For principled model fitting in mathematical biology

The mathematical models used to capture features of complex, biological systems are typically non-linear, meaning that there are no generally valid simple relationships between their outputs and the data that might be used to validate them. This invalidates the assumptions behind standard statistical methods such as linear regression, and often the methods used to parameterise biological models from data are ad hoc. In this perspective, I will argue for an approach to model fitting in mathematical biology that incorporates modern statistical methodology without losing the insights gained through non-linear dynamic models, and will call such an approach principled model fitting. Principled model fitting therefore involves defining likelihoods of observing real data on the basis of models that capture key biological mechanisms.Comment: 7 pages, 3 figures. To appear in Journal of Mathematical Biology. The final publication is available at Springer via http://dx.doi.org/10.1007/s00285-014-0787-

Non-Markovian stochastic epidemics in extremely heterogeneous populations

A feature often observed in epidemiological networks is significant heterogeneity in degree. A popular modelling approach to this has been to consider large populations with highly heterogeneous discrete contact rates. This paper defines an individual-level non-Markovian stochastic process that converges on standard ODE models of such populations in the appropriate asymptotic limit. A generalised Sellke construction is derived for this model, and this is then used to consider final outcomes in the case where heterogeneity follows a truncated Zipf distribution.Comment: 10 pages, 1 figur

Algebraic moment closure for population dynamics on discrete structures

Moment closure on general discrete structures often requires one of the following: (i) an absence of short closed loops (zero clustering); (ii) existence of a spatial scale; (iii) ad hoc assumptions. Algebraic methods are presented to avoid the use of such assumptions for populations based on clumps, and are applied to both SIR and macroparasite disease dynamics. One approach involves a series of approximations that can be derived systematically, and another is exact and based on Lie algebraic methods.Comment: 12 pages, 4 figure

Lie algebra solution of population models based on time-inhomogeneous Markov chains

Many natural populations are well modelled through time-inhomogeneous stochastic processes. Such processes have been analysed in the physical sciences using a method based on Lie algebras, but this methodology is not widely used for models with ecological, medical and social applications. This paper presents the Lie algebraic method, and applies it to three biologically well motivated examples. The result of this is a solution form that is often highly computationally advantageous.Comment: 10 pages; 1 figure; 2 tables. To appear in Applied Probabilit

Effective action of (massive) IIA on manifolds with SU(3) structure

In this paper we consider compactifications of massive type IIA supergravity on manifolds with SU(3) structure. We derive the gravitino mass matrix of the effective four-dimensional N = 2 theory and show that vacuum expectation values of the scalar fields naturally induce spontaneous partial supersymmetry breaking. We go on to derive the superpotential and the Kaehler potential for the resulting N = 1 theories. As an example we consider the SU(3) structure manifold SU(3)/U(1)xU(1) and explicitly find N = 1 supersymmetric minima where all the moduli are stabilised at non-trivial values without the use of non-perturbative effects.Comment: 25 pages, 2 figures. References added and typos corrected to match published versio

Epidemic prediction and control in clustered populations

There has been much recent interest in modelling epidemics on networks, particularly in the presence of substantial clustering. Here, we develop pairwise methods to answer questions that are often addressed using epidemic models, in particular: on the basis of potential observations early in an outbreak, what can be predicted about the epidemic outcomes and the levels of intervention necessary to control the epidemic? We find that while some results are independent of the level of clustering (early growth predicts the level of ‘leaky’ vaccine needed for control and peak time, while the basic reproductive ratio predicts the random vaccination threshold) the relationship between other quantities is very sensitive to clustering

Household structure and infectious disease transmission

One of the central tenets of modern infectious disease epidemiology is that an understanding of heterogeneities, both in host demography and transmission, allows control to be efficiently optimized. Due to the strong interactions present, households are one of the most important heterogeneities to consider, both in terms of predicting epidemic severity and as a target for intervention. We consider these effects in the context of pandemic influenza in Great Britain, and find that there is significant local (ward-level) variation in the basic reproductive ratio, with some regions predicted to suffer 50% faster growth rate of infection than the mean. Childhood vaccination was shown to be highly effective at controlling an epidemic, generally outperforming random vaccination and substantially reducing the variation between regions; only nine out of over 10 000 wards did not obey this rule and these can be identified as demographically atypical regions. Since these benefits of childhood vaccination are a product of correlations between household size and number of dependent children in the household, our results are qualitatively robust for a variety of disease scenarios