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E-raamat: Stochastic Epidemic Models with Inference

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Stochastic Epidemic Models with InferenceSuurem pilt
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Focussing on stochastic models for the spread of infectious diseases in a human population, this book is the outcome of a two-week ICPAM/CIMPA school on "Stochastic models of epidemics" which took place in Ziguinchor, Senegal, December 5–16, 2015. The text is divided into four parts, each based on one of the courses given at the school: homogeneous models (Tom Britton and Etienne Pardoux), two-level mixing models (David Sirl and Frank Ball), epidemics on graphs (Viet Chi Tran), and statistics for epidemic models (Catherine Larédo).

The CIMPA school was aimed at PhD students and Post Docs in the mathematical sciences. Parts (or all) of this book can be used as the basis for traditional or individual reading courses on the topic. For this reason, examples and exercises (some with solutions) are provided throughout.


Part I Stochastic Epidemics in a Homogeneous Community
Tom Britton
Etienne Pardoux
Introduction
3(2)
1 Stochastic Epidemic Models
5(16)
1.1 The Stochastic SEIR Epidemic Model in a Closed Homogeneous Community
5(4)
1.1.1 Model Definition
5(1)
1.1.2 Some Remarks, Submodels and Model Generalizations
6(2)
1.1.3 Two Key Quantities: Ro and the Escape Probability
8(1)
1.2 The Early Stage of an Outbreak
9(5)
1.3 The Final Size of the Epidemic in Case of No Major Outbreak
14(3)
1.4 Vaccination
17(4)
2 Markov Models
21(22)
2.1 The Deterministic SEIR Epidemic Model
21(4)
2.2 Law of Large Numbers
25(9)
2.3 Central Limit Theorem
34(6)
2.4 Diffusion Approximation
40(3)
3 General Closed Models
43(16)
3.1 Exact Results for the Final Size in Small Communities
43(4)
3.2 The Sellke Construction
47(2)
3.3 LLN and CLT for the Final Size of the Epidemic
49(5)
3.3.1 Law of Large Numbers
50(2)
3.3.2 Central Limit Theorem
52(2)
3.4 The Duration of the Stochastic SEIR Epidemic
54(5)
4 Open Markov Models
59(38)
4.1 Open Populations: Time to Extinction and Critical Population Size
59(4)
4.2 Large Deviations and Extinction of an Endemic Disease
63(34)
4.2.1 Introduction
63(1)
4.2.2 The Rate Function
64(4)
4.2.3 The Lower Bound
68(6)
4.2.4 The Upper Bound
74(8)
4.2.5 Time of Extinction in the SIRS Model
82(10)
4.2.6 Time of Extinction in the SIS Model
92(1)
4.2.7 Time of Extinction in the SIR Model with Demography
93(4)
Appendix
97(22)
A.1 Branching Processes
97(5)
A.1.1 Discrete Time Branching Processes
97(2)
A.1.2 Continuous Time Branching Processes
99(3)
A.2 The Poisson Process and Poisson Point Process
102(3)
A.3 Cramer's Theorem for Poisson Random Variables
105(2)
A.4 Martingales
107(3)
A.4.1 Martingales in Discrete Time
107(2)
A.4.2 Martingales in Continuous Time
109(1)
A.5 Tightness and Weak Convergence in Path Space
110(1)
A.6 Pontryagin's Maximum Principle
111(2)
A.7 Semi- and Equicontinuity
113(1)
A.8 Solutions to Selected Exercises
113(6)
References for Part I
119(4)
Part II Stochastic SIR Epidemics in Structured Populations
Frank Ball
David Sirl
Introduction
123(2)
1 Single Population Epidemics
125(18)
1.1 Standard SIR Epidemic Model
126(1)
1.2 Random Graph Representation of Epidemic
126(2)
1.3 Symmetric Sampling Procedures
128(2)
1.4 Gontcharoff Polynomials
130(2)
1.5 Susceptibility Sets
132(2)
1.6 Total Size
134(2)
1.7 Total Size and Severity
136(2)
1.8 Epidemics with Outside Infection
138(2)
1.9 Mean Final Size of a Multitype Epidemic
140(3)
2 The Households Model
143(16)
2.1 Introduction and Definition
143(2)
2.2 Early Stages
145(2)
2.3 Final Outcome of a Major Outbreak
147(1)
2.4 Vaccination
148(6)
2.4.1 Modelling Vaccination
148(2)
2.4.2 Threshold Parameter
150(2)
2.4.3 Vaccination Schemes
152(2)
2.5 Other Measures of Epidemic Impact
154(1)
2.6 Exercises
155(4)
3 A General Two-Level Mixing Model
159(56)
3.1 Definition and Examples
160(3)
3.1.1 Standard SIR Model
161(1)
3.1.2 Households Model
161(1)
3.1.3 Households-workplaces Model
161(1)
3.1.4 Great Circle Model
162(1)
3.1.5 Network Model with Casual Contacts
162(1)
3.2 Local Infectious Clumps and Susceptibility Sets
163(2)
3.3 Early Stages of an Epidemic
165(16)
3.3.1 Introduction
165(1)
3.3.2 Heuristics
165(3)
3.3.3 The Basic Reproduction Number R0
168(2)
3.3.4 Uniform Vaccination
170(4)
3.3.5 Exponential Growth Rate
174(1)
3.3.6 Formal Results and Proofs
175(6)
3.4 Final Outcome of a Global Outbreak
181(11)
3.4.1 Heuristics
181(3)
3.4.2 'Rigorous' Argument and Central Limit Theorem
184(8)
3.5 Applications to Special Cases
192(51)
3.5.1 Standard SIR Model
193(1)
3.5.2 Households Model
194(3)
3.5.3 Households-workplaces Model
197(6)
3.5.4 Great Circle Model
203(4)
3.5.5 Network Model with Casual Contacts
207(8)
Solutions
215(20)
References for Part II
235(6)
Part III Stochastic Epidemics in a Heterogeneous Community
Viet Chi Tran
Introduction
241(2)
1 Random Graphs
243(8)
1.1 Definitions
243(2)
1.2 Classical Examples of Random Graphs
245(4)
1.3 Sequences of Graphs
249(1)
1.4 Definition of the SIR Model on a Random Graph
250(1)
2 The Reproduction Number R0
251(12)
2.1 Homogeneous Mixing
252(1)
2.2 Configuration Model
252(3)
2.3 Stochastic Block Models
255(1)
2.4 Household Structure
256(1)
2.5 Statistical Estimation of R0 for SIR on Graphs
256(1)
2.6 Control Effort
257(6)
3 SIR Epidemics on Configuration Model Graphs
263(36)
3.1 Moment Closure in Large Populations
263(4)
3.2 Volz and Miller Approach
267(4)
3.2.1 Dynamics of θ(t)
269(1)
3.2.2 Miller's Equations
270(1)
3.3 Measure-valued Processes
271(28)
3.3.1 Stochastic Model for a Finite Graph with N Vertices
271(1)
3.3.2 Dynamics and Measure-valued SDEs
272(5)
3.3.3 Rescaling
277(2)
3.3.4 Large Graph Limit
279(2)
3.3.5 Ball-Neal and Volz Equations
281(3)
3.3.6 Degree Distribution of the "Initial Condition"
284(3)
3.3.7 Proof of the Limit Theorem
287(12)
4 Statistical Description of Epidemics Spreading on Networks: The Case of Cuban HIV
299(16)
4.1 Modularity and Assortative Mixing
300(2)
4.2 Visual-mining
302(2)
4.3 Analysis of the "Giant Component"
304(3)
4.4 Descriptive Statistics for Epidemics on Networks
307(24)
4.4.1 Estimating Degree Distributions
308(3)
4.4.2 Joint Degree Distribution of Sexual Partners
311(1)
4.4.3 Computation of Geodesic Distances and Other Connectivity Properties
312(3)
Appendix: Finite Measures on Z+
315(4)
References for Part III
319(8)
Part IV Statistical Inference for Epidemic Processes in a Homogeneous Community
Catherine Laredo
Introduction
327(4)
1 Observations and Asymptotic Frameworks
331(12)
1.1 Various Kinds of Observations and Asymptotic Frameworks
331(4)
1.1.1 Observations
332(1)
1.1.2 Various Asymptotic Frameworks
332(2)
1.1.3 Various Estimation Methods
334(1)
1.2 An Example Illustrating the Inference in these Various Situations
335(8)
1.2.1 A Simple Model for Population Dynamics: AR(1)
335(1)
1.2.2 Ornstein-Uhlenbeck Diffusion Process with Increasing Observation Time
336(4)
1.2.3 Ornstein-Uhlenbeck Diffusion with Fixed Observation Time
340(1)
1.2.4 Ornstein-Uhlenbeck Diffusion with Small Diffusion Coefficient
341(1)
1.2.5 Conclusions
342(1)
2 Inference for Markov Chain Epidemic Models
343(20)
2.1 Markov Chains with Countable State Space
343(9)
2.1.1 Greenwood Model
345(2)
2.1.2 Reed-Frost Model
347(1)
2.1.3 Birth and Death Chain with Re-emerging
348(2)
2.1.4 Modeling an Infection Chain in an Intensive Care Unit
350(2)
2.2 Two Extensions to Continuous State and Continuous Time Markov Chain Models
352(1)
2.2.1 A Simple Model for Population Dynamics
352(1)
2.2.2 Continuous Time Markov Epidemic Model
352(1)
2.3 Inference for Branching Processes
353(10)
2.3.1 Notations and Preliminary Results
353(2)
2.3.2 Inference when the Offspring Law Belongs to an Exponential Family
355(3)
2.3.3 Parametric Inference for General Galton-Watson Processes
358(3)
2.3.4 Examples
361(1)
2.3.5 Variants of Branching Processes
361(2)
3 Inference Based on the Diffusion Approximation of Epidemic Models
363(54)
3.1 Introduction
363(2)
3.2 Diffusion Approximation of Jump Processes Modeling Epidemics
365(8)
3.2.1 Approximation Scheme Starting from the Jump Process Q-matrix
365(3)
3.2.2 Diffusion Approximation of Some Epidemic Models
368(5)
3.3 Inference for Discrete Observations of Diffusions on [ 0,T]
373(8)
3.3.1 Assumptions, Notations and First Results
374(2)
3.3.2 Preliminary Results
376(5)
3.4 Inference Based on High Frequency Observations on [ 0,T]
381(14)
3.4.1 Properties of the Estimators
381(3)
3.4.2 Proof of Theorem 3.4.1
384(11)
3.5 Inference Based on Low Frequency Observations
395(6)
3.5.1 Preliminary Result on a Simple Example
395(1)
3.5.2 Inference for Diffusion Approximations of Epidemics
396(5)
3.6 Assessment of Estimators on Simulated Data Sets
401(3)
3.6.1 The SIR Model
402(2)
3.6.2 The SIRS Model
404(1)
3.7 Inference for Partially Observed Epidemic Dynamics
404(13)
3.7.1 Inference for High Frequency Sampling of Partial Observations
406(5)
3.7.2 Assessment of Estimators on Simulated and Real Data Sets
411(6)
4 Inference for Continuous Time SIR models
417(30)
4.1 Introduction
417(1)
4.2 Maximum Likelihood in the SIR Case
418(9)
4.2.1 MCMC Estimation
421(2)
4.2.2 EM Algorithm for Discretely Observed Markov Jump Processes
423(4)
4.3 ABC Estimation
427(8)
4.3.1 Main Principles of ABC
428(2)
4.3.2 Comparisons Between ABC and MCMC Methods for a Standard SIR Model
430(1)
4.3.3 Comparison Between ABC with Full and Binned Recovery Times
431(4)
4.4 Sensitivity Analysis
435(12)
4.4.1 A Non-parametric Estimator of the Sobol Indices of Order 1
437(2)
4.4.2 Statistical Properties
439(8)
Appendix
447(20)
A.1 Some Classical Results in Statistical Inference
447(2)
A.1.1 Heuristics on Maximum Likelihood Methods
447(1)
A.1.2 Miscellaneous Results
448(1)
A.2 Inference for Markov Chains
449(8)
A.2.1 Recap on Markov Chains
449(4)
A.2.2 Other Approaches than the Likelihood
453(3)
A.2.3 Hidden Markov Models
456(1)
A.3 Results for Statistics of Diffusions Processes
457(3)
A.3.1 Continuously Observed Diffusions on [ 0,T]
457(1)
A.3.2 Discrete Observations with Sampling δ on a Time Interval [ 0,T]
458(1)
A.3.3 Inference for Diffusions with Small Diffusion Matrix on [ 0, T]
459(1)
A.4 Some Limit Theorems for Martingales and Triangular Arrays
460(4)
A.4.1 Central Limit Theorems for Martingales
460(1)
A.4.2 Limit Theorems for Triangular Arrays
461(3)
A.5 Inference for Pure Jump Processes
464(3)
A.5.1 Girsanov Type Formula for Counting Processes
464(1)
A.5.2 Likelihood for Markov Pure Jump Processes
465(1)
A.5.3 Martingale Properties of Likelihood Processes
465(2)
References for Part IV
467
Tom Britton is professor at the Department of Mathematics at Stockholm University. His research focuses on stochastic modelling, and inference procedures, for biological and medical problems, in particular models for the spread of infectious diseases, networks and phylogenetics. He is the author of more than 100 publications, and two monographs about models and analysis of infectious disease spreading.

Etienne Pardoux is professor emeritus at the Institut de Mathématiques de Marseille, within Aix Marseille Univ. His research has covered several topics of stochastic analysis, in particular stochastic partial differential equations, backward stochastic differential equations and homogenization. More recently, he has turned his interests towards evolutionary biology and modeling of infectious diseases. He is the author of more than 160 publications, including four books.
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