Population Biology and Criticality

The following sections are included:

• Preface

• Contents

The following sections are included:

• Basic Probability Theory: The Tool Box

• Stochastic Description of a Deterministic System: The Ulam Map

• A Fully Stochastic Dynamic System: The AR(1)-Process

• From Perron-Frobenius to Master Equation

• A First Example of a Master Equation: The Linear Infection Model
  • Solving the first example of a master equation
  • Solution of the linear infection model
  • Mean value and its dynamics
  • Mean dynamics

• The Birth and Death Process, a Non-Linear Stochastic System

• Solution of the Birth-Death ODE Shows Criticality
  • Numerical integration shows power law at criticality
  • Temporal correlation length diverges at criticality

The following sections are included:

• The Spatial Master Equation
  • A first inspection of the spatial birth-death process

• Clusters and their Dynamics
  • Time evolution of marginals and local expectations

• Moment Equations
  • Mean field behavior
  • Pair approximation

• The SIS Dynamics under Pair Approximation

• Conclusions and Further Reading

The following sections are included:

• The Glauber Model: Stochastic Dynamics for the Ising Model
  • A first glance at the dynamic Ising model

• The Ising Model, a Paradigm for Equilibrium Phase Transitions
  • Distribution of magnetization and Gibbs free energy

• Equilibrium Distribution around Criticality
  • Distribution of magnetization
  • External magnetic field
  • The maximum of the total magnetization distribution
    • Approximation with Lagrange polynomials
    • The maximum magnetization with changing parameters

• Mean Field Theory and its Exponents
  • Mean field self-consistency equation
  • Mean field quantities obtained from the self-consistency equation
    • Critical coupling strength in mean field approximation
    • The first critical exponent in mean field approximation
  • Universal scaling function and exponent δ in mean field
  • The magnetic susceptibility or second moment of the magnetization diverges
  • State equation in mean field

• Critical Exponents of the Ising Model beyond Mean Field

The following sections are included:

• A Model with Partial Immunization: SIRI

• Local Quantities

• Dynamics Equations for Global Pairs
  • The SIRI dynamics under pair approximation
  • Balance equations for means and pairs

• Mean Field Model: SIRI with Reintroduced Susceptibles
  • Pair dynamics for the SIRI model

• Fruitful Transfer between Equilibrium and Non-Equilibrium Systems

The following sections are included:

• Introduction

• Real Space Renormalization in One-Dimensional Lattice Gas

• Directed Percolation and Path Integrals
  • Master equation of the birth—death process
  • Schrödinger-like equation
  • δ-Bosons for hard-core particles
  • Path integral for hard-core particles in a birth—death process

• Series Expansions
  • The SIS epidemic model revisited
  • Perturbation analysis gives critical threshold

• Generalization to the SIRI Epidemic Model
  • The SIRI epidemic model
  • Transitions in the SIRI model

The following sections are included:

• Measles around Criticality

• The SIR Model
  • The SIR model with vaccination
  • Stationary states and vaccination threshold
  • Definition and expression for the reproduction number
  • Vaccination level at criticality v_c
  • Realistic parameters for measles epidemics

• Stochastic Simulations
  • Stochastic bifurcation diagram for vaccine uptake
  • Large outbreaks during decreasing vaccine uptake

The following sections are included:

• Introduction

• Models in Genetics
  • The Moran model
  • The probability of fixation

• Mean Time until Fixation

The following sections are included:

• Accidental Pathogens

• Modeling Infection with Accidental Pathogens
  • The meningitis model: SIRYX
  • The invasion dynamics of mutant strains
  • Divergent fluctuations for vanishing pathogenicity

• Evolution toward Criticality
  • Fast and slow time scales in the meningitis model
  • Analytics for evolution toward criticality
  • Simulation for evolution toward criticality
  • Spatial modeling: Outbreak clusters without direct disease contacts
  • Mean pathogenicity goes toward zero

• Empirical Data Show Fast Epidemic Response and Long-Lasting Fluctuations
  • Modeling fast epidemic response Finds long-lasting fluctuations
  • Data with fast epidemic response and long-lasting fluctuations

The following sections are included:

• Appendix A Invariant Density of the Ulam Map
  • Time Evolution Equation of the Density
  • Conjugation of Ulam Map and Tent Map
  • Exponential Divergence in the Ulam Map

• Appendix B Parameter Estimation for the Autoregressive AR(1)-Process Gives Least Squares Estimators
  • Likelihood and its Maximization
  • Bayesian Parameter Estimation
    • Conditional posteriors
    • Gibbs sampler

• Appendix C From Stochastic Epidemics to Parameter Estimation
  • Likelihood for Simple Stochastic Epidemic Model
  • Calculation of the Likelihood Function
  • Confidence Intervals via Inverse Fisher Matrix
  • Improving Confidence Intervals

• Appendix D Product Space for Spin 1/2 Many-Particle Systems

• Appendix E Path Integral Using Coherent States for Hard-Core Bosons

• Appendix F Analytical Power Laws in the Meningitis Model
  • Distribution of Total Number of Cases
  • Scaling
  • Solution of Size Distribution of the Epidemic
  • Size Distribution of the Epidemic for ε Zero

• Bibliography

• Index