on 29-Nov-2019 (Fri)

It is also useful to recall that weather forecasting went through a similar transition 60 y ago, from primarily statistical time series predictions to mechanistic models numerically solving the hydrodynamic and thermodynamic equations ruling atmospheric evolution. This transition took a careful understanding of the performances of different models, optimization of the spatial scales most appropriate for different components of the forecasts, and improvement in the quality and quantity of weather observations available to calibrate models (17, 18)

Although a body of theoretical work [25,26,30] has demonstrated the importance of incorporating realistic distributions of latent and infectious periods into models of endemic disease, few studies have considered the effects associated with making predictions for an emerging disease [42].

The large discrepancies between estimates of R₀ from the exponentially distributed and gamma-distributed fits reiterate the importance of accurately determining the precise distributions of latent and infectious periods. Although the data required for such a task are often available from post hoc analyses of epidemics they are certainly lacking for a novel emerging infection. Instead, the uncertainty surrounding assumptions about the distributions should be incorporated into quantitative predictions made from epidemiological models, especially since this may well be greater than any uncertainty that arises from noise in the data. Of course, more sophisticated fitting methods than those used in this paper exist [43–46], but if the underlying structure of the model is inappropriate, the method of parameterization is largely irrelevant.

Chretien et al. (2014)'s review suggests need for:

• use of good practices in influenza forecasting (e.g., sensitivity analysis);
• direct comparisons of diverse approaches;
• assessment of model calibration;
• integration of subjective expert input;
• operational research in pilot, real-world applications; and
• improved mutual understanding among modelers and public health officials

ILI is a medical diagnosis of possible influenza with a set of common symptoms: cough and measured or reported fever >38˚C, with onset within the last 10 days [21]

Attributes likely to be essential components of any GCBR-level pathogen include:

• efficient human-to-human transmissibility,
• an appreciable case fatality rate,
• the absence of an effective or widely available medical countermeasure,
• an immunologically naïve population,
• virulence factors enabling immune system evasion, and
• respiratory mode of spread.

Additionally, the ability to transmit during incubation periods and/or the occurrence of mild illnesses would further augment spread.

Most classes of microbe could evolve or be manipulated in ways that would cause a catastrophic risk to humans. However, viruses—especially RNA viruses—are the most likely class of microorganism to have this capacity.

Likelihood: The likelihood for θ given observations x is L_X(θ; x) = f_X(x |θ), θ ∈ Θ regarded as a function of θ for fixed x.

The weak likelihood principle: If X = x and X = y are two observations for the experiment E_X = {X, Θ, f_X(x |θ)} such that L_X(θ; y) = c(x, y)L_X(θ; x) for all θ ∈ Θ then the inference about θ should be the same irrespective of whether X = x or X = y was observed.

Point estimators map from the sample space X to a point in the parameter space Θ.

Set estimators map from the sample space X to a set in Θ.

A drawback with the bias is that it is not, in general, transformation invariant. For example, if T is an unbiased estimator of θ then T⁻¹ is not, in general, an unbiased estimator of θ⁻¹ as E(T⁻¹ | θ) ≠ 1/E(T | θ) = θ⁻¹ .

For an estimator T, a better criterion to being unbiased is that T has small mean square error (MSE)

If we accept, as our working hypothesis, that one of the elements in the family of distributions is true (ie: that there is a θ∗ ∈ Θ which is the true value of θ) then the corresponding predictive distribution f_{Y |X}(y |x, θ∗ ) is the true predictive distribution for Y . The classical solution is to replace θ∗ by plugging-in an estimate based on x.

E(θ |X), the posterior expectation, minimises the posterior expected square error and the minimum value of this error is Var(θ |X), the posterior variance.

Scoring rules (also called scoring functions) are the key measures for the evaluation of probabilistic forecasts.

Scoring rules assign a numerical score based on the predictive density f(y) for the unknown quantity and on the true value y_obs , that has later materialised.

Scoring rules are called proper, if they do not provide any incentive to the forecaster to digress from her true belief, and strictly proper if any such digress results in a penalty, i. e. the forecaster is encouraged to quote her true belief rather than any other predictive distribution.

Note that in the literature inappropriate scoring methods are still often used, e. g. correlation coefficients between point predictions and observations [8, 10]. It is well known in the medical literature that high correlation does not necessarily imply good agreement [36] and therefore a very poor forecasting method may have high correlation.

usage of metrics incorporating the whole probabilistic forecast (rather than using only point predictions) is still rare [6].

denote by Y the predictive distribution which we compare with the actual observation y_obs

The logarithmic score [37] is strictly proper and defined as LogS(Y, y_obs) = − log f(y_obs ),

A strictly proper alternative is the ranked probability score [19], which can be written for count data as RPS(Y, y obs ) = ∞ X k=0 {Pr(Y ≤ k) − 1(y obs ≤ k)} 2 , the sum of the Brier scores for binary predictions at all possible thresholds k ∈ {0, 1, . . .} [20]. An equivalent definition is RPS(Y, y obs ) = E |Y − y obs | − 1 2 E |Y − Y 0 |, (10) here Y and Y 0 are independent realisations from f(y) [19].

a z-statistic z = RPS − E 0 (RPS) Var 0 (RPS) 1/2 a ∼ H 0 N(0, 1) can be computed where the sign of z indicates if the observations are over-/underdispersed relative to the predictions (+/− sign of z) [23, 39]. A (two-sided) P -value can be computed to quantify the evidence for miscalibration.

One possibility for a proper scoring rule is the multivariate Dawid-Sebastiani score [38] mDSS(Y , y obs ) = log|Σ| + (y obs − µ) > Σ −1 (y obs − µ), (11) that depends only on the mean vector µ and the covariance matrix Σ of the predictive distribution. The first term in (11) involves the determinant |Σ| of the covariance matrix Σ, here Σ is a d × d matrix. Transformed to DS = |Σ| 1/(2d) , (12) this is known as the determinant sharpness (DS) and recommended as a multivariate measure of sharpness [24], with smaller values corresponding to sharper predictions.

evaluation of the Dawid-Sebastiani score based on Monte Carlo estimates of the first two moments is not recommended, since the determinant |Σ| is known to be very sensitive to Monte Carlo sampling error [25].

The branching process approximation is a CTMC

the distribution of the estimator T is known as the sampling distribution

Pathogen fitness is traits such as such as replication rate, transmissibility, and immune recognition

Although sequence data are extremely valuable, to link these data fully to disease dynamics, it will be important to determine how sequence changes affect functions related to pathogen fitness

Molecular epidemiological studies often treat pathogen genetic variation as simply reflecting the underlying transmission process, whereas in reality such variation may play an important role in determining transmission dynamics, as exemplified by escape from herd immunity by influenza A virus

Hill et al. (2019) used data subsequent to the 2009 pandemic for England, and developed a dynamic multi-strain SEIR-type transmission model for seasonal influenza, explicitly incorporating immunity propagation mechanisms between influenza seasons.

The posterior expectation minimises the posterior expected square error

The large discrepancies between estimates of R₀ from the exponentially distributed and gamma-distributed fits reiterate the importance of accurately determining the precise distributions of latent and infectious periods.

the uncertainty surrounding assumptions about the distributions of the latent and infectious periods should be incorporated into quantitative predictions made from epidemiological models, especially since this may well be greater than any uncertainty that arises from noise in the data

if the underlying structure of the model is inappropriate, the method of parameterization is largely irrelevant