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**|**PLoS Pathog**|**v.9(6); 2013 June**|**PMC3680036

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PLoS Pathog. 2013 June; 9(6): e1003277.

Published online 2013 June 6. doi: 10.1371/journal.ppat.1003277

PMCID: PMC3680036

Glenn F. Rall, Editor^{}

Laboratoire MIVEGEC (UMR CNRS 5290, IRD 224, UM1, UM2), Montpellier, France

(The Fox Chase Cancer Center, United States of America)

* E-mail: matthew.hartfield/at/ird.fr

The authors have declared that no competing interests exist.

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

This article has been cited by other articles in PMC.

When a pathogen is rare in a host population, there is a chance that it will die out because of stochastic effects instead of causing a major epidemic. Yet no criteria exist to determine when the pathogen increases to a risky level, from which it has a large chance of dying out, to when a major outbreak is almost certain. We introduce such an outbreak threshold (*T _{0}*), and find that for large and homogeneous host populations, in which the pathogen has a reproductive ratio

With the constant risk of pathogens emerging [1], such as Severe Acute Respiratory Syndrome (SARS) or avian influenza virus in humans, foot-and-mouth disease virus in cattle in the United Kingdom [2], or various plant pathogens [3], it is imperative to understand how novel strains gain their initial foothold at the onset of an epidemic. Despite this importance, it has seldom been addressed how many infected individuals are needed to declare that an outbreak is occurring: that is, when the pathogen can go extinct due to stochastic effects, to when it infects a high enough number of hosts such that the outbreak size increases in a deterministic manner (Figure 1A). Generally, the presence of a single infected individual is not sufficient to be classified as an outbreak, so how many infected individuals need to be present to cause this deterministic increase? Understanding at what point this change arises is key in preventing and controlling nascent outbreaks as they are detected, as well as determining the best course of action for prevention or treatment.

The classic prediction for pathogen outbreak is that the pathogen's reproductive ratio (*R _{0}*), the number of secondary infections caused by an infected host in a susceptible population, has to exceed one [4], [5]. This criterion only strictly holds in deterministic (infinite population) models; in finite populations, there is still a chance that the infection will go extinct by chance rather than sustain itself [4]–[6]. Existing studies usually consider random drift affecting outbreaks in the context of estimating how large a host population needs to be to sustain an epidemic (the “Critical Community Size” [4], [7], [8]), calculating the outbreak probability in general [9]–[12], or ascertaining whether a sustained increase in cases over an area has occurred [13]. Here we discuss the fundamental question of how many infected individuals are needed to almost guarantee that a pathogen will cause an outbreak, as opposed to the population size needed to maintain an epidemic once it has appeared (Critical Community Size; see also Box 1). We find that only a small number of infected individuals are often needed to ensure that an epidemic will spread.

- The
**Basic Reproductive Ratio**(*R*) is the number of secondary infections caused by a single infected individual, in a susceptible population. It is classically used to measure the rate of pathogen spread. In infinite-population models, a pathogen can emerge if_{0}*R*>1. In a finite population, the pathogen can emerge from a single infection with probability 1-1/_{0}*R*if_{0}*R*>1, otherwise extinction is certain._{0} - The
**Critical Community Size**(CCS) is defined as the total population size (of susceptible and infected individuals, or others) needed to sustain an outbreak once it has appeared. This idea was classically applied to determining what towns were most likely to maintain measles epidemics [7], so that there would always be some infected individuals present, unless intervention measures were taken. - The
**Outbreak Threshold**(*T*) has a similar definition to the CSS, but is instead for use at the onset of an outbreak, rather than once it has appeared. It measures how many infected individuals (not the total population size) are needed to ensure that an outbreak is very unlikely to go extinct by drift. Note that the outbreak can still go extinct in the long term, even if_{0}*T*is exceeded, if there are not enough susceptible individuals present to carry the infection afterwards._{0}

We introduce the concept of the outbreak threshold (denoted *T _{0}*), which we define as the number of infected individuals needed for the disease to spread in an approximately deterministic manner.

This basic result can be modified to consider more realistic or precise cases, and *T _{0}* can be scaled up if an exact outbreak risk is desired. For example, for the pathogen extinction probability to be less than 1%, there needs to be at least 5/Log(

So far we have only considered homogenous outbreaks, where on average each individual has the same pathogen transmission rate. In reality, there will be a large variance among individual transmission rates, especially if “super-spreaders” are present [17]. This population heterogeneity can either be deterministic, due to differences in immune history among hosts or differences in host behavior, or stochastic, due to sudden environmental or social changes. Spatial structure can also act as a form of heterogeneity, if each region or infected individual is subject to different transmission rates, or degree of contact with other individuals [18]. In such heterogeneous host populations, the number of secondary cases an infected individual engenders is jointly captured by *R _{0}* and a dispersion parameter

In a heterogeneous host population (see the main text for the bases of this heterogeneity), it has been shown that the number of secondary infections generated per infected individual can be well described by a negative binomial distribution with mean *R _{0}* and dispersion parameter

Although in this case it is not possible to find a strict analytical form for the outbreak threshold, progress can be made if we measure the ratio of the heterogeneous and homogeneous thresholds. This function yields values that are independent of a strict cutoff probability (Material S.3 in Text S1). By investigating this ratio, we first found that for a fixed *R _{0}*, a function of order 1/

(1)

As in the homogeneous case, *T _{0}* only provides us with an order of magnitude and it can be multiplied by −Log(

The outbreak threshold *T _{0}* of an epidemic, which we define as the number of infected hosts above which there is very likely to be a major outbreak, can be estimated using simple formulae. Currently, to declare that an outbreak has occurred, studies choose an arbitrary low or high threshold depending, for instance, on whether they are monitoring disease outbreaks or modeling probabilities of emergence. We show that the outbreak threshold can be defined without resorting to an arbitrary cutoff. Of course, the generality of this definition has a cost, which is that the corresponding value of

These results are valid if there are enough susceptible individuals present to maintain an epidemic in the initial stages, as assumed in most studies on emergence [6], [11]–[13], otherwise the pathogen may die out before the outbreak threshold is reached (Box 3 and Material S.2 in Text S1). Yet the key message generally holds that while the number of infections lies below the threshold, there is a strong chance that the pathogen will vanish without causing a major outbreak. From a biological viewpoint, unless *R _{0}* is close to one, these thresholds tend to be small (on the order of 5 to 20 individuals). This contrasts with estimates of the Critical Community Size, which tend to lie in the hundreds of thousands of susceptible individuals [3], [7], [8]. Therefore, while only a small infected population is needed to trigger a full-scale epidemic, a much larger pool of individuals are required to maintain an epidemic, once it appears, and prevent it from fading out. This makes sense, since there tends to be more susceptible hosts early on in the outbreak than late on.

The basic result for the homogeneous population, *T _{0}*~1/Log(

is less than 1/Log(*R _{0}*), then the threshold cannot be reached. Since this maximum is dependent on the population size, outbreaks in smaller populations are less likely to reach the outbreak threshold. For example, if

Estimates of *R _{0}* and

A smallpox outbreak (Variola minor) was initiated in Birmingham, United Kingdom in 1966 due to laboratory release. We calculate a threshold such that the chance of extinction is less than 0.1%, which means that *T _{0}* is equal to 7 times Equation 1. With an estimated

The SARS outbreak in Singapore in 2003 is an example of an outbreak with known super-spreaders [21], with an estimated initial *R _{0}* of 1.63 and a low

Overall, very early measures were necessary to successfully prevent a smallpox outbreak due to its rapid spread. In theory, it should have been “easier” to contain the SARS outbreak, as its threshold is three times greater than that for smallpox due to higher host heterogeneity (*k*). However, the first reported infected individual was a super-spreader, who infected at least 21 others. This reflects that in heterogeneous outbreaks, although the emergence probability is lower, the disease spread is faster (compared to homogeneous infections) once it does appear [11]. Quick containment of the outbreak was difficult to achieve since SARS was not immediately recognised, as well as the fact that the incubation period is around 5 days, by which point it had easily caused more secondary cases. However, in subsequent outbreaks super-spreaders might not be infected early on, allowing more time to contain the spread.

For newly-arising outbreaks, *T _{0}* can be applied in several ways. If the epidemic initially spreads slowly, then

PDF file containing the following: **Material S.1:** Full derivations of outbreak threshold formulae for a homogeneous outbreak; **Material S.2:** Calculations of limitations due to small host population sizes; **Material S.3:** Finding solutions for outbreak threshold formulae for heterogeneous outbreaks.

(PDF)

Click here for additional data file.^{(762K, pdf)}

Same as Text S1, but in Mathematica format.

(ZIP)

Click here for additional data file.^{(257K, zip)}

We would like to thank Yannis Michalakis, Minus van Baalen, Vincent Jansen, Roland Regoes, Troy Day, Helen Alexander, and Marc Choisy for comments and discussion on this paper.

MH was funded by an ATIP-Avenir grant from CNRS and INSERM to SA. SA and MH acknowledge additional support from the CNRS and the IRD. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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