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JMD designed the study, analyzed the data, and wrote the paper.

The author has declared that no competing interests exist.

Early warning systems for outbreaks of infectious diseases are an important application of the ecological theory of epidemics. A key variable predicted by early warning systems is the final outbreak size. However, for directly transmitted diseases, the stochastic contact process by which outbreaks develop entails fundamental limits to the precision with which the final size can be predicted.

I studied how the expected final outbreak size and the coefficient of variation in the final size of outbreaks scale with control effectiveness and the rate of infectious contacts in the simple stochastic epidemic. As examples, I parameterized this model with data on observed ranges for the basic reproductive ratio (_{0}) of nine directly transmitted diseases. I also present results from a new model, the simple stochastic epidemic with delayed-onset intervention, in which an initially supercritical outbreak (_{0} > 1) is brought under control after a delay.

The coefficient of variation of final outbreak size in the subcritical case (_{0} < 1) will be greater than one for any outbreak in which the removal rate is less than approximately 2.41 times the rate of infectious contacts, implying that for many transmissible diseases precise forecasts of the final outbreak size will be unattainable. In the delayed-onset model, the coefficient of variation (CV) was generally large (CV > 1) and increased with the delay between the start of the epidemic and intervention, and with the average outbreak size. These results suggest that early warning systems for infectious diseases should not focus exclusively on predicting outbreak size but should consider other characteristics of outbreaks such as the timing of disease emergence.

Using a mathematical model, John Drake shows that early warning systems for infectious diseases should not focus exclusively on predicting outbreak size but should consider other characteristics of outbreaks such as the timing of disease emergence.

Early warning systems that are used to look for outbreaks of infectious diseases are important in public-health planning. One of the most important things that such early warning systems try to predict is the final size of the outbreak. However, for diseases transmitted directly from person to person (rather than via a mosquito, for example), the precision with which the final size can be predicted is often very low.

This researcher wanted to study how predictable the final outbreak size of an epidemic is if the effectiveness of control measures and the average number of infectious contacts are known.

He developed a mathematical model that took into account the variation in the infectiousness of nine well-studied infectious diseases. He found that for any outbreak that increases slowly, precise forecasts of the final outbreak size will be impossible. This result was especially true for epidemics in which there was a substantial delay in intervention after infection occurred, and the precision of the forecast got worse as the delay between the start of the epidemic and intervention increased, and with the average outbreak size.

These results suggest that early warning systems for infectious diseases should not focus just on trying to predict outbreak size because this estimate may be inaccurate, but rather they should instead try to predict other characteristics of outbreaks. These results will be of use to people trying to plan for infectious disease outbreaks, but will not affect how patients are managed individually.

Based in the United States, the Centers for Disease Control and Prevention (CDC) has a Web site that gives background on how the CDC investigates disease outbreaks, along with details of individual diseases:

The World Health Organization has interesting information on early warning systems:

In the United Kingdom, the Health Protection Agency has a similar function and gives details on investigations of infectious diseases:

The epidemiological responsibility to forecast disease outbreaks is an onerous one. Because of the devastating consequences and high costs of disease, predicting outbreaks is a chief goal for public-health planning and emergency preparedness. Thus, quantitative forecasting and development of early warning systems (EWSs) for disease outbreak is a high priority for research and development [

The reason that final outbreak size is generally not predictable is that the eventual dynamics of the outbreak are highly sensitive to the seemingly random sequence of infectious contacts and removal of infectious individuals in the early, typically unobserved stages of the outbreak [_{0}) together with initial conditions. Deterministic models of epidemics have provided insight into such important topics as the design of vaccination campaigns and the effect of age structure on epidemic dynamics [

In contrast, the stochastic theory of epidemics represents the population as a statistical ensemble with constant or regular average properties but probabilistic changes in disease status for individuals. As a result, properties of the ensemble, such as the final epidemic size, are probabilistic as well [

The stochastic theory of epidemics can therefore be used to understand the theoretical limits to forecasting precision for disease outbreaks, including EWSs or forecasts based on the developing epidemic curve as case reports accumulate. I studied how precision in the forecasted final outbreak size for transmissible diseases depends on two dynamical features of outbreaks: the contact rate (β) and the rate of removal (γ) in the simple stochastic epidemic. Next, I developed models of forecast precision for nine outbreak-prone diseases (chicken pox, diphtheria, measles, mumps, poliomyelitis, rubella, scarlet fever, smallpox, and whooping cough) and used removal rate as a control parameter to relate intervention effectiveness to final outbreak size and forecast precision. Finally, I developed a new model to understand how delays in implementing interventions affect final outbreak size and forecast prevision.

The simplest realistic model for outbreaks with a small number of initially infectious individuals is the simple stochastic epidemic with contact rate β and removal rate γ, which do not change appreciably over the time scale of the outbreak [

and

Properties of the final size distribution for other classes of epidemics can be found in [

The solution given by equations _{0} < 1, or (2) public health policy is applied consistently so that intervention is constant and under policy conditions _{0} < 1. For many emerging diseases this is not the case. Rather, initially _{0} > 1, but through intervention that was established some measurable time after the outbreak started, the reproductive ratio is reduced below the epidemic threshold (e.g., severe acute respiratory syndrome [SARS]). This case is considerably more complicated and, to my knowledge, no simple formulas have been obtained for the mean and variance of the final outbreak size. However, it is reasonably straightforward to solve the equations computationally, and a range of conditions can be studied. Below, I consider a case that is more applicable to forecasting emerging diseases, the simple stochastic epidemic with delayed-onset intervention in which there is a constant rate of infectious contacts (β) and a removal rate (γ) that depends on the time since the outbreak began. Specifically, at the start of the outbreak the removal rate is some value less than the rate of infectious contacts and remains constant until some intervention is applied a time _{0} later, after which the removal rate is some constant value greater than β, i.e., γ(_{1}^{*}) + γ_{2}^{*})
, where

A measure of precision should quantify the relative magnitude of deviations from an expected value. The coefficient of variation _{0}^{−1} and not on the individual parameter values, to study how forecast precision depends on outbreak characteristics and to estimate forecast precision for nine infectious diseases under different levels of control, represented by increasing γ (see

Although every outbreak will be different as a result of evolution of the etiological agent, changes in social behavior, timing, and the ecological and geographical context in which the outbreak starts, many epidemic parameters (most famously _{0}), are reasonably conserved across outbreaks of the same disease. Here, I treat the removal rate γ as a control parameter because it is crucially related to interventions, and estimate β, which is assumed to depend on uncontrollable aspects of the outbreak. The variable β, which is the individual rate of infectious contacts [_{0}) by the equation β = β_{0}_{0} by the equation:

Where removal results from recovery of the diseased individual, we can estimate γ from the duration of the incubation (τ_{1}), latent (τ_{2}), and infectious (τ_{3}) periods with the equation γ = (τ_{1}+τ_{2}+τ_{3})^{−1}
. Estimates of _{0} have been obtained for numerous directly transmitted diseases [

Given that reported values for these variables vary somewhat, we put an upper bound on β by choosing the highest reported value of _{0} and the lowest reported values for the different τs, whereas a lower bound is obtained from the lowest reported value of _{0} and the highest reported values for the different τs. As a central estimate, I used the center of the reported interval for each variable. Estimates of the ranges of these quantities for several directly transmitted diseases were compiled by Anderson and May ([

I also considered the delayed-onset intervention model wherein initially β > γ (the supercritical case in which epidemic occurs with high probability), but after a time _{0} intervention increases the removal rate γ so β < γ (the subcritical case in which the outbreak is brought under control). This model is a more realistic representation of many emerging outbreaks (e.g., SARS, Foot-and-Mouth disease, and Marburg virus). The solution to the simple stochastic epidemic with delayed-onset intervention can be obtained using generating functions for the probability distribution of the size of the outbreak [_{0}. First, I studied the situation with β = 0.5 and γ_{1} = 0.25 (_{0} = 2). Second, I studied the situation with β = 0.5 and γ_{1} = 0.45 (_{0} ≈ 1.1). In both cases, γ_{2} (the removal rate after intervention) was one, so that post-intervention reproductive ratio was 0.5.

The ratio γ/β, the rate of removal compared with the rate of infection, represents the relative effectiveness of interventions. In the simple stochastic epidemic, the relative effectiveness of intervention is always greater than one because we assume that the outbreak is eventually controlled, i.e., the assumption β < γ above.

The expected final outbreak size (solid line) and CV in the final outbreak size (dashed line) are shown as a function of intervention effectiveness (the ratio of the removal rate and contact rate γ/β) for the simple stochastic epidemic. The light horizontal line designates the benchmark where CV = 1.

The expected final outbreak size (_{0}.

The CV in final outbreak size (_{0}. The horizontal line indicates CV = 1.

Numerical analysis of the delayed-onset intervention model showed that (1) the average outbreak size increased with the delay between the start of the outbreak and the start of intervention (_{0} had a lower average outbreak size (_{0} < 1 and _{0} close to 1, respectively) were less predictable (have lower CV) than supercritical (_{0} >> 1) outbreaks of comparable size.

Effect of time delay until intervention on outbreak size is contrasted for outbreaks with _{0} = 2 (solid lines) and _{0} ≈ 1.1 (dashed lines).

(A) Average outbreak size (

(B) CV in outbreak size (

(C) CV in outbreak size (_{0}.

Using theoretical models, I found that unless controls are extremely effective, limits to forecast precision result in highly uncertain estimates of final outbreak size. Specifically, for the simple stochastic epidemic (subcritical case), unless the removal rate is greater than approximately 2.41 times the effective contact rate, the CV of final outbreak size will be greater than one. Imprecision in the delayed-onset intervention model was typically even greater.

Reliable forecasts of outbreaks based on initial cases and/or EWSs could potentially save many lives by increasing preparedness for outbreaks when and where they are most likely or most severe. According to the World Health Organization, forecasts will be most useful when they accurately predict the final size of the outbreak [

This result does not apply to diseases that are not directly transmitted (e.g., vector-borne illnesses) or to diseases in which parameters change as the outbreak progresses (e.g., SARS [

Generally, these violations of the simple stochastic epidemic must be considered on a case-by-case basis. We studied one realistic example (the simple stochastic epidemic with delayed-onset intervention) in which an initially supercritical outbreak (_{0} > 1) is controlled by public health measures that increase the rate at which infectious individuals are removed from the population to a level ensuring the outbreak will eventually die out. This is a reasonably realistic model for dynamics of emerging infections with a short incubation period. For two representative examples, we found that the average outbreak size scaled approximately exponentially with the delay between the start of the outbreak and the implementation of intervention (note the log scale of the _{0} was high the average outbreak size increased faster than when _{0} was low. We also found that the CV in the final outbreak size increased with the lag between initial infection and control, but was smaller in the case with high _{0} than in the case with low _{0}. Indeed, for the delayed-onset case with relatively high _{0} (_{0} = 2) the CV seemed to level off at a delay of around 15–20 d, although this was not shown in the case with lower _{0} (

In conclusion, the fundamental limit to forecasting precision obtained here represents only variation that results from the stochastic contact process and not from uncertainty about the underlying model or parameter values (compare [

I was unable to obtain a simple relation for the coefficient of variation (CV) in the outbreak size of the subcritical simple stochastic epidemic in terms of the basic reproductive ratio _{0}. Numerical results confirm that the CV of final outbreak size depends only on the ratio β and γ (i.e., on _{0}). This plot represents the information in _{0}. The value _{0} ≤ _{0}* are predictable while outbreaks with _{0} > _{0}* are unpredictable.

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The research was conducted while the author was a Postdoctoral Associate at the National Center for Ecological Analysis and Synthesis, a Center funded by the National Science Foundation (Grant #DEB-0072909), the University of California, and the Santa Barbara campus. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

coefficient of variation

early warning system

severe acute respiratory syndrome