Birth and death processes with mutation and selection, structured population dynamics, lineages and environment.
Whereas classically, observations were obtained on large populations and led to deterministic mathematical models, single
cell observations created the necessity to develop stochastic models, called individual-based models, taking into account the random behavior of each individual. The individuals are characterized by genetic or phenotypic information through a quantitative parameter called trait. The evolution of the trait distribution results from several mechanisms: heredity (transmission of the ancestral trait to the offspring) except when a mutation occurs, generating variability in the trait values, and selection. Selection can be genetic (an individual with higher reproduction ability or survival probability will spread through the population over time) or ecological (competition for resources or for sharing habitats). Bacteria can also exchange genes (plasmids) by contact. This horizontal gene transfer has a main role in the evolution of virulence and is considered as the primary reason for bacterial antibiotic resistance.
The mathematical model describes the stochastic times at which events occur and their nature: cell divisions with possible mutation, cell deaths. The model can also include the cell growth and their age or location and also external information as dynamics of resources or environmental changes (temperature, humidity, antibiotics or chemotherapy). The model can be studied in forward time to answer questions such: will the population die out? Will the mutant fix in the population? (fundamental questions if the cell population is a pathogen population or of the mutant is a cancer cell). One can also study the branching population model in backward time to trace individual lineages. This can be crucial face to a changing environment, to know which types of individuals can survive over time.
Mathematical questions
The stochastic individual-based models are consistent, under large population assumption, with usual deterministic models. Nevertheless, the large scale behavior cannot be really understood without a hierarchy of finer scales, down to the individual behavior. One needs to investigate the impact of various time scales on macroscopic approximations. Preliminary numerical simulations indicate that these models should exhibit many surprising asymptotic behaviours such as cyclic behaviours that need to be derived rigorously. The study of the lineages of sampled individuals at a given observation time by determining mathematically their time reversal paths is also of particular relevance when taking into account the effect of time dependent environments, in which case the survival of individuals may only be explained by a very small number of initial individuals
