The Impact of Non-Homogeneous Mixing on Demography and Epidemics - Fabio Milner
The majority of mathematical models in gender-structured populations and epidemics assume homogeneous mixing between groups that lead to new relevant individuals for the model (e.g. between male and female singles that lead to new births in demography, between infectious and susceptible individuals in epidemics that lead to new infections). The reality is that mixing is usually far from homogeneous and there are models that account for heterogeneity in at least two different ways: introducing several subgroups of individuals each of which has a different (but constant) in-group contact rate, or introducing a single group of individuals who do not participate in the generation of new relevant individuals (e.g. sexually abstained in two-sex models who do not ever have progeny, or permanently immune individuals in the case of an epidemic).
The aim of this project will be to formulate models that incorporate both ideas together and study the impact of temporary isolation from the generation of new relevant individuals (i.e. temporary abstinence from reproduction in demography, temporary immunity in epidemics) and address the following questions:
Students should have some ability to program in a language such as C/C++ or Matlab. Familiarity with some elementary dynamical systems theory would be helpful, as would some knowledge of population dynamics or epidemics. This project would be ideal for a small group that included both biology and mathematics majors.
Evolutionary Genetics of Antigen Repertoires - Jay Taylor
Parasites that cause chronic infections in vertebrates use a variety of strategies to evade the adaptive immune responses of their hosts. One strategy, called antigenic variation, plays an important role in some vector-transmitted diseases and occurs when the parasite genome encodes multiple surface antigens, only a few of which are expressed at a time. Although host immune responses will eventually develop against expressed antigen types, the infection can persist if small numbers of parasites are able to randomly switch expression to antigen types that have not yet been seen by the host. Familiar examples of human pathogens that engage in antigenic variation include the bacterium Borrelia hermsii, which causes tickborne relapsing fever, and the parasitic protozoa Trypanosoma brucei and Plasmodium falciparum, which cause African sleeping sickness and malaria, respectively.
Although genome sequencing projects are providing important insights into the makeup of antigen gene repertoires, little mathematical theory has been developed to understand how these gene families evolve. In part, this is because diversification of antigen repertoires depends on a hierarchy of processes operating at three levels. Within each parasite genome, the antigen repertoire is subject to mutation, recombination, gene conversion, and duplication or deletion of entire genes. This leads to variation between antigen repertoires that are subject to natural selection and demographic stochasticity both within individual infections as well as during transmission between hosts. The aim of this project will be to formulate a stochastic model that incorporates these genomic and epidemiological processes and to use simulations to investigate some of the following questions:
Students should have some ability to program in a language such as C/C++ or Matlab. Familiarity with some elementary probability theory (at the level of Ross' Introduction to Probability Theory for example) would be helpful, as would some knowledge of population genetics or evolutionary biology. This project would be ideal for a small group that included both biology and mathematics majors.