In epidemiology, we want to count individuals in different disease states, and yet the non-linear pure-jump processes that we might want to use for modelling and inference are often highly computationally demanding, or even intractable. I will consider a set of approximating stochastic processes to these full models.

The emergence of networks as a modelling paradigm has allowed to capture and model contact with an unprecedented level of detail, leading to a major shift in how epidemics are investigated. However, the propensity of contacts to cluster (i.e., neighbours of a node being also likely to be connected) continues to pose major modelling and analytical challenges. Apart from some theoretical results on idealised networks there is a limited number of general theoretical results. Mean-field models, such as pairwise models, have been used successfully in approximating the average behaviour of epidemics on networks with no clustering and have led to significant results such as an analytical epidemic threshold and final epidemic size for susceptible-infected-recovered (SIR) epidemics. While equivalents of the pairwise model for clustered networks are available, there are limited analytic or semi-analytic results. In this talk I show that it is possible to use the correlation structure at early times to work out a semi-analytic epidemic threshold using pairwise models for clustered networks. Furthermore, I present a systematic comparison between two pairwise models for clustered networks and the pairwise model for networks with no clustering. This comparison shows that for SIR epidemics, these models only differ over a limited range of the average degree and transmission rates. Although clustering in densely connected networks has little effect on epidemics, especially for the simplest epidemic dynamics, its effect may be more profound for dynamics such as complex contagion where the higher-order structure of the network, i.e., how clustering is achieved, has a more marked effect on the dynamics unfolding on it. This may indicate that choosing models for clustered networks may strongly depend on the dynamics considered.

This is joint work with Rosanna C. Barnard, Luc Berthouze, Nicos Georgiou, and Simon L. Péter.

The classical many-sources asymptotic regime of the single server queue is widely known and exhaustively studied. However, it is limited in that it does not enable the study of different traffic regimes, and how they interact with each other.

In this talk, we present a new paramerization of the many-sources asymptotic regime, indexed by $\alpha$ and $\beta$. The parameter $\alpha$ is used to control the threshold size scaling, while the parameter $\beta$ controls the excess service rate above the arrival rate. We also present new large deviations bounds for threshold overflow probabilities for different values of $\alpha$ and $\beta$.

The aim of this talk is to show an alternative to the analysis of the chemical master equation (CME) when studying biological systems from a stochastic point of view. In particular, we propose to analyse probabilistic descriptors that allow us to obtain information of the process under study, sidestepping the analysis of the CME. We illustrate this approach in a stochastic model for the interaction between the vascular endothelial growth factor receptor 2 (VEGFR-2), with the vascular endothelial growth factor VEGF-A in human vascular endothelial cells. Receptor VEGFR-2 and ligand VEGF-A interact through their binding, forming signalling bound complexes on the cell surface. The interest here is in the analysis of the time scales to reach any signal threshold. We quantitatively measure this time as a random variable that can be identified as a first-passage time in the theory of stochastic processes, and the computation of its different order moments can be reduced to the solution of a series of systems of linear equations, that can be efficiently solved through a matrix formalism. Our methodology is illustrated by carrying out a series of numerical results, where we focus on the effect that different hypothesis about signal formation have on this time.

I will present a flexible framework for deriving and quantifying the accuracy of Gaussian process approximations to non-linear stochastic individual-based models of infectious diseases. This will be demonstrated with a simple infectious disease model, although it is straight-froward to extend to more complex models. I will then show how it can be used with real outbreak data on the number of new cases of norovirus on a cruise ship to quickly infer the underlying parameters of the epidemiological model and the unobserved processes.