### Research, as of 2011

#### Mathematical epidemiology and ecology

We have long been involved in the mathematical modelling of the spread of epidemics. Our current focus is on the analysis of host-parasite systems, with particular emphasis on those stochastic models which can be approximated by infinite systems of ODE's. The systems which appear in practice frequently have no useful monotonicity properties, so that even questions of existence and uniqueness of solutions over finite time intervals can be surprisingly awkward to handle. Equilibria are of particular interest, since they should correspond to the stochastic quasi-equilibria which describe what is observed in nature; for many of the models, finding the possible equilibria and establishing conditions for convergence to them are problems which are still entirely open. Our practical interest has been in applications to modelling the spread of Echinococcus granulosus, using data collected in Kirgistan.We have a recent project of more general importance in population biology, concerning the conditions under which a quasi-equilibrium may exist in populations which, in the far distance future, are certain to become extinct. A further aim is to find computational means of determining these distributions. We are also involved in a project, in which we model the way in which multicellularity may evolve in nature.

We have active collaboration with the Institute of Parasitology at UZH (P. Torgerson, P. Deplazes) and the Department of Evolution and Environment at UZH (H. Bagheri), as well as with G. Reinert (Oxford), A. Pugliese (Trento), P.K. Pollett (Queensland) and M. Luczak (LSE).

#### Approximation by Stein's method

Stein's method, first proposed in (1970), has become a central tool in many areas of probability theory. Our research is particularly concerned with Poisson and compound Poisson approximation, always with explicit error bounds, together with applications in graph theory, combinatorics and reliability theory. However, the generality of the method is such that it also appears as an essential component in the proofs of many of the theorems appearing under the other headings.

Both of my Ph.D. students are involved in this project. We also work with many colleagues at other universities: G. Reinert (Oxford), Ch. Stein (Stanford), L.H.Y. Chen (Singapore), S. Janson (Uppsala), T. Lindvall (Göteborg), S. Utev (Nottingham), A. Xia (Melbourne), O. Chryssaphinou (Athens), V. Cekanavicius (Vilnius), I. Kontoyiannis (Athens).

#### Combinatorial probability

A large number of classical decomposable combinatorial structures, such as permutations, mappings and polynomials over a finite field, turn out to have widely similar asymptotic behaviour. We have been able to show that this is because they all satisfy two rather elementary conditions, and have thus been able greatly to extend the range of structures belonging to the class. Our most recent work is concerned with the properties of (some natural generalizations of) additive functions defined on such structures, and with an extension of the class of objects studied in our book, known as quasi-logarithmic structures.This work is in collaboration with R. Arratia (U. Southern California) and S. Tavaré (Cambridge).

#### Computational biology

The group is currently involved in joint projects with the Bioinformatics group at UZH (A. Wagner), with the Molecular Cancer Institute at UZH (E. Lacko, J. Jiricny), and with the Hutchinson Cancer Centre in Cambridge (S. Tavaré). The projects are connected with understanding the evolution of tumours, with the analysis of data on tumour progressions, and with the population biology of transposable elements.