Threshold parameters for a model of epidemic spread among households and workplaces

The basic reproduction number R 0 is one of the most important concepts in modern infectious disease epidemiology. However, for more realistic and more complex models than those assuming homogeneous mixing in the population, other threshold quantities can be defined that are sometimes more useful and easily derived in terms of model parameters. In this paper, we present a model for the spread of a permanently immunizing infection in a population socially structured into households and workplaces/schools, and we propose and discuss a new household-to-household reproduction number R H for it. We show how R H overcomes some of the limitations of a previously proposed threshold parameter, and we highlight its relationship with the effort required to control an epidemic when interventions are targeted at randomly selected households.

[1]  C. Fraser,et al.  Reducing the impact of the next influenza pandemic using household-based public health interventions. , 2006, Hong Kong medical journal = Xianggang yi xue za zhi.

[2]  H. Andersson,et al.  Stochastic Epidemic Models and Their Statistical Analysis , 2000 .

[3]  A. Nizam,et al.  Containing pandemic influenza with antiviral agents. , 2004, American journal of epidemiology.

[4]  N. Stollenwerk,et al.  Measles Outbreaks in a Population with Declining Vaccine Uptake , 2003, Science.

[5]  A. Flahault,et al.  Estimating the impact of school closure on influenza transmission from Sentinel data , 2008, Nature.

[6]  Frank Ball,et al.  The threshold behaviour of epidemic models , 1983, Journal of Applied Probability.

[7]  F. Ball,et al.  Epidemics with two levels of mixing , 1997 .

[8]  N. Grassly,et al.  Mathematical models of infectious disease transmission , 2008, Nature Reviews Microbiology.

[9]  J. Radcliffe,et al.  The Mathematical Theory of Infectious Diseases and its Applications , 1977 .

[10]  M. G. Roberts,et al.  A new method for estimating the effort required to control an infectious disease , 2003, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[11]  I. Longini,et al.  Household and community transmission parameters from final distributions of infections in households. , 1982, Biometrics.

[12]  D. Cummings,et al.  Strategies for containing an emerging influenza pandemic in Southeast Asia , 2005, Nature.

[13]  W. Edmunds,et al.  Who mixes with whom? A method to determine the contact patterns of adults that may lead to the spread of airborne infections , 1997, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[14]  Frank Ball,et al.  A general model for stochastic SIR epidemics with two levels of mixing. , 2002, Mathematical biosciences.

[15]  C. Fraser.  Estimating Individual and Household Reproduction Numbers in an Emerging Epidemic , 2007, PloS one.

[16]  M. Elizabeth Halloran,et al.  Containing Pandemic Influenza at the Source , 2005, Science.

[17]  F. Ball.  A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models , 1986, Advances in Applied Probability.

[18]  Frank Ball,et al.  Stochastic multitype epidemics in a community of households: Estimation of threshold parameter R* and secure vaccination coverage , 2004 .

[19]  F. Ball,et al.  Optimal vaccination policies for stochastic epidemics among a population of households. , 2002, Mathematical biosciences.

[20]  K Dietz,et al.  Threshold parameters for epidemics in different community settings. , 1995, Mathematical biosciences.

[21]  R. May,et al.  Infectious Diseases of Humans: Dynamics and Control , 1991, Annals of Internal Medicine.

[22]  N. Becker,et al.  Optimal vaccination strategies for a community of households. , 1997, Mathematical biosciences.

[23]  M G Roberts,et al.  An integral equation model for the control of a smallpox outbreak. , 2005, Mathematical biosciences.

[24]  M. Keeling,et al.  Household structure and infectious disease transmission , 2008, Epidemiology and Infection.

[25]  D. Cummings,et al.  Strategies for mitigating an influenza pandemic , 2006, Nature.

[26]  Donald Ludwig,et al.  Final size distribution for epidemics , 1975 .

[27]  K Dietz,et al.  The effect of household distribution on transmission and control of highly infectious diseases. , 1995, Mathematical biosciences.

[28]  M G Roberts,et al.  Modelling strategies for minimizing the impact of an imported exotic infection , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[29]  E Ackerman,et al.  Stochastic two-agent epidemic simulation models for a community of families. , 1971, American journal of epidemiology.

[30]  N. Ferguson,et al.  Planning for smallpox outbreaks , 2003, Nature.

[31]  N Wilson,et al.  A model for the spread and control of pandemic influenza in an isolated geographical region , 2007, Journal of The Royal Society Interface.

[32]  J. Staněk,et al.  Stochastic Epidemic Models , 2006 .

[33]  Kathryn Glass,et al.  Controlling emerging infectious diseases like SARS. , 2005, Mathematical biosciences.

[34]  S. Riley,et al.  Smallpox transmission and control: Spatial dynamics in Great Britain , 2006, Proceedings of the National Academy of Sciences.

[35]  Frank Ball,et al.  Network epidemic models with two levels of mixing. , 2008, Mathematical biosciences.

[36]  A. Nizam,et al.  Containing Pandemic Influenza at the Source , 2005, Science.

[37]  Klaus Dietz,et al.  Reproduction Numbers and Critical Immunity Levels for Epidemics in a Community of Households , 1996 .

[38]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[39]  S. Utev,et al.  The effect of community structure on the immunity coverage required to prevent epidemics. , 1998, Mathematical biosciences.

[40]  M. Newman.  Spread of epidemic disease on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  A. Langworthy,et al.  An influenza simulation model for immunization studies. , 1976, American journal of epidemiology.

[42]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[43]  J A P Heesterbeek,et al.  The type-reproduction number T in models for infectious disease control. , 2007, Mathematical biosciences.

[44]  R. Bartoszynski.  On a certain model of an epidemic , 1972 .

[45]  C. Macken,et al.  Mitigation strategies for pandemic influenza in the United States. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[46]  Lorenzo Pellis,et al.  The relationship between real-time and discrete-generation models of epidemic spread. , 2008, Mathematical biosciences.