revised: Let ε = 5 log N/ √ N and let Pr [ E ext ] be the probability of early extinction.

Original: If β < γ − ε , then Pr [ E ext ] ≥ 1 − o (1) .
Revised: If β < γ − ε , then Pr [ E ext ] ≥ 1 − o (1) .

Original: If β > γ + ε , then Pr [ E ext ] = (1 / R 0 ) s ± o (1) .
Revised: If β > γ + ε , then Pr [ E ext ] = (1 / R 0 ) s ± o (1) .

Original: For the case where β > γ and s = 1 the probability of early extinction is essentially 1 / R 

Revised: For the case where β > γ and s = 1 the probability of early extinction is essentially 1 / R

Original: = γ/β .
Revised: = γ/β .

Original: If s > 1 the probability decreases exponentially in s .
Revised: If s > 1 the probability decreases exponentially in s .

Original: If β < γ , then the process is very likely to reach an early extinction.
Revised: If β < γ , then the process is very likely to reach an early extinction.

Original: We leave the case β = γ open but note that ε will be arbitrary small with growing N .
Revised: We leave the case β = γ open but note that ε will be arbitrary small with growing N .

Original: The challenging part in the proof of
Revised: The challenging part in the proof of

Original: Theorem 4 is non-linear update rule of our process.
Revised: Theorem 4 is non-linear update rule of our process.

Original: Note that rewriting (1)-(1) by replacing R t = N − S t − I t reveals the non-linearity.
Revised: Note that rewriting (1)-(3) by replacing R t = N − S t − I t reveals the non-linearity.

Original: To overcome these challenges, we use a series of couplings, allowingwhich allows us to relate our SIR process to a biased random walk.
Revised: To overcome these challenges, we use a series of couplings, allowing us to relate our SIR process to a biased random walk.

Original: Note that in the above result E ext is defined as the event of having √ N many infected individuals at the same time.
Revised: Note that in the above result E ext is defined as the event of having √ N many infected individuals at the same time.

Original: Hence, it is still possible that R ∞ =
Revised: Hence, it is still possible that R ∞ =

Original: Ω( N ) , i.e., at the end of the process the number of recovered individuals (which equals the total number of infections) is linear in
Revised: Ω( N ) , i.e., at the end of the process the number of recovered individuals (which equals the total number of infections) is linear in

Original: N .
Revised: N .

Original: In Proposition 1 (see Section 3.1) we show that this is not the case; in the event of E ext the number of recovered individuals never exceeds √ N log N , w.h.p.
Revised: In Proposition 1 (see Section 3.1) we show that this is not the case; in the event of E ext the number of recovered individuals never exceeds √ N log N , w.h.p.

Original: This means that our bounds on the the early extinction still hold even if the definition was changed to requiring R ∞ ≤ √ N log N .
Revised: This means that our bounds on the the early extinction still hold even if the definition was changed to requiring R ∞ ≤ √ N log N .

Original: The next theorem shows a bound on the the expected number of total infections
Revised: The next theorem shows a bound on the the expected number of total infections

Original: E [ R ∞ ] , expressed in terms of the Lambert function W .
Revised: E [ R ∞ ] , expressed in terms of the Lambert function W .

Original: Note that W ( x ) < 0 for x ∈ [ − 1 /e, 0) .
Revised: Note that W ( x ) < 0 for x ∈ [ − 1 /e, 0) .

Original: As mentioned before, many approaches like first-order methods, mean field approaches and ordinary differential equations
Revised: As mentioned before, many approaches like first-order methods, mean field approaches and ordinary differential equations

Original: ODE (see e.g., Kröger and Schlickeiser [2020], Bicher and
Revised: ODE (see e.g., Kröger and Schlickeiser [2020], Bicher and

Original: Popper [2013]), cannot account for early extinction due to the underlying determinism.
Revised: Popper [2013]), cannot account for early extinction due to the underlying determinism.

Original: Instead, they obtain bounds of the kind E (cid:2) r ∞ | E ext (cid:3) .
Revised: Instead, they obtain bounds of the kind E (cid:2) r ∞ | E ext (cid:3) .

Original: There has also been been some work based on stochastic differential equations (e.g., Allen [2008], Williams et al.
Revised: There has also been been some work based on stochastic differential equations (e.g., Allen [2008], Williams et al.

Original: [2012]), complemented with simulations results.
Revised: [2012]), complemented with simulations results.

Original: We are not aware of results based on SDEs that obtain closed-form results for E [ r ∞ ] .
Revised: We are not aware of results based on SDEs that obtain closed-form results for E [ r ∞ ] .

Original: Theorem 2.
Revised: Theorem 2.

Original: Assume that | β − γ | ≥ ε for an arbitrarily small constant ε .
Revised: Assume that | β − γ | ≥ ε for an arbitrarily small constant ε .

Original: Assume N = N ( ε ) is large enough.
Revised: Assume N = N ( ε ) is large enough.

Original: Let
Revised: Let

Original: W : [ − 1 /e, ∞ ) −→ [ − 1 , ∞ ) be the Lambert function.
Revised: W : [ − 1 /e, ∞ ) −→ [ − 1 , ∞ ) be the Lambert function.

Original: Consider the single country model starting with I 

Revised: Consider the single country model starting with I

Original: = s ≤ √
Revised: = s ≤ √

Original: N 2log N loglog N infections.
Revised: N 2log N loglog N infections.

Original: If β < γ − ε , then the expected total number of infections is sublinear, i.e., E [ r ∞ ]
Revised: If β < γ − ε , then the expected total number of infections is sublinear, i.e., E [ r ∞ ]

Original: = o (1) .
Revised: = o (1) .

Original: If β > γ + ε, then 
Revised: If β > γ + ε, then E

Original: [Equation]
Revised: [Equation]

Original: See Fig.
Revised: See Fig.

Original: 4 in the Appendix for a depiction.
Revised: 1 for a depiction.

Original: The following theorem estimates the expected number of infections conditioned on not having early extinction.
Revised: The following theorem estimates the expected number of infections conditioned on not having early extinction.

Original: It is used to show Theorem 2.
Revised: It is used to show Theorem 2.

Original: The following theorem yields a result similar to the one obtained through first-order methods used by Allen [2008], Williams et al.
Revised: The following theorem yields a result similar to the one obtained through first-order methods used by Allen [2008], Williams et al.

Original: [2012], but adapted to our population based model.
Revised: [2012], but adapted to our population based model.

Original: In Theorem 6 (in Section 4) we provide a generalisation of the following theorem.
Revised: In Theorem 6 (in Section 4) we provide a generalisation of the following theorem.

Original: Theorem 3.
Revised: Theorem 3.

Original: Consider the single country model.
Revised: Consider the single country model.

Original: Assume that β > γ are constants.
Revised: Assume that β > γ are constants.

Original: Let W : [ − 1 /e, ∞ ) −→ [ − 1 , ∞ ) be the Lambert function.
Revised: Let W : [ − 1 /e, ∞ ) −→ [ − 1 , ∞ ) be the Lambert function.

Original: Then, 
Revised: Then, for large n ,

Original: [Equation]
Revised: [Equation]

Original: Furthermore, r ∞ is concentrated around its expected value.
Revised: Furthermore, r ∞ is concentrated around its expected value.

Original:  EARLY EXTINCTION
Revised:  EARLY EXTINCTION

Original: The SIR model described in Section 2, often results in herdimmunity, where a large fraction of the population was infected such that only very few susceptible individuals remain.
Revised: The SIR model described in Section 2, often results in herdimmunity, where a large fraction of the population was infected such that only very few susceptible individuals remain.

Original: From then on, the number of infections decreases slowly until it finally reaches zero.
Revised: From then on, the number of infections decreases slowly until it finally reaches zero.

Original: However, in some case, even when β > γ , it can happen that the number of infections remains low and the virus vanishes very suddenly.
Revised: However, in some case, even when β > γ , it can happen that the number of infections remains low and the virus vanishes very suddenly.

Original: This is what we refer to as an early extinction and which is the focal point of this section.
Revised: This is what we refer to as an early extinction and which is the focal point of this section.

Original: We will derive bounds on the probability of an early extinction in terms of the parameters γ and β .
Revised: We will derive bounds on the probability of an early extinction in terms of the parameters γ and β .

Original: The formal statement is given in the next theorem and implies the theorem given in Theorem 1.
Revised: The formal statement is given in the next theorem and implies the theorem given in Theorem 1.

Original: Theorem 4.
Revised: Theorem 4.

Original: Consider the single country SIR model as described in Section 2.
Revised: Consider the single country SIR model as described in Section 2.

Original: Define τ = √ N log N , ε = 5 log N/ √ N , and ε ′ =
Revised: Define τ = √ N log N , ε = 5 log N/ √ N , and ε ′ =

Original: ε ( N − τ ) /N , and assume I 

Revised: ε ( N − τ ) /N , and assume I

Original: = s for any s ≤ √ N 2log N loglog N .
Revised: = s for any s ≤ √ N 2log N loglog N .

Original: Then we have for β > γ + ε 
Revised: Then we have for β > γ + ε (cid:16) [Equation] (cid:17)

Original: [Equation]
Revised: [Equation]

Revised: −

Original:  and for β < γ − ε
Revised: [Equation] Pr [ E ext ] ≤ (cid:16) [Equation] − sτ (cid:17) (cid:16) γβ (cid:17) s + 2 N and for β < γ − ε

Original: [Equation]
Revised: [Equation]

Original: To show Theorem 4 our approach is as follows.
Revised: To show Theorem 4 our approach is as follows.

Original: Ideally, we would like to bound the number of infections as a (biased) random walk over the number of infected individuals.
Revised: Ideally, we would like to bound the number of infections as a (biased) random walk over the number of infected individuals.

Original: The probability that such a random walk reaches either of its end points can be limited by a bound like the one given in Theorem 7.
Revised: The probability that such a random walk reaches either of its end points can be limited by a bound like the one given in ?? .

Original: Unfortunately, here the probability to ’walk’ from I t to I t − 1{+
+} or I t + 1 is a function of both I t and S t .
Revised: Unfortunately, here the probability to ’walk’ from I t to I t − {+
+} or I t +1 is a function of both I t and S t .

Original: We avoid the dependency on I t by considering active steps only.
Revised: We avoid the dependency on I t by considering active steps only.

Original: Recall, these are steps in which the number of infected individuals either increases by one or decreases by one.
Revised: Recall, these are steps in which the number of infected individuals either increases by one or decreases by one.

Original: This allows us to drop the terms I t /N in the transition probabilities.
Revised: This allows us to drop the terms I t /N in the transition probabilities.

Original: We circumvent the dependency on S t by defining a family of processes and by coupling them  which each other (see Section 3.1).
Revised: We circumvent the dependency on S t by defining a family of processes and by coupling them with which each other (see Section 3.1).

Original: The processes will be defined and motivated later in this section.
Revised: The processes will be defined and motivated later in this section.

Original: Once both problems are solved, we are left with four biased random walks (upper vs lower bounds and β < γ vs β > γ ) that we analyse using
Revised: Once both problems are solved, we are left with four biased random walks (upper vs lower bounds and β < γ vs β > γ ).

Original: Theorem 7 (see Appendix A). This allows us in Theoremto derive almost tight bounds on Pr [ E ext ] .
Revised: This allows us in Theorem 4 to derive almost tight bounds on Pr [ E ext ] .

Original: 3.1 COUPLINGS
Revised: 3.1 COUPLINGS

Original: In this section our goal is to show Theorem 5 which upper and lower bounds the probability of early extinction of our process by the extinction probability of two biased random walk process.
Revised: In this section our goal is to show Theorem 5 which upper and lower bounds the probability of early extinction of our process by the extinction probability of two biased random walk process.

Original: Theorem 5 shows that the approximation error is tiny and has only a second-order impact on the extinction probability.
Revised: Theorem 5 shows that the approximation error is tiny and has only a second-order impact on the extinction probability.

Original: In order to prove Theorem 5, we will define two intermediate processes and provide three pairwise couplings.
Revised: In order to prove Theorem 5, we will define two intermediate processes and provide three pairwise couplings.

Original: As a first step, in the entire section, unless stated otherwise, we only consider active steps, i.e., steps where I t +

Revised: In this section, unless stated otherwise, we only consider active steps, i.e., steps where I t +

Original: ̸
Revised: ̸

Original: = I t .
Revised: = I t .

Original: The subscript t counts now only the active steps.
Revised: The subscript t counts now only the active steps.

Original: Before defining the processes we give some definitions. Recall that E ext (early extinction) is the event that the number of infections reaches zero before ever having √ N infected individuals at the same time.
Revised: Recall that E ext (early extinction) is the event that the number of infections reaches zero before ever having √ N infected individuals at the same time.

Original: Define stopping time as T =
Revised: Define stopping time as T =

Original: min t (cid:110) I t = 

Revised: min t (cid:110) I t =

Original: or I t = √ N (cid:111) .
Revised: or I t = √ N (cid:111) .

Original: Hence, T refers to the number of active steps until there are either zero or √ N many infected individuals, and E ext is the event that I T = 0 .
Revised: Hence, T refers to the number of active steps until there are either zero or √ N many infected individuals, and E ext is the event that I T = 0 .

Original: For any SIR process P , we denote by ξ ( P ) t the configuration of P at time t , which is given by the triplet ξ ( P ) t = (cid:16) S ( P ) t , I ( P ) t , R ( P ) t (cid:17) .
Revised: For any SIR process P , we denote by ξ ( P ) t the configuration of P at time t , which is given by the triplet ξ ( P ) t = (cid:16) S ( P ) t , I ( P ) t , R ( P ) t (cid:17) .

Original: We define τ = √ N log N and a second stopping time T τ = min t { I t = 0 , I t = √ N or t = τ } =
Revised: We define τ = √ N log N and a second stopping time T τ = min t { I t = 0 , I t = √ N or t = τ } =

Original: min { T, τ } .
Revised: min { T, τ } .

Original: Compared to the stopping time T , here the process does not only stop if the number of infected individuals reaches zero or √ N but also if that does not happen during the first τ (active) time steps.
Revised: Compared to the stopping time T , here the process does not only stop if the number of infected individuals reaches zero or √ N but also if that does not happen during the first τ (active) time steps.

Original: All our processes are stopped at time either T or T τ .
Revised: All our processes are stopped at time either T or T τ .

Original: We assume that for every t larger that the stopping time we have ξ ( P ) t =
Revised: We assume that for every t larger that the stopping time we have ξ ( P ) t =

Original: ξ ( P ) T or ξ ( P ) t = ξ ( P ) T τ , respectively.
Revised: ξ ( P ) T or ξ ( P ) t = ξ ( P ) T τ , respectively.

Original: Now we are ready to define our processes, which we call P, P τ, ∗ , P ∗ ,S , P τ,S .
Revised: Now we are ready to define our processes, which we call P, P τ, ∗ , P ∗ ,S , P τ,S .

Original: P is our original process with the stopping time T . 2.
Revised: no time limit time limit τ S not fixed P [Figure]

Original: P τ, ∗ is our original process with the stopping time T τ . 3.
Revised: S fixed [Figure] [Figure] [Table]

Original: For S ∈ { N
Revised: [Figure] [Table]

Original:  − τ, N } , we define the following biased random walk process P ∗ ,S .
Revised: [Figure] P is our original process with the stopping time T . 2. P τ, ∗ is our original process with the stopping time T τ . Fixing S t over time to a constant S ∈ { N − τ, N } , we define the following biased random walk process P ∗ ,S .

Original: At each round the following is done.
Revised: At each round the following is done.

Original: (a) Draw random numbers
Revised: (a) Draw random numbers

Original: X I , X S , X a i.i.d.
Revised: X I , X S , X a i.i.d.

Original: and u.a.r.
Revised: and u.a.r.

Original: from [0 , 1] (b) If X I ∈ [0 , I ( P ∗ ,S ) t /N ) , then i. if X a ∈ [0 , β ) and X s ∈ [0 , S/N ] , increase I ( P ∗ ,S ) t ; ii.
Revised: from [0 , 1] (b) If X I ∈ [0 , I ( P ∗ ,S ) t /N ) , then i. if X a ∈ [0 , β ) and X s ∈ [0 , S/N ] , increase I ( P ∗ ,S ) t ; ii.

Original: if X a ∈ [ β, β + γ ) , decrease I ( P ∗ ,S ) t .
Revised: if X a ∈ [ β, β + γ ) , decrease I ( P ∗ ,S ) t .

Original: The nice property about process P ∗ ,S is that (since we only consider active time steps) the process behaves exactly like a biased random walk.
Revised: The nice property about process P ∗ ,S is that (since we only consider active time steps) the process behaves exactly like a biased random walk.

Original: The increase probability of I t is βS/N and the decreas probability is γ .
Revised: The increase probability of I t is βS/N and the decreas probability is γ .

Original: For this process we apply the stopping time T . 
Revised: For this process we apply the stopping time T . 4.

Original: For S ∈ { N − τ, N } , we define the process P τ,S .
Revised: For S ∈ { N − τ, N } , we define the process P τ,S .

Original: P τ,S behaves like P ∗ ,
Revised: P τ,S behaves like P ∗ ,

Original: S
Revised: S

Original: but we apply the stopping time T τ .
Revised: but we apply the stopping time T τ .

Original: Table 1 sums up the four processes, while Fig.
Revised: Table 1 sums up the four processes, while Fig.

Original: 1 shows the couplings we are going to prove in the following sections.
Revised: 2 shows the couplings we are going to prove in the following sections.

Original: In this section we will show the following result.
Revised: In this section we will show the following result.

Original: no time limit time limit τ

Original: S not fixed

Original: P

Original: [Figure]

Original: fixed

Original: [Figure]

Original: P

Original: [Figure]

Original: [Figure]

Original: Theorem 5.
Revised: Theorem 5.

Original: Consider the processes P and P ∗ ,S .
Revised: Consider the processes P and P ∗ ,S .

Original: We have
Revised: We have

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: From this it follows that we can analyse the early extinction time of P ∗ ,S instead of P .
Revised: From this it follows that we can analyse the early extinction time of P ∗ ,S instead of P .

Original: To show Theorem 5 we use 3 couplings: Proposition 1 couples processes P and P τ, ∗ ,
Revised: To show Theorem 5 we use 3 couplings: Proposition 1 couples processes P and P τ, ∗ ,

Original: Proposition 2 couples P τ, ∗ and P τ,S , and Propositionconnects P τ,S and P ∗ ,S .
Revised: Proposition 2 couples P τ, ∗ and P τ,S , and Propositionconnects P τ,S and P ∗ ,S .

Original: Proposition 1.
Revised: Proposition 1.

Original: With prob 1 − 1 /N we have I ( P ) T
Revised: With prob 1 − 1 /N we have I ( P ) T

Original: [Equation]
Revised: = I ( P τ, ∗ ) T τ .

Original: Proposition 2. Consider the processes P τ, ∗ and P τ,S for
Revised: Proposition 2. Consider the processes P τ, ∗ and P τ,S for

Original: S ∈ { N, N − τ } .
Revised: S ∈ { N, N − τ } .

Original: Let P =
Revised: Let P =

Original: Pr (cid:104) I
Revised: Pr (cid:104) I

Original: ( P τ, ∗ ) T τ [Equation] N (cid:105) .
Revised: ( P τ, ∗ ) T τ = √ N (cid:105) .

Original: We have
Revised: We have

Revised: Pr

Original: [Equation]
Revised: [Equation]

Original: Proofs can be found in Appendix C and Appendix D, resp.

Original: Proposition 3.
Revised: Proposition 3.

Original: Consider the processes P τ,S and P ∗ ,S for
Revised: Consider the processes P τ,S and P ∗ ,S for

Original: S ∈ { N − τ, N } .
Revised: S ∈ { N − τ, N } .

Original: Then we have
Revised: Then we have

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: This proof is similar to proof of Proposition 1.
Revised: This proof is similar to proof of Proposition 1.

Original: Putting all three couplings together yields Theorem 5.
Revised: Putting all three couplings together yields Theorem 5.

Original: From this we are finally able to conclude Theorem 4, since now we can analyse a biased random walk instead.
Revised: From this we are finally able to conclude Theorem 4, since now we can analyse a biased random walk instead.

Original: The proof distinguishes between two cases, depending on whether β or γ is larger and builds on Theorem 7.
Revised: The proof distinguishes between two cases, depending on whether β or γ is larger and builds on ?? .

Original: It can be found in the supplementary material.
Revised: It can be found in the supplementary material.

Original:  TOTAL NUMBER OF INFECTIONS
Revised:  TOTAL NUMBER OF INFECTIONS

Original: In this section we analyse the total number of infections for the SIR model for one country.
Revised: In this section we analyse the total number of infections for the SIR model for one country.

Original: The proofof the next result can be found in Appendix B.
Revised: The proof can be found in the full version.

Original: Theorem 6.
Revised: Theorem 6.

Original: Consider the single country model.
Revised: Consider the single country model.

Original: Assume that β − γ ≥ ε for an arbitrarily small constant ε .
Revised: Assume that β − γ ≥ ε for an arbitrarily small constant ε .

Original: Assume N = N ( ε ) is large enough.
Revised: Assume N = N ( ε ) is large enough.

Original: Let W : [ − 1 /e, ∞ ) −→ [ − 1 , ∞ ) be the Lambert function.
Revised: Let W : [ − 1 /e, ∞ ) −→ [ − 1 , ∞ ) be the Lambert function.

Original: Consider a time step t ∗ = ω (1) .
Revised: Consider a time step t ∗ = ω (1) .

Original: Let E ′ be the event I t ∗ ∈
Revised: Let E ′ be the event I t ∗ ∈

Original: [ ω (1) , o ( N )] and
Revised: [ ω (1) , o ( N )] and

Original: R t ∗ =
Revised: R t ∗ =

Original: o ( N ) .
Revised: o ( N ) .

Original: Then, 
Revised: Then, for large n ,

Original: [Equation]
Revised: [Equation]

Original: Furthermore, r ∞ is concentrated around its expected value.
Revised: Furthermore, r ∞ is concentrated around its expected value.

Revised: Moreover, E [ r ∞ | E ′ ] is concave for R 0 .

Original:  SIMULATIONS FOR TWO COUNTRIES
Revised:  SIMULATIONS FOR TWO COUNTRIES

Original: In this section, we study the impact of travelling and its relationship to early extinction on the total number of infections.
Revised: In this section, we study the impact of travelling and its relationship to early extinction on the total number of infections.

Original: Recall that p travel is defined as the travelling rate.
Revised: Recall that p travel is defined as the travelling rate.

Original: Without loss of generality we set γ = 0 , 2 in all our simulations.
Revised: Without loss of generality we set γ = 0 , 2 in all our simulations.

Original: 2 We set the initial size of each country to be
Revised: 2 We set the initial size of each country to be

Original: N (1) = N (2) = 2000 .
Revised: N (1) = N (2) = 2000 .

Original: Note that these size vary to a small extent throughout the process due to traveling individuals.
Revised: Note that these size vary to a small extent throughout the process due to traveling individuals.

Original: The total population is N = N (1)0 + N (2)
Revised: The total population is N = N (1)0 + N (2)

Original: = 4000 .
Revised: = 4000 .

Original: Initially, there is one infection per country ( I (1) = I (2) = 1 ) and the rest of the population is susceptible.
Revised: Initially, there is one infection per country ( I (1) = I (2) = 1 ) and the rest of the population is susceptible.

Original: In our simulations, we vary the travelling rate p travel and the infection rates β (1) and β (2) .
Revised: In our simulations we vary the travelling rate p travel and the infection rates β (1) and β (2) .

Original: W.l.o.g.
Revised: W.l.o.g.

Original: we assume β (1)= 0 ≤ β (2)= 0 .
Revised: we assume β (1) ≤ β (2) .

Original: For each value of p travel , β (1) , and β (2) , we output the average of[-
-]
Revised: For each value of p travel , β (1) , and β (2) , we output the average of[-
-]

Original:  . 000 iterations.
Revised: 10 . 000 iterations.

Original: The outcome of the simulations can be found in Fig.
Revised: The outcome of the simulations can be found in Fig.

Original: 2 and Fig.
Revised: 3 and Fig.

Original: For each value of p travel we plot the average number of infections (cid:100) R (1) ∞ in country one in light green and in light blue we plot the average number of infections in country two (cid:100) R (2) ∞ .
Revised: For each value of p travel we plot the average number of infections (cid:100) R (1) ∞ in country one in light green and in light blue we plot the average number of infections in country two (cid:100) R (2) ∞ .

Original: Note that we write (cid:100) R ∞ instead of E [ R ∞ ] to emphasise the difference between the average and the expected value.
Revised: Note that we write (cid:100) R ∞ instead of E [ R ∞ ] to emphasise the difference between the average and the expected value.

Original: The total number of infections across both countries (cid:100) R ∞ = (cid:100) R (1) ∞ + (cid:100) R (2) ∞ is plotted in red.
Revised: The total number of infections across both countries (cid:100) R ∞ = (cid:100) R (1) ∞ + (cid:100) R (2) ∞ is plotted in red.

Original: We plot the smoothed version of (cid:100) R ∞ in black.
Revised: We plot the smoothed version of (cid:100) R ∞ in black.

Original: Let R ∞ ( p travel ) denote the total number of infections for a given travelling rate p travel .
Revised: Let R ∞ ( p travel ) denote the total number of infections for a given travelling rate p travel .

Original: Finally, the corresponding standard deviation is plotted using vertical bars.
Revised: Finally, the corresponding standard deviation is plotted using vertical bars.

Original: First, note that the standard deviation appears large: this is unavoidable and due to the fact that early extinctions (in this case, when (cid:100) R ∞ ≤ √ 2 N = √ 4000 ≈ 63 ) occur with constant probability.
Revised: First, note that the standard deviation appears large: this is inherent to the process and due to the fact that early extinctions (in this case, when (cid:100) R ∞ ≤ √ 2 N = √ 4000 ≈ 63 ) occur with constant probability.

Revised: Indeed, the plots are the result of averaging among all the iterations, included the ones resolving in early extinction.

Original: Excluding runs with an early extinction, the standard deviation becomes negligible.
Revised: By excluding runs with an early extinction, the standard deviation becomes negligible.

Original: Another effect of this exclusion is that the values of (cid:100) R ∞ turn up to be visibly smaller (see Fig. 2).
Revised: Another effect of this exclusion is that the values of (cid:100) R ∞ turn up to be visibly smaller (see Fig. 3).

Revised: Another interesting detail with respect to this is shown in Fig. 4a.

Revised: In there, the error bars representing the variance significantly decrease with increasing p travel .

Revised: This tells us that (cid:100) R ∞ becomes small because the percentage of runs that terminate with early extinction increases significantly, compared to the case where no or little travel is present.

Revised: This to the point that these runs do not constitute the variance of the process anymore, but the most likely outcome, hence the reduction in the variance.

Original: It is perhaps natural to assume that increasing p travel also increases (cid:100) R ∞ .
Revised: It is perhaps natural to assume that increasing p travel also increases (cid:100) R ∞ .

Original: While this is true for some values of the β s, there are also values for which (cid:100) R ∞ first increases monotonically and then decreases monotonically.
Revised: While this is true for some values of the β s, there are also values for which (cid:100) R ∞ first increases monotonically and then decreases monotonically.

Original: In addition, for[Table] large p travel the number of total infections can drop below the  where no travelling occurs, i.e., (cid:100) R ∞ (1) <
Revised: In addition, for large p travel the number of total infections can drop below the value where no travelling occurs, i.e., (cid:100) R ∞ (1) <

Original: (cid:92) R ∞ (0) .
Revised: (cid:100) R ∞ (0) .

Original: See Fig. 3a and Fig. 3b.
Revised: Table 2, Fig. 3 and Fig. 4.

Original: Our simulation results can be found in Table 2, Fig. 2 and Fig.

Original: Simulation Result 1.
Revised: Simulation Result 1.

Original: Assume that γ = 0 .
Revised: Assume that γ = 0 .

Original: β (1) = 0 and β (2) = 0 .
Revised: β (1) = 0 and β (2) = 0 .

Original: 3 results in R ∞ ( p travel ) being non-monotone.
Revised: 3 results in R ∞ ( p travel ) being non-monotone.

Original: Moreover, R ∞ (1 / 2) =
Revised: Moreover, R ∞ (1 / 2) =

Original: O (1) (Fig. 3a).
Revised: O (1) (Fig. 4a).

Original: β (1) = 0 and β (2) = 0 .
Revised: β (1) = 0 and β (2) = 0 .

Original: 5 results in R ∞ ( p travel ) being non-monotone.
Revised: 5 results in R ∞ ( p travel ) being non-monotone.

Original: (Fig. 3b).
Revised: (Fig. 4b).

Original: β ( = 0 .
Revised: β (1) = 0 .

Original: 5 and β (2) = 0 .
Revised: 5 and β (2) = 0 .

Original: 8 results in R ∞ ( p travel ) increasing monotonically.
Revised: 8 results in R ∞ ( p travel ) increasing monotonically.

Original: (Table 2, and also Fig. 3c).
Revised: (Table 2, and also Fig. 4c).

Original: See further examples in Appendix H Based on this, we believe that the simulation results can be generalised to the following conjecture.
Revised: Based on this, we believe that the simulation results can be generalised to the following conjecture.

Original: Conjecture 1.
Revised: Conjecture 1.

Original: For all β (1) , β (2) and γ we have (i) β (1) , β (2) < 
Revised: For all β (1) , β (2) and γ we have (i) β (1) , β (2) < γ :

Original: γ : E [ R ∞ ( p travel )]
Revised: E [ R ∞ ( p travel )]

Original: = O (1) for all p travel ∈ [0 , 
Revised: = O (1) for all p travel ∈ [0 , 1] . [Figure] [Table]

Revised: We round to the nearest integer.

Original: (ii) β (1) <
Revised: (ii) β (1) <

Original: γ < β (2) is non-monotone.
Revised: γ < β (2) is non-monotone.

Original: (iii) β (1) , β (2) > γ : E [ R ∞
Revised: (iii) β (1) , β (2) > γ : E [ R ∞

Original: ( p travel )] increases monotonically.
Revised: ( p travel )] increases monotonically.

Original: Note that Fig.
Revised: Note that Fig.

Original: 3c suggests that it is important for the β s to be strictly larger than γ in order for E [ R ∞ ( p travel )] to increase monotonically.
Revised: 4c suggests that it is important for the β s to be strictly larger than γ in order for E [ R ∞ ( p travel )] to increase monotonically.

Original: At the core of the conjecture is the following observation.
Revised: At the core of the conjecture is the following observation.

Original: As p travel increases, both β (1) and β (2) are blended together resulting  a linear combination of both values.
Revised: As p travel increases, both β (1) and β (2) are blended together resulting in a linear combination of both values.

Original: Consider the following thought experiment where the β s fully blend until the “effective” β of both cities is the average β = ( β (1) + β (2) ) / 2 .
Revised: Consider the following thought experiment where the β s fully blend until the “effective” β of both cities is the average β = ( β (1) + β (2) ) / 2 .

Original: If we ignore the effect of travelling all statements of the conjecture follow from the Theorem 2.
Revised: If we ignore the effect of travelling, all statements of the conjecture follow from the Theorem 2.

Original: We see in Fig.
Revised: On the other hand, for high values of p travel , then we can look at β and see in Fig.

Original: 3a that if β is below γ , then the total number of infections R ∞ is close to 0 .For β larger than γ , R ∞ is
Revised: 4a that if β is below γ , then the total number of infections R ∞ is close to 0 .

Original: [Figure]
Revised: We give the following explanation. When travel becomes more likely, infected agents might be selected to travel from one country to the other one.

Original: [Figure]
Revised: In such starting country, then, there will be less infected agents, making the infection step even more less likely than the recovery step. This further reduces the number of infected agents. On top of this, recovered or susceptible agents can be selected to travel too, lowering the ratio of infected agents in their destination country. Since the process starts with a small number of infected agents, travelling will undermine their influence significantly more than the influence of susceptible or recovered agents.

Original: [Figure] concave.
Revised: For β larger than γ , R ∞ is concave (see Theorem 6).

Original: The concavity is the key to the understanding.
Revised: The concavity is the key to the understanding.

Original: For example, when both original β s were above γ , then having β in both countries increases R ∞ , due to the concavity.
Revised: For example, when both original β s were above γ , then having β in both countries increases R ∞ , due to the concavity.

Original: Let us consider the cases of the conjecture one by one.
Revised: Let us consider the cases of the conjecture one by one.

Original: Part (i) follows since β < γ and any blending of the β s will yield “effective” β s less than γ .
Revised: Part (i) follows since β < γ and any blending of the β s will yield “effective” β s less than γ .

Original: For part (ii) note that when p travel = 0 and no blending of the β s occurs, then we have that in one city R 0 > 1 (expected pandemic) and in the other R 0 < 1 (expected early extinction).
Revised: For part (ii) note that when p travel =and no blending of the β s occurs, then we have that in one city R 0 > 1 (expected pandemic) and in the other R 0 <(expected early extinction).

Original: When p travel is slightly above 0 the the blending of the β s effectively decreases β (2) a little bit, the number of infectable individuals also doubles once p travel > 0 .
Revised: When p travel is slightly abovethen the blending of the β s effectively decreases β (2) a little bit, the number of infectable individuals also doubles once p travel > 0 .

Original: This becomes clear when one considers the setting where β (1)
Revised: This becomes clear when one considers the setting where β (1)

Original: = 

Revised: =

Original: and β (2)
Revised: and β (2)

Original: = 1 .
Revised: = 1 .

Original: Here E [ R ∞ ] > N and thus clearly some individuals of country 1 contribute.
Revised: Here E [ R ∞ ] > N and thus clearly some individuals of country 1 contribute.

Original: Part (iii) follows immediately from the concavity.
Revised: Part (iii) follows immediately from the concavity.

Original: The simulations presented here are only a small fraction of the number of our simulations covering the large variety of different values for β (1) , β (2) and γ . 
Revised: The simulations presented here are only a small fraction of the number of our simulations covering the large variety of different val- [Figure] c) β (1) , β (2) > γ . The total number of

Revised: [Figure]

Original:  All our simulations confirmed our conjecture.
Revised: R ∞ ( p travel ) appears to be monotonically increasing. See also Table 2. ues for β (1) , β (2) and γ . All our simulations confirmed our conjecture.

Original: Nonetheless, we were not able to turn the arguments above into a rigorous proof.
Revised: Nonetheless, we were not able to turn the arguments above into a rigorous proof.

Original: On one hand, considering one country only is already immensely complicated and a second country increases the number of variables and their decencies even further.
Revised: On one hand, considering one country only is already immensely complicated and a second country increases the number of variables and their decencies even further.

Original: On the other hand, Pr [ E ext ] and further considerations have to be accounted for.
Revised: On the other hand, Pr [ E ext ] and further considerations have to be accounted for.

Original: Indeed, one cannot simply ignore p travel and assume that both countries have β .
Revised: Indeed, one cannot simply ignore p travel and assume that both countries have β .

Original: The simulations suggest that we have essentially two values for R ∞ ; one being R ∞ (0) and one being R ∞ ( p ) , for any p > 0 ( See Appendix H).
Revised: The simulations suggest that we have essentially two values for R ∞ ; one being R ∞ (0) and one being R ∞ ( p ) , for any p > 0 .

Original: The reason for this is that if p travel = 0 , then it can happen that one city has an early extinction.
Revised: The reason for this is that if p travel = 0 , then it can happen that one city has an early extinction.

Original: When p travel > 0 , we believe that whenever only one of the countries has an early extinction, then the other country will eventually send infected individuals over and rekindle the infection until it succeeds.
Revised: When p travel > 0 , we believe that whenever only one of the countries has an early extinction, then the other country will eventually send infected individuals over and rekindle the infection until it succeeds.

Original: While the above shows the influence of effects when p travel is small, when believe that for large values of p travel our arguments capture the behaviour of the process.A APPENDIX: PROOF OF THEOREM 4PROBABILITY OF EARLY EXTINCTION
Revised: While the above shows the influence of effects when p travel is small, ww believe that for large values of p travel our arguments capture the behaviour of the process.

Original:  In this section we prove Theorem 4 using the coupling result from Theorem 5 which allow us bound the probability of the the early extinction time of our process by Pr (cid:104) E ( P ∗ ,S ) ext (cid:105) , the early extinction time of P ∗ ,S for S = N and S = N − τ. The nice property about P ∗ ,S is that, restricted to active time steps, the process behaves exactly like a biased random walk. In the following we will omit the superscript P ∗ ,S and write I t , S t and R t for the infected, susceptible and recovered individuals.
Revised:  FUTURE WORK

Original: When clear from the context, we will omit the index t .

Original: Hence, the probability to increase I t by 1 is βS/N and the probability to decrease I t is γ .

Original: In more detail

Original: [Equation] With this is mind, we consider the ratio 1 − pp , which is equal to [Equation] We can apply Theorem 7 in Appendix G, and derive
Revised: In this work we adopted one of the many standard models for disease spreading assuming pairwise interactions.

Original: Pr

Original: [Equation]

Original: ext ]

Original: [Equation]

Original: Pr [ I

Original: [Equation]

Original: 0] =

Original: [Equation]

Original: [-where I 
-]
Revised: [-where I 
-]

Original: = s .

Original: We separate the proof in two cases, depending on whether the quantity (cid:16) γNβS (cid:17) is smaller or bigger than 1 .

Original: These cases are, respectively:

Original: β > γ + ε ′ and β < γ − ε .

Original: We define ε ′ =

Original: ε ( N − τ ) N , as reported in the theorem statement.

Original: Case 1: β > γ + ε ′ .

Original: From Theorem 5 we have

Original: [Equation]

Original: The left-hand side of Eq.

Original: (5) can be lower-bounded as

Original: Pr

Original: [Equation]

Original: while the right-hand side as

Original: [Equation]

Original: We get that

Original: [Equation]

Original: for x = τ/ ( N − τ ) .

Original: We have (1 + x ) s ≤ 1 + sx 1 − sx + x for all x ∈ [ − 1 , 1 / ( s − 1)] and s > 1 .

Original: Moreover,

Original: [Equation]

Original: − 1) .

Original: Using x = τ/ ( N − τ ) we get

Original: [Equation]

Original: Case 2: β < γ − ε .

Original: The left-hand side of Eq.

Original: (5) can be lower-bounded by

Original: [Equation]

Original: B APPENDIX: PROOF OF THEOREM

Original: Let Z t

Original: [-= I t + R t = N − S t be the total number of infections after t steps with Z 
-]
Revised: [-= I t + R t = N − S t be the total number of infections after t steps with Z 
-]

Original: = I 0 + R 0 .

Original: We can write E [ I t +1 | I t ] as E [ I t +1 | I t ]

Original: = I t · (cid:104) 1 + βN · S t N − γN (cid:105) .

Original: [-For the analysis we consider active steps only, i. e., steps t in which the number of infected particles either increases by one (new infection: I t +
-]
Revised: [-For the analysis we consider active steps only, i. e., steps t in which the number of infected particles either increases by one (new infection: I t +
-]

Original: =

Original: I t +1 ), or decreases by one (new recovery:

Original: I t +1 =

Original: I t − 1 , R t +1 =

Original: R t + 1 ).

Original: In the first case S t +1 =

Original: S t − 1 and Z t +1 =

Original: Z t + 1 ; in the second case, S t +1 =

Original: S t and Z t +1 =

Original: Z t .

Original: Let L be the number of individual infected during the whole process, not counting the initial I 0 infections (i.e., R ∞ =

Original: I 0 + R 0 + L ), and for j = 0 , 1 , . . .

Original: , L , let T j ≥ 0 be the length (in active steps) of the j -th recovery run.

Original: That is, T j is the number of recoveries between the j -th and ( j + 1) - st infections (or, for j = L , the number of the remaining recoveries after the very last infection in the process).

Original: Let

Original: T ( j ) =

Original: [-T 
-]
Revised: [-T 
-]

Original: + T 1 + · · · + T j .

Original: For j = 0 , 1 , . . .

Original: , L , after the j -th recovery run, that is, after T ( j ) + j active steps (which are T ( j ) recoveries plus one infection after each recovery run except the last one), the parameters S , I and R are

Original: S ( j ) =

Original: S 0 − j, I ( j ) =

Original: [-I 
-]
Revised: [-I 
-]

Original: + j − T ( j ) , R ( j ) =

Original: R 0 + T ( j ) .

Original: After the last recovery run, I ( L ) = 0 (no infected individuals left), so I 0 + R 0 + L − T ( L ) = 0 , and we want to use this equation to derive L (and hence also R ∞ ).

Original: If 0 ≤ j ≤ L , then the length T j of the j -th recovery run is almost a geometric process: While the number of susceptible individuals stays at S ( j ) , we are waiting for the next infection (’success’), which comes at each active step independently with probability

Original: [Equation]

Original: It is not exactly a geometric process, since we stop waiting for a success also when all remaining infected individuals happen to recover (no next infection).

Original: More precisely, for 0 ≤ j ≤ L , T j = min (cid:110) (cid:101) T j , I ( j − 1) + 1 (cid:111) where (cid:101) T j + 1 are independent geometric variables with the parameters p j , and I ( j − 1) + 1 is the number of infected individuals at the beginning of the j -th recovery run.

Original: (Set

Original: I ( − 1) =

Original: I 0 − 1 , to make T j work also for j = 0 ).

Original: We derive the expectation and concentration of L by referring to the equation

Original: I 0 + L − T ( L ) = 0 and by finding k such that

Original: [-I 
-]
Revised: [-I 
-]

Original: + k − E (cid:104) (cid:101) T ( k ) (cid:105) =

Original: O (1) and k =

Original: Θ( N ) (11), where (cid:101) T ( k ) =

Original: [-(cid:101) T 
-]
Revised: [-(cid:101) T 
-]

Original: + (cid:101) T 1 + · · · + (cid:101) T k .

Original: Let k r be the solution to (11).

Original: We will prove that k r ∼ r.h.s.

Original: of (2).

Original: Before showing this, we first argue that | L − k r | =

Original: O ( √ N log 3 N ) with either (polynomially) high probability (w.h.p.) or with probability at least 1 − exp ( − Ω( I 0 )) , which ever is smaller.

Original: This implies that E [ R ∞ ] ∼ E [ L ] ∼ k r (since k r =

Original: Θ( N ) and

Original: L = O ( N ) ) and R ∞ is concentrated around its expectation.

Original: While solving (11), we will observe that for k ≤ k r such that k r − k

Original: [-= o ( N ) , I 
-]
Revised: [-= o ( N ) , I 
-]

Original: +

Original: [Equation]

Original: Based on this, we take k ′ =

Original: k r − Θ( √ N log 3 N ) such that I 0 + k ′ − E (cid:104) (cid:101) T ( k ′ )

Original: (cid:105) =

Original: (1 + o (1)) √ N log 3 N .

Original: Informally, we are not looking at infection k r , which might not even occur, but instead at an earlier infection k ′ <

Original: k r , ensuring that at this point, while the process is dying out, there are still Θ( √

Original: N log 3 N ) infections left to clear.

Original: We distinguish between two cases based on I 0 .

Original: First assume

Original: I 0 ≥ 2 √ N log 3 N .

Original: [-For k increasing from 0 to k ′ , the function k − E (cid:104) (cid:101) T ( k ) (cid:105) is bitonic (first increasing and then decreasing), so for each 0 ≤ k ≤ k ′ , I 
-]
Revised: [-For k increasing from 0 to k ′ , the function k − E (cid:104) (cid:101) T ( k ) (cid:105) is bitonic (first increasing and then decreasing), so for each 0 ≤ k ≤ k ′ , I 
-]

Original: + k − E (cid:104) (cid:101) T

Original: [Equation]

Original: k ) (cid:105) ≥ √

Original: N log 3 N .

Original: As (cid:101) T ( k ) is the sum of fewer than N independent geometric random variables, each with the probability of success at least some positive constant, it can be shown that the probability that (cid:101) T ( k ) differs from its expectation by at least √ N log 3 N is at least e − Ω(log 2 N ) (for example, by using the Azuma–Hoeffding inequality, considering separately the case when the value of one of the geometric variables is greater than log 2 N ).

Original: Thus, by the union bound, with high probability (w.h.p.), for each 0 ≤ k ≤ k ′ ,

Original: [-I 
-]
Revised: [-I 
-]

Original: [-+ k − (cid:101) T ( k ) > 0 , and also I 
-]
Revised: [-+ k − (cid:101) T ( k ) > 0 , and also I 
-]

Original: + k ′ − (cid:101) T ( k ′ )

Original: ≤ 2 √ N log 3 N .

Original: This implies that, w.h.p.

Original: for each 0 ≤ k ≤ k ′ , T k = (cid:101) T ( k ) , so, w.h.p.

Original: L ≥ k ′ and the number of active infections after the k ′ recovery run is I ( k ′ )

Original: =

Original: [-I 
-]
Revised: [-I 
-]

Original: + k ′ − T ( k ′ ) =

Original: I 0 + k ′ − (cid:101) T ( k ′ )

Original: =

Original: O ( √ N log 3 N ) .

Original: The probability of infection at each active step after the k ′ infections is a constant less than 1 / 2 ( p k r is a constant less than 1 /and p k ′ =

Original: p k r + o (1) ) so the process finishes w.h.p. within additional O ( √ N log 3 N ) active steps, that is, L = k ′ + O ( √ N log 3 N ) = k r + O ( √ N log 3 N ) .
Revised: Incorporating super spreaders is a great idea for further work.

Original: For I 0 = ω (1) but less than 2 √ N log 3 N , we reduce to the previous case by observing that with probability at least 1 − exp − Ω

Original: ( I 0 ) , the number of active infections increases to 2 √ N log 3 N , while the number of recoveries increases by Θ( √ N log 3 N ) .

Original: This can easily be done by using the proof idea of Section 3.

Original: We now come back to solving (11) for k .

Original: The expectation of (cid:101) T j is equal to 1 /p j − 1 , so the expectation of (cid:101) T ( k ) is

Original: [Equation]

Original: The r.h.s.

Original: above is equal to

Original: [-I 
-]
Revised: [-I 
-]

Original: + k when

Original: I

Original: [Equation]

Original: , yielding

Original: [Equation]

Original: [-From this equation, putting κ = k/N , using the assumption that S 
 = N − o ( N ) , and considering N increasing-] to infinity, we get 1 − [Equation]
Revised: [-From this equation, putting κ = k/N , using the assumption that S 
 = N − o ( N ) , and considering N increasing-]One way to incorporate super spreaders is to assume an underlying social network with high degree nodes.

Original: =

Original: [Equation]

Original: =

Original: [Equation]

Original: , so − ( β/γ )( − κ ) e − ( β/γ )(1 − κ ) = − ( β/γ ) e − ( β/γ ) (cid:0) 1 + O (cid:0) I 0 N (cid:1)(cid:1) , giving a solution κ r = 1 + γβ W (cid:0) − ( β/γ ) e − ( β/γ ) (cid:1) + O (cid:0) I 0 N (cid:1) .

Original: Thus k r

Original: = κ r N ∼ r.h.s.

Original: of (2).

Original: C APPENDIX: PROOF OF PROPOSITION

Original: Proof.

Original: To show this result we will use an identity coupling. Hence, we assume that both processes activate the same individuals and make the same random decisions. It iseasy to see that the result holds if T = T τ (both processes stopped for the same reason).
Revised: Another one is to first activate a random individual and then to choose a random number of interaction partners using a suitable distribution.

Original: Hence, to proof the lemma we have to estimate the probability that T τ < T .

Original: In the following we will consider the first τ time steps of P and P τ, ∗ .

Original: We consider two cases: β > γ + ε or β < γ − ε .

Original: Case β > γ + ε : We consider a slightly different process

Original: P ′ in which we always keep at least one infected individual (if there is only one infected individual we do not allow that individual to recover).

Original: For P ′ we show that the probability of hitting √ N many infected individuals during the first τ time steps is at least 1 − 1 /N .

Original: Note that this event ( P ′ hitting √ N many infected individuals during the first τ steps) implies that process P hits either √ N or 0 many infected individuals during the first τ time steps, so we will conclude that Pr [ T τ = T ] ≥ 1 − 1 /N , or equivalently that

Original: Pr [ T τ < T ] ≤ 1 /N .

Original: It remains to show that the probability for P ′ to hit √ N many infected individuals in the first τ time steps is at least 1 − 1 /N .

Original: Since we only consider the first τ time steps of P ′

Original: we know that S t ≥ N − τ .

Original: Fix an active step t ′ ≤ τ .

Original: Define δ =

Original: ε 4 β .

Original: The probability that the number of infections is increased by one is

Original: [Equation]

Original: where the last inequality holds for sufficiently large N .

Original: Hence, in τ ′ = √ N log N many steps the expected number of infections is increased at least (cid:0) 12 + δ (cid:1) √ N log N many times.

Original: Similarly, since the probability that the number of infections is decreased by one in step t ′ is at most 12 − δ , in τ ′ =

Original: √ N log N many steps the expected number of decreases is at most (cid:0) 12 − δ (cid:1) √ N log N .

Original: Using the

Original: Chernoff bounds from Lemma 1, together with the union bound, we can show that with a probability of 1 − 1 /N

Original: the number of increases is at least (cid:0) 12 + δ 2 (cid:1) √ N log N and the number of decreases is at most (cid:0) 12 −

Original: δ 2 (cid:1) √ N log N .

Original: Hence with probability at least 1 − 1 /N

Original: we have I τ ≥ √ N.

Original: Case β < γ − ε : This proof is identical to that of Case 1.

Original: The only difference is now that we show that it is very likely to have much more decreases such that I τ = 0 .

Original: Proposition 4.

Original: Let E ′ be the event that R T τ ≤ √ N log N/and I

Original: T τ = √ N .

Original: Then,

Original: [Equation]

Original: Moreover,

Original: E

Original: [Equation]

Original: r ∞ | E

Original: [Equation]

Original: ] = o

Original: [Equation]

Original: [Equation]

Original: [Equation]

Original: Proof of Proposition 4.

Original: Note that

Original: I ( P ) T

Original: =

Original: I

Original: ( P τ, ∗ ) T τ implies that R ( P

Original: [Equation]

Original: ≤

Original: [Equation]

Original: N log N.

Original: By Proposition 1,

Original: Pr (cid:104)

Original: I

Original: [Equation]

Original: P

Original: [Equation]

Original: ≥ 1 − 1 /N.

Original: Thus,

Original: [Equation]

Original: since we assume β and γ to be bounded by constants.

Original: From this, we get

Original: [Equation]

Original: Note that E (cid:2) r ∞ | E ext , E ′ (cid:3) ≤ 1 .

Original: Thus, asymptotically, we derive

Original: [Equation] Using Similar arguments, one can show that E [ r ∞ | E ext ] = o (1) . D APPENDIX: PROOF OF PROPOSITION
Revised: Both models need a different analysis method compared to the analysis in this paper.

Original: Proof. First we consider the case S = N and show [Equation] = √ N (cid:105) .
Revised: We believe it is very interesting to study them in future work.

Original: Via induction we show that for all t ≤ T τ we have | I ( P τ, ∗ )

Original: T τ | ≤ | I ( P τ,N ) T τ | .

Original: The result follows directly form the induction.

Original: [-Since both processes have exactly one infected individual at t = 
-]
Revised: [-Since both processes have exactly one infected individual at t = 
-]

Original: the base case certainly holds.

Original: Now we have to show that | I ( P τ, ∗ ) t +1 | ≤ |

Original: I ( P τ,N ) t + | .

Original: Note that, since T τ = √ N log N , both processes have susceptible individuals to turn into infected individuals.

Original: Hence,

Original: [Equation]

Original: (cid:105) .

Original: If I ( P τ, ∗ )

Original: [Equation]

Original: [Equation]

Original: then both processes have the same probability

Original: I ( Pτ, ∗ ) t N · γ to decrease the number of infections by one.

Original: If I ( P τ,N ) t <

Original: I ( P τ,N ) t the probability for a decrease is slightly larger for

Original: P τ,N

Original: but we still have | I ( P τ, ∗ ) t +1 | ≤ | I ( P τ,N ) t + | .

Original: In both cases we have | I ( P τ, ∗ ) t +1 | ≤ |

Original: I ( P τ,N ) t + | .

Original: Now we are ready for the case S =

Original: N − τ .

Original: Again, via induction we show that for all t ≤ T τ we have | I ( P τ,N ) T τ | ≤ | I ( P τ,N ) T τ | .

Original: The base case trivially holds again.

Original: Since S ( Pτ, ∗ ) t N ≥ N − τ we get

Original: Pr

Original: [Equation]

Original: If

Original: I ( P

Original: [Equation]

Original: ,N −

Original: [Equation]

Original: ) t

Original: = I ( P

Original: [Equation]

Original: t then both processes have the same probability to decrease the number of infections by one.

Original: If I ( P τ,N − τ ) t < I ( P τ,N ) t the probability for a decrease is slightly larger for P τ,N , but we still have | I ( P τ, ∗ ) t +1 | ≤ |

Original: I ( P τ,N ) t + | .

Original: E APPENDIX: MISSING PROOFS

Original: Proof of Theorem 2.

Original: By law of total expectation, we can bound the term E [ r ∞ ] as follows

Original: [Equation]

Original: The bound follows by using Proposition 4 and plugging in the value for E [ r ∞ | E ′ ]

Original: (Theorem 6) and the value for

Original: Pr (cid:2) E ext (cid:3) (Theorem 4).

Original: Proof of Theorem 3.

Original: The result follows from Theoremand Proposition 4.

Original: F APPENDIX: CONVEXITY OF R

Original: See Fig. 4.

Original: G APPENDIX: AUXILIARY CLAIMS

Original: Theorem 7 (Chapter XIV.2, XIV.3 in Feller [1968]) .

Original: Let p ∈

Original: (0 , 1) \ { 1 / 2 } and b, s ∈ N .

Original: Consider a discrete time

Original: Markov chain ( Z t ) t ≥ 0 with state space Ω = [0 , b ] where

Original: • Z 0 = s ∈ [0 , b ] • Pr

Original: [ Z t = i | Z t − 1 =

Original: i − 1]

Original: = p for i ∈ [1 , b − 1] , t ≥ • Pr

Original: [ Z t = i | Z t − 1 =

Original: i + 1] = 1 − p for i ∈ [1 , b − 1] , t ≥

Original: [Figure]

Original: [Equation]

Original: [Figure]

Original: [Equation]

Original: [Figure]

Original: [Equation]

Original: [Figure]

Original: [Equation]

Original: for

Original: [Equation]

Original: Let T =

Original: min

Original: t ≥ 0 | Z t ∈ { 0 , b }} .

Original: Then,

Original: Pr

Original: [Equation]

Original: (cid:16)

Original: [Equation]

Original: − pp (cid:17) s −(cid:16) 1 − pp (cid:17) b − 1 and

Original: Pr [ Z T = 0] = (cid:16) 1 − pp (cid:17) b − (cid:16) 1 − pp (cid:17) s (cid:16) 1 − pp (cid:17)

Original: b −.

Original: Moreover,

Original: E

Original: [Equation]

Original: =

Original: [Equation]

Original: (cid:16)

Original: [Equation]

Original: (cid:17) s

Original: [Equation]

Original: (cid:16)

Original: [Equation]

Original: Lemma 1 (Chernoff bounds) .

Original: Let X 1 . . .

Original: X n be independent random variables taking values in [0 , 1] .

Original: Let X = (cid:80) ni

Original: =1 X i .

Original: Then for all λ ≥

Original: Pr [ X

Original: [Equation]

Original: [ X ] + λ ] ≤ e

Original: [Equation]

Original: [Equation]

Original: 2 N .

Original: [Equation]

Original: H APPENDIX: FURTHER SIMULATIONS PLOTS

Original: Simulation Environment: or all the simulations, the language used is C++.

Original: Simulations were run on a DELL Latitude 7400 laptop, with Intel Core i7-8665U, vPro - 8th

Original: Gen CPU and 16GB RAM.

Original: We plot the mean value and the standard deviation.

Original: [Figure]