A within-host birth–death and time–dose–response model for Legionnaires’ disease

Understanding the dose–response relationship for infectious diseases is important for quantitative microbial risk assessment studies to mitigate risk. To capture the dose–response dynamics, understanding the pathogenesis of the infectious agent is desirable. Typically, attempting to understand the dose–response dynamics would involve within-host mathematical modelling and fitting dose–response curves to experimental data. No mathematical model exists that describes the within-host dynamics that occur within an individual infected with Legionnaires’ disease. Further, most dose–response models are based either on a single-hit or threshold hypothesis for the cause of illness. Here, we derive a model to explain within-host dynamics post-infection with Legionnaires’ disease that incorporates heterogeneity at the cellular and population levels. We develop a new dose–response model that allows for either of two hypotheses for the cause of illness, adding a new level of flexibility not currently seen in the literature. We extend the dose–response model to incorporate time as we develop a dose-dependent incubation-period model that is based on biological mechanisms. Our within-host models provide an ID50 of between eight and nine Legionella and median incubation periods close to four days, which is consistent with evidence obtained from animal experiments and human outbreaks in the literature.