The Burr distribution as a model for the delay between key events in an individual's infection history

Understanding the temporal relationship between key events in an individual's infection history is crucial for disease control. Delay data between events, such as infection and symptom onset times, is doubly censored because the exact time at which these key events occur is generally unknown. Current mathematical models for delay distributions are derived from heuristic justifications. Here, we derive a new model for delay distributions, specifically for incubation periods, motivated by bacterial-growth dynamics that lead to the Burr family of distributions being a valid modelling choice. We also incorporate methods within these models to account for the doubly censored data. Our approach provides biological justification in the derivation of our delay distribution model, the results of fitting to data highlighting the superiority of the Burr model compared to currently used models when the mode of the distribution is clearly defined or when the distribution tapers off. Under these conditions, our results indicate that the derived Burr distribution is a better-performing model for incubation-period data than currently used methods, with the derived Burr distribution being 13 times more likely to be a better-performing model than the gamma distribution for Legionnaires' disease based on data from a known outbreak.