Present mathematical challenges for Theoretical Immunology cover different areas and range between pure and applied mathematics and statistics. We propose a research programme that concentrates on four major themes in Immunology, which present significant immunological and mathematical challenges. Our first theme, T cell receptor antigen recognition and activation (Theme 1), involves challenges for the techniques of large deviation theory, exact asymptotics, Monte- Carlo sampling, cell signalling networks and multi-scale modelling. Although mathematical models (mostly coupled ODE models) have proved invaluable for understanding our second theme, homeostasis and control of lymphocyte proliferation and death (Theme 2), there are still major mathematical challenges to be resolved, such as parameter identifiability, non- Markovian branching processes, calculating, or even demonstrating the existence of, the quasi-stationary distributions of birth-death processes with absorbing states, multi-variate competition processes and large N expansions. The rapid and extended proliferation that follows recognition of antigen by našve T cells (the subset of T cells that have i not encountered their corresponding cognate antigens) is also accompanied by lymphocyte differentiation (Theme 3) and the acquisition of effector and/or memory phenotypes. This has parallels with developmental processes in many areas of biology; the response of T cells appears to have a strong genetic programming component, but cell fates are also clearly influenced by environmental factors and interactions with other cell types. Integrating models across scales, from genes, sig- nal transduction, inter-cellular interactions to organisation of lymphoid tissues, constitutes a challenge, as the mathematical models must incorporate experimental data of gene expression, signalling events and inter-cellular communication.
Recent immunological studies examine how the cells of the immune system interact in vivo in the lymph nodes. These two-photon imaging studies do not reveal the underlying mechanism of the observed behaviour or the relevance of the lymph node environment in which the cell-cell interactions take place. Our fourth theme cell-cell interactions (Theme 4) will address challenges such as continuum versus lattice-based models of cell motility, energy versus force based models of cell motility, deterministic versus stochastic description of cell motility and interactions, and mathematical analysis of the mean time to collision in different geometries.