A two-scale model of Legionnaires disease to predict incubation periods and risk of symptomatic disease

Abstract

Legionella pneumophila is an intracellular pathogen that causes Legionnaires disease, a severe pneumonia acquired primarily through contaminated water systems. Public health interventions rely on accurate estimates of the incubation period and dose-response (DR) relationship, yet currently used approaches in the literature underestimate incubation periods by assuming Markovian rupture times for infected macrophages. Here, we develop the first non-Markovian, two-scale within-host framework for Legionnaires disease, coupling stochastic intracellular replication in individual macrophages with extracellular Legionella-macrophage population dynamics. At the cellular level, we model intracellular replication using a stochastic logistic birth-death (SLBD) process, coupled with non-Markovian rupture-time distributions (Erlang and Burr). The Erlang distribution preserves tractability via the method of separation, whereas the Burr distribution captures heavy-tailed rupture times consistent with experimental data. Simulations are implemented using a renewal-based non-Markovian Gillespie algorithm. At the host level, successive infection and rupture events describe population-scale infection dynamics, enabling estimation of DR curves and incubation-period distributions. Across six model variants, DR predictions remain robust, with ID50 estimates narrowly ranging between 8.79 and 8.94 Legionella, consistent with guinea pig challenge data. In contrast, incubation-period estimates show strong dependence on rupture-time assumptions: non-Markovian models predict median incubation periods of 5-6 days, correcting the previous 2-3 day underestimation and aligning with human outbreak data (2-10 days, up to 13 days). Sensitivity analysis identifies rupture size, phagocytosis rates, and threshold effects as key determinants of incubation-period results. By relaxing exponential assumptions, our framework provides biologically realistic within-host dynamics that improve epidemiological predictions. These results refine the quantitative basis for outbreak investigations and environmental risk assessment and are generalizable to other intracellular pathogens such as Coxiella burnetii and Francisella tularensis. Author summaryLegionella pneumophila causes Legionnaires disease, a serious pneumonia often linked to contaminated water systems, but key quantities such as the incubation period remain difficult to estimate accurately. Existing models assume that infected immune cells rupture at random times with no memory, an assumption that simplifies mathematics but does not reflect experimental observations. We developed a model that follows bacterial growth inside individual macrophages and connects these cellular events to infection dynamics within a host. Unlike previous approaches, our model allows rupture times to follow more realistic, non-exponential patterns that better match laboratory data. Using simulations, we show that commonly used assumptions systematically underestimate the incubation period of Legionnaires disease. Our results predict incubation periods of 5-6 days, consistent with human outbreak data, while leaving estimates of infectious dose largely unchanged. This work improves the biological realism of within-host infection models and provides a stronger quantitative foundation for outbreak investigation and environmental risk assessment. The modelling framework can be adapted to study other intracellular pathogens that replicate inside host immune cells.