Results from a mathematical model for human monocytic ehrlichiosis

Mathematical model of Ehrlichia chaffeensis transmission dynamics in dogs

Ehrlichia chaffeensis is a tick-borne disease transmitted by ticks to dogs. Few studies have mathematical modelled such tick-borne disease in dogs, and none have developed models that incorporate different ticks' developmental stages (discrete variable) as well as the duration of infection (continuous variable). In this study, we develop and analyze a model that considers these two structural variables using integrated semigroups theory. We address the well-posedness of the model and investigate the existence of steady states. The model exhibits a disease-free equilibrium and an endemic equilibrium. We calculate the reproduction number (T 0 ). We establish a necessary and sufficient condition for the bifurcation of an endemic equilibrium. Specifically, we demonstrate that a bifurcation, either backward or forward, can occur at T 0 = 1 , leading to the existence, or not, of an endemic equilibrium even when T 0 < 1 . Finally, numerical simulations are employed to illustrate these theoretical findings.

Long-term transmission dynamics of tick-borne diseases involving seasonal variation and co-feeding transmission

Co-feeding is a mode of pathogen transmission for a wide range of tick-borne diseases where susceptible ticks can acquire infection from co-feeding with infected ticks on the same hosts. The significance of this transmission pathway is determined by the co-occurrence of ticks at different stages in the same season. Taking this into account, we formulate a system of differential equations with tick population dynamics and pathogen transmission dynamics highly regulated by the seasonal temperature variations. We examine the global dynamics of the model systems, and show that the two important ecological and epidemiological basic reproduction numbers can be used to fully characterize the long-term dynamics, and we link these two important threshold values to efficacy of co-feeding transmission.

Dynamics of two pathogens in a single tick population

A mathematical model for a two-pathogen, one-tick, one-host system is presented and explored. The model system is based on the dynamics of Amblyomma americanum, Rickettsia parkeri, and Rickettsia amblyommatis. The goal of this model is to determine how long an invading pathogen, R. parkeri, persists within a tick population, A. americanum, in which a resident pathogen, R. amblyommatis, is already established. The numerical simulations of the model demonstrate the parameter ranges that allow for coexistence of the two pathogens. Sensitivity analysis highlights the importance of vector-borne, tick-to-host, transmission rates on the invasion reproductive number and persistence of the pathogens over time. The model is then applied to a case study based on a reclaimed swampland field site in south-eastern Virginia using field and laboratory data. The results pinpoint the thresholds required for persistence of both pathogens in the local tick population. However, R. parkeri, is not predicted to persist beyond 3 years. Understanding the persistence and coexistence of tick-borne pathogens will allow public health officials increased insight into tick-borne disease dynamics.

Metapopulation models in tick-borne disease transmission modelling

Human monocytic ehrlichiosis (Ehrlichia chaffeensis), or HME, is a tick-transmitted, ricksettisal disease with growing impact in the United States. Risk of a tick-borne disease such as HME to humans can be estimated using the prevalence of that disease in the tick population. A deterministic model for HME is explored to investigate the underlying dynamics of prevalence in tick populations, particularly when spatial considerations are allowed. The dynamics of HME in a single spatial patch are considered first to determine which model components are most important to predicting disease dynamics in a local ecology. The model is then expanded to spatially-explicit patches on which patch connectivity, the surrounding environment and boundary effects are studied. The results of this investigation show that predicting risk of this disease to humans is determined by many complicated interactions. Areas that would be endemic in isolation may or may not sustain the disease depending on the surrounding habitat. Similarly, control efforts are shown to be far more effective when applied in wooded habitats than in neighboring grassy habitats. Boundary assumptions which describe the reality of increasing habitat fragmentation additionally play a large role in predicting the endemicity of an HME outbreak. Overall, HME and all tick-borne diseases are complex, nonlinear systems that have just begun to be explored.