Forecasting infectious disease emergence subject to seasonal forcing
Miller et al. Theoretical Biology and Medical Modelling
Forecasting infectious disease emergence subject to seasonal forcing
Paige B. Miller 1 2
Eamon B. O'Dea 1 2
Pejman Rohani 0 1 2
John M. Drake 1 2
0 Department of Infectious Diseases, University of Georgia , Athens , USA
1 Center for the Ecology of Infectious Diseases, University of Georgia , Athens , USA
2 University of Georgia, Odum School of Ecology , 140 E. Green Street, Athens , USA
Background: Despite high vaccination coverage, many childhood infections pose a growing threat to human populations. Accurate disease forecasting would be of tremendous value to public health. Forecasting disease emergence using early warning signals (EWS) is possible in non-seasonal models of infectious diseases. Here, we assessed whether EWS also anticipate disease emergence in seasonal models. Methods: We simulated the dynamics of an immunizing infectious pathogen approaching the tipping point to disease endemicity. To explore the effect of seasonality on the reliability of early warning statistics, we varied the amplitude of fluctuations around the average transmission. We proposed and analyzed two new early warning signals based on the wavelet spectrum. We measured the reliability of the early warning signals depending on the strength of their trend preceding the tipping point and then calculated the Area Under the Curve (AUC) statistic. Results: Early warning signals were reliable when disease transmission was subject to seasonal forcing. Wavelet-based early warning signals were as reliable as other conventional early warning signals. We found that removing seasonal trends, prior to analysis, did not improve early warning statistics uniformly. Conclusions: Early warning signals anticipate the onset of critical transitions for infectious diseases which are subject to seasonal forcing. Wavelet-based early warning statistics can also be used to forecast infectious disease.
Disease forecasting; Seasonality; Critical transition
Improvement of methods for forecasting infectious disease dynamics is of tremendous
value to public health and the global economy [
]. Disease forecasting is challenging
because transmission is subject to complex interactions among a variety of independent
actors, non-linearity, noise, and seasonality driven by age  and spatial structure [
susceptible depletion [
], environmental variability , and behavior [
]. Despite these
challenges, a number of approaches have been developed along distinctly different lines:
(i) sequential Monte-Carlo methods (e.g., ), (ii) data assimilation techniques inspired
by numerical weather prediction [
], and (iii) “Wisdom of the Crowd” approaches
 among others. Monte-Carlo methods aim to estimate unobserved states in Markov
chains (such as the size of the infectious population) while simultaneously estimating
unknown parameters [
]. In contrast, ensemble Kalman adjusted filters are commonly
used in data assimilation and take advantage of incoming information by iteratively
updating an ensemble of models to contract variance [
]. Lastly, “Wisdom of the
Crowd” approaches incorporate experts’ experience and knowledge . These studies
demonstrate that accurately forecasting the trajectory of infectious disease incidence is
possible. In contrast, we are in the earlier stages of theory and method development for
anticipating the emergence of infectious disease [
Emerging infectious diseases of the past century include three major influenza
pandemics, Ebola, HIV, and Zika. The increasing frequency of emergence by novel pathogens
is usually attributed to changes in socio-economic, environmental and ecological
]. All of these are now combated with highly developed diagnostics, therapeutics,
and behavioral interventions. If they could have been anticipated, behavioral
interventions could have been deployed earlier, preventing loss of life and reducing spread even
while medical countermeasures were under development. Additionally, some countries
have reported resurgence of vaccine preventable diseases (e.g., pertussis, measles) despite
high rates of vaccination [
]. Many candidate hypotheses have been advanced to explain
the resurgence of vaccine preventable diseases including changing vaccination schedule
and composition, changing immune acquisition, and evolution of the pathogen (reviewed
]). The economic and health benefits of the increased preparedness that could be
achieved by more advanced warning of disease emergence and re-emergence would be
In the past decade, researchers have studied an alternative, model-independent
forecasting approach which summarizes the patterns of fluctuations in a system to infer
whether it is close to a critical transition [
]. A critical transition is a sudden and
large change in the state of a dynamical system . Statistical signals primarily result
from a phenomenon known as critical slowing down (CSD), which is a signature of a
second-order phase transition and refers to the increased relaxation time for a
nearcritical system to reach equilibrium following a perturbation compared with a system
that is far from critical [
]. Critical slowing down occurs in a wide range of natural
tipping points, including large-scale climate shifts [
], population collapse [
], and lake
]. O’Regan and Drake [
] proposed that statistical early warning
signals (EWS) may also precede the onset of sustained transmission in a contagion process
where the approach to disease emergence is marked by a slow drift to criticality. Because
of the key assumption of slow drift, critical slowing down has primarily been
considered in non-periodic systems [
]. But, many diseases exhibit periodic forcing due to
seasonal variations in climate [
], human behavior , or immune function [
Because seasonal forcing results in periodic sojourns near criticality, we wondered if
standard approaches to detecting critical slowing down might therefore fail in systems such
as resurgent childhood infections forced by the annual calendar of school terms [
We performed a simulation study to investigate how periodic forcing might interfere
with the detection of critical slowing down in directly-transmitted disease systems. A
recent review of resilience in ecological systems noted that early warning signals are
difficult to detect in systems with intrinsic cycles [
]. In this case, [
first removing periodic trends in the time series through seasonal detrending and then
performing analysis on the residuals. However, prior to reaching criticality,
transmission systems may not exhibit a reliably periodic pattern. Thus, it’s possible that periodic
detrending or differencing may corrupt the signal of CSD and unintentionally reduce the
reliability of the early warning signals. Nonetheless, the robustness of differencing and
detrending have never been validated in the context of early warning signals of epidemic
transitions. To fill this gap, we studied the detectability of CSD prior to disease emergence
in simulations. To address the issue of periodic forcing, we analyzed our data both with
and without first removing seasonal trends via seasonal detrending and differencing.
Additionally, we propose and examine two new statistics based on components of
the frequency domain as alternatives to seasonal detrending and differencing. In the
frequency domain of time series without periodic cycles, CSD manifests as spectral
reddening, or the dominance of low frequencies prior to a transition [
]. Spectral reddening
of the Fourier spectra was shown to be a reliable indicator of climate tipping points [
and housing market changes [
]. Our concern is that for a periodic time series, the
periodic signal will overwhelm that of spectral reddening such that CSD is difficult to
detect. Wavelet analysis, a non-parametric approach to identifying periodic components
of the system over time, might allow us to identify the frequencies that are most
sensitive to CSD. In this paper, we show that signals of CSD can still be found in systems with
periodic behavior and present methods for forecasting emergence of seasonally forced
infectious diseases based on spectral reddening of the wavelet spectrum.
To simulate the dynamics of an immunizing pathogen with seasonal transmission
(e.g., pertussis, measles) [
], we considered a seasonally-forced stochastic
SusceptibleInfected-Recovered (SIR) model. Transitions between states occur with rates shown in
Table 1 where N = S + I + R and the mean field equations are
d(S/N ) = −β(t)(S/N )(I/N ) − ξ(S/N ) + μ(1 − S/N )
d(Id/tN ) = β(t)(S/N )(I/N ) + ξ(S/N ) − γ (I/N ) − μ(I/N ) (1)
d(R/N ) = γ (I/N ) − μ(R/N ).
Parameter definitions and values are given in Table 2. We modeled the contagion
process as a Markov chain using Gillespie’s direct method [
]. A Markov chain is a stochastic
process with the property that the future state of the system is dependent only on the
present state of the system and conditionally independent of all past states (i.e., the system
has a memoryless property).
Our model is slowly forced through a transition from subcritical (average R0 during
a year less than 1) to supercritical (average R0 during a year greater than 1) by linearly
increasing the average transmission rate, β0, over time. Increasing transmission, as
formulated in this model, can be envisioned as changes in behavior between individuals
in a population leading to higher contact rates over time or pathogen adaptation. We
assumed frequency-dependent transmission but note that in this case, the difference
between density-dependent and frequency-dependent transmission is negligible because
the population size fluctuates minimally (due to demographic stochasticity).
In our model, susceptible individuals are infected through contact with infected
individuals within the population at a time-varying rate β(t). To represent a realistic scenario
such that low levels of contact with outside populations also occur, we assumed that
susceptible individuals can also became infected at a constant rate ξ [
]. This assumption
signifies that the equilibrium size of the infectious population is non-zero, allowing for
stuttered chains of transmission even when the system is sub-critical (i.e., when R0 < 1).
Sinusoidal transmission follows the function of time β(t) = β0(t)+β1 sin(2π t/365) where
t is in units of days. We assume the time-localized average rate of transmission (β0) begins
to increase linearly ten years later (i.e., tstart = 10 · 365 days), so that
βk, t < tstart
βk + b (t − tstart) , t ≥ tstart
where βk = 0.04 is the initial transmissibility and b = 1.7242 × 10−5/day is the daily
rate at which transmission increases. These parameters cause the critical value of R0 to
be reached at year fifteen (five years after tstart). Before tstart, the average R0, calculated
as β/(γ + μ), is approximately 0.56 (β0 = 0.04). To explore the effect of seasonality, we
varied β1 linearly from 0 to 0.04 with 50 levels and simulated each level 250 times. For
sinusoidal seasonal forcing, we present and discuss intensity of relative fluctuations, i.e.
β1/β0, for ease of interpretation. For example, if β1/β0 = 1 then seasonality would cause
R0 to fluctuate between 0 and twice that of baseline transmission (Fig. 1).
We note that seasonality in transmission is often modeled as β = β0(1 + β1 cos(2π t/p))
where β1 can be interpreted as the amplitude of fluctuations relative to the average rate
of transmission (see [
]). In this formulation, though, if β0 were to increase (i.e., as
formulated in our model) then the size of oscillations around the increasing average
transmission, β0, would increase as well. By contrast, our parameterization induces a constant
amplitude for seasonality. The relative amplitude of fluctuation in our parameterization
is simply given by β1/β0.
For validation of our results with sinusoidal forcing, we also analyzed the impact of
seasonal transmission on EWS using square wave forcing, term-time forcing [
monthly averaged rates [
]. These other seasonality functions were defined by a set of 52
repeating deviations from the mean. The magnitude of seasonality for these models was
controlled by scaling the deviations such that the deviation below the mean was at most
equal to a given fraction of the mean.
We allowed a burn-in time of ten years before analyzing simulated incidence to remove
the effects of transient behavior. For comparison with typical incidence data, cases were
aggregated as number of individuals entering the removed class each week denoted by Xt.
We assumed no under-reporting. Because we assumed individuals within the population
could become infected from those outside the population, sparking transmission with
initially infected individuals was not necessary. Thus, our initial conditions for state variables
were S(0) = N (0), I(0) = 0, R(0) = 0. All simulations were performed in R using
packages spaero [
] and pomp [
]. Code to reproduce the simulations is available online
Calculation of early warning signals
We defined the null and test intervals to compare trends in EWS when the system is and
is not approaching criticality. The null interval is defined when β0 remains constant as
compared with the test interval when β0 is increasing and therefore approaching
criticality (Fig. 1). Following separation into the null and test intervals, we calculated early
warning statistics both with and without first removing seasonal trends via (1) seasonal
decomposition by Lowess [
] and (2) seasonal differencing where the time series at Xt is
subtracted from Xt−52 when observations are measured weekly.
We analyzed two statistics for anticipating disease emergence based on the wavelet
spectrum: (1) wavelet filtered reddening and (2) wavelet spectral reddening. We defined
wavelet filtered reddening, W¯ t2,j1j2 , as any quantity proportional to the variance of the time
series after it is filtered to include only its components in a wavelet transform’s frequency
domain with scales ranging from sj1 to sj2 . We calculated it according to
W¯ t2,j1j2 =
0.34T δt j2 |Wt(sj)|2
where T is the total number of observations in the times series, δt is the difference in time
between observations, sj is a wavelet scale, Wt(sj) is a wavelet transform of the observation
time series X with localized time index t and scale sj, and λ is the ratio of the Fourier
period of the wavelet function to its scale. The Fourier period of the wavelet function is
the period at which the Fourier transform of the wavelet function peaks.
To see that (3) is proportional to the average variance of the time series within a
frequency band, note that the right hand side of (3) differs from that of Eq. 24 of Ref. [
by a constant factor only. Equation (3) may also be viewed as a partial summation
over elements of the bias-corrected power spectrum of Ref. [
] multiplied by a
constant factor. Thus in our plot of the wavelet power spectrum, we plot the values of
[ 0.34T δt|Wt(sj)|2] /[ sjλσ 2] for all scales sj and times t, where σ 2 is the estimated
variance of the time series X. Because we look at trends in the wavelet power spectrum, the
constant factors in our equations have no effect on our results and are included simply
because they are included in the calculations of the software package we used.
Next we explain exactly how we calculated the wavelet transform. The wavelet
transform was calculated to satisfy
Xt − X¯
(t − t)δt ,
ψ0(η) = π −1/4eiω0ηe−η2/2
X¯ is the mean of the time series, and ψ0∗ is the complex conjugate of the Morlet wavelet
function. The Morlet wavelet function satisfies
where ω0 is the nondimensional frequency. We used typical parameters to calculate the
wavelet transform. We set ω0 to 6, and the sequence of scales used started at 2δt and
increased geometrically with 12 equal steps per octave until it reached about one third the
duration of the time series. These parameters correspond to the defaults in the R package
], which we used for all wavelet calculations. To choose the parameters j1
and j2 in Eq. (3), we conducted a grid search and used AUC to determine which j1 and j2
resulted in good separation of the distributions of W¯ t2,j1j2 calculated from emergence and
Our second wavelet statistic, wavelet spectral reddening, was defined as the median
scale of the wavelet spectral density at a given time index t. We denote this statistic as
smedian(t), and we calculate it according to
smedian(t) = min sj such that W¯ t2,1j/W¯ t2,1jmax > 0.5 ,
where the indices of the increasing wavelet scales sj run from 1 to jmax. A spectral
reddening statistic of the Fourier power spectrum was calculated in an analogous manner in
]. This statistic is intended to quantify any increasing dominance of the low
frequency components of the power spectrum associated with the decreasing stability of an
The other early warning statistics were computed according to the formulas in Table 3,
in which the weights wt,t depend on the smoothing kernel and bandwidth. We used both
a uniform kernel and a Gaussian kernel. The equation for the weights using the uniform
j=max(1,t−b+1) 1. The equation for the weights
1/Nt, |t − t | < b,
where the normalization constant Nt =
using the Gaussian kernel is
wt,t = f (t − t , b)/Nt
where f satisfies f (x, b) = exp −x2/(2b2) / b√2π and Nt = tT=1 f (t − t , b).
17, 22, 40, 41
] the other early warning statistics were computed according to
the formulas in Table 3. Moving window statistics of time series X (centered on index t)
were calculated with a uniform kernel and the bandwidth (parameter b in Eq. 7) was 100.
If data were detrended using STL , the equations for the moving window statistics
were slightly modified. The STL estimate of the trend was used as meant in Table 3. Then,
the STL estimate of the irregular component of the time series was substituted in for the
residuals (Xt − meant) in Table 3.
We determined the performance of each EWS using the AUC statistic as follows. We
calculated the association between each indicator time series and time using Kendall’s
rank correlation coefficients. Since we had 250 simulations, we generated a
distribution of correlation coefficients for each indicator when the system was and was
not approaching disease emergence (i.e., from the test or null interval). The amount
of overlap between the distributions was calculated with the AUC statistic: AUC =
[rtest − (ntest + 1)/2] /(ntestnnull) where rtest is the sum of the ranks of test coefficients in
a combined set of test and null coefficients (where the lower numbers have lower ranks),
ntest is the number of test coefficients, and nnull is the number of null coefficients. The
AUC of an EWS is the probability that a randomly chosen test coefficient is higher than a
randomly chosen null coefficient [
We documented signals of critical slowing down when transmission was subject to
periodic variation, addressing a current gap in studies of early warning signals for critical
transitions (Fig. 2). The AUCs of most early warning statistics were negatively associated
with increasing seasonal amplitude (Figs. 2 and 3). When transmission was not subject to
seasonal forcing, the most reliable statistics (mean, variance-based, and wavelet filtering),
achieved AUCs above 0.85 regardless of how data were pre-processed. For simulations
with the highest amplitude of seasonal transmission, AUCs for the mean, variance,
autocovariance, and wavelet filtering decreased by 0.05, 0.07, 0.06, and 0.02 compared to the
non-seasonal simulations, respectively.
Skewness and kurtosis were poor indicators of disease emergence compared with
variance-based statistics but detrending the time series prior to analysis improved the
AUCs slightly when transmission was subject to moderate or high levels of seasonal
forcing (Right panel, Fig. 2). For other EWS (i.e., decay time, autocorrelation, and index of
dispersion), pre-processing the time series prior to analysis worsened or had little impact
on the AUCs (Middle and left panels, Fig. 2).
In simulations with non-seasonal transmission, wavelet reddening and filtering
performed similarly to variance-based early warning statistics (Figs. 2 and 3). However, as the
amplitude of seasonality increased, the AUC of wavelet reddening decreased drastically
(from 0.72 to 0.35) with increasing seasonal amplitude whereas the AUCs of wavelet
filtering remained relatively constant as seasonal amplitude increased. For all levels of
seasonality, the AUCs of wavelet filtering were highest when the frequency band ranged
from 5 to 90 weeks (Fig. 4).
Early warning signals performed similarly regardless of the function of seasonal
forcing (sinusoidal, term-time [
], monthly averaged [
], and square-wave seasonal forcing)
(Additional file 1). For each type of seasonal forcing, the most reliable early
warning statistics were variance, variance convexity, autocovariance, and wavelet filtering
(Additional file 1). Conventional early warning signals were sensitive to the choice of
bandwidth. When the bandwidth was 150 weeks long, rather than 100, early warning
signal reliability decreased slightly (Additional file 1). The choice of window shape (i.e.,
uniform or Gaussian) was less important and resulted in similar AUC values for moving
A principal challenge in infectious disease epidemiology is predicting epidemics.
Prediction of epidemics is made difficult by a combination of non-linearities driving the
system. Early warning signals for disease emergence are potentially useful for
predicting emerging infectious diseases because they do not require a detailed understanding
of the system’s drivers. For infectious diseases without a seasonal trend (e.g., STDs),
generic statistical signals are present before critical transitions [
]. However, seasonal
variation alters the spread and persistence of many diseases [
] and has been noted
as a limitation to using early warning signals [
]. Understanding how seasonal forcing,
a common feature of disease dynamics, affects the trends in generic leading indicators
of stability is therefore important for accurately inferring whether a disease is at risk of
We conducted a simulation study of early warning signals for emergence of infectious
diseases. To understand the relationship between seasonality and the predictability of
disease emergence, we varied the amplitude of seasonal transmission. We showed
numerically that critical slowing down anticipates transitions in periodic systems. Additionally,
we proposed and measured the performance of two new leading indicators, wavelet
spectral reddening and filtering, compared to conventional early warning signals. Wavelet
filtering, calculated by summing specific coefficients of wavelets across the time series,
was among the top performing EWS. Our second statistic, wavelet reddening, was less
reliable than wavelet filtering as the level of seasonal transmission increased. The
advantage to using wavelet-based statistics is that they do not require a choice of bandwidth.
Although wavelet filtering requires a choice of frequency band, we showed that the
optimal choice of the frequency band was invariant to three levels of seasonal forcing.
Furthermore, the AUCs for wavelet filtering were generally highest with the largest frequency
bands. This suggests that all periods might be used and therefore this statistic would be
Our study’s purpose was to characterize the impact of seasonality on early warning
signals for disease emergence of an immunizing pathogen such as childhood infections. To
do this, we varied only the amplitude and held constant the frequency of the periodic
component of transmission. We showed numerically that early warning signals can be
reliable even in systems with highly seasonal transmission. A recent study [
special methods for calculating early warning signals when the time scale of the system is
similar to that of the forcing. In our model, the time scale of the system could be
characterized by the decay time of the number of cases per week and the time scale of the forcing is
the period of the seasonal cycle in R0. To characterize the timescale of the model as a
function of R0, we calculated the ensemble decay time based on the ensemble of Xt comprising
the simulation replicates of our model. We used simulations with a relative fluctuations
of zero and an initial β of 0.04. We found that the estimated decay time averaged 0.017
years in the null interval and rose to 0.14 years over the course of the test interval.
Therefore, within the test interval the timescale of the system was similar to that of the annual
forcing. There are many differences between our model and the model analyzed in
]. For example, stochastic effects (e.g., imported cases) are more important in our
model and the periodic forcing does not affect our observations as directly. Further
analysis of models similar to ours could clarify when seasonality of infectious diseases could
pose a major problem for EWS of disease emergence.
For our model, estimating and removing a periodic trend prior to EWS analysis did not
improve prediction uniformly among statistics. This was not entirely surprising because
the seasonal signal was not apparent in many time series. Rather, sporadically spaced
small outbreaks comprised the dynamics. Therefore, periodic detrending and
differencing introduced artificial patterns in the time series. In summary, even in systems where
transmission is highly seasonal, and a seasonal trend is apparent in a part of the
observation window, seasonal detrending is often disadvantageous for forecasting approaches
using early warning signals.
In our study, we held constant the period of forcing and other parameters that
sometimes exhibit periodicity (e.g., rate of imported infectious individuals, recovery
rate, or demographic rates). We also kept the rate of emergence constant among all
simulations. A recent study examining the effects that rate of forcing has on the
strength of EWS found that trends in EWS were more difficult to assess in
systems with faster rates of change [
]. Thus, we would expect that EWS be less
reliable for quickly emerging diseases. Here, we focused on periodicity in
transmission because it is the driving variable causing emergence and the other parameters
(e.g., demographic and recovery rates) are not typically periodic in human
populations. Lastly, as is typical in seasonally forced models of infectious diseases [
studies of periodicity in natural systems [
], we focused on seasonality using a sine
function. However, we showed that our results were largely invariant to different
types of seasonality functions (term-time [
], monthly-averaged [
], and square-wave
Variance and early warning signals based on spectral properties are beginning to
emerge as the most reliable indicators of upcoming transitions. In O’Regan and Drake
], approach to disease endemicity was much more difficult to forecast than
elimination but variance was the best predictor in SIR and SIS systems with
immigration. In vector-borne disease transmission, elimination was best predicted by variance
and coefficient of variation [
]. Additionally, disturbances in food chain dynamics
causing trophic cascades (i.e., the release of prey without suppression by predators)
were strongly associated with changes in variance and spectral density of
population sizes [
]. These experimental and theoretical results are beginning to reveal the
most robust, generic indicators across a wide range of ecological and epidemiological
Forecasting emerging infectious diseases is of tremendous value to public health and
society. Advances in surveillance systems and data science have generated the
expectation that early warning systems are not only feasible but necessary tools to combat the
spread of infectious diseases [
]. Critical slowing down was previously shown to
signal disease emergence without seasonal transmission [
]. Here, we showed that critical
slowing down also precedes disease emergence with seasonal transmission. Additionally,
we found that statistics based on the wavelet spectrum are robust signals of disease
emergence. Future research should aim to document critical slowing down in
experimental or surveillance data to estimate the potential forecast horizon that early warning
Additional file 1: Supplementary analysis showing effects of different forms of seasonal forcing, window size, and
window shape on performance of early warning signals. (PDF 151 kb)
This work was done on the Olympus High Performance Compute Cluster located at the Pittsburgh Supercomputing
Center at Carnegie Mellon University, which is supported by National Institute of General Medical Sciences Modeling
Infectious Disease Agent Study (MIDAS) Informatics Services Group grant 1U24GM110707.
Research funded by the National Institute of General Medical Sciences of the National Institutes of Health (Award
Availability of data and materials
All code for simulations and analysis will be made available at https://github.com/project-aero following publication.
All authors jointly formulated the mathematical modelling approach. PM and EO analyzed the data. PM drafted the
manuscript and all authors revised the manuscript. All other authors gave comments on the revised manuscript and
approved the final version of the manuscript.
Ethics approval and consent to participate
This study does not involve human participants, human data or human tissue.
Consent for publication
This study does not contain any individual person’s data in any form.
The authors declare that they have no competing interests.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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