文／Andrei R. Akhmetzhanov

　　When discussing the mathematical modeling of infectious diseases, many people envision a system of differential equations representing a susceptible-infected-removed (SIR) model. Each individual in that model falls into one of three compartments: either susceptible to infection ("S" compartment), infected ("I" compartment), or recovered or removed from consideration ("R" compartment"). Formulating a model aims to describe the temporal changes in the count of cases within each compartment and how individuals move between compartments. There have been a number of successful applications of the SIR model and its more complex variants. These applications range from describing a small outbreak of seasonal influenza at a school[1] to detailing the first months of the COVID-19 pandemic.[2] Among the objectives of investigating the SIR model is to determine what would happen if the outbreak turned the other way. A counter-factual scenario, for example, may include imagining what would happen if the outbreak was recognized sooner or if the vaccination campaign had been initiated sooner or later. These targets are often the subject of so-called scenario modeling.

　　In parallel, real-time modeling of infectious diseases has emerged as a distinct field. Its primary aim is to forecast future disease incidence during ongoing outbreaks, despite limited knowledge and incomplete data. Early examples include the FluSight forecast challenge where various research teams competed against each other to predict the outcome of the upcoming influenza season.[3] In the absence of access to all counts, each team analyzed only a snapshot of the data. Not only was the future incidence unknown, but the case counts for the preceding one-two weeks were incomplete since some cases may have been infected, possibly experienced some symptoms, but have yet to be reported or confirmed. To forecast future incidence, the research teams needed to first forecast those cases, i.e., the number of cases that were not yet reported, and only then build forecasts of future incidence. Stakeholders were interested in predicting the peak of influenza season, i.e., the maximum number of influenza patients and when it will peak, as these are both important for assessing future burden and improving preparedness. When the Western Africa Ebola outbreak struck in 2023, another challenge was designed to predict the dynamics of the outbreak in affected countries.[4] Similar challenges to forecasting respiratory infections such as COVID-19 and RSV have also been implemented in recent years.[5,6]

　　The process of forecasting infectious diseases can be linked to weather forecasting, particularly in predicting the trajectory of a typhoon. When a tropical depression first forms, forecasting its future path is challenging due to the limited data and the complex, dynamic nature of the weather system. Similarly, in the early stages of an infectious disease outbreak, forecasting is difficult due to low counts of cases and the unknown circumstances surrounding the outbreak. As the typhoon strengthens and its eye forms, forecasting becomes more accurate, much like how forecasts for infectious diseases improve as more data becomes available and the outbreak progresses. However, even as a typhoon weakens, it can still pose significant threats, such as re-strengthening or causing severe flooding. Likewise, even after the peak of an epidemic, accurate forecasting remains crucial for managing the end of the outbreak and preparing for potential secondary waves. This analogy demonstrates the importance of timing and data availability in both weather and infectious disease forecasts. Just as weather forecasts are most valuable when they provide early warnings, infectious disease forecasts are most beneficial when they predict the onset and peak of an outbreak. However, the accuracy of these forecasts is often limited in the early stages, similar to how early typhoon forecasts can be uncertain. As the outbreak progresses, forecasts become more reliable, but their use diminishes for stakeholders who need to make early decisions. Therefore, determining the optimal time to build forecasts is crucial for researchers in this field.

　　The applications of real-time modeling raise several critical questions. What value will such forecasts have in the event of a rapid outbreak and a rapid change in circumstances? Acute diseases, for example, the COVID-19 pandemic, can spread rapidly, causing a sharp surge in incidence and overwhelming the healthcare system. As a result, it is essential to develop capabilities for providing short-term forecasts in a timely manner. These forecasts would not predict what would happen in several months, as is the case with the FluSight challenge, but they would predict incidence in the next few weeks, thereby enabling more efficient allocation of resources and human resources. Secondly, even if we were able to provide such forecasts, how accurate can they be? As outlined above, early-stage forecasts often have significant uncertainties, whereas accuracy improves post-peak. Determining the optimal timing for forecasts remains a key challenge. As forecasting tools constantly develop and evolve, the ultimate goal is to provide real-time infectious disease predictions with the same reliability as modern weather forecasts.

References 

1. Avilov KK, Li Q, Lin L, Demirhan H, Stone L, He D. The 1978 English boarding school influenza outbreak: where the classic SEIR model fails. Journal of the Royal Society Interface. 2024;21(220):20240394. https://doi.org/10.1098/rsif.2024.0394 
2. Grinsztajn L, Semenova E, Margossian CC, Riou J. Bayesian workflow for disease transmission modeling in Stan. Statistics in medicine. 2021;40(27):6209-34. https://doi.org/10.1002/sim.9164 
3. Reich NG, Brooks LC, Fox SJ, Kandula S, McGowan CJ, Moore E, Osthus D, Ray EL, Tushar A, Yamana TK, Biggerstaff M. A collaborative multiyear, multimodel assessment of seasonal influenza forecasting in the United States. Proceedings of the National Academy of Sciences. 2019;116(8):3146-54.  https://doi.org/10.1073/pnas.1812594116 
4. Champredon D, Li M, Bolker BM, Dushoff J. Two approaches to forecast Ebola synthetic epidemics. Epidemics. 2018 Mar 1;22:36-42. https://doi.org/10.1016/j.epidem.2017.02.011 
5. Cramer EY, Huang Y, Wang Y, Ray EL, Cornell M, Bracher J, Brennen A, Rivadeneira AJ, Gerding A, House K, Jayawardena D. The United States COVID-19 forecast hub dataset. Scientific data. 2022;9(1):462. https://doi.org/10.1038/s41597-022-01517-w 
6. Hansen CL, Lee L, Bents SJ, Perofsky AC, Sun K, Starita LM, Adler A, Englund JA, Chow EJ, Chu HY, Viboud C. Scenario Projections of Respiratory Syncytial Virus Hospitalizations Averted Due to New Immunizations. JAMA network open. 2025;8(6):e2514622-. https://doi.org/10.1001/jamanetworkopen.2025.14622 

------------------------------------------------------------------------------

公衛四季：臺大公衛電子報第018期【2025夏季號】文章 延伸閱讀

• 換一種速度生活
• 《健康老化：你我都該來一份的人生套餐》從科技到社會支持，打造不孤單的高齡生活藍圖
• 健康老化的新思維—新孝道共好時代
• 植物性飲食與健康老化