Contents -- Preface -- Chapter 1 From Deterministic to Stochastic Dynamics -- 1.1 Basic Probability Theory: The Tool Box -- 1.2 Stochastic Description of a Deterministic System: The Ulam Map -- 1.3 A Fully Stochastic Dynamic System: The AR(1)-Process -- 1.4 From Perron-Frobenius to Master Equation -- 1.5 A First Example of a Master Equation: The Linear Infection Model -- 1.5.1 Solving the first example of a master equation -- 1.5.2 Solution of the linear infection model -- 1.5.3 Mean value and its dynamics -- 1.5.4 Mean dynamics -- 1.6 The Birth and Death Process, a Non-Linear Stochastic System -- 1.7 Solution of the Birth-Death ODE Shows Criticality -- 1.7.1 Numerical integration shows power law at criticality -- 1.7.2 Temporal correlation length diverges at criticality -- Chapter 2 Spatial Stochastic Birth-Death Process or SIS-Epidemics -- 2.1 The Spatial Master Equation -- 2.1.1 A first inspection of the spatial birth-death process -- 2.2 Clusters and their Dynamics -- 2.2.1 Time evolution of marginals and local expectations -- 2.3 Moment Equations -- 2.3.1 Mean field behavior -- 2.3.2 Pair approximation -- 2.4 The SIS Dynamics under Pair Approximation -- 2.5 Conclusions and Further Reading -- Chapter 3 Criticality in Equilibrium Systems -- 3.1 The Glauber Model: Stochastic Dynamics for the Ising Model -- 3.1.1 A first glance at the dynamic Ising model -- 3.2 The Ising Model, a Paradigm for Equilibrium Phase Transitions -- 3.2.1 Distribution of magnetization and Gibbs free energy -- 3.3 Equilibrium Distribution around Criticality -- 3.3.1 Distribution of magnetization -- 3.3.2 External magnetic field -- 3.3.3 The maximum of the total magnetization distribution -- 3.3.3.1 Approximation with Lagrange polynomials -- 3.3.3.2 The maximum magnetization with changing parameters -- 3.4 Mean Field Theory and its Exponents.

3.4.1 Mean field self-consistency equation -- 3.4.2 Mean field quantities obtained from the self-consistency equation -- 3.4.2.1 Critical coupling strength in mean field approximation -- 3.4.2.2 The first critical exponent in mean field approximation -- 3.4.3 Universal scaling function and exponent δ in mean field -- 3.4.4 The magnetic susceptibility or second moment of the magnetization diverges -- 3.4.5 State equation in mean field -- 3.5 Critical Exponents of the Ising Model beyond Mean Field -- Chapter 4 Partial Immunization Models -- 4.1 A Model with Partial Immunization: SIRI -- 4.2 Local Quantities -- 4.3 Dynamics Equations for Global Pairs -- 4.3.1 The SIRI dynamics under pair approximation -- 4.3.2 Balance equations for means and pairs -- 4.4 Mean Field Model: SIRI with Reintroduced Susceptibles -- 4.4.1 Pair dynamics for the SIRI model -- 4.5 Fruitful Transfer between Equilibrium and Non-Equilibrium Systems -- Chapter 5 Renormalization and Series Expansion: Techniques to Study Criticality -- 5.1 Introduction -- 5.2 Real Space Renormalization in One-Dimensional Lattice Gas -- 5.3 Directed Percolation and Path Integrals -- 5.3.1 Master equation of the birth-death process -- 5.3.2 Schrödinger-like equation -- 5.3.3 δ-Bosons for hard-core particles -- 5.3.4 Path integral for hard-core particles in a birth-death process -- 5.4 Series Expansions -- 5.4.1 The SIS epidemic model revisited -- 5.4.2 Perturbation analysis gives critical threshold -- 5.5 Generalization to the SIRI Epidemic Model -- 5.5.1 The SIRI epidemic model -- 5.5.2 Transitions in the SIRI model -- Chapter 6 Criticality in Measles under Vaccination -- 6.1 Measles around Criticality -- 6.2 The SIR Model -- 6.2.1 The SIR model with vaccination -- 6.2.2 Stationary states and vaccination threshold -- 6.2.3 Definition and expression for the reproduction number R.

6.2.4 Vaccination level at criticality vc -- 6.2.5 Realistic parameters for measles epidemics -- 6.3 Stochastic Simulations -- 6.3.1 Stochastic bifurcation diagram for vaccine uptake -- 6.3.2 Large outbreaks during decreasing vaccine uptake -- Chapter 7 Genetics and Criticality -- 7.1 Introduction -- 7.2 Models in Genetics -- 7.2.1 The Moran model -- 7.2.2 The probability of fixation -- 7.3 Mean Time until Fixation -- Chapter 8 Evolution to Criticality in Meningococcal Disease -- 8.1 Accidental Pathogens -- 8.2 Modeling Infection with Accidental Pathogens -- 8.2.1 The meningitis model: SIRYX -- 8.2.2 The invasion dynamics of mutant strains -- 8.2.3 Divergent fluctuations for vanishing pathogenicity -- 8.3 Evolution toward Criticality -- 8.3.1 Fast and slow time scales in the meningitis model -- 8.3.2 Analytics for evolution toward criticality -- 8.3.3 Simulation for evolution toward criticality -- 8.3.4 Spatial modeling: Outbreak clusters without direct disease contacts -- 8.3.5 Mean pathogenicity goes toward zero -- 8.4 Empirical Data Show Fast Epidemic Response and Long-Lasting Fluctuations -- 8.4.1 Modeling fast epidemic response finds long-lasting fluctuations -- 8.4.2 Data with fast epidemic response and long-lasting fluctuations -- Appendix A Invariant Density of the Ulam Map -- A.1 Time Evolution Equation of the Density -- A.2 Conjugation of Ulam Map and Tent Map -- A.3 Exponential Divergence in the Ulam Map -- Appendix B Parameter Estimation for the Autoregressive AR(1)-Process Gives Least Squares Estimators -- B.1 Likelihood and its Maximization -- B.2 Bayesian Parameter Estimation -- B.2.1 Conditional posteriors -- B.2.2 Gibbs sampler -- Appendix C From Stochastic Epidemics to Parameter Estimation -- C.1 Likelihood for Simple Stochastic Epidemic Model -- C.2 Calculation of the Likelihood Function.

C.3 Confidence Intervals via Inverse Fisher Matrix -- C.4 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 -- F.1 Distribution of Total Number of Cases -- F.2 Scaling -- F.3 Solution of Size Distribution of the Epidemic -- F.4 Size Distribution of the Epidemic for Є Zero -- Bibliography -- Index.