The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae. 1. Introduction. A

This paper is concerned with a stochastic model for the spread of an SIR (susceptible ! infective ! removed) epidemic among a closed, finite population that contains several types of individuals and is partitioned into households. Previously obtained probabilistic and inferential results for the model are used to estimate the threshold parameter R , which determines whether or not a major outbreak can occur, both before and after vaccination. It turns out that R cannot be estimated consistently from final outcome data, so a Perron-Frobenius argument is used to obtain sharp lower and upper bounds for R , which can be estimated consistently. Determining the allocation of vaccines that reduces the upper bound for R to its threshold England. E-mail: Owen.Lyne@maths.nottingham.ac.uk value of one with minimum vaccine coverage is shown to be a linear programming problem. The estimates of R , before and after vaccination, and of the secure vaccination coverage (i.e. the proportion of individuals that have to be vaccinated to reduce the upper bound for R to 1, assuming an optimal vaccination scheme), are equipped with standard errors, thus yielding conservative confidence bounds for these key epidemiological parameters. The methodology is illustrated by application to data on influenza outbreaks in Tecumseh, Michigan.

Abstract. In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p−coin tosses. Spatial variants of these models are considered, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for these models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift. The corresponding scaling limits are super-Brownian motions and Dawson-Watanabe processes with killing, respectively. 1.

Why do we seek to quantify and understand disease dynamics? • Prevention and control of emerging infectious diseases (SARS, HIV/AIDS, H5N1 influenza “bird flu”) • Understanding the development and spread of drug resistant strains (malaria, tuberculosis, MRSA “the hospital super-bug”) Ed Ionides Infectious disease dynamics: a statistical perspective 4 Disease dynamics: epidemiology or ecology, or both? • Environmental host/pathogen dynamics are close to predator/prey relationships which are a central topic of ecology. • Analysis of diseases as ecosystems complements more traditional epidemiology (risk factors etc). • Ecologists typically seek to avoid extinctions, whereas epidemiologists typically seek the reverse. Things are not always this simple... – Helicobacter pylori bacteria used to live in the stomach of most humans. Some strains cause stomach ulcers and cancer. It is almost extinct in the developed world due to widespread use of

Why do we seek to quantify and understand disease dynamics? • Prevention and control of emerging infectious diseases (SARS, HIV/AIDS, H5N1 “bird flu ” influenza, H1N1 “swine flu ” influenza) • Understanding the development and spread of drug resistant strains (malaria, tuberculosis, MRSA) Ed Ionides Infectious disease dynamics: a statistical perspective 4 Disease dynamics: epidemiology or ecology, or both? • Environmental host/pathogen dynamics are close to predator/prey relationships which are a central topic of ecology. • Analysis of diseases as ecosystems complements more traditional epidemiology (risk factors etc). • Ecologists typically seek to avoid extinctions, whereas epidemiologists typically seek the reverse. Ed Ionides Infectious disease dynamics: a statistical perspective 5 Infectious disease transmission: the statistical challenge • Time series data of sufficient quantity and quality to support investigations of disease dynamics are increasingly available.

Bootstrap percolation on the random graph Gn,p is a process of spread of “activation ” on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least r ≥ 2 active neighbours become active as well. We study the size A ∗ of the final active set. The parameters of the model are, besides r (fixed) and n (tending to ∞), the size a = a(n) of the initially active set and the probability p = p(n) of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model the final size of activation with a high probability is either n − o(n) or it is o(n). We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for A ∗ ; we also prove a central limit theorem for A ∗ in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops.

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To Piet Groeneboom, on the occasion of his 39th birthday. Abstract: In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p−coin tosses. Spatial variants of these models are proposed, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for both the mean-field and spatial models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a sizedependent drift. 1. Stochastic epidemic models 1.1. Mean-field models The simplest and most thoroughly studied stochastic models of epidemics are meanfield

In this paper we establish a relation between the spread of infectious diseases and the dynamics of so called M/G/1 queues with processor sharing. The in epidemiology well known relation between the spread of epidemics and branching processes and the in queueing theory well known relation between M/G/1 queues and birth death processes will be combined to provide a framework in which results from queueing theory can be used in epidemiology and vice versa. In particular, we consider the number of infectious individuals in a standard SIR epidemic model at the moment of the first detection of the epidemic, where infectious individuals are detected at a constant per capita rate. We use a result from the literature on queueing processes to show that this number of infectious individuals is geometrically distributed. 1