Introducing the Outbreak Threshold in Epidemiology

Summary

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 (T₀), and find that for large and homogeneous host populations, in which the pathogen has a reproductive ratio R₀, on the order of 1/Log(R₀) infected individuals are needed to prevent stochastic fade-out during the early stages of an epidemic. We also show how this threshold scales with higher heterogeneity and R₀ in the host population. These results have implications for controlling emerging and re-emerging pathogens.

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 (R₀), 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.

We introduce the concept of the outbreak threshold (denoted T₀), which we define as the number of infected individuals needed for the disease to spread in an approximately deterministic manner. T₀ can be given by simple equations. Indeed, if the host population is homogeneous (that is, where there is no individual variability in reproductive rates) and large enough so that depletion of the pool of susceptible hosts is negligible, then the probability of pathogen extinction if I infected hosts are present is (1/R₀)^I ([6], details in Material S.1 in Text S1). By solving this equation in the limit of extinction probability going to zero, we find that on the order of 1/Log(R₀) infected hosts are needed for an outbreak to be likely (black thick curve in Figure 1B), a result that reflects similar theory from population genetics [14]–[16]. Note that this result only holds in a finite population, as an outbreak in a fully susceptible infinite population is certain if R₀>1 ([4], see also Material S.1 in Text S1).

This basic result can be modified to consider more realistic or precise cases, and T₀ 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(R₀) infected individuals. More generally, the pathogen extinction probability is lower than a given threshold c if there are at least −Log(c)/Log(R₀) infected individuals. Furthermore, if only a proportion p<1 of all infected individuals are detected, then the outbreak threshold order is p/Log(R₀). Also, if there exists a time-lag τ between an infection occurring and its report, then the order of T₀ is e^−τ(β-δ)/Log(R₀), where β is the infection transmission rate and 1/δ the mean duration of the infectious period (Material S.1 in Text S1). Finally, we can estimate how long it would take, on average, for the threshold to be reached and find that, if the depletion in susceptible hosts is negligible, this duration is on the order of 1/(β-δ) (Material S.1 in Text S1).

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 R₀ and a dispersion parameter k (see Box 2). This dispersion parameter controls the degree of variation in individual transmission rates, while fixing the average R₀. The consequence of this model is that the majority of infected hosts tend to cause few secondary infections, while the minority behave as super-spreaders, causing many secondary infections. Host population heterogeneity (obtained with lower values of k) increases the probability that an outbreak will go extinct, as the pathogen can only really spread via one of the dwindling super-spreading individuals. In this heterogeneous case, we can find accurate values of T₀ numerically. As shown in Figure 1B, if R₀ is close to 1, host heterogeneity (k) does not really matter (T₀ tends to be high). However, if the pathogen has a high R₀ and thus spreads well, then host heterogeneity strongly affects T₀. Additionally, we find that the heterogeneous threshold simply scales as a function of k and R₀² (see Box 2). As an example, if k = 0.16, as estimated for SARS infections [11], the number of infected individuals needed to guarantee an outbreak increases 4-fold compared to a homogeneous population (Material S.3 in Text S1).

Box 2. Heterogeneous Outbreak Threshold

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 R₀ and dispersion parameter k [11]. The dispersion parameter determines the level of variation in the number of secondary infections: if k = 1, we have a homogeneous outbreak, but heterogeneity increases as k drops below 1; that is, it enlarges the proportion of infected individuals that are either “super-spreaders” or “dead-ends” (those that do not transmit the pathogen). Lloyd-Smith et al. [11] showed how to estimate R₀ and k from previous epidemics through applying a maximum-likelihood model to individual transmission data.

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 R₀, a function of order 1/k fitted the numerical solutions very well. By measuring these curves for different R₀ values, we further found that a function of order 1/R₀² provided a good fit to the coefficients. By fitting a function of order 1/kR₀² to the numerical data using least-squares regression in Mathematica 8.0 [19], we obtained the following adjusted form for the outbreak threshold T₀ in a heterogeneous population:

As in the homogeneous case, T₀ only provides us with an order of magnitude and it can be multiplied by −Log(c) to find the number of infected hosts required for there to be a probability of outbreak equal to 1-c. A sensitivity analysis shows that Equation 1 tends to be more strongly affected by changes in R₀ than in k (Material S.3 in Text S1).

The outbreak threshold T₀ 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 T₀ is only an order of magnitude. Modifications are needed to set a specific cutoff value or to capture host heterogeneity in transmission or incomplete sampling.

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 R₀ 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.

Box 3. Effect of Limiting Host Population Size

The basic result for the homogeneous population, T₀∼1/Log(R₀), assumes that during the time to pathogen outbreak, there are always enough susceptible individuals available to transmit to, so R₀ remains approximately constant during emergence. This assumption can be violated if R₀ is close to 1, or if the population size is small. More precisely, if the maximum outbreak size in a Susceptible-Infected-Recovered (SIR) epidemic, which is given by

is less than 1/Log(R₀), 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 N = 10,000 then R₀ needs to exceed 1.06 for 1/Log(R₀) to be reached; this increases to 1.34 if N decreases to 100. Further details are in Material S.2 in Text S1.

Estimates of R₀ and k from previous outbreaks can be used to infer the approximate size of this threshold, to determine whether a handful or hundreds of infected individuals are needed for an outbreak to establish itself. Box 4 outlines two case studies (smallpox in England and SARS in Singapore), estimates of T₀ for these, and how knowledge of the threshold could have aided their control. These examples highlight how the simplicity and rigorousness of the definition of T₀ opens a wide range of applications, as it can be readily applied to specific situations in order to determine the most adequate policies to prevent pathogen outbreaks.

Supporting Information

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