Containment strategy for an epidemic based on fluctuations in the SIR model

Philip Bittihn1 and Ramin Golestanian1, 2

1Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Gottingen, Germany2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom

(Dated: April 14, 2020)

Building on the observation that the spread of an infection is subject to stochastic fluctuationswhen infection numbers are small, we propose a strategy of containment that falls in between rel-atively mild social distancing measures and maximally restrictive lockdown strategies. The keyinnovation of our proposed strategy is its ability to recruit stochastic effects such as spontaneousextinction and fluctuations in timing to contain the epidemic, even when infection numbers in thepopulation as a whole are not small and their spatial distribution is unknown. It involves parti-tioning the population into smaller isolated sub-populations within which, while social distancingis practiced as much as possible to reduce the contact rate, a relatively normal lifestyle is main-tained. As a rule of thumb, the optimal size of the sub-populations can be obtained by dividing thetotal population size by the best estimate of the number of infected individuals at the time of theimplementation of this containment strategy.

a. Introduction.— Serious epidemics like the cur-rent outbreak of the novel Corona virus compel govern-ments to react with containment measures that aim atslowing down or, ideally, stopping the spread of the dis-ease [1]. When infections increase and quarantining con-firmed infected individuals is not sufficient to preventspreading, lowering the overall contact rate of individ-uals in the population might seem like the only viableoption, which, depending on the severity of the restric-tions, comes at a substantial economic and social cost.Here, we propose a containment strategy that involvesisolation of smaller sub-populations, which boosts thepossibility for stochastic effects to spontaneously end in-fection chains in local communities via a process calledextinction. We quantify the resulting reduction in thepeak number of infected individuals for given epidemicparameters due to local extinction of the epidemic anddesynchronization between sub-populations (see Figs. 1aand 1b). These effects can then lead to a reduction in theintensity of the outbreak, even when, deterministically,infections grow exponentially. This could be the case ei-ther near the beginning of an outbreak, when numbersare low and drastic measures have not been activated,or as an exit strategy, when drastic measures such aslockdowns have brought the number of infections down,but gradually lifting restrictions might reinitiate the ex-ponential growth. Although our proposed strategy alsorepresents a type of confinement (namely confinement ofindividuals to local communities), it is in many aspectscomplementary to these contact reduction measures.

Extensive prior work exists on the spread of infectionsthrough populations of various topological structure; see,e.g. [2] and references therein. While many details aboutthe biology and modes of infection of a specific diseaseare important for its dynamics and advanced models [3],and control schemes have considered stochastic effects [4]as well as certain features of the underlying contact net-works [5, 6], one distinguishing feature of our strategy isthat it does not depend on many of these details.

b. Model.— We consider a population of N individ-uals with SIR dynamics [7]

S + Ib−−→ I + I, I

k−−→ R, (1)

with S, I and R referring to susceptible, infected and re-moved individuals, respectively, where removal with per-capita rate k happens due to recovery, quarantine ordeath. The rate b corresponds to the number of contactsper unit time an individual has with a random other in-dividual in the population, multiplied by the probabilitythat a contact between a susceptible and an infected in-dividual leads to transmission. These rates are related

FIG. 1: Stochastic effects lower the peak in subdivided pop-ulations. (a) Time course for a population of N = 1,000,000with I0 = 10 initially infected individuals for Ns = 1 largepopulation (red) and a population split into Ns = 10 sub-populations (turquoise), b = 0.2, k = 0.14. Shading indi-cates ±25% confidence intervals across 100 simulations. Sub-populations are shown from one simulation with Ns = 10.(b) Enlarged plot of the initial phase for the same traces asin panel a. (c) Distribution of peak times in sub-populationsfor Ns = 10. Occurrence fraction indicates fraction of sub-populations across all simulations. Dashed line indicates an-alytical approximation, Eq. (8), with uniform n = I0/Ns = 1.

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to the basic reproduction number R0 = b/k, which iscommonly used to quantify the intensity of an outbreak[2], and the epidemic threshold above which an outbreakoccurs is R0 = 1. The population is subject to thetotal constraint N = S(t) + I(t) + R(t), where we de-note the number of individuals in each state by the sameletters. All numerical results shown below are obtainedfrom stochastic simulations of Eq. (1) using the Gillespiealgorithm [8].

c. Deterministic behaviour.— The dynamics resultsin the deterministic mean-field equations

dS

dt= − b

NS I, (2a)

dI

dt=

b

NS I − k I, (2b)

dR

dt= k I, (2c)

which give rise to two regimes in the dynamics. Duringthe initial regime, I starts off from an initial value I(0) =I0, rises exponentially ∼ I0e

(b−k)t, and saturates to apeak value

Imax ≡ γN ≈(

1− k

b

[1 + log(b/k)

])N, (3)

where the approximation for the maximum fraction of in-fected individuals 0 < γ < 1 is valid as long as the entirepopulation is initially susceptible, i.e. S(0) ≈ N [9]. Inthe secondary regime when the recovery dynamics dom-inates, I decays to zero exponentially, as the number ofsusceptibles decreases below the value necessary to sus-tain spreading.

d. Small numbers.— Deterministic behaviour onlyapplies if S and I are both large, particularly only afterthe number of infected people I has risen to apprecia-ble levels. If I is still low, stochastic effects determinewhether I will “take off” and develop exponential be-haviour, even if b > k. This problem was already consid-ered shortly after Kermack and McKendrick introducedthe original SIR model [10], and we briefly recapitulatethe important result here, as the specific stochastic pro-cess we consider as an analogy during the initial phase ofthe epidemic will be used for derivations throughout thisstudy. During this phase, we can assume that S ≈ N andthat I follows a simple birth-death process with rates bfor birth and k for death. From the theory of branchingprocesses, it is well known that even an exponentiallygrowing population that starts from an initial conditionof I(0) = 1 has a finite extinction probability of

P0(t) =k

b· e(b−k)t − 1

e(b−k)t − k/b. (4)

which asymptotically approaches k/b at long times; seethe derivation in Appendix A. This means that withprobability pext

1 = k/b [11] the dynamics never entersthe exponentially growing deterministic regime, but de-cays back to zero due to number fluctuations. For two

independent lineages in the same population, the extinc-tion probability is therefore pext

2 = (k/b)2, and, similarly,pextn = (k/b)n, as long as the total population is suffi-

ciently large such that the lineages do not interfere witheach other.

e. Isolated sub-populations.— In a population withN individuals, where the number of infected cases hasalready left the stochastic regime, the peak number ofinfected individuals will be given by Eq. (3). In con-trast, if the population is split up intoNs sub-populationsof equal size N/Ns and the infected individuals are dis-tributed among them, some sub-populations can expe-rience extinction of the outbreak due to the low num-ber of initially infected individuals. The fact that thesize of the sub-populations is also smaller than the orig-inal population is irrelevant in the early stages of theepidemic, as the extinction probabilities above do notdepend on the population size. Note that, from a de-terministic standpoint, this subdivision would have noeffect, since Eq. (2) remains invariant when scaling S, I,R and N by a common factor. This also means that thesubdivision is not analogous to cutting links in a con-tact network, since we assume that the contact rate b re-mains unchanged. While, in reality, b might decrease dueto such subdivision and deterministically reduce R0 andtherefore Imax, we intentionally keep it constant here toextract the effects of stochasticity. An example for pop-ulations of N = 1,000,000 individuals split into Ns = 10sub-populations is shown in Figs. 1a and 1b, along withthe expected dynamics of a single large population (redcurve). In one example set of 10 sub-populations, onlythree sub-populations (blue, yellow, and green curves)experience a significant outbreak, and they are desyn-chronized. Spontaneous extinction and desynchroniza-tion lead to an average behaviour across 100 simulationswith a significantly reduced peak (turquoise curve). Notethat, on average, both the undivided large populationand the sum of the smaller sub-populations initially ex-hibit exponential growth in the number of infected indi-viduals (Fig. 1b). The effect of the extinction events insome sub-populations is only seen later during the satu-ration phase.

To obtain an estimate for the effect of extinctionand the distribution of infected individuals, we add upthe maximum numbers of infected individuals in thesub-populations. Each of these peaks is approximatelyγN/Ns, but only if the infection does not stochasticallybecome extinct during the initial stages. For large pop-ulation sizes and values of b/k that result in a signif-icant peak, extinction usually happens well before thepeak is reached in other sub-populations (Appendix B),such that these populations do not contribute. There-fore, on average, the contribution of each sub-populationwill be Is,max(n) = γ (1− pext

n )N/Ns, where n indicatesthe number of initial infected individuals in the sub-population and pext

n is the probability that they go ex-tinct without entering deterministic growth as discussedabove. Therefore, the total peak number of infected in-

3

0 100 200 300 400 500t (days)

0

1

2

3

4

5

6

% infe

cte

dN

s = 1

Ns = 100

Ns = 500

0 2 4 6

observed con

(%)

0

0.2

0.4

0.6

0.8

1

occurr

ence fra

ction

250 300 350 400termination time (days)

0

0.1

0.2

0.3

0.4

0.5

occurr

ence fra

ction

100 150 200 250peak time (days)

0

0.2

0.4

0.6

0.8

occurr

ence fra

ction

a) b)

c) d)

FIG. 2: Epidemics for different subdivisions of the popula-tion. N = 8,000,000, b = 0.2, k = 0.14, three different valuesof Ns. 20, 200 and 40 individual simulations for Ns = 1,Ns = 100 and Ns = 500, respectively. (a) Time courses (solidlines) and 2.5/97.5 percentiles (shading). (b) Distributionsof observed peak percentage γcon (in the whole population).Occurrence fraction indicates fraction of simulations. Analyt-ical estimates for γext (dashed lines), Eq. (6), and γcon (solidlines), Eq. (9), assume gn according to a binomial distribu-tion. (c) Distribution of peak times in the sub-populations.Inset provides an enlarged y-axis. Dashed lines indicate ana-lytical approximation, Eq. (8), assuming a uniform n = I0/Ns

for each case. (d) Distribution of termination times, definedas the time when I in the total population drops below I0.

dividuals in all the sub-populations due to extinction isgiven by Iext

max =∑n gnIs,max(n), where gn is the number

of sub-populations with n initially infected individuals.Note that Ns =

∑n gn. Combining the above equations,

we obtain

Iextmax = γ

N

Ns

∑n

(1− pext

n

)gn = γN

[1−

∑n gn(k/b)n∑

n gn

].

(5)

The above result manifestly shows that

γext ≡ Iextmax

N= γ

[1−

∑n

gnNs

(k/b)n

]< γ (6)

holds. Note that this reduction is exclusively due toextinction and the simple summation of the individualmaxima neglects the possible desynchronization betweensub-populations, which we will consider further below.

For example, for the ideal case where each sub-population only contains at most one infected individual,we have

γext1 = γ

I0Ns

(1− k

b

), (7)

where g1 = I0 is the total number of initially infected in-dividuals in the large population (for this to make sense,Ns ≥ I0 is required). Since γ corresponds to the casewhere the population was not split up, the peak numberof infected can therefore be reduced by increasing thenumber of sub-populations Ns or by bringing b closer tok. Note that this is in addition to the decrease in the de-terministic peak fraction γ of infected, which naturallyresults when b approaches k (cf. Eq. (3)).

The independent summation of maxima in differentsub-populations is a conservative estimate, since fluctu-ations can lead to stochastic desynchronization and thusto a further reduction of the peak value. The distribu-tion of peak times in the sub-populations from the pre-vious example is shown in Fig. 1c. The temporal shiftbetween the different sub-populations can be attributedentirely to stochastic fluctuations in the initial phase ofthe dynamics. Assuming that this time shift accumulateswhile the dynamics can still be modeled as a pure birth-death process without saturation effects, we can derivethe probability distribution for the deviation from themean peak time ∆tpeak ≡ tpeak − 〈tpeak〉 as

P (∆tpeak) = k(1− k/b)[1− (k/b)n]

× exp

(−(b− k) (τ + ∆tpeak)− k

be−(b−k)(τ+∆tpeak)

),

(8)

where n is the initial number of infected individuals inthe sub-population and τ = ln (γk/b) /(b− k) with γ be-ing the exponential of the Euler constant (see AppendixC for details). Note that n here was only used to incor-porate the extinction probability, while the shape of thedistribution is based on a single initially infected indi-vidual. Nevertheless this result is in excellent agreementwith the measured distribution for randomly distributedinfected individuals (see dashed line in Fig. 1c).

We can then use this distribution to obtain a quan-titative estimate for the additional peak reduction dueto desynchronization. For this purpose, we approximatethe deterministic time evolution of I in the vicinity ofthe peak as I(t) ≈ Nγ exp

(− 1

2 bkγ (t− tpeak)2), which is

valid as long as S(t) remains of order ∼ N (Appendix D),i.e. b/k is not too large. In the limit of many superim-posed peaks of this shape, with the variability of tpeak

given by Eq. (8), the peak is reduced by an additionalfactor α−1:

γcon =γext

α, α =

√1 +

π2[R0 − 1− log(R0)]

6(R0 − 1)2. (9)

The peak number of infected individuals, with bothstochastic effects of the confinement taken into account,similarly becomes Icon

max = Nγcon = Iextmax/α. It is inter-

esting to note that this reduction factor is bounded frombelow by limR0→1 α

−1 =√

12/(12 + π2) ≈ 0.7407. Thedesynchronization effect is therefore much more limitedthan the extinction effect.

4

f. Discussion.— Standard strategies to contain anepidemic, such as quarantine for infected individuals andsocial distancing, are typically aimed at reducing the in-fectious contact rate b or increasing the removal rate k,leading to an overall “flattening of the curve” and a de-crease of the deterministic peak fraction of infected, γ. Inthis situation, the epidemic still grows exponentially, al-beit with a smaller initial rate b−k (and the time scale ofsaturation is increased substantially). A similar situationcan occur when drastic measures to reduce contact ratesuch as lockdowns are successful in bringing b below k,such that the number of infected individuals actually de-clines, but then these measures are slowly lifted to avoidirreparable damage to the economy and the psychologi-cal well-being of the population. In both situations, thenumber of recovered/immune individuals in the popula-tion will still be too small to prevent exponential growthwhen b is close to, but still (or again) larger than k.

The above analysis shows that the isolation of smallsub-populations can reduce the overall peak number ofinfected people (and therefore strain on the healthcaresystem) by an additional factor of up to I0/Ns · (1−k/b)when I0/Ns < 1. One contribution in this ideal case ofat most one infected individual per sub-population comesfrom the communities which have no infections and arenow protected (I0/Ns), while another contribution comesfrom the possibility that an infection chain in the localcommunity stochastically ends due to fluctuations (k/b).Stochastic desynchronization further reduces the peak byup to about 25% according to Eq. (9). However, as ourestimates show, a reduction can be achieved regardlessof the distribution of infected individuals across the sub-populations. It is also worth noting that, in contrast toflatten-the-curve approaches, the time scale of the out-break is not increased by this strategy.

We consider as an example a region with a popula-tion of 8,000,000 and 500 infected individuals (I0/N ∼6 · 10−5), and assume a removal rate of k = 0.14, corre-sponding to a realistic mean removal time of 1/k ≈ 7 daysfor the recent epidemic [12] (particularly if symptomaticindividuals are quickly removed from the infectious poolthrough quarantining). Let us further assume that theinfectious contact rate is b = 0.2 (> k). This correspondsto a substantial reduction ofR0 from its initial value of 2–2.5 [13] through mild measures such as social distancing,although the epidemic would still spread exponentially,with infection numbers doubling about every 12 days. Ifthis population is allowed to mix homogeneously, the dy-namics will evolve according the deterministic predictionwith a peak around 5% infected individuals (blue datain Figs. 2). If instead, the population is split up and the500 infected people are distributed randomly across thesub-populations, the peak percentage of infected individ-uals decreases to around 3% (for 100 sub-populations of80,000 people) or 1% (for 500 sub-populations of 16,000people) on average (red and yellow, respectively). In allcases, the analytical estimate which only considers theextinction effect, Eq. (6), is only an upper bound for

the peak percentage of infected individuals in the totalpopulation, while also considering desynchronization ac-cording to Eq. (9) yields a good estimate the typical peakvalues. The peak time distributions for the three differ-ent ways of splitting up the population shown in Fig. 2calso agree with the analytical estimate of Eq. (8). Notethat these distributions are not normalized since a signif-icant fraction of sub-populations experience extinction ofthe epidemic and therefore do not exhibit a peak. Thereis also a subtle, non-monotonic effect on the termina-tion time of the epidemic (Fig. 2d), whose distribution isbroader when the population is split up, but does notchange position appreciably. Note that the reductionfor Ns = 500 sub-populations in Fig. 2 is comparable(or even slightly lower) than the case where the 500 in-fected individuals are not distributed randomly acrossthe sub-populations, but each sub-population containsexactly one infected individual. In this case (see Fig. 4in Appendix E), there are no sub-populations with ini-tially zero infected individuals, implying that the reduc-tion in peak value compared to the large homogeneouspopulation is strictly due to extinction and desynchro-nization, which are again well predicted by the analyticalestimates.

To examine the validity of our approximations acrossdifferent parameters, we varied the contact rate b andcarried out numerical simulations for values of R0

ranging between 1.14 and 2. We analyzed the resultingpeak magnitudes to extract the individual contributionsof extinction and desynchronization, which are inexcellent agreement with our predictions of Eq. (6) and(9), as shown in Fig. 3. The contribution of extinctionalone was estimated numerically by summing maximain different sub-populations, regardless of their timing.

0.16 0.2 0.24 0.28b

0

5

10

15

, co

n (

%)

Ns = 1

Ns = 500, ext.

Ns = 500, full

extinction

desynchronization

0.16 0.2 0.24 0.28b

0.06

0.1

0.2

0.4

1

co

n/

a) b)

FIG. 3: Peak reduction for different values of b. N =8,000,000, I0 = 500, Ns = 500 with random distribution of in-dividuals across sub-populations. Ns = 1 corresponds to onelarge populations. k = 0.14. Data points correspond to 100simulations each. (a) Peak fraction of infected individuals.Symbol colour indicates reduction due to extinction (red) orboth extinction and desynchronization (green) as measuredin simulations. Red/green shading and solid lines indicateanalytical predictions from Eq. (6) and Eq. (9), respectively.Black line indicates deterministic estimate from Eq. (3). (b)Same as panel (a), plotted logarithmically and normalized bytheoretical γ, Eq. (3). Black dots mark data points affectedby additional effects (see Appendix B).

5

Overall, the simulations confirm the relative importanceof the extinction effect, whereas the additional reductionby desynchronization plays a smaller role. For low valuesof b very close to k, deviations from the theory beginto appear, as the time scale of the extinction processbecomes comparable to that of the deterministic SIRdynamics, and the distinction between an initial stochas-tic phase and take-off/extinction becomes increasinglyblurred (Appendix B). In particular, this affects theestimation of the extinction contribution (marked byblack dots).

In summary, we have shown that, even without achiev-ing b < k, isolating smaller communities can signifi-cantly reduce the peak number of infected individualsof an epidemic outbreak. The benefits of such a con-tainment strategy are obvious even from a determinis-tic standpoint in the case where many regions initiallycontain no infected individuals. However, our analysisshows that this advantage persists due to stochastic ex-tinction events and desynchronization even if many orall sub-populations initially contain infections, as longas I0/Ns ∼ 1 (of course, increasing Ns further is alwaysbeneficial due to the above-mentioned deterministic effectbut might not be realistic in practice). While extinctionhas been widely considered for SIR-type models [10] andhas been related to a minimum number of infections nec-essary to cause a “major” outbreak [4], we have shownhere that, even if the large population has already leftthe stochastic regime and the outbreak is well underway,it is possible to resurrect these effects for the purpose ofcontainment by artificially sub-dividing the population.This is based on the fact that extinction only depends onthe number of infections, but not on the population sizeitself (as long as the population is sufficiently large). Dueto the exponential dependence of the extinction probabil-ity on n (see Eq. (5)) it is important to obtain a conser-vative estimate for I0, for example by adjusting the num-ber of reported cases by an appropriate factor related tothe detection probability. In contrast to extinction, thedesynchronization effect does not reveal itself on the levelof a single population (except as a difference in timing)

and is therefore an emergent property of our subdivisionscenario (albeit having a smaller effect).

We expect that individuals will not compensate for allavoided contacts outside the local community with con-tacts within it, as we have a conservatively assumed bykeeping b constant. Instead, isolation of sub-populationswill naturally lead to a reduction in b, akin to cuttinglinks in the spreading network [5], so that the effect ofsuch containment measures will be even greater in prac-tice. In addition, given data about the geographic distri-bution of infections, it should be possible to isolate the“right” communities and adjust their size much more in-telligently than we have done here (where the infected in-dividuals were distributed randomly or uniformly acrosssub-populations). This should further increase the likeli-hood of stochastic termination of the epidemic in manysmall communities (even if there are undetected cases).Note that the additional desynchronization effect couldbe relevant in practice, as it would allow non-peak re-gions to provide medical support and hospital space topeak regions, further improving the management of theepidemic. The approach is complementary to many othercontainment measures, such as social distancing and elec-tronic contact tracing [12], which still allow for a func-tioning economy and public life. However, it also doesnot preclude the activation of more drastic individualconfinement measures in regions beginning to show de-terministic exponential behaviour, while still sparing themajority of regions from them.

Acknowledgments

We have benefited from discussions with J. Agudo-Canalejo, A. Bahrami, H. Bickeboeller, E. Bodenschatz,W. Brck, H. Chate, R. Fleischmenan, T. Friede, T.Geisel, H. Grubmller, R. Jahn, B. Mahault, V. Priese-mann, T. Richter, S. Scheithauer, A. Vilfan, M. Wilczek,and R. Yahyapour. The research was supported by theMax-Planck-Gesellschaft.

[1] 72th WHO situation report on COVID-19, Sub-ject in Focus: Public Health and Social Measures,https://www.who.int/docs/default-source/coronaviruse/situation-reports/20200401-sitrep-72-covid-19.pdf

[2] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, andA. Vespignani, Rev. Mod. Phys. 87, 925 (2015).

[3] H. Heesterbeek et al., Science 347, aaa4339 (2015).[4] L.J.S. Allen, Infect. Disease Mod. 2, 128-142 (2017).[5] J.T. Matamalas, A. Arenas, S. Gmez, Science Advances

4, eaau4212 (2018).[6] L. Hufnagel, D. Brockmann and T. Geisel, Proc. Natl.

Acad. Sci. 101, 151249 (2004).[7] W. O. Kermack and A. G. McKendrick, Proc. R. Soc. A

115, 700-721 (1927).[8] D.T. Gillespie, J. Phys. Chem. 81, 2340-2361 (1977).[9] H. Weiss. The SIR model and the Foundations of Public

Health. ISSN 1887-1097, MATerials MATematics 3, 117,(2013).

[10] P. Whittle, Biometrika, 42, 116122 (1955).[11] Note that, for simplicity of presentation, we are using

the long-time limit of the extinction probability to definepext1 rather than P0(t). This simplification implies thatextinction happens fast enough on the time scale relevantfor our problem. It can be justified by comparing the timescale of the extinction process to that of the infectionpeak in the SIR model (see further below in the main

6

text and [? ]).[12] L. Ferretti, C. Wymant, et al., Science, eabb6936 (2020).[13] Q. Li et al., N. Engl. J. Med., 382, 1199-1207 (2020).[14] M.M. Desai and D.S. Fisher, Genetics 176, 1759 (2007).

Appendix A: Exact solution of the birth-deathprocess

Consider a population of the infected individuals I thatcan undergo the following two processes:

Ib−→ I + I, I

k−→ ∅, (A1)

i.e., each I can give birth to another I with rate b, or, itcan die with rate k, at any time. Ignoring the stochastic-ity, the average behaviour of the system is described byexponential birth and death. The population n(t) can bedetermined as follows:

dn(t)

dt= (b− k)n(t) ⇒ n(t) = e(b−k)t, (A2)

where we have assumed that the initial size of the popula-tion is one. As this is a one-step process, the probabilityof finding n copies of I in the sample at time t satisfiesthe following Master equation

dPn(t)

dt= k(n+1) Pn+1(t)+b(n−1) Pn−1(t)−(k + b)nPn(t),

(A3)The factor of n is needed because the birth or death couldhappen to anyone. Equation (A3) can be solved by anansatz of the form Pn ∼ fn for n ≥ 1, which togetherwith the initial condition Pn(0) = δn,1 gives us the solu-tion as

Pn(t) =n(1− k/b)2

(n− 1)(n− k/b)

(n− 1

n− k/b

)n. (A4)

The distribution can be used to calculate the first twomoments

〈n(t)〉 =

∞∑n

nPn(t) = n(t) = e(b−k)t, (A5)

∆n2 =⟨

[n− 〈n〉]2⟩

=

(b+ k

b− k

)e(b−k)t

[e(b−k)t − 1

],

(A6)

which reveal more interesting features about the system.First, it is reassuring that the average population sizebehaves according to the mean-field description abovethat predicted exponential growth or decay. A quantityof interest is

∆n2

n=

(b+ k

b− k

)[e(b−k)t − 1

], (A7)

which probes whether number fluctuations follow a char-acteristic Poisson behaviour. In the long time limit, we

have

∆n2

n=

∞ ; b > k,

k + b

k − b; b < k,

(A8)

which shows that while a decaying population that cor-responds to b < k has a Poisson behaviour, a growingpopulation corresponding to b > k has giant number fluc-tuations, which can be characterized via

∆n

n=

√b+ k

b− k√

1− e−(b−k)t, (A9)

which leads to

∆n

n=

√b+ k

b− k, (A10)

in the long time limit. In other words, the fluctuationsscale with the average population size when b > k, andwith the square root of the average population size whenb < k.

The above solution allows us to calculate the extinctionprobability of the population P0(t), which is an absorbingstate. We find

P0(t) = 1−∞∑n=1

Pn(t) =k

b· e(b−k)t − 1

e(b−k)t − k/b. (A11)

which is a very interesting result. When k > b, n→ 0 atlong times, and we obtain P0 = 1. It is no surprise thatextinction at long times is a certainty when the deathrate is larger than the birth rate. However, when k < b,n → ∞ at long times, and we obtain P0 = k/b; a resultthat is in contradiction with the prediction of the averagebehaviour of the system, which is exponential growth.So, number fluctuations could completely annihilate anexponentially growing population.

7

Appendix B: Time scale of the extinction processand accuracy of maxima detection in

sub-populations

Here, we derive quantitative estimates that allow us tocompare the time scale of the extinction process to thatof the deterministic peak in the SIR model. This is con-ceptually interesting in its own right, but it also allowsus to meaningfully differentiate between “real” maximaand random transient peaks in the number of infected in-dividuals in sub-populations that experience extinction.

In the pure birth-death process, the fraction of extinc-tion events, 0 ≤ φx ≤ 1, that have already happened bytime t can easily be calculated from Eq. (A11) as

φx(t) =P0(t)

limt→∞ P0(t)= 1− 1− k/b

e(b−k)t − k/b. (B1)

This equation can be inverted to yield the time tx bywhich a fraction φx of extinction events have happened:

tx(φx) =1

b− klog

(1− φxk/b

1− φx

). (B2)

On the other hand, we can also estimate the fraction ofnon-extinct populations, 0 ≤ φc ≤ 1, that will still bebelow a cutoff size nc at time t:

φc(t, nc) =

nc−1∑n=1

Pn(t),

= 1− e(b−k)t − k/be(b−k)t − 1

(1− 1− k/b

e(b−k)t − k/b

)nc

.

(B3)

Evaluating φc(tx(φx), nc) therefore yields the fraction ofpopulations still below nc when a fraction φx of extinctionevents have already happened. This expression can beinverted to yield the simple relationship

nc(φx, φc) = 1 +log(1− φc)

log φx, (B4)

giving the number of infected individuals below whicha fraction φc of non-extinct populations will still be, atthe time when a fraction φx of populations destined forextinction have already reached the extinct state.

In order to estimate the effect of extinction in ournumerical simulations (cf. Fig. 4), we detect themaximum number of infected individuals in each sub-population (independent of their timing), and comparethe sum of these numbers to our estimate Iext

max fromthe main text. In the sub-populations that experiencerandom extinction of the epidemic, the detected numeri-cal maxima will in reality be transient fluctuations be-fore extinctions. These contribute more and more asR0 = b/k → 1, when the deterministic peak valueNγ = N

(1− (1 + logR0)/R0

)[9] decreases and the ex-

tinction probability 1/R0 increases. Using the estimates

above, we can exclude these false maxima based on theirtiming, by only considering those maxima for which

tmax > tx(φx), (B5)

and simultaneously ensuring that

nc(φx, φc) < Nγ, (B6)

is fulfilled. φx and φc play the role of accuracy param-eters. The first condition ensures that false maxima areexcluded with probability φx, while the second one en-sures that a pure birth-death process would not havereached the deterministic SIR peak by the same time withprobability φc. Note that the latter is a conservative esti-mate, as growth in the SIR model is significantly slowedbefore reaching its peak compared to a pure birth-deathprocess. In Fig. 4, we use a value of φx = φc = 0.99 toexclude 99% of false maxima and still detect more than99% of deterministic SIR maxima, except for the datapoints marked as unreliable, for which Eq. (B6) is notfulfilled and therefore the extinction process and the de-terministic SIR peak are not clearly separated in time.Conversely, this also means that for all other parameters(i.e. larger R0 = b/k), extinction usually happens wellbefore the deterministic SIR dynamics reaches its peak.

It is worth emphasizing that, in the limit b → kand small populations, the distinction between an initialstochastic phase and a deterministic time course becomesmeaningless, since γ eventually becomes order ∼ 1/Nand the mean extinction time diverges. At this point,the dynamics throughout will be dominated by randomgrowth of the number of infected individuals and stochas-tic fluctuations will continue to contribute, even as thenumber of susceptibles decreases, eventually ending theepidemic (i.e. during and beyond the maximum). In ad-dition, the assumption that there is no depletion of sus-ceptibles in the early phase (and thus the equivalence to apure birth-death process) breaks down. However, in thisstudy, we are interested in the regime where even sub-populations are still large, and while b is sufficiently closeto k to yield a significant extinction probability k/b, itis large enough to lead to a significant deterministic out-break peak. Therefore, we do not investigate this regime.

Appendix C: Analytical approximation of therelative peak time distribution

The fact that the early phase of the dynamics in theSIR model (when S ≈ N and I is small) correspondsto a simple birth-death process also allows us to obtainan analytical estimate for the peak time distributions ofthe sub-populations. This can be readily adapted froma similar calculation performed on an equivalent prob-lem in evolution, where the dynamics of a small mutantsub-population with a given selective advantage can like-wise be understood as a birth-death branching process

8

[14], for which the transition from the initial stochas-tic regime where extinction is still possible to the deter-ministic regime of exponential growth corresponds to theestablishment of the mutation in the population (whichprecedes fixation).

We obtain an approximation for the establishmenttime distribution of the disease in a sub-population as

P estSIR(τ) = k (1− k/b) exp

(−(b− k)τ − k

be−(b−k)τ

),

(C1)where we have corrected for an additional minus signmissing from Ref. [14]. The variation in the timing of thelater deterministic dynamics is due entirely to fluctua-tions in this initial stochastic phase. To compare this an-alytical approximation with our simulation results for thepeak time in the main text, we plot the non-normalized,unconditional distribution

P (tpeak) = [1− (k/b)n]P estSIR (tpeak + τ − 〈tpeak〉) , (C2)

which is diminished by a factor [1 − (k/b)n] (from Eq.(C1)) accounting for the probability of extinction in apopulation with initially n infected individuals, and hasits mean shifted to the measured mean peak time 〈tpeak〉.Here

τ ≡ 〈τ〉 =1

b− kln

(γk

b

), (C3)

where γ = 1.7810724 · · · is the exponential of Euler’sconstant.

We note that simply shifting the mean of the distribu-tion is justified because the dynamics is predominantlyidentical in different sub-populations once they arein the deterministic regime, while only lagging by arandom time span τ . This simple argument dependson the assumption that stochastic fluctuations can beignored before deviations from exponential behaviour(i.e. saturation effects) have to be considered for thedeterministic dynamics. This is true for the scenarios weconsider in the SIR model, since our sub-populations stillconsist of thousands of individuals and we are explic-itly focusing on cases where b is not arbitrarily close to k.

Appendix D: Estimating the effect of sub-populationdesynchronization

For estimating the peak reduction effect due to desyn-chronization of the sub-populations, it is convenient towork with the normalized equations for s = S/N andi = I/N , which read

s = −bsi, (D1a)

i = bsi− ki. (D1b)

When i reaches its peak γ = i(tpeak), new infections andrecovery balance according to Eq. (D1b) and s(tpeak) =

k/b. Based on this known value, we use the followingansatz for s

s(t) =k

b

(1 + ε(t)

), (D2)

with ε(tpeak) = 0. Since we are interested in the regimewhere there is a substantial extinction probability k/b,s(tpeak) is also still of order 1. Together with the fact

that i(tpeak) = 0 by definition, we expect from Eq. (D1a)that the lowest (linear) order of ε will suffice to describethe dynamics around the peak, i.e. ε(t) ≈ ε1 · (t− tpeak)(conversely, we expect this approximation to break downwhen b � k). Substituting the ansatz into Eq. (D1a)yields ε1 = −bγ, or

ε(t) ≈ −bγ(t− tpeak). (D3)

With this, we can obtain an approximation for i aroundthe peak. From Eq. (D1b), we know that

d log(i)

dt= bs(t)− k = kε(t), (D4)

which can easily be solved. Together with the conditioni(tpeak) = γ, we obtain

i(t) ≈ γ exp

(−1

2bkγ(t− tpeak)2

). (D5)

Now that we have an approximation for i(t) near thepeak, we can calculate how these time courses add upacross individual sub-populations, by assuming that theyall have the shape (D5), with the peak time tpeak stochas-tically distributed according to Eq. (C2). Defining

i(t) = limNs→∞

1

Ns

Ns∑j=1

i(tjpeak; t), (D6)

where each i(tjpeak; t) represents a time course as in

Eq. (D5) with tjpeak drawn from the distribution (C2)for each j, we obtain an average superposition of manysub-populations in the limit Ns →∞:

i(t) =

∫dtpeak P

estSIR(tpeak + τ) i(tpeak; t). (D7)

Note that (as compared to Eq. (C2)) we use here the nor-malized distribution, without the diminishing factor dueto extinction, in order to extract the reduction strictlydue to desynchronization. We have also set 〈tpeak〉 to0 without loss of generality, as a different value wouldsimply shift i(t) by the corresponding time.

The integral in Eq. (D7) cannot be integrated in closedform. We therefore replace it by a normal distributionN (0, σ2) with the same variance σ2 = π2/[6(b− k)2]. Itis useful to note that, as for the normal distribution, thevariance completely determines the shape of the Gumbel

9

distribution in Eq. (C1), which means that the system-atic error introduced by this replacement is parameterindependent. Finally, we can calculate

i(t) =

∫dtpeak

exp(−t2peak/(2σ

2))

√2πσ

i(tpeak; t),

=γ

αexp

(−1

2bkγ

t2

α2

), (D8)

with α =

√1 +

π2bkγ

6(b− k)2.

The maximum of the resulting time course occurs att = 0 (due to our arbitrary choice of the mean fortpeak) and is i(0) = γ/α. Since the expected peak valuewithout desynchronization is γ, desynchronization re-duces this peak value by a factor of α−1. According toEq. (D8), α itself depends on γ, which in turn is a func-tion of R0 = b/k. Using the well known approximationγ = 1 − [1 + log(R0)]/R0 [9], which is valid as long asS ≈ N initially, we rewrite α as

α =

√1 +

π2[R0 − 1− log(R0)]

6(R0 − 1)2. (D9)

While we expect the quantitative estimate to be less ac-curate towards higher R0 (see above), we note that theimportant limits

limR0→∞

1

α= 1, (D10)

limR0→1

1

α=

√12

12 + π2≈ 0.7407, (D11)

exist. The first one signifies that there is no peak re-duction due to desynchronization for R0 → ∞, consis-tent with the disappearance of the stochastic phase atthe beginning of the dynamics. The second limit indi-cates a finite reduction by a factor ≈ 0.7407 towards

R0 = b/k = 1. Since the time scales of both the stochas-tic fluctuations and the deterministic peak behaviour di-verge for R0 → 1 (and are ill-defined for R0 = 1),this means that they must exhibit identical scaling be-haviour in order for neither of them to dominate. Inbetween the two extremes, 1/α increases monotonicallywithR0, which implies that the maximum reduction thatcan be achieved by desynchronization is about 26% andis reached close to R0 = 1. It is important to note, how-ever, that several assumptions even about the determin-istic time course (for example, the value of γ) break downwhen R0 is so close to 1 that γ becomes of order 1/N , soa fully stochastic treatment would be needed to fully cap-ture this regime. This does not limit the validity of theresults in the regime we are interested in, i.e. where sub-populations still exhibit clear deterministic outbreaks (orextinction).

Appendix E: Uniform distribution of infectedindividuals

0 100 200 300 400 500t (days)

0

0.5

1

1.5%

infe

cte

d

0 1 2 3

observed con

(%)

0

0.2

0.4

0.6

0.8

1

occurr

ence fra

ction

a) b)

FIG. 4: Plots analogous to Figs. 2a and 2b for the caseNs = 500, but with exactly one initially infected individ-ual in each sub-population instead of a random distribution.Analytical estimates, Eq. (6) (dashed line) and Eq. (9) (solidline), accordingly use g1 = I0 = 500.