Dynamical Systems in Biology: Persistence and Models of Bacterial and Viral Infection and Treatment

The mathematical theory of persistence is an important tool for the analysis of biological models. It provides a mathematically rigorous method to establish the long term persistence or viability of a population (or chemical species) which, at the same time, is user friendly. Epidemiological models are especially rich sources of issues that can be decided by the theory: Can a disease drive the host population to extinction? This is a question of host persistence. Does the disease become endemic in a population? This is the question of disease persistence. However, to-date, the theory has been difficult to apply to discrete-time dynamical systems arising in epidemiology and population dynamics because of the complexity of the boundary attractor. PI aims to use the theory of lyapunov exponents applied to the matrix cocyle induced by a nonlinear system to obtain persistence results. An important part of the proposed work is to develop methods of computing/estimating the key normal lyapunov exponent. Previous work of the PI and collaborators on the mathematical modeling of generic bacterial infections of mammalian tissues including host immune response and antibiotic therapy will be extended to include: (1) treating microbial populations as heterogeneous with respect to virulence and susceptibility to antibiotic treatment, (2) consideration of both specific and nonspecific host immune response, (3) include transfer of antibiotic resistance between bacterial strains by mutation, plasmid transfer, and non-genetic modes of acquiring resistance via passing to non-growing, quiescent state, and (4) allow general pharmacokinetics involving both continuous and discrete antibiotic delivery modes. The proposed extensions should lead to better understanding of the infection and treatment processes, allowing computer simulation of processes which are difficult to monitor in vivo. The PI intends to construct detailed mathematical models of virus-host cell interaction that extend current models by including (1) host-cells structured by the number of attached virus, (2) infected host cells structured by time since infection, (3) general progeny release rates that accommodate both continuous budding from a live cell and discontinuous burst of progeny at cell lysis, and (4) distinguishing multiply infected cells. Such models should allow more accurate determination of the progression of virus infections than current models which neglect these features. Mathematical modeling can predict and confirm features of virus-host cell dynamics that are difficult to monitor in animal models.