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Mathematical Tools for Understanding Infectious Disease Dynamics

Book Description

Mathematical modeling is critical to our understanding of how infectious diseases spread at the individual and population levels. This book gives readers the necessary skills to correctly formulate and analyze mathematical models in infectious disease epidemiology, and is the first treatment of the subject to integrate deterministic and stochastic models and methods.

Mathematical Tools for Understanding Infectious Disease Dynamics fully explains how to translate biological assumptions into mathematics to construct useful and consistent models, and how to use the biological interpretation and mathematical reasoning to analyze these models. It shows how to relate models to data through statistical inference, and how to gain important insights into infectious disease dynamics by translating mathematical results back to biology. This comprehensive and accessible book also features numerous detailed exercises throughout; full elaborations to all exercises are provided.

  • Covers the latest research in mathematical modeling of infectious disease epidemiology
  • Integrates deterministic and stochastic approaches
  • Teaches skills in model construction, analysis, inference, and interpretation
  • Features numerous exercises and their detailed elaborations
  • Motivated by real-world applications throughout

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
    1. A brief outline of the book
  6. I The bare bones: Basic issues in the simplest context
    1. 1 The epidemic in a closed population
      1. 1.1 The questions (and the underlying assumptions)
      2. 1.2 Initial growth
      3. 1.3 The final size
      4. 1.4 The epidemic in a closed population: summary
    2. 2 Heterogeneity: The art of averaging
      1. 2.1 Differences in infectivity
      2. 2.2 Differences in infectivity and susceptibility
      3. 2.3 The pitfall of overlooking dependence
      4. 2.4 Heterogeneity: a preliminary conclusion
    3. 3 Stochastic modeling: The impact of chance
      1. 3.1 The prototype stochastic epidemic model
      2. 3.2 Two special cases
      3. 3.3 Initial phase of the stochastic epidemic
      4. 3.4 Approximation of the main part of the epidemic
      5. 3.5 Approximation of the final size
      6. 3.6 The duration of the epidemic
      7. 3.7 Stochastic modeling: summary
    4. 4 Dynamics at the demographic time scale
      1. 4.1 Repeated outbreaks versus persistence
      2. 4.2 Fluctuations around the endemic steady state
      3. 4.3 Vaccination
      4. 4.4 Regulation of host populations
      5. 4.5 Tools for evolutionary contemplation
      6. 4.6 Markov chains: models of infection in the ICU
      7. 4.7 Time to extinction and critical community size
      8. 4.8 Beyond a single outbreak: summary
    5. 5 Inference, or how to deduce conclusions from data
      1. 5.1 Introduction
      2. 5.2 Maximum likelihood estimation
      3. 5.3 An example of estimation: the ICU model
      4. 5.4 The prototype stochastic epidemic model
      5. 5.5 ML-estimation of α and β in the ICU model
      6. 5.6 The challenge of reality: summary
  7. II Structured populations
    1. 6 The concept of state
      1. 6.1 i-states
      2. 6.2 p-states
      3. 6.3 Recapitulation, problem formulation and outlook
    2. 7 The basic reproduction number
      1. 7.1 The definition of R0
      2. 7.2 NGM for compartmental systems
      3. 7.3 General h-state
      4. 7.4 Conditions that simplify the computation of R0
      5. 7.5 Sub-models for the kernel
      6. 7.6 Sensitivity analysis of R0
      7. 7.7 Extended example: two diseases
      8. 7.8 Pair formation models
      9. 7.9 Invasion under periodic environmental conditions
      10. 7.10 Targeted control
      11. 7.11 Summary
    3. 8 Other indicators of severity
      1. 8.1 The probability of a major outbreak
      2. 8.2 The intrinsic growth rate
      3. 8.3 A brief look at final size and endemic level
      4. 8.4 Simplifications under separable mixing
    4. 9 Age structure
      1. 9.1 Demography
      2. 9.2 Contacts
      3. 9.3 The next-generation operator
      4. 9.4 Interval decomposition
      5. 9.5 The endemic steady state
      6. 9.6 Vaccination
    5. 10 Spatial spread
      1. 10.1 Posing the problem
      2. 10.2 Warming up: the linear diffusion equation
      3. 10.3 Verbal reflections suggesting robustness
      4. 10.4 Linear structured population models
      5. 10.5 The nonlinear situation
      6. 10.6 Summary: the speed of propagation
      7. 10.7 Addendum on local finiteness
    6. 11 Macroparasites
      1. 11.1 Introduction
      2. 11.2 Counting parasite load
      3. 11.3 The calculation of R0 for life cycles
      4. 11.4 A ‘pathological’ model
    7. 12 What is contact?
      1. 12.1 Introduction
      2. 12.2 Contact duration
      3. 12.3 Consistency conditions
      4. 12.4 Effects of subdivision
      5. 12.5 Stochastic final size and multi-level mixing
      6. 12.6 Network models (an idiosyncratic view)
      7. 12.7 A primer on pair approximation
  8. III Case studies on inference
    1. 13 Estimators of R0 derived from mechanistic models
      1. 13.1 Introduction
      2. 13.2 Final size and age-structured data
      3. 13.3 Estimating R0 from a transmission experiment
      4. 13.4 Estimators based on the intrinsic growth rate
    2. 14 Data-driven modeling of hospital infections
      1. 14.1 Introduction
      2. 14.2 The longitudinal surveillance data
      3. 14.3 The Markov chain bookkeeping framework
      4. 14.4 The forward process
      5. 14.5 The backward process
      6. 14.6 Looking both ways
    3. 15 A brief guide to computer intensive statistics
      1. 15.1 Inference using simple epidemic models
      2. 15.2 Inference using ‘complicated’ epidemic models
      3. 15.3 Bayesian statistics
      4. 15.4 Markov chain Monte Carlo methodology
      5. 15.5 Large simulation studies
  9. IV Elaborations
    1. 16 Elaborations for Part I
      1. 16.1 Elaborations for Chapter 1
      2. 16.2 Elaborations for Chapter 2
      3. 16.3 Elaborations for Chapter 3
      4. 16.4 Elaborations for Chapter 4
      5. 16.5 Elaborations for Chapter 5
    2. 17 Elaborations for Part II
      1. 17.1 Elaborations for Chapter 7
      2. 17.2 Elaborations for Chapter 8
      3. 17.3 Elaborations for Chapter 9
      4. 17.4 Elaborations for Chapter 10
      5. 17.5 Elaborations for Chapter 11
      6. 17.6 Elaborations for Chapter 12
    3. 18 Elaborations for Part III
      1. 18.1 Elaborations for Chapter 13
      2. 18.2 Elaborations for Chapter 15
  10. Bibliography
  11. Index