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The authors have declared that no competing interests exist.

Emerging and re-emerging pathogens exhibit very complex dynamics, are hard to model and difficult to predict. Their dynamics might appear intractable. However, new statistical approaches—rooted in dynamical systems and the theory of stochastic processes—have yielded insight into the dynamics of emerging and re-emerging pathogens. We argue that these approaches may lead to new methods for predicting epidemics. This perspective views pathogen emergence and re-emergence as a “critical transition,” and uses the concept of noisy dynamic bifurcation to understand the relationship between the system observables and the distance to this transition. Because the system dynamics exhibit characteristic fluctuations in response to perturbations for a system in the vicinity of a critical point, we propose this information may be harnessed to develop early warning signals. Specifically, the motion of perturbations slows as the system approaches the transition.

Outbreaks of re-emerging pathogens are among the most unpredictable threats to public health and global security [

The approach we propose anticipates disease re-emergence through a

In contrast to most research on critical transitions, which concern the catastrophic shifts associated with saddle-node bifurcations, epidemic transitions are associated with a transcritical bifurcation, which is not abrupt, but piecewise continuous. That is, when the mean prevalence moves above zero in response to a small change in parameters, it nonetheless remains close to zero. Therefore, there can be no rapid shift to a distant equilibrium. However, even these incremental increases in theoretical prevalence are important due to stochastic effects when the pathogen is introduced at a low rate from an external source. This is because even if each infectious case can, on average, generate more than one additional case, there is a non-zero probability of the chain of infections going extinct before an epidemic occurs [

Our goal is to find statistical methods to anticipate epidemics. The point in connecting outbreak potential with bifurcations is that the theory of near-critical systems predicts that dynamics will exhibit consistent features, stemming from the phenomenon of

Critical slowing down is illuminated by a closer look at a theoretical contagion process. We start with the standard model of mathematical epidemiology, the general epidemic or SIR model [_{S}(_{I}(

Despite their simplicity, such models have provided profound insight into many features of infectious disease dynamics and contagion processes in general [

Interestingly, the SEIR model predicts with remarkable accuracy the period of oscillations apparent in time series of disease incidence for a number of well-known infectious diseases [

Infection | Setting | Calculated | Observed |
---|---|---|---|

Measles | England and Wales, 1948–68 | 2 | 2 |

Aberdeen, Scotland, 1883–1902 | 2 | 2 | |

Baltimore, USA, 1900–27 | 2 | 2 | |

Paris, France, 1880–1910 | 2 | 2 | |

Yaounde, Cameroun, 1968–75 | 1–2 | 1 | |

Rubella | Manchester, UK, 1916–83 | 4–5 | 3.5 |

Glasgow, Scotland, 1929–64 | 4–5 | 3.5 | |

Mumps | England and Wales, 1948–82 | 3 | 3 |

Baltimore, USA, 1928–73 | 3–4 | 2–4 | |

Poliomyelitis | England and Wales, 1948–65 | 4–5 | 3–5 |

Smallpox | India, 1868–1948 | 4–5 | 5 |

Chickenpox | New York City, USA, 1928–72 | 3–4 | 2–4 |

Glasgow, Sotland, 1929–72 | 3–4 | 2–4 | |

Scarlet fever | England and Wales, 1897–1978 | 4–5 | 3–6 |

Diphtheria | England and Wales, 1897–1979 | 4–5 | 4–6 |

Pertussis | England and Wales, 1948–85 | 3–4 | 3–4 |

Theoretical and observed inter-epidemic periods (in years) of some common infections (from Table 6.1 of Ref.

For our theory, the essential link between the dynamics of the disease and the proximity to control by vaccination lies in the stability of its equilibrium. For trajectories that begin in the neighborhood of the equilibrium, we simplify the equations by linearizing them [

Disease prevalence (_{c} ≈ 0.941; green dashed line) is approached. Oscillatory dynamics occur at another immunization level (^{5} y^{−1}, ^{−1}, ^{−1}, ^{−5} y^{−1}, _{0} = 17. To write the potential function in terms of prevalence, we scaled the deviations of the linearized system by the square root of the equilibrium population size (

Insight into the dynamics of the SIR model (_{S}, _{I}), solve the system of linear equations,
_{S}, one obtains
_{I} is found by integrating the restorative force, _{I} will oscillate as it approaches zero. The equation for _{I} is _{I} (_{1} exp(λ_{1}_{2} exp(λ_{2}_{1} and λ_{2} are the eigenvalues of _{1} and _{2} are determined by the initial values of _{I} and _{S}.

Equations for the eigenvalues of the Jacobian provide a precise statement of the relationship between the distance of the vaccine uptake to the immunization threshold and the speed of the dynamics. This fact might be useful in practical applications where the distance to the threshold is unknown. For the SIR model, it is instructive to consider the equations in the limiting case that the sparking rate _{0}(1 − _{0} = _{0}(1 − _{0}(1 − _{0}(1 − _{0}, (_{0}(1 − _{0}(1 − _{0}(1 − _{0}(1 − _{0}(1 − ^{1/2} = det(^{1/2}. The modulus summarizes both the rate of contraction and the frequency of oscillation of the inward spiral to the equilibrium. In summary, equations for the eigenvalues show that the speed at which perturbations return to the equilibrium increases with |_{0}(1 −

If the trajectories that result from perturbations can be directly observed, as might be the case for data from a large population, these relationships can be used directly to determine the distance to an immunization threshold [_{0}(1 − _{0}(1 − ^{1/2}. Also, for small _{0}(1 − ^{−1} [^{−1}, the expected value of the infectious period, we then have (^{−1/2} for the imaginary part of the eigenvalue. Thus we have recovered the expression for the period of 2^{1/2}. As the vaccination rate increases and approaches the threshold, the period 2^{1/2} increases.

Dynamics of the deterministic component of the SIR model (

Likewise, the real part of the eigenvalues for our SIR model at the endemic equilibrium can be approximated as −_{0}(1 − _{0}(1 −

Thus, it is now clear how the change of eigenvalues corresponds to the more general slowing of the dynamics as the distance to the immunization threshold |_{0}(1 −

In addition to the distance-speed relationship across populations, there is also some work validating the distance-speed relationship within populations. For example, other work by Anderson and May [

The aim of the present article is to suggest ways by which these behaviors in near-critical systems illuminate the approach to epidemic transitions [

One potentially useful distributional property is the variance, which arises from the balance between the random noise that generates perturbations (the Brownian motion terms in the model) and the rate at which perturbations decay toward the equilibrium [

The variance of

Critical slowing down does not lead to an increase in the variance of this variable as the immunization threshold is approached. However, critical slowing down can still be observed from the decrease in the frequency of oscillations in the autocorrelation function (ACF). Vaccine uptake

One might also expect that in a multivariate model a multivariate summary of variance would peak near the threshold, based on the fact that the variance should be very high along the eigendirection associated with the eigenvalue that approaches zero, which should dominate a multivariate summary of the variance. Because the stationary distribution of the linearized model is bivariate normal [

The ellipses indicate the area containing the deviations 95% of the time. The area of the ellipse is largest in the vicinity of the threshold immunization level, which is consistent with the common result that critical slowing down leads to increases in variance. Parameters are as in

A second potentially useful distributional property is the autocorrelation. The autocovariance of small deviations from the equilibrium at a given lag is given by the product of the covariance matrix and the matrix exponential of the Jacobian times the lag [

Another case of interest is when the disease is endemic but vaccine uptake is rising. In this case, the system is often underdamped, and the autocorrelation exhibits damped oscillations as the lag increases. The imaginary part of the eigenvalues can then be estimated by the frequency of the oscillation and the real part of the eigenvalue by the damping rate. Observations of either the

The variance and autocorrelation are but two of a growing number of distributional properties that are expected to change in predictable ways as a threshold is approached [

Many factors can result in the emergence and re-emergence of infectious diseases, including collective changes in individual movement patterns, population-level birth rates, pathogen evolution and, most poignantly, declines in vaccine uptake [

For systems in which transmission is relenting (_{0} tending to 1 from above), documenting paths to disease elimination is valuable and has been identified as a critical component of elimination of several tropical diseases [

_{0}and Transmission Heterogeneity from the Size Distribution of Stuttering Chains