KEY POINTS * The last decade has seen considerable advances in statistical, mathematical and computational techniques that are available for the analysis of outbreak data. These advances

have greatly increased our capacity to generate meaningful epidemiological information from relatively small numbers of cases of an infection by better representation of the stochastic

nature of the outbreak events and improved methods for estimating parameters from such data. * This review focuses on the application of such models, which capture the highly variable

dynamics of infection spread amongst small numbers of individuals, in the following areas: quantifying the basic reproduction ratio _R_0 in the early stages of an outbreak (for

(antibiotic-resistant nosocomial infections). * An important development is the integration of clinical, surveillance and contact-tracing data into the modelling process. This leads to

models that are better able to capture the underlying variability in the transmission dynamics and can do so with minimal assumptions. In particular, heterogeneities in the number of

secondary infections generated by an infected case (as exemplified by the super-spreaders of the SARS outbreak) mean that contact tracing is essential for a proper quantification of

uncertainty in the reproduction ratio. * Advances in the analysis of outbreak data in the near future will probably come from the further development of molecular techniques (to assist

contact tracing) and from the better integration of disease data with demographic and environmental information. ABSTRACT Recent major disease outbreaks, such as severe acute respiratory

syndrome and foot-and-mouth disease in the UK, coupled with fears of emergence of human-to-human transmissible variants of avian influenza, have highlighted the importance of accurate

quantification of disease threat when relatively few cases have occurred. Traditional approaches to mathematical modelling of infectious diseases deal most effectively with large outbreaks

in large populations. The desire to elucidate the highly variable dynamics of disease spread amongst small numbers of individuals has fuelled the development of models that depend more

directly on surveillance and contact-tracing data. This signals a move towards a closer interplay between epidemiological modelling, surveillance and disease-management strategies. Access

through your institution Buy or subscribe This is a preview of subscription content, access via your institution ACCESS OPTIONS Access through your institution Subscribe to this journal

Receive 12 print issues and online access $209.00 per year only $17.42 per issue Learn more Buy this article * Purchase on SpringerLink * Instant access to full article PDF Buy now Prices

may be subject to local taxes which are calculated during checkout ADDITIONAL ACCESS OPTIONS: * Log in * Learn about institutional subscriptions * Read our FAQs * Contact customer support

SIMILAR CONTENT BEING VIEWED BY OTHERS SPATIO-TEMPORAL PREDICTIVE MODELING FRAMEWORK FOR INFECTIOUS DISEASE SPREAD Article Open access 24 March 2021 EVALUATION OF THE NUMBER OF UNDIAGNOSED

INFECTED IN AN OUTBREAK USING SOURCE OF INFECTION MEASUREMENTS Article Open access 11 February 2021 USING MECHANISTIC MODEL-BASED INFERENCE TO UNDERSTAND AND PROJECT EPIDEMIC DYNAMICS WITH

TIME-VARYING CONTACT AND VACCINATION RATES Article Open access 28 November 2022 REFERENCES * Vinten-Johansen, P., Brody, H., Paneth, N., Rachman, S. & Rip, M. _Cholera, Chloroform and

the Science of Medicine; A Life of John Snow_ (Oxford University Press, Oxford, 2003). Google Scholar  * Ross, R. _The Prevention of Malaria_ (John Murray, London, 1911). Google Scholar  *

Kermack, W. O. & McKendrick, A. G. Contributions to the mathematical theory of epidemics — I. 1927. _Bull. Math. Biol._ 53, 33–55 (1991). CAS  PubMed  Google Scholar  * Macdonald, G. The

analysis of equilibrium in malaria. _Trop. Dis. Bull._ 49, 813–829 (1952). CAS  PubMed  Google Scholar  * Dietz, K. in _Epidemiology_ (eds Ludwig, D. & Cooke, K. L.) 104–121 (Society

for Industrial and Applied Mathematics, Philadelphia, 1975). Google Scholar  * Heesterbeek, J. A. P. A brief history of R0 and a recipe for its calculation. _Acta Biotheor._ 50, 189–204

(2002). Article  CAS  Google Scholar  * Anderson, R. M. & May, R. M. _Infectious Diseases of Humans: Dynamics and Control_ (Oxford University Press, Oxford, 1991). Google Scholar  *

Fine, P. E. M. Herd immunity: history, theory, practice. _Epidemiol. Rev._ 15, 265–302 (1993). Article  CAS  Google Scholar  * Gay, N. J. The theory of measles elimination: implications for

the design of elimination strategies. _J. Infect. Dis._ 189, S27–S35 (2004). Article  Google Scholar  * Yorke, J. A., Hethcote, H. W. & Nold, A. Dynamics and control of transmission of

gonorrhea. _Sex. Transm. Dis._ 5, 51–56 (1978). Article  CAS  Google Scholar  * Bartlett, M. S. The critical community size for measles in the United States. _J. R. Stat. Soc. Ser. A_ 123,

37–44 (1960). Article  Google Scholar  * Keeling, M. J. Modelling the persistence of measles. _Trends Microbiol._ 5, 513–518 (1997). Article  CAS  Google Scholar  * Swinton, J., Harwood, J.,

Grenfell, B. T. & Gilligan, C. A. Persistence thresholds for phocine distemper virus infection in harbour seal _Phoca vitulina_ metapopulations. _J. Anim. Ecol._ 67, 54–68 (1998).

Article  Google Scholar  * Becker, N. G. _Analysis of Infectious Disease Data_ (Chapman and Hall, London, 1989). Google Scholar  * Bailey, N. T. J. _The Mathematical Theory of Infectious

Diseases and its Applications_ (Charles Griffin, London, 1975). Google Scholar  * Bartlett, M. S. _Stochastic Population Models in Ecology and Epidemiology_ (Methuen, London, 1960). Google

Scholar  * Cauchemez, S., Carrat, F., Viboud, C., Valleron, A. J. & Boelle, P. Y. A Bayesian MCMC approach to study transmission of influenza: application to household longitudinal data.

_Stat. Med._ 23, 3469–3487 (2004). Article  CAS  Google Scholar  * Gibson, G. J. Markov chain Monte Carlo methods for fitting spatiotemporal stochastic models in plant epidemiology. _J. R.

Stat. Soc. Ser. C_ 46, 215–233 (1997). Article  Google Scholar  * Gibson, G. J., Kleczkowski, A. & Gilligan, C. A. Bayesian analysis of botanical epidemics using stochastic compartmental

models. _Proc. Natl Acad. Sci. USA_ 101, 12120–12124 (2004). Article  CAS  Google Scholar  * O'Neill, P. D., Balding, D. J., Becker, N. G., Eerola, M. & Mollison, D. Analyses of

infectious disease data from household outbreaks by Markov chain Monte Carlo methods. _J. R. Stat. Soc. Ser. C_ 49, 517–542 (2000). Article  Google Scholar  * O'Neill, P. D. A tutorial

introduction to Bayesian inference for stochastic epidemic models using Markov chain Monte Carlo methods. _Math. Biosci._ 180, 103–114 (2002). A USEFUL INTRODUCTION TO THE USE OF BAYESIAN

INFERENCE AND MCMC METHODS FOR THE ANALYSIS OF INFECTIOUS DISEASE OUTBREAK DATA. Article  Google Scholar  * Chowell, G., Fenimore, P. W., Castillo-Garsow, M. A. & Castillo-Chavez, C.

SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism. _J. Theor. Biol._ 224, 1–8 (2003). Article  CAS  Google Scholar  * Wang, W. D.

& Ruan, S. G. Simulating the SARS outbreak in Beijing with limited data. _J. Theor. Biol._ 227, 369–379 (2004). Article  Google Scholar  * Riley, S. et al. Transmission dynamics of the

etiological agent of SARS in Hong Kong: impact of public health interventions. _Science_ 300, 1961–1966 (2003). ILLUSTRATES THE USE OF A COMBINATION OF MODELLING APPROACHES: THE AUTHORS USE

A STOCHASTIC COMPARTMENTAL METAPOPULATION MODEL TO CAPTURE BOTH STOCHASTIC FLUCTUATIONS WHEN NUMBERS OF CASES ARE LOW AND A METAPOPULATION STRUCTURE TO SIMULATE THE VARIATION IN SARS

INCIDENCE ACROSS GEOGRAPHICAL DISTRICTS. Article  CAS  Google Scholar  * Li, Y. G. et al. Predicting super spreading events during the 2003 severe acute respiratory syndrome epidemics in

Hong Kong and Singapore. _Am. J. Epidemiol._ 160, 719–728 (2004). Article  Google Scholar  * Boelle, P. Y., Cesbron, J. Y. & Valleron, A. J. Epidemiological evidence of higher

susceptibility to vCJD in the young. _BMC Infect. Dis._ 4, 26 (2004). Article  Google Scholar  * Hunter, N. PrP genetics in sheep and the implications for scrapie and BSE. _Trends

Microbiol._ 5, 331–334 (1997). Article  CAS  Google Scholar  * Woolhouse, M. E. J., Etard, J. F., Dietz, K., Ndhlovu, P. D. & Chandiwana, S. K. Heterogeneities in schistosome

transmission dynamics and control. _Parasitology_ 117, 475–482 (1998). Article  Google Scholar  * Donaldson, A. I., Alexandersen, S., Sorensen, J. H. & Mikkelsen, T. Relative risks of

the uncontrollable (airborne) spread of FMD by different species. _Vet. Rec._ 148, 602–604 (2001). Article  CAS  Google Scholar  * _Health Canada_ [online]

<www.hc-sc.gc.ca/english/protection/warnings/sars/learning/EngSe30_ch2.htm> (2004). * Svoboda, T. et al. Public health measures to control the spread of the severe acute

respiratory syndrome during the outbreak in Toronto. _N. Engl. J. Med._ 350, 2352–2361 (2004). Article  CAS  Google Scholar  * Lipsitch, M. et al. Transmission dynamics and control of severe

acute respiratory syndrome. _Science_ 300, 1966–1970 (2003). USES A BAYESIAN METHOD TO ESTIMATE _R_ FOR THE SARS OUTBREAK. Article  CAS  Google Scholar  * Renshaw, E. _Modelling Biological

Populations in Space and Time_ (Cambridge University Press, Cambridge, 1993). Google Scholar  * Gibson, G. J. & Renshaw, E. Likelihood estimation for stochastic compartmental models

using Markov chain methods. _Statistics and Computing_ 11, 347–358 (2001). Article  Google Scholar  * Wong, C. W. et al. Tracking the evolution of the SARS coronavirus using high-throughput,

high-density resequencing arrays. _Genome Res._ 14, 398–405 (2004). Article  CAS  Google Scholar  * Anderson, R. M. et al. Epidemiology, transmission dynamics and control of SARS: the

2002–2003 epidemic. _Philos. Trans. R. Soc. Lond. B_ 359, 1091–1105 (2004). Article  Google Scholar  * Donnelly, C. A. et al. Epidemiological determinants of spread of causal agent of severe

acute respiratory syndrome in Hong Kong. _Lancet_ 361, 1761–1766 (2003). Article  Google Scholar  * Eames, K. T. D. & Keeling, M. J. Contact tracing and disease control. _Proc. R. Soc.

Lond. B_ 270, 2565–2571 (2003). Article  Google Scholar  * FitzGerald, M. R., Thirlby, D. & Bedford, C. A. The outcome of contact tracing for gonorrhoea in the United Kingdom. _Int. J.

STD AIDS_ 9, 657–660 (1998). Article  CAS  Google Scholar  * Fraser, C., Riley, S., Anderson, R. M. & Ferguson, N. M. Factors that make an infectious disease outbreak controllable.

_Proc. Natl Acad. Sci. USA_ 101, 6146–6151 (2004). Article  CAS  Google Scholar  * Ferguson, N. M., Donnelly, C. A. & Anderson, R. M. The foot-and-mouth epidemic in Great Britain:

pattern of spread and impact of interventions. _Science_ 292, 1155–1160 (2001). Article  CAS  Google Scholar  * Keeling, M. J. et al. Dynamics of the 2001 UK foot and mouth epidemic:

stochastic dispersal in a heterogeneous landscape. _Science_ 294, 813–817 (2001). STOCHASTIC MICROSIMULATION MODEL OF THE SPATIAL AND TEMPORAL SPREAD OF FOOT-AND-MOUTH DISEASE THAT

EXPLICITLY REPRESENTS INDIVIDUAL LIVESTOCK HOLDINGS. Article  CAS  Google Scholar  * Morris, R. S., Wilesmith, J. W., Stern, M. W., Sanson, R. L. & Stevenson, M. A. Predictive spatial

modelling of alternative control strategies for the foot-and-mouth disease epidemic in Great Britain, 2001. _Vet. Rec._ 149, 137–144 (2001). Article  CAS  Google Scholar  * Haydon, D. T. et

al. The construction and analysis of epidemic trees with reference to the 2001 UK foot-and-mouth outbreak. _Proc. R. Soc. Lond. B_ 270, 121–127 (2003). USES A CASE-BY-CASE RECONSTRUCTIVE

APPROACH TO ESTIMATE _R_ DURING THE UK FOOT-AND-MOUTH EPIDEMIC. Article  CAS  Google Scholar  * Woolhouse, M. E. J., Shaw, D. J., Matthews, L., Chase-Topping, M. & St Rose, S.

Epidemiology and control of foot-and-mouth disease in the UK. In _Proceedings of the 6th International Congress of Veterinary Virology_ (2003). Google Scholar  * Woolhouse, M. E. J. et al.

Epidemiology. Foot-and-mouth disease under control in the UK. _Nature_ 411, 258–259 (2001). Article  CAS  Google Scholar  * WHO _Estimating the impact of the next influenza pandemic:

enhancing preparedness_ [online] <www.who.int/csr/disease/influenza/preparedness2004_12_08/en/> (2004). * Trampuz, A., Prabhu, R. M., Smith, T. F. & Baddour, L. M. Avian

influenza: a new pandemic threat? _Mayo Clin. Proc._ 79, 523–530 (2004). Article  Google Scholar  * Taubenberger, J. K., Reid, A. H., Krafft, A. E., Bijwaard, K. E. & Fanning, T. G.

Initial genetic characterization of the 1918 “Spanish” influenza virus. _Science_ 275, 1793–1796 (1997). Article  CAS  Google Scholar  * Ferguson, N. M., Fraser, C., Donnelly, C. A., Ghani,

A. C. & Anderson, R. M. Public health risk from the avian H5N1 influenza epidemic. _Science_ 304, 968–969 (2004). USES DISTRIBUTION OF OUTBREAK SIZES TO ASSESS WHETHER AN ANOMALOUSLY

LARGE CLUSTER HAS ARISEN. Article  CAS  Google Scholar  * Farrington, C. P., Kanaan, M. N. & Gay, N. J. Branching process models for surveillance of infectious diseases controlled by

mass vaccination. _Biostatistics_ 4, 279–295 (2003). Article  CAS  Google Scholar  * Jansen, V. A. et al. Measles outbreaks in a population with declining vaccine uptake. _Science_ 301, 804

(2003). USES CHANGING OUTBREAK SIZES TO ASSESS THE RISK OF RE-EMERGENCE OF MEASLES IN THE UK. Article  CAS  Google Scholar  * Cox, D. R. & Miller, H. D. _The Theory of Stochastic

Processes_. (Chapman and Hall, London, 1997). Google Scholar  * De Serres, G., Gay, N. J. & Farrington, C. P. Epidemiology of transmissible diseases after elimination. _Am. J.

Epidemiol._ 151, 1039–1048 (2000). Article  CAS  Google Scholar  * Keeling, M. J., Brooks, S. P. & Gilligan, C. A. Using conservation of pattern to estimate spatial parameters from a

single snapshot. _Proc. Natl Acad. Sci. USA_ 101, 9155–9160 (2004). Article  CAS  Google Scholar  * Matthews, L. et al. Super-shedding cattle and the transmission dynamics of _Escherichia

coli_ O157. _Epidemiol. Infect._ (in the press). USES DISTRIBUTION OF PREVALENCES ACROSS CATTLE POPULATIONS TO QUANTIFY WITHIN-HERD TRANSMISSION DYNAMICS AND IDENTIFY IMPORTANT

HETEROGENEITIES IN THE TRANSMISSION DYNAMICS. * Woolhouse, M. E. J. et al. Heterogeneities in the transmission of infectious agents: implications for the design of control programs. _Proc.

Natl Acad. Sci. USA_ 94, 338–342 (1997). Article  CAS  Google Scholar  * Cohen, M. L. Epidemiology of drug resistance — implications for a post-antimicrobial era. _Science_ 257, 1050–1055

(1992). Article  CAS  Google Scholar  * Austin, D. J., Bonten, M. J. M., Weinstein, R. A., Slaughter, S. & Anderson, R. M. Vancomycin-resistant enterococci in intensive-care hospital

settings: transmission dynamics, persistence, and the impact of infection control programs. _Proc. Natl Acad. Sci. USA_ 96, 6908–6913 (1999). Article  CAS  Google Scholar  * Pelupessy, I.,

Bonten, M. J. M. & Diekmann, O. How to assess the relative importance of different colonization routes of pathogens within hospital settings. _Proc. Natl Acad. Sci. USA_ 99, 5601–5605

(2002). USES A STOCHASTIC, INDIVIDUAL-BASED MODEL TO DESCRIBE THE TIME EVOLUTION OF THE DISTRIBUTION OF THE NUMBER OF INFECTED PATIENTS TO ASSESS RELATIVE RATES OF EXOGENOUS AND ENDOGENOUS

ACQUISITION OF ANTIBIOTIC RESISTANCE. Article  CAS  Google Scholar  * Cooper, B. S., Medley, G. F. & Scott, G. M. Preliminary analysis of the transmission dynamics of nosocomial

infections: stochastic and management effects. _J. Hosp. Infect._ 43, 131–147 (1999). Article  CAS  Google Scholar  * Sebille, V. & Valleron, A. J. A computer simulation model for the

spread of nosocomial infections caused by multidrug-resistant pathogens. _Comput. Biomed. Res._ 30, 307–322 (1997). Article  CAS  Google Scholar  * Cooper, B. & Lipsitch, M. The analysis

of hospital infection data using hidden Markov models. _Biostatistics_ 5, 223–237 (2004). Article  Google Scholar  * Gamerman, D. _Markov Chain Monte Carlo: Stochastic Simulation for

Bayesian Inference_ (Chapman and Hall, London, 1997). Google Scholar  * O'Neill, P. D. & Marks, P. J. Bayesian model choice and infection route modelling in an outbreak of

Norovirus. _Stat. Med._ 7 Apr. 2005 [epub ahead of print]. USES MCMC TO MODEL AND DETERMINE RISK FACTORS FOR THE ACQUISITION OF NOROVIRUS INFECTION IN SCHOOLCHILDREN. * Marks, P. J. et al.

Evidence for airborne transmission of Norwalk-like virus (NLV) in a hotel restaurant. _Epidemiol. Infect._ 124, 481–487 (2000). Article  CAS  Google Scholar  * Marks, P. J. et al. A school

outbreak of Norwalk-like virus: evidence for airborne transmission. _Epidemiol. Infect._ 131, 727–736 (2003). Article  CAS  Google Scholar  * Hufnagel, L., Brockmann, D. & Geisel, T.

Forecast and control of epidemics in a globalized world. _Proc. Natl Acad. Sci. USA_ 101, 15124–15129 (2004). Article  CAS  Google Scholar  * Eubank, S. et al. Modelling disease outbreaks in

realistic urban social networks. _Nature_ 429, 180–184 (2004). Article  CAS  Google Scholar  * Woolhouse, M. E. J., Haydon, D. T. & Antia, R. Emerging pathogens: the epidemiology and

evolution of species jumps. _Trends Ecol. Evol._ (in the press). * _Northumberland Report: Report of the Committee of Inquiry on Foot-and–Mouth Disease_ (Her Majesty's Stationary

Office, London, 1969). * Tinline, R. Lee wave hypothesis for initial pattern of spread during 1967–68 foot and mouth epizootic. _Nature_ 227, 860–862 (1970). Article  CAS  Google Scholar  *

Haydon, D. T., Woolhouse, M. E. J. & Kitching, R. P. An analysis of foot-and-mouth–disease epidemics in the UK. _IMA J. Math. Appl. Med. Biol._ 14, 1–9 (1997). Article  CAS  Google

Scholar  Download references ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of the Wellcome Trust through the International Partnership Research Award in Veterinary

Epidemiology consortium, the support of the Department for Environment, Food and Rural Affairs through the Veterinary Research Training Initiative and a Mathematical Biology Research

Training Fellowship to L.M. The authors also thank M. Chase-Topping and S. St Rose for their valuable contributions to the work described here. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS *

Veterinary Epidemiology Group, Centre for Tropical Veterinary Medicine, University of Edinburgh, Easter Bush Veterinary Centre, Roslin, EH25 9RG, Midlothian, Scotland Louise Matthews & 

Mark Woolhouse Authors * Louise Matthews View author publications You can also search for this author inPubMed Google Scholar * Mark Woolhouse View author publications You can also search

for this author inPubMed Google Scholar CORRESPONDING AUTHOR Correspondence to Louise Matthews. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing financial interests.

RELATED LINKS RELATED LINKS DATABASES ENTREZ _Escherichia coli_ O157 CDC INFECTIOUS DISEASE INFORMATION cholera influenza measles mumps Norovirus rubella SARS vancomycin-resistant

enterococci GLOSSARY * DETERMINISTIC A process that does not contain an element of chance. Deterministic models are often used to describe the progress of an epidemic through large

populations, in which small fluctuations at the individual level are assumed not to have an important effect on the dynamics. * STOCHASTIC A process that incorporates an element of chance;

every realization of the process can produce a different outcome. Stochastic effects are particularly important when the numbers involved are small, for example, at the start of, or during,

the 'tail' of an epidemic, when there are few infectious individuals. * COMPARTMENTAL MODEL A model in which discrete subsets of the host population are defined according to their

infection status. Commonly used compartments are susceptible, latent infected, infectious and recovered or removed. * METAPOPULATION MODEL A model comprising a set of epidemiologically

linked subpopulations. * EFFECTIVE REPRODUCTION RATIO The average number of secondary infections that is produced by one infected individual when that individual is introduced into a

population that might have been previously exposed to infection, contain vaccinated individuals or be subject to control measures to limit transmission. * GENERATION TIME The mean time

interval between an individual becoming infected and an individual that they infect becoming infected. * MICROSIMULATION MODEL A stochastic model in which each individual in the population

is represented explicitly, as opposed to tracking the number of individuals in each of a set of compartments. * CASE REPRODUCTION RATIO The average number of secondary infections that is

produced by a single individual infected at time t. It is typically smaller than the basic reproduction ratio, as factors such as depletion of susceptible individuals or implementation of

control measures will reduce the number of secondary cases generated. * PARAMETER SPACE The range of biologically plausible values that can be taken by the parameters of a model. RIGHTS AND

PERMISSIONS Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Matthews, L., Woolhouse, M. New approaches to quantifying the spread of infection. _Nat Rev Microbiol_ 3, 529–536

(2005). https://doi.org/10.1038/nrmicro1178 Download citation * Issue Date: 01 July 2005 * DOI: https://doi.org/10.1038/nrmicro1178 SHARE THIS ARTICLE Anyone you share the following link

with will be able to read this content: Get shareable link Sorry, a shareable link is not currently available for this article. Copy to clipboard Provided by the Springer Nature SharedIt

content-sharing initiative