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2019 | Buch

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Quantitative Methods for Investigating Infectious Disease Outbreaks

verfasst von: Dr. Ping Yan, Gerardo Chowell

Verlag: Springer International Publishing

Buchreihe : Texts in Applied Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book provides a systematic treatment of the mathematical underpinnings of work in the theory of outbreak dynamics and their control, covering balanced perspectives between theory and practice including new material on contemporary topics in the field of infectious disease modelling. Specifically, it presents a unified mathematical framework linked to the distribution theory of non-negative random variables; the many examples used in the text, are introduced and discussed in light of theoretical perspectives.

The book is organized into 9 chapters: The first motivates the presentation of the material on subsequent chapters; Chapter 2-3 provides a review of basic concepts of probability and statistical models for the distributions of continuous lifetime data and the distributions of random counts and counting processes, which are linked to phenomenological models. Chapters 4 focuses on dynamic behaviors of a disease outbreak during the initial phase while Chapters 5-6 broadly cover compartment models to investigate the consequences of epidemics as the outbreak moves beyond the initial phase. Chapter 7 provides a transition between mostly theoretical topics in earlier chapters and Chapters 8 and 9 where the focus is on the data generating processes and statistical issues of fitting models to data as well as specific mathematical epidemic modeling applications, respectively.

This book is aimed at a wide audience ranging from graduate students to established scientists from quantitatively-oriented fields of epidemiology, mathematics and statistics. The numerous examples and illustrations make understanding of the mathematics of disease transmission and control accessible. Furthermore, the examples and exercises, make the book suitable for motivated students in applied mathematics, either through a lecture course, or through self-study. This text could be used in graduate schools or special summer schools covering research problems in mathematical biology.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Infectious diseases ranging from respiratory (influenza, common cold, tuberculosis, the respiratory syncytial virus), vector-borne (plague, malaria, dengue, chikungunya, and Zika) to sexually transmitted (the human immunodeficiency virus, syphilis) have historically affected the human population in profound ways. For example, the Great Plague, well known as the Black Death, was caused by the bacterium Yersinia pestis and killed up to 200 million people in Eurasia and about 30–60% of Europe’s population during a 5-year span in the fourteenth century. At the time, the plague infection was thought to be due to some “bad air”, but it was not discovered that bites of infected fleas were behind the pandemic until late 1890s. If the human civilization had known about the transmission mechanisms behind the plague infections, the epidemic’s impact on morbidity and mortality could have been mitigated through basic public health interventions. This is to say that knowledge of the transmission processes and the natural history of infectious diseases in different environments represents invaluable actionable information for thwarting the spread of infectious diseases.

Ping Yan, Gerardo Chowell

Chapter 2. Shapes of Hazard Functions and Lifetime Distributions

Abstract

The main focus of this book is to address phenomenological questions regarding the spread of infectious diseases at the population level. Examples of such questions include:

If one or a few infected individuals are “seeded” in a large and completely susceptible population, will it only lead to a handful of infected individuals and the (small) outbreak burns out; or will it lead to an “explosive” (large) outbreak that results in a significant proportion of the population infected?

(a)

If the outcome is the former, what is the expected total number of infected individuals and what is the expected time to extinction?

(b)

If the outcome is the latter, how fast will it grow?

In a large outbreak, can we predict the peak burden of the disease and the timing of the peak? How about the long-term outcomes? Will it simply go away after a single wave or a few repeated waves, or will it settle down at some equilibrium state and the epidemic becomes endemic?

What about the effects of control measures, such as public health interventions including quarantine, isolation, or pharmaceutical treatments and vaccination?

Ping Yan, Gerardo Chowell

Chapter 3. Random Counts and Counting Processes

Abstract

We now turn our attention to the population level dynamics and ask phenomenological questions. First, many important measures in the study of infectious diseases are count variables N, taking integer values n = 0, 1, 2, ….

Ping Yan, Gerardo Chowell

Chapter 4. Behaviors of a Disease Outbreak During the Initial Phase and the Branching Process Approximation

Abstract

We consider that at the beginning, t = 0, there is no disease. We call the system at this condition the disease-free equilibrium. We assume that the entire population is susceptible. The size of the susceptible population is denoted by m.

Ping Yan, Gerardo Chowell

Chapter 5. Beyond the Initial Phase: Compartment Models for Disease Transmission

Abstract

We start with simple models that describe the dynamics of disease transmission over time t in a constant population of size m and investigate the long-term epidemic dynamics as t →∞. In these simple models, we assume there is no replacement of susceptible individuals due to demographic input of susceptible newborns. The population is partitioned into compartments, with at least one compartment representing the prevalence of individuals who are susceptible to infection and at least one compartment representing the prevalence of individuals who are infectious (at time t).

Ping Yan, Gerardo Chowell

Chapter 6. More Complex Models and Control Measures

Abstract

We have seen that, under suitable assumptions such as homogeneous mixing, the basic reproduction number R ₀, defined at the start of the epidemic and given by (4.2) in Chap. 4, transcends to the asymptotic equilibrium (t →∞) outcomes such as the final size (5.32) in a closed population or the endemic equilibrium \(x(\infty )=\lim _{t\rightarrow \infty }S_{d}(t)/m\rightarrow R_{0}^{-1}\) in a constant population. Meanwhile, we have also seen that, in compartment transmission models of the SEIRS type (in Chap. 5) with exponentially distributed durations, R ₀ is expressed as a function of parameters representing rates in these models, such as R ₀ = β∕γ in SEIRS models without mortality or other in-flow and out-flow of the population, or (5.70) in SEIRS models with mortality or other in-flow and out-flow of the population.

Ping Yan, Gerardo Chowell

Chapter 7. Some Statistical Issues

Abstract

All the models presented in the previous chapters are parametric. They belong to different types and serve different purposes.

Ping Yan, Gerardo Chowell

Chapter 8. Characterizing Outbreak Trajectories and the Effective Reproduction Number

Abstract

Emerging and re-emerging infectious diseases pose major challenges to public health worldwide. Fortunately mathematical and statistical inference and simulation approaches are part of the toolkit for guiding prevention and response plans. As the recent 2013–2016 Ebola epidemic exemplified, an unfolding infectious disease outbreak often forces public health officials to put in place control policies in the context of limited data about the outbreak and in a changing environment where multiple factors positively or negatively impact local disease transmission. Hence, the development of public health policies could benefit from mathematically rigorous and computationally efficient approaches that comprehensively assimilate data and model uncertainty in real time in order to (1) estimate transmission rates, (2) assess the impact of control interventions (vaccination campaigns, behavior changes), (3) test hypotheses relating to transmission mechanisms, (4) evaluate how behavior changes affect transmission dynamics, (5) optimize the impact of control strategies, and (6) generate forecasts to guide interventions in the short and long terms.

Ping Yan, Gerardo Chowell

Chapter 9. Mechanistic Models with Spatial Structures and Reactive Behavior Change

Abstract

As we have emphasized in Chaps. 4 and 5, simple homogeneous models of transmission or growth dynamics often yield an early exponential epidemic growth phase even when the population is stratified into different groups (e.g., age, gender, regions). However, recent work has highlighted the presence of early sub-exponential growth patterns in case incidence from empirical outbreak data. This suggests that integrating detailed and often unobserved heterogeneity into simple mechanistic models could open the door to a new and exciting research area to better understand the role of heterogeneity on key transmission parameters, epidemic size, stochastic extinction, the effects of interventions, and disease forecasts.

Ping Yan, Gerardo Chowell

Backmatter

Titel: Quantitative Methods for Investigating Infectious Disease Outbreaks
verfasst von: Dr. Ping Yan
Gerardo Chowell
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-21923-9
Print ISBN: 978-3-030-21922-2
DOI: https://doi.org/10.1007/978-3-030-21923-9

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