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

Understanding the dose–response (DR) relationship for infectious diseases is important for quantitative microbial risk assessment studies to mitigate risk. To capture the DR dynamics, understanding the pathogenesis of the infectious agent is desirable. Typically, attempting to understand the DR dynamics would involve within-host mathematical modelling and fitting DR 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 DR 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 DR 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 DR 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 4 days, which is consistent with evidence obtained from animal experiments and human outbreaks in the literature.