Nonlinear Stochastic Modelling of Antimicrobial resistance in Bacterial Populations

This thesis applies mathematical modelling and statistical methods to investigate the dynamics and mechanisms of bacterial evolution. More specifically it is concerned with the evolution of antibiotic resistance in bacteria populations, which is an increasing problem for the treatment of infections in humans and animals. To prevent the evolution and spread of resistance, there is a need for further understanding of its dynamics. A grey-box modelling approach based on stochastic differential equations is the main and innovative method applied to study bacterial systems in this thesis. Through the stochastic differential equation approach, knowledge of continuous dynamical systems can be combined with strong statistical methods. Hereby, important tools for model development, parameter estimation, and model validation are provided when in connection with data. The data available for the model development consist mainly of optical density measurements of bacterial concentrations. At high cell densities the optical density measurements will be effected by shadow effects from the bacteria leading to an underestimation of the concentration. To circumvent this problem a exponential calibration curve has been applied for all the data. This new curve was found to performthe best calibration in a comparison with other earlier suggested curves. In this thesis a new systematic framework for model improvement based on the grey-box modelling approach is proposed, and applied to find a model for bacterial growth in an environment with multiple substrates. Models based on stochastic differential equations are also used in studies of mutation and conjugation. Mutation and conjugation are important mechanisms for the development of resistance. Earlier models for conjugation have described systems where the substrate is present in abundant amounts, but in this thesis a model for conjugation in exhaustible media has been proposed. The role of mutators for bacterial evolution is another topic studied in this thesis. Mutators are characterized by having a high mutation rate and are believed to play an important role for the evolution of resistance. When growing under stressed conditions, such as in the presence of antibiotics, mutators are considered to have an advantages in comparison to non-mutators. This has been supported by a mathematical model for competing growth between a mutator and a non-mutator population. The growth rates of the two populations were initially compared by a maximum likelihood approach and the growth rates were found to be equal. Thereafter a model for the competing growth was developed. The models showthat mutatorswill obtain a higher fitness by adapting faster to an environment with antibiotics than the non-mutators. In another study a new hypothesis for the long term role of mutator bacteria is tested. This model suggests that mutators can work as "genetic work stations", where multiple mutations occur and subsequently are transmitted to the non-mutator population by conjugation. Another study in this thesis is concerned with the spread of colonization with resistant bacteria between patients in a hospital and people in the related catchment population. The resistance considered is extended-spectrumbeta-lactamases, and it is the first time a model has been developed for the spread of this type of resistance. Different transfer mechanisms are studied and quantified with the model. Simulations of the model indicates that cross-transfer of resistance between patients is the most important mechanism of transfer. The mathematical models developed in this thesis have helped to an improved understanding of the evolution and spread of resistance. They are thus a prime example of the strength of combining microbiology and experiments with modelling.