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

We apply fuzzy modeling to derive a mathematical model for a biological phenomenon: the regulation of population size in the Australian sheep-blowfly Lucilia cuprina. This behavior was described by several ethologists and fuzzy modeling allows us to transform their verbal descriptions into a well-defined mathematical model. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature. We believe that the fuzzy modeling approach demonstrated here may supply a suitable framework for biomimicry, that is, the design of artificial systems based on mimicking natural behavior.

Mathematical population models are constructed based on plausible explicit and implicit biological assumptions. While it is easy to incorporate explicit assumptions correctly in the models, those implicit ones are often ill treated or forgotten. Indeed, this happens to some well known models in the literature and examples of such will be mentioned and discussed. For a model to be logically credible, we must do our best to ensure that all assumptions are incorporated correctly and consistently. To this end, we exam a simple set of criteria proposed by Arditi and Michalski in 1996.

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

Inference for static parameters in state space models (i.e., unknown model parameters that do not vary in time) • Numerical issues have led to a considerable literature on this topic. • Iterated filtering maximizes the likelihood function via taking an average of filtered “local ” parameter estimates obtained by adding noise to the static parameters (which regularizes the numerical issues). This process is recursively repeated while reducing the added noise. • Iterated filtering, implemented by basic SMC, has a plug-and-play property: it requires simulation from the state process but not transition densities. Ed Ionides The theory and practice of iterated filtering 3 Plug-and-play methods for state space models • Statistical methods are plug-and-play if they require simulation from the dynamic model but not explicit likelihood ratios. • Bayesian plug-and-play: 1. Artificial parameter evolution (Liu and West, 2001)

What is a “mechanistic ” approach to time series analysis? • Write down equations, based on scientific understanding of a dynamic system, which describe how it evolves with time. • Further equations describe the relationship of the state of the system to available observations on it. • Mechanistic time series analysis concerns drawing inferences from the available data about the hypothesized equations. • Questions of general interest: Are the data consistent with a particular model? If so, for what range of values of model parameters? Does one mechanistic model describe the data better than another? • A defining principle: the model structure should be chosen based on scientific considerations, rather than statistical convenience. Ed Ionides Time series analysis via mechanistic models 3 Example: why quantify biological population dynamics? • Conservation. Mankind is increasingly responsible for managing ecosystems. This requires a quantitative understanding of population behavior. • Public health. Pathogens are also biological populations. Despite successes of vaccination and medical treatment, new diseases are emerging (SARS, HIV/AIDS) and old ones re-emerging due to drug resistant strains (malaria, tuberculosis). Treating the pathogen as part of an ecosystem is one approach to understanding and controlling emergent and re-emergent diseases. • Basic scientific interest. Ed Ionides Time series analysis via mechanistic models 4 Time series data of sufficient quantity and quality to justify mechanistic modeling are increasingly available: Two recent examples • King, Ionides, Pascual and Bouma. Inapparent infections and cholera dynamics. To appear in Nature.

Inference for partially observed Markov process models has been a longstanding methodological challenge with many scientific and engineering applications. Iterated filtering algorithms maximize the likelihood function for partially observed Markov process models by solving a recursive sequence of filtering problems. We present new theoretical results pertaining to the convergence of iterated filtering algorithms implemented via sequential Monte Carlo filters. This theory complements the growing body of empirical evidence that iterated filtering algorithms provide an effective inference strategy for scientific models of nonlinear dynamic systems. The first step in our theory involves studying a new recursive approach for maximizing the likelihood function of a latent variable model, when this likelihood is evaluated via importance sampling. This leads to the consideration of an iterated importance sampling algorithm which serves as a simple special case of iterated filtering, and may have applicability in its own right. 1. Introduction. Partially observed Markov process (POMP) models are

Inference for partially observed Markov process models has been a longstanding methodological challenge with many scientific and engineering applications. Iterated filtering algorithms maximize the likelihood function for partially observed Markov process models by solving a recursive sequence of filtering problems. We present new theoretical results pertaining to the convergence of iterated filtering algorithms implemented via sequential Monte Carlo filters. This theory complements the growing body of empirical evidence that iterated filtering algorithms provide an effective inference strategy for scientific models of nonlinear dynamic systems. The first step in our theory involves studying a new recursive approach for maximizing the likelihood function of a latent variable model, when this likelihood is evaluated via importance sampling. This leads to the consideration of an iterated importance sampling algorithm which serves as a simple special case of iterated filtering, and may have applicability in its own right. 1

modeling and inference for ecological systems

Time series analysis for nonlinear dynamical systems with applications to modeling of infectious diseases by