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 rely solely on heuristic justifications for their applicability. 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 in the literature. Our results indicate that the derived Burr distribution is 13 times more likely to be a better-performing model to incubation-period data than currently used methods. Further, we show that incorporating methods for handling the censoring issue results in the mean of the underlying continuous incubation-period model being reduced by a whole day, compared to the mean obtained under alternative modelling techniques in the literature.