Time Series in Epidemiology: Surveillance, Forecasting, and Interventions

Epidemiological time series are not simply counts indexed by date. Observed cases are generated by at least two processes:

A rise in reported cases can reflect increased transmission. It can also reflect testing expansion, reporting backlog, changes in case definition, healthcare-seeking behavior, or administrative delays. A useful time-series model must distinguish those mechanisms when the scientific question requires it.

Incidence, prevalence, admissions, and deaths are different series

Common epidemiological series include:

• incident cases;
• prevalent cases;
• hospital admissions;
• emergency presentations;
• laboratory positives;
• deaths;
• syndromic indicators.

They have different delays and denominators. For example, daily reported deaths are a lagged consequence of earlier infections. Modeling them as if they were contemporaneous measures of transmission can distort interpretation.

Counts need count models

If

is a weekly case count, Gaussian errors may be a poor default when counts are small or variance changes with the mean. A Poisson model begins with

while a negative-binomial model allows overdispersion,

The time-series dependence then enters the model for

This is often more natural than forcing the count series into a Gaussian ARIMA framework.

Seasonality

Respiratory infections frequently show strong seasonal structure. A simple regression can include periodic terms,

where $s$ is the seasonal period. Other options include seasonal ARIMA, splines, Gaussian processes, or state-space models. The goal is not merely to remove seasonality. Seasonality may itself be epidemiologically meaningful.

Reporting delay and nowcasting

Recent observations are often incomplete because reports arrive late. Let

denote events occurring at time $t$ but reported with delay $d$. The finally observed count is

At the present time, only some delays have arrived. A nowcast estimates the current but not-yet-complete value. This is different from forecasting future incidence. Confusing nowcasting and forecasting can make model performance look better than it is.

Outbreak detection

Surveillance systems often ask whether the current count is unusually high relative to an expected baseline. A generic alarm rule is

where $q_{1-\alpha,t}$ is a time-specific upper quantile under the baseline model. CUSUM, Farrington-type algorithms, control charts, and Bayesian surveillance models all implement variants of this logic. The baseline must account for seasonality and historical trend. Otherwise ordinary seasonal peaks become false alarms.

False alarms and delayed alarms

Outbreak detection is a sequential decision problem. Repeated testing over time means that a nominal one-day false-alarm probability does not equal the long-run probability of ever issuing a false alert. Performance should include:

• false alerts per unit time;
• detection delay;
• probability of detection;
• alert duration;
• stability under reporting revisions.

A detector that is sensitive but alarms every week is operationally poor.

Forecasting

Forecasting asks for a future distribution such as

The horizon $h$ matters. A model can be excellent at one-week forecasts and poor at eight-week forecasts. Evaluation should therefore report horizon-specific scores. For probabilistic forecasts, useful metrics include log score, interval coverage, and continuous ranked probability score rather than only point RMSE.

Mechanistic epidemic models

For infectious disease, compartmental models encode transmission structure. A simple SIR model is

This model is crude, but it distinguishes susceptible depletion and infectious duration from generic autocorrelation. Time-series and mechanistic models answer different questions. Hybrid models can combine both.

Interventions are causal questions

Suppose cases fall after a vaccination campaign or policy change. A before-after time-series pattern does not by itself identify a causal effect. The counterfactual question is

versus

the outcome trajectory with and without the intervention. Interrupted time series, synthetic controls, difference-in-differences, or mechanistic counterfactual models may be appropriate depending on design. Forecasting accuracy alone does not identify intervention effects.

Nonstationarity is often the phenomenon

In epidemiology, changing transmission is often exactly what matters. Trying to mechanically difference every series into stationarity can remove interpretable structure. A changing reproduction number, intervention, new variant, immunity shift, or case-definition change creates nonstationarity for substantive reasons. State-space models and time-varying parameter models can represent that evolution directly.

Data revisions matter

Public-health time series are often revised retrospectively. A model evaluated using the final revised data may appear better than a model that would have been available in real time. For operational surveillance, evaluation should use vintage data when possible:

denotes the value for event time $t$ as known at vintage $v$. This prevents hindsight from leaking into the validation set.

Conclusion

Epidemiological time-series analysis contains several distinct tasks:

Each requires a different target and often a different model. The strongest analyses model not only disease dynamics but also the measurement and reporting system through which those dynamics become data.

References

• Held, L., Höhle, M., & Hofmann, M. (2005). A statistical framework for the analysis of multivariate infectious disease surveillance counts. Statistical Modelling, 5(3), 187–199.
• Höhle, M. (2007). Surveillance: An R package for the monitoring of infectious diseases. Computational Statistics, 22, 571–582.
• Cori, A., Ferguson, N. M., Fraser, C., & Cauchemez, S. (2013). A new framework and software to estimate time-varying reproduction numbers during epidemics. American Journal of Epidemiology, 178(9), 1505–1512.