No abstract

A key issue in the dynamical modelling of epidemics is the synthesis of complex mathematical models and data by means of time series analysis. We report such an approach, focusing on the particularly well-documented case of measles. We propose the use of a discrete time epidemic model comprising the infected and susceptible class as state variables. The model uses a discrete time version of the susceptible±exposed±infected±recovered type epidemic models, which can be ®tted to observed disease incidence time series. We describe a method for reconstructing the dynamics of the susceptible class, which is an unobserved state variable of the dynamical system. The model provides a remarkable ®t to the data on case reports of measles in England and Wales from 1944 to 1964. Morever, its systematic part explains the well-documented predominant biennial cyclic pattern. We study the dynamic behaviour of the time series model and show that episodes of annual cyclicity, which have not previously been explained quantitatively, arise as a response to a quicker replenishment of the susceptible class during the baby boom, around 1947.

Before the development of mass-vaccination campaigns, measles exhibited persistent fluctuations (endemic dynamics) in large British cities, and recurrent outbreaks (episodic dynamics) in smaller communities. The critical community size separating the two regimes was ϳ300 000-500 000. We develop a model, the TSIR (Time-series Susceptible-Infected-Recovered) model, that can capture both endemic cycles and episodic outbreaks in measles. The model includes the stochasticity inherent in the disease transmission (giving rise to a negative binomial conditional distribution) and random immigration. It is thus a doubly stochastic model for disease dynamics. It further includes seasonality in the transmission rates. All parameters of the model are estimated on the basis of time series data on reported cases and reconstructed susceptible numbers from a set of cities in England and Wales in the prevaccination era . The 60 cities analyzed span a size range from London (3.3 ϫ 10 6 inhabitants) to Teignmouth (10 500 inhabitants). The dynamics of all cities fit the model well. Transmission rates scale with community size, as expected from dynamics adhering closely to frequency dependent transmission (''true mass action''). These rates are further found to reveal strong seasonal variation, corresponding to high transmission during school terms and lower transmission during the school holidays. The basic reproductive ratio, R 0 , is found to be invariant across the observed range of host community size, and the mean proportion of susceptible individuals also appears to be constant. Through the epidemic cycle, the susceptible population is kept within a 3% interval. The disease is, thus, efficient in ''regulating'' the susceptible population-even in small cities that undergo recurrent epidemics with frequent extinction of the disease agent. Recolonization is highly sensitive to the random immigration process. The initial phase of the epidemic is also stochastic (due to demographic stochasticity and random immigration). However, the epidemic is nearly ''deterministic'' through most of the growth and decline phase.

A stochastic discrete-time susceptible-exposed-infectious-recovered (SEIR) model for infectious diseases is developed with the aim of estimating parameters from daily incidence and mortality time series for an outbreak of Ebola in the Democratic Republic of Congo in 1995. The incidence time series exhibit many low integers as well as zero counts requiring an intrinsically stochastic modeling approach. In order to capture the stochastic nature of the transitions between the compartmental populations in such a model we specify appropriate conditional binomial distributions. In addition, a relatively simple temporally varying transmission rate function is introduced that allows for the effect of control interventions. We develop Markov chain Monte Carlo methods for inference that are used to explore the posterior distribution of the parameters. The algorithm is further extended to integrate numerically over state variables of the model, which are unobserved. This provides a realistic stochastic model that can be used by epidemiologists to study the dynamics of the disease and the effect of control interventions.

Transcriptional reprogramming forms a major part of a plant's response to pathogen infection. Many individual components and pathways operating during plant defense have been identified, but our knowledge of how these different components interact is still rudimentary. We generated a high-resolution time series of gene expression profiles from a single Arabidopsis thaliana leaf during infection by the necrotrophic fungal pathogen Botrytis cinerea. Approximately one-third of the Arabidopsis genome is differentially expressed during the first 48 h after infection, with the majority of changes in gene expression occurring before significant lesion development. We used computational tools to obtain a detailed chronology of the defense response against B. cinerea, highlighting the times at which signaling and metabolic processes change, and identify transcription factor families operating at different times after infection. Motif enrichment and network inference predicted regulatory interactions, and testing of one such prediction identified a role for TGA3 in defense against necrotrophic pathogens. These data provide an unprecedented level of detail about transcriptional changes during a defense response and are suited to systems biology analyses to generate predictive models of the gene regulatory networks mediating the Arabidopsis response to B. cinerea.