A branching model for the spread of infectious animal diseases in varying environments

Estimation in branching processes with restricted observations

We consider an epidemic model where the spread of the epidemic can be described by a discrete-time Galton-Watson branching process. Between times n and n + 1, any infected individual is detected with unknown probability π and the numbers of these detected individuals are the only observations we have. Detected individuals produce a reduced number of offspring in the time interval of detection, and no offspring at all thereafter. If only the generation sizes of a Galton-Watson process are observed, it is known that one can only estimate the first two moments of the offspring distribution consistently on the explosion set of the process (and, apart from some lattice parameters, no parameters that are not determined by those moments). Somewhat surprisingly, in our context, where we observe a binomially distributed subset of each generation, we are able to estimate three functions of the parameters consistently. In concrete situations, this often enables us to estimate π consistently, as well as the mean number of offspring. We apply the estimators to data for a real epidemic of classical swine fever.

Use of epidemiologic models in the control of highly pathogenic avian influenza

In the past decades, mathematical models have become more and more accepted as a tool to develop surveillance programs and to evaluate the efficacy of intervention measures for the control of infectious diseases such as highly pathogenic avian influenza. Predictive models are used to simulate the effect of various control measures on the course of an epidemic; analytical models are used to analyze data from outbreaks or from experiments. A key parameter in both types of models is the reproductive ratio, which indicates whether virus can be transmitted in the population, resulting in an epidemic, or not. Parameters obtained from real data using the analytical models can subsequently be used in predictive models to evaluate control strategies or surveillance programs. Examples of the use of these models are described here.

A useful relationship between epidemiology and queueing theory: the distribution of the number of infectives at the moment of the first detection

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 relation between the spread of epidemics and branching processes, which is well known in epidemiology, and the relation between M/G/1 queues and birth death processes, which is well known in queueing theory, 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.

Reproduction numbers for epidemics on networks using pair approximation

One way to describe the spread of an infection on a network is by using the method of pair approximation. This method is a deterministic pair-based variant of the usual methods used to describe the progress of an epidemic in randomly mixing populations. However, although the ideas of pair approximation are intuitively clear, it is not straightforward to make all assumptions used explicit. Furthermore, in literature problems arise in defining basic quantities like the basic reproduction number R(0) and the real-time epidemic growth rate parameter r. We formulate the pair approximations and the needed assumptions explicitly. We discuss problems inherent to this method. Furthermore, we define a new reproduction number, similar to R(0) and a new real-time growth rate parameter similar to r. We illustrate the methods of the paper by an example for which we can compare the approximation of the reproduction number with exact results.

Characterizing Transmission and Control of the SARS Epidemic: Novel Stochastic Spatio-Temporal Models

Severe Acute Respiratory Syndrome (SARS), the first epidemic of the 21st century, has an outbreak history of more than 2 years till today and caused tremendous damage to the human society. Accordingly, many studies on modeling the SARS epidemic have been reported, whereas deficiencies were still lying in those models because of their separate space/time methodology. In this paper, we propose novel comprehensive stochastic spatio-temporal models from both of the macro aspect and individual aspect for characterizing transmission and control of the SARS disease. Based on a new SARS spread process in consideration of "suspicious" population, we firstly establish the stochastic temporal models from two different aspects: the macro model is described with birth-death process and the individual Markov model is described with probability transition matrix (PTM). And then, we amalgamate the deterministic/stochastic population-flow model with the stochastic temporal models together to set up the comprehensive stochastic spatio-temporal models. Simulations on computer have evaluated the effect of various realistic parameters and control policies, and also have testified the accuracy and efficacy of the new models. Additionally, particular studies on the cases of Tsinghua University and Beijing City are presented. The comprehensive stochastic spatio-temporal models have considerably reduced the complexity plus errors as compared with previous works and will be able to characterize other various epidemics, e.g. Avian Flu.

Seasonal infectious disease epidemiology

Seasonal change in the incidence of infectious diseases is a common phenomenon in both temperate and tropical climates. However, the mechanisms responsible for seasonal disease incidence, and the epidemiological consequences of seasonality, are poorly understood with rare exception. Standard epidemiological theory and concepts such as the basic reproductive number R0 no longer apply, and the implications for interventions that themselves may be periodic, such as pulse vaccination, have not been formally examined. This paper examines the causes and consequences of seasonality, and in so doing derives several new results concerning vaccination strategy and the interpretation of disease outbreak data. It begins with a brief review of published scientific studies in support of different causes of seasonality in infectious diseases of humans, identifying four principal mechanisms and their association with different routes of transmission. It then describes the consequences of seasonality for R0, disease outbreaks, endemic dynamics and persistence. Finally, a mathematical analysis of routine and pulse vaccination programmes for seasonal infections is presented. The synthesis of seasonal infectious disease epidemiology attempted by this paper highlights the need for further empirical and theoretical work.

Societal learning in epidemics: intervention effectiveness during the 2003 SARS outbreak in Singapore

BACKGROUND

Rapid response to outbreaks of emerging infectious diseases is impeded by uncertain diagnoses and delayed communication. Understanding the effect of inefficient response is a potentially important contribution of epidemic theory. To develop this understanding we studied societal learning during emerging outbreaks wherein patient removal accelerates as information is gathered and disseminated.

METHODS AND FINDINGS

We developed an extension of a standard outbreak model, the simple stochastic epidemic, which accounts for societal learning. We obtained expressions for the expected outbreak size and the distribution of epidemic duration. We found that rapid learning noticeably affects the final outbreak size even when learning exhibits diminishing returns (relaxation). As an example, we estimated the learning rate for the 2003 outbreak of severe acute respiratory syndrome (SARS) in Singapore. Evidence for relaxation during the first eight weeks of the outbreak was inconclusive. We estimated that if societal learning had occurred at half the actual rate, the expected final size of the outbreak would have reached nearly 800 cases, more than three times the observed number of infections. By contrast, the expected outbreak size for societal learning twice as effective was 116 cases.

CONCLUSION

These results show that the rate of societal learning can greatly affect the final size of disease outbreaks, justifying investment in early warning systems and attentiveness to disease outbreak by both government authorities and the public. We submit that the burden of emerging infections, including the risk of a global pandemic, could be efficiently reduced by improving procedures for rapid detection of outbreaks, alerting public health officials, and aggressively educating the public at the start of an outbreak.