The main thrust of my theoretical work is to understand the role that spatial heterogeneity and stochasticity play in biological (ecological, epidemiological and evolutionary) systems. As such my research involves computationally intensive simulations and the search for simple intuitive models that can capture the salient features of such simulations. Listed below are some of the main themes of my interest together with links to collaborators, useful websites and important researchers in that field.
Spatial heterogeneity is vitally important to the dynamics and persistence of many biological populations. Often this spatial heterogeneity can be a reflection of variations in the underlying distribution of resources. However, a secondary cause of spatial heterogeneity is the interaction between organisms leading to spatial correlations; a simple example of which might be the aggregation of fish into schools.
Traditionally, such spatial structure was only reproduced by complex spatial models, more akin to simulations than mathematical equations. In recent years however a series of tools has been developed to directly model the spatial correlations that can arise when discrete organisms interact with their local environment. One of the simplest type of such models is the pair-wise model, where instead of modelling the number of individuals, the number of interacting pairs is modelled by a set of differential equations.
This framework is particularly useful for understanding the spread of diseases, where a contact network determines who interacts (and therefore can infect) whom. A mixture of analytical and numerical studies have shown that including the correlations that develop within this network structure can have strong quantitative (as well as qualitative) effects.
Future work intends to apply this research to the spread of STDs (sexually transmitted diseases) where the network of contacts is well defined, and the low number of average connections means that there will be large deviations away from standard (well-mixed or mean-field) models.
People Minus van Baalen, Ben Bolker, Ulf Dieckman, Joao Filipe, Richard Law, David Murrell, Chris Bauch
Moment closure is a technique by which higher order moments are approximated in terms of lower order ones, such that a closed system of equations can be developed. As such the pair-wise models explained above are an example of moment closure as triples are approximated in terms of pairs. A second form of moment closure is to consider distributions of population sizes, and approximate the third order cumulate (the skew) in terms of the second order cumulate (the variance). Such models allow us to extend standard deterministic models so as to approximate the amount of variance that occurs due to the presence of demographic stochasticity or environmental noise. I have applied this theory to two different situations,
1) For an infinite set of habitats, with rapid global movement of individuals, it is obvious that the population distributions should be close to Poisson. The moment-closure technique allowed us to relax this rapid mixing limit, and hence consider the stability of natural enemy systems (the Lotka-Volterra and Nicholson-Bailey models) as heterogeneities begin to develop. This work showed that heterogeneities could be either stabilising or destabilising, and that there were significant differences between the two models studied.
2) The technique was extended further by considering a small number of subpopulations, such that there is variance at both the habitat and meta-population scale. For these models, the closure technique assumed that the distributions were close to log-normals. In this way the extinction risk of a variety of models could be theoretically calculated; single species models persisted best when movement between patches was large, whereas natural-enemy models persisted best for intermediate levels of coupling.
I believe that moment-closure techniques will become increasingly important in understanding biological phenomenon, as the roles played by spatial correlations and temporal variability receive more recognition.
People Steve Pacala, Ingemar Nåssel, Tom Briton, Eric Renshaw, Gavin Gibbon.
Metapopulations are an incredibly intuitive and powerful means of representing a stochastic spatial system that is composed of many separate habitats. In its simplest formulation, the Levins metapopulation models each habitat is either occupied (with probability p) or empty; with the behaviour of p given by,
Stochasticity occurs from two basic sources; demographic stochasticity due to the random nature of events and the individuality of populations, and environmental stochasticity due to the irregular or noisy dynamics of some process outside the biological system (such as the weather). Much of my research concentrates exclusively on the well defined notion of demographic stochasticity, although in reality both processes are usually present. One of the most interesting facets of moving from a deterministic to a stochastic framework is the non-uniqueness of this transition; there are many different biologically-meaningful stochastic processes that have the same underlying deterministic set of equations. Moment closure techniques offer a good means of understanding the role and magnitude of stochasticity.
I am interested in the role the stochasticity has to play in population dynamics, in particular understanding why measles and whooping cough react so differently to demographic stochasticity. The use of master-equations to determine the entire landscape of population distributions may prove to be a powerful tool in understand stochasticity, although computational limitations are still restrictive.
A mathematically appealing issue is how demographic stochasticity is able to generate apparent non-linear power-laws in simple population models (Taylor's power-law). I believe this is caused by the interaction of two time-scales - fast dynamics when the population is large and near its carrying capacity, and a slow time scale when the population is small and close to extinction.
People Pej Rohani, Eric Renshaw, Rich Durrett, Per Bak,
Simple spatial models, such as cellular automaton offer an interesting opportunity to study in some detail the effects of space without the many complications that are associated with studying realistic models. As such not only do this sort of model provide a useful tool for understanding specific phenomena, they also provide the ideal models with which to test many of the approximation techniques listed above. In many cases, understanding the behaviour of these toy models is a very active area of research in itself. Work in progress will address whether non-linear power-law interactions in mean-field models are capable of approximating the dynamics of spatial systems.
Other work, in collaboration with Steve Brookes and Chris Gilligan, is using a combination of MCMC techniques and moment calculations to estimate parameters for spatial disease systems. Initially this methodology is being tested using simple spatial models of epidemics, but we hope to construct a general framework that can be applied to any spatial disease data-set.
People Rich Durrett, Per Bak, Simon Levin, Claudia Neuhauser, Mercedes Pascual
Sexually transmitted diseases are on the increase, both world-wide and in the UK; the AIDS epidemic of the past two decades is a prime example of how devastating such diseases can be. Whereas most air-borne diseases have a large number of ill-defined potential contacts to which the disease can be spread, for STDs the contacts are more defined and fewer. This means that correlations rapidly develop in the network of contacts, and techniques such as pair-approximations become very useful. Current work (with my PhD student Ken Eames) is attempting to make these pair-wise models more applicable and to use the network data available from different studies.
People Ken Eames, Geoff Garent, Alden Klovdahl, Mark Williams, Azra Ghani, Chris Bauch.
Foot and Mouth
The 2001 Foot-and-Mouth epidemic demonstrated how even in a (supposedly) well resourced society a disease can rapidly reach epidemic proportions, and how difficult it can be to subsequently control. The foot-and-mouth epidemic was unique for a number of reasons, such as the fact that epidemiologists were involved from an early stage and helped to shape the control policy, but equally important is the vast amount of spatio-temporal data that was generated. Despite some obvious biases, we have data on the size, species composition and location of even livestock farm in Britain, and a day by day record of the reporting of infection, serological confirmation and subsequent slaughter. This provides a superb data-set in which individual heterogeneities are known and can be explicitly modelled. Even though the epidemic is now over, research still proceeds in an attempt to understand this outbreak and to be better prepared for any future incursion.
People Mark Woolhouse, Darren Shaw, Louise Mathews, Neil Ferguson, Roland Kao, DEFRA
Plague (and other zoonoses)
Zoonoses are defined as any disease, which is primarily of animals, but that can be transmitted to humans. Obviously, Bubonic Plague is a prime example, but other such diseases include Hanta virus, Ebola, West-Nile fever and Lyme's disease. With zoonoses it is important to consider the disease dynamics within the primary animal host, as well as within the human population. Work with Chris Gilligan on Bubonic Plague has highlighted some interesting historical and contempory issues. The occasional epidemics of bubonic plague may not be due to irregular imports of infection, but can be generated by the disease entering a low prevelence endemic state in the rat population. Secondly, control by eradication of rats may be problematic, as this can often release many infected fleas into the environment.
Future work will examine the incidence of Bubonic Plague in India, for which there is good spatio-temporal data from 1890-1950. I also want to develop an intuitive understanding of zoonoses in general, using simple equations for fluctuating animal populations.
People Chris Gilligan, Mike Begon,
Evolution is one of the few underlying principles in biology. It is most rapid when generations are short and conditions for weakness are most harst - these are the precise conditions suffered by infectious diseases. It seems clear that most diseases have evolved to expoilt their host population, changing infectious period and transmissibility to optimise their fitness. I am therefore interested in understanding the evolutionary dynamics of diseases, and how these are influenced by the underlying population structure and interactions. Simple models suggest that transmission rates should increase indefinitely, but stochastic spatial models predict that there is an ESS (evolutionary stable stratergy) at intermediate levels of transmission and infectious period. This work has the potential to be linked to simple within-host models of disease dynamics to determine a relationship between transmission, infectious period and virulence.
People Jon Read.Mike Boots
It is becoming increasingly appreciated that to understand and predict the extinction-risk threatening a species, requires models that can incorporate both spatial heterogeneities and stochastic dynamics. The heterogeneities usually arise do to how easily an area can be reached and its population exploited. Conservation methods that utilise spatial structure (such as the development of marine-reserves or fishing exclusion-zones off the Australian coast) have been highly successful. Previous work has considered two species of wild-pig on the island of Sulawesi; here, heterogeneities arise due to the presence of the only market in the extreme North-East of the region and the poor road access to many regions. The spatial model, which incorporated simple economics, predicted that one species - the Babirusa - was likely to be driven to extinction throughout most of its current range.
We are currently seeking funding for a project studying the Bornean Bearded Pigs; this species undergoes dramatic population fluctuations driven by mast-fruiting of the diptriocarp forests, and performs large scale migrations in search of food. The interaction between pigs, forests and El Nino is of prime concern.
People EJ Milner-Gulland