# Distribution Theory, Stochastic Processes And Infectious Disease Modelling

P. Yan

Published 2008 · Computer Science

The occurrence of a major outbreak, the shape of the epidemic curves, as well as the final sizes of outbreaks, are realizations of some stochastic events with some probability distributions. These distributions are manifested through some stochastic mechanisms. This chapter divides a typical outbreak in a closed population into two phases, the initial phase and beyond the initial phase. For the initial phase, this chapter addresses several aspects: the invasion (i.e. the risk of a large outbreak); quantities associated with a small outbreak; and characteristics of a large outbreak. In a large outbreak beyond the initial phase, the focus is on its final size. After a review of distribution theories and stochastic processes, this chapter separately addresses each of these issues by asking questions such as: Are the latent period and/or the infectious period distributions playing any role? What is the role of the contact process for this issue? Is the basic reproduction number R 0 sufficient to address this issue? How many stochastic mechanisms may manifest observations that may resemble a power-law distribution, and how much detail is really needed to address this specific issue? etc. This chapter uses distribution theory and stochastic processes to capture the agent–host–environment interface during an outbreak of an infectious disease. With different phases of an outbreak and special issues in mind, modellers need to choose which detailed aspects of the distributions and the stochastic mechanisms need to be included, and which detailed aspects need to be ignored. With these discussions, this chapter provides some syntheses for the concepts and models discussed in some proceeding chapters, as well as some food for thought for following chapters on case studies.

This paper references

10.1016/0025-5564(75)90119-4

Final size distribution for epidemics

D. Ludwig (1975)

10.1017/S0001867800015196

ASYMPTOTIC FINAL-SIZE DISTRIBUTION FOR SOME CHAIN-BINOMIAL PROCESSES

G. Scaliatomba (1985)

Collective epidemic processes: a general modelling approach to the final outcome of SIR infectious diseases

C Lefèvre (1995)

10.1016/j.jtbi.2004.07.026

Network theory and SARS: predicting outbreak diversity

L. Meyers (2005)

10.2307/2980979

An Introduction To Probability Theory And Its Applications

F. William (1950)

10.2307/2985209

Stochastic Population Models in Ecology and Epidemiology.

M. Bulmer (1961)

10.1007/978-3-540-78911-6_3

An Introduction to Stochastic Epidemic Models

L. Allen (2008)

10.1098/rsif.2006.0161

Comparative estimation of the reproduction number for pandemic influenza from daily case notification data

G. Chowell (2006)

10.1198/tech.2006.s421

Univariate Discrete Distributions

J. W. Davis (2006)

10.1007/s11538-005-9047-7

Generality of the Final Size Formula for an Epidemic of a Newly Invading Infectious Disease

Junling Ma (2006)

implications for the transmission of sexually transmitted infections

F. Liljeros (2003)

10.1093/biomet/67.1.191

On the spread of a disease with gamma distributed latent and infectious periods

D. Anderson (1980)

10.2307/2314395

A First Course on Stochastic Processes

S. Karlin (1966)

10.1239/JAP/1032265215

The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier

A. Martin-Löf (1998)

predicting outbreak diverity

L. Meyers (2005)

Papers in Honour of Sir David Cox on his 80th Birthday, ed

Isham (2005)

10.1126/SCIENCE.1086616

Transmission Dynamics and Control of Severe Acute Respiratory Syndrome

M. Lipsitch (2003)

10.1016/S1286-4579(02)00058-8

Sexual networks: implications for the transmission of sexually transmitted infections.

Fredrik Liljeros (2003)

10.1007/978-3-540-78911-6_4

An Introduction to Networks in Epidemic Modeling

F. Brauer (2008)

their Structure and Relation to Data, ed

Näsell (1995)

10.2307/2343842

The Generalized Waring Distribution Applied to Accident Theory

J. Irwin (1968)

Mixed Poisson distributions. International Statistical Review

D Karlis (2005)

a general modelling approach to the final outcome of SIR infectious diseases

C. Lefèvre (1995)

10.1093/acprof:oso/9780198566540.003.0002

Stochastic Models for Epidemics

V. Isham (2006)

10.1126/SCIENCE.286.5439.509

Emergence of scaling in random networks

Barabási (1999)

10.2307/3214172

SYMMETRIC SAMPLING PROCEDURES, GENERAL EPIDEMIC PROCESSES AND THEIR THRESHOLD LIMIT THEOREMS

A. Martin-Löf (1986)

An examination of the Reed-Frost theory of epidemics.

H. Abbey (1952)

10.2307/2532277

Univariate Discrete Distributions.

S. Kocherlakota (1993)

Contributions to the mathematical theory of epidemics, part I

W. O. Kermack (1927)

10.1371/journal.pmed.0020174

Appropriate Models for the Management of Infectious Diseases

Helen J Wearing (2005)

The Mathematical Theory of Infectious Diseases and Its Applications

A. Grey (1977)

10.2307/2341080

An Inquiry into the Nature of Frequency Distributions Representative of Multiple Happenings with Particular Reference to the Occurrence of Multiple Attacks of Disease or of Repeated Accidents

M. Greenwood (1920)

10.1524/STRM.2000.18.4.349

INFERENCE BASED ON THE EMPIRICAL PROBABILITY GENERATING FUNCTION FOR MIXTURES OF POISSON DISTRIBUTIONS

Remillard Bruno (2000)

model building, analysis and interpretation

O. Diekmann (2000)

10.2307/2342709

A Unified Derivation of Some Well‐Known Frequency Distributions of Interest in Biometry and Statistics

J. Irwin (1955)

10.1142/4243

Stochastic Processes in Epidemiology: Hiv/Aids, Other Infectious Diseases and Computers

C. Mode (2000)

10.1016/j.jtbi.2007.11.027

Separate roles of the latent and infectious periods in shaping the relation between the basic reproduction number and the intrinsic growth rate of infectious disease outbreaks.

P. Yan (2008)

10.1098/rspa.1927.0118

A Contribution to the Mathematical Theory of Epidemics

W. O. Kermack (1927)

On the incidence of industrial accidents upon individuals with special reference to multiple accidents

M. Greenwood (1919)

The univariate generalized Waring distribution in relation to accident theory: proness, spells or contagion? Biometrics

E. A. Xekalaki (1983)

Final size distributions for epidemics

D. Ludwig (1975)

Deterministic and Stochastic Epidemics in Closed Populations

D. Kendall (1956)

10.1090/S0002-9947-1984-0756039-5

The evolution of random graphs

B. Bollob'as (1984)

10.1016/0167-5699(93)90204-X

Infectious diseases of humans: Dynamics and control

P. Kaye (1993)

10.1111/J.1751-5823.2005.TB00250.X

Mixed Poisson Distributions

D. Karlis (2005)

10.1103/PhysRevE.66.016128

Spread of epidemic disease on networks.

M. Newman (2002)

10.1112/jlms/s1-42.1.367

A FIRST COURSE IN STOCHASTIC PROCESSES

J. Kingman (1967)

Some problems in the theory of infectious disease transmission and control

K. Dietz (1995)

10.4324/9781351291521-31

An essay on the principle of population, as it affects the future improvement of society

T. Malthus (1986)

10.2307/1266435

An Introduction to Probability Theory and Its Applications, Volume II

F. E. Grubbs (1971)

An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to occurence of multiple attacks of diseases or of repeated accidents

M. Greenwood (1920)

10.2307/1426600

Threshold limit theorems for some epidemic processes

Bengt von Bahr (1980)

10.2307/2532920

Epidemic Models: Their Structure and Relation to Data.

D. Mollison (1996)

Celebrating Statistics: Papers in honour of Sir David Cox on his 80th birthday

A. C. Davison (2005)

10.21236/ada171415

Mixtures, generalized convexity and balayages

J. Lynch (1986)

10.2307/2282020

Stochastic Population Models in Ecology and Epidemiology.

N. Bailey (1961)

10.1007/978-0-8176-4626-4_2

The Polygonal Distribution

D. Karlis (2008)

10.1098/rspb.2003.2369

An assessment of preferential attachment as a mechanism for human sexual network formation

J. Jones (2003)

The threshold concept in stochastic and endemic models

I Näsell (1995)

10.2307/2347630

Stochastic Processes with Applications

I. G. MacKenzie (1992)

Their Structure and Relation to Data, ed

Dietz (1995)

10.1016/S0025-5564(98)10048-2

The effect of random vaccine response on the vaccination coverage required to prevent epidemics.

N. Becker (1998)

10.1007/BF00048405

The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections

J. Metz (1978)

10.2307/2531324

The univariate generalized Waring distribution in relation to accident theory: proneness, spells or contagion?

E. Xekalaki (1983)

10.1214/aop/1176988177

Poisson Approximation for the Final State of a Generalized Epidemic Process

C. Lefèvre (1995)

10.1090/s0002-9947-1984-0756039-5

On the evolution of random graphs

P. Erdős (1984)

The Mathematical Theory of Infectious Diseases and its applications

N. Ling (1978)

10.1017/cbo9780511546594

Random Graph Dynamics

R. Durrett (2006)

This paper is referenced by

Bridging the COVID-19 Data and the Epidemiological Model using Time Varying Parameter SIRD Model

C. Çakmaklı (2020)

10.1016/j.jtbi.2011.10.039

Estimating the transmission potential of supercritical processes based on the final size distribution of minor outbreaks

H. Nishiura (2012)

10.1007/S11009-013-9325-Z

Applications of the Variance of Final Outbreak Size for Disease Spreading in Networks

Lilia L. Ramírez-Ramírez (2014)

10.1145/3178876.3186108

SIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Populations

Marian-Andrei Rizoiu (2018)

Infinitely Stochastic Micro Forecasting

Mat'uvs Maciak (2019)

10.1145/3178876.3186108

SIR-Hawkes: on the Relationship Between Epidemic Models and Hawkes Point Processes

Marian-Andrei Rizoiu (2017)

10.1145/3289600.3291601

Linking Epidemic Models and Hawkes Point Processes for Modeling Information Diffusion

Quyu Kong (2019)

10.1145/3336191.3371821

Modeling Information Cascades with Self-exciting Processes via Generalized Epidemic Models

Quyu Kong (2020)

10.1101/2020.07.28.20163535

The basic reproduction number of SARS-CoV-2: a scoping review of available evidence

A. Barber (2020)

10.3390/ijerph17113763

An Integrated Approach for Spatio-Temporal Cholera Disease Hotspot Relation Mining for Public Health Management in Punjab, Pakistan

Fatima Khalique (2020)

10.1101/768853

Will an outbreak exceed available resources for control? Estimating the risk from invading pathogens using practical definitions of a severe epidemic

R.N. Thompson (2020)

10.1109/IJCNN.2018.8489509

Pareto cascade modeling of diffusion networks

Christopher Ma (2018)

10.1073/pnas.2006520117

The challenges of modeling and forecasting the spread of COVID-19

A. Bertozzi (2020)

10.3182/20090812-3-DK-2006.0099

Analysing Outbreak Data in a Heterogeneous Population with Migration

M. Wolkewitz (2009)

Statistical Inference on Stochastic Graphs

Yasaman Hosseinkashi (2011)

10.1155/2017/7481210

Low Utilization of Insecticide-Treated Bed Net among Pregnant Women in the Middle Belt of Ghana

G. Manu (2017)

10.1007/978-3-030-21923-9_4

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

Ping Yan (2019)

10.1016/j.tpb.2009.11.003

The impact of network clustering and assortativity on epidemic behaviour.

J. Badham (2010)

A Study Of Computational Problems In Computational Biology And Social Networks: Cancer Informatics And Cascade Modelling

Christopher Ma (2018)

10.1016/j.ijid.2020.02.033

Estimation of the reproductive number of novel coronavirus (COVID-19) and the probable outbreak size on the Diamond Princess cruise ship: A data-driven analysis

S. Zhang (2020)

10.1016/S0252-9602(09)60087-4

ON THE BASIC REPRODUCTION NUMBER OF GENERAL BRANCHING PROCESSES

Lan Guo-lie (2009)

10.1002/MMA.3424

Stage‐structured population systems with temporally periodic delay

Xiaotian Wu (2015)

10.1016/j.idm.2018.08.001

A primer on the use of probability generating functions in infectious disease modeling

J. Miller (2018)