A Boundary Value Approach to Optimization with an Application to Salmonella Competition

Fixation in the stochastic Lotka-Volterra model with small fitness trade-offs

We study the probability of fixation in a stochastic two-species competition model. By identifying a naturally occurring fast timescale, we derive an approximation to the associated backward Kolmogorov equation that allows us to obtain an explicit closed form solution for the probability of fixation of either species. We use our result to study fitness tradeoff strategies and show that, despite some tradeoffs having nearly negligible effects on the corresponding deterministic dynamics, they can have large implications for the outcome of the stochastic system.

Spatial dynamics of inflammation-causing and commensal bacteria in the gastrointestinal tract

In recent years, new research programmes have been initiated to understand the role of gut bacteria in health and disease, enabled in large part by the emergence of high-throughput sequencing. As new genomic and other data emerge it will become important to explain observations in terms of underlying population mechanisms; for instance, it is of interest to understand how resident bacteria interact with their hosts and pathogens, and how they play a protective role. Connecting underlying processes with observed patterns is aided by the development of mathematical models. Here, we develop a spatial model of microbial populations in the gastrointestinal tract to explore conditions under which inflammation-causing bacteria can invade the gut and under which such pathogens become persistent. We find that pathogens invade both small and large intestine from even a relatively small inoculum size but are usually eliminated by the host response. When the immune response is weak, the pathogen is able to persist for a long period. Spatial structure affects these dynamics by creating moving refugia which facilitate bouts of pathogen resurgence and inflammation in persistent infections. Space also plays a role in repopulation by commensals after infection. We further find that the rate of decay of inflammation has a stronger effect on outcomes than the initiation of inflammation or other parameters. Finally, we explore the impact of partially inflammation-resistant commensals on these dynamics.

Modelling the effect of birth and feeding modes on the development of human gut microbiota

The human gut microbiota is transmitted from mother to infant through vaginal birth and breastfeeding. Bifidobacterium, a genus that dominates the infants' gut, is adapted to breast milk in its ability to metabolize human milk oligosaccharides; it is regarded as a mutualist owing to its involvement in the development of the immune system. The composition of microbiota, including the abundance of Bifidobacteria, is highly variable between individuals and some microbial profiles are associated with diseases. However, whether and how birth and feeding practices contribute to such variation remains unclear. To understand how early events affect the establishment of microbiota, we develop a mathematical model of two types of Bifidobacteria and a generic compartment of commensal competitors. We show how early events affect competition between mutualists and commensals and microbe-host-immune interactions to cause long-term alterations in gut microbial profiles. Bifidobacteria associated with breast milk can trigger immune responses with lasting effects on the microbial community structure. Our model shows that, in response to a change in birth environment, competition alone can produce two distinct microbial profiles post-weaning. Adding immune regulation to our competition model allows for variations in microbial profiles in response to different feeding practices. This analysis highlights the importance of microbe-microbe and microbe-host interactions in shaping the gut populations following different birth and feeding modes.