Abstract
In this paper we present a novel and coherent modelling framework for the characterisation of the real-time growth rate in SIR models of epidemic spread in populations with social structures of increasing complexity. Known results about homogeneous mixing and multitype models are included in the framework, which is then extended to models with households and models with households and schools/workplaces. Efficient methods for the exact computation of the real-time growth rate are presented for the standard SIR model with constant infection and recovery rates (Markovian case). Approximate methods are described for a large class of models with time-varying infection rates (non-Markovian case). The quality of the approximation is assessed via comparison with results from individual-based stochastic simulations. The methodology is then applied to the case of influenza in models with households and schools/workplaces, to provide an estimate of a household-to-household reproduction number and thus asses the effort required to prevent an outbreak by targeting control policies at the level of households. The results highlight the risk of underestimating such effort when the additional presence of schools/workplaces is neglected. Our framework increases the applicability of models of epidemic spread in socially structured population by linking earlier theoretical results, mainly focused on time-independent key epidemiological parameters (e.g. reproduction numbers, critical vaccination coverage, epidemic final size) to new results on the epidemic dynamics.
Article PDF
Similar content being viewed by others
References
Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74(1): 47–97
Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University Press, Oxford
Andersson H, Britton T (2000) Stochastic epidemic models and their statistical analysis. In: Lecture notes in statistics, vol 151. Springer, New York
Ball F (1986) A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv Appl Probab 18(2): 289–310
Ball FG, Britton T (2007) An epidemic model with infector-dependent severity. Adv Appl Probab 39(4): 949
Ball F, Neal P (2002) A general model for stochastic SIR epidemics with two levels of mixing. Math Biosci 180: 73–102
Ball FG, Neal P (2008) Network epidemic models with two levels of mixing. Math Biosci 212(1): 69–87
Ball F, Mollison D, Scalia-Tomba G (1997) Epidemics with two levels of mixing. Ann Appl Probab 7(1): 46–89
Bansal S, Grenfell BT, Meyers LA (2007) When individual behaviour matters: homogeneous and network models in epidemiology. J R Soc Interface 4(16): 879
Bartoszyński R (1972) On a certain model of an epidemic. Appl Math 13(2): 139–151
Becker NG, Dietz K (1995) The effect of household distribution on transmission and control of highly infectious diseases. Math Biosci 127: 207–219
Cauchemez S, Carrat F, Viboud C, Valleron AJ, Boelle PY (2004) A Bayesian MCMC approach to study transmission of influenza: application to household longitudinal data. Stat Med 23(22): 3469–3487
Cauchemez S, Valleron AJ, Boelle PY, Flahault A, Ferguson NM (2008) Estimating the impact of school closure on influenza transmission from Sentinel data. Nature 452(7188): 750–754
Cauchemez S, Donnelly CA, Reed C, Ghani AC, Fraser C, Kent CK, Finelli L, Ferguson NM (2009) Household transmission of 2009 pandemic influenza a (h1n1) virus in the united states. N Engl J Med 361(27): 2619–2627
Colizza V, Barrat A, Barthelemy M, Valleron AJ, Vespignani A (2007) Modeling the worldwide spread of pandemic influenza: Baseline case and containment interventions. PLoS Med 4(1): e13
Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley series in mathematical and computational biology. Wiley, Chichester
Diekmann O, Heesterbeek JAP, Roberts MG (2010) The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface 7(47): 873–885
Edmunds WJ, O’Callaghan CJ, Nokes DJ (1997) Who mixes with whom? a method to determine the contact patterns of adults that may lead to the spread of airborne infections. Proc R Soc Lond Ser B 264(1384): 949–957
Ferguson NM, Cummings DAT, Cauchemez S, Fraser C, Riley S, Meeyai A, Iamsirithaworn S, Burke DS (2005) Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature 437: 209–214
Ferguson NM, Cummings DAT, Fraser C, Cajka JC, Cooley PC, Burke DS (2006) Strategies for mitigating an influenza pandemic. Nature 442(7101): 448
Fraser C (2007) Estimating individual and household reproduction numbers in an emerging epidemic. PLoS ONE 2(8): e758
Fraser C, Donnelly CA, Cauchemez S, Hanage WP, Van Kerkhove MD, Hollingsworth TD, Griffin J, Baggaley RF, Jenkins HE, Lyons EJ, Jombart T, Hinsley WR, Grassly NC, Balloux F, Ghani AC, Ferguson NM, Rambaut A, Pybus OG, Lopez-Gatell H, Alpuche-Aranda CM, Chapela IB, Zavala EP, Guevara DME, Checchi F, Garcia E, Hugonnet S, Roth C, Collaboration TWRPA (2009) Pandemic potential of a strain of influenza A (H1N1): early findings. Science 324(5934): 1557–1561
Goldstein E, Paur K, Fraser C, Kenah E, Wallinga J, Lipsitch M (2009) Reproductive numbers, epidemic spread and control in a community of households. Math Biosci 221(1): 11–25
Grassly NC, Fraser C (2008) Mathematical models of infectious disease transmission. Nat Rev Microbiol 6(6): 477–487
Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42(4): 599–653
Kendall WS, Saunders IW (1983) Epidemics in competition II: the general epidemic. J R Stat Soc Ser B 45(2): 238–244
Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond Ser A 115: 700–721
Ludwig D (1975) Final size distributions for epidemics. Math Biosci 23: 33
Neal P (2006) Multitype randomized reed-frost epidemics and epidemics upon random graphs. Ann Appl Probab 16(3): 1166–1189
Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45(2): 167–256
Pellis L (2009) Mathematical models for emerging infections in socially structured populations: the presence of households and workplaces. PhD thesis, Imperial College London. http://hdl.handle.net/10044/1/4696
Pellis L, Ferguson NM, Fraser C (2008) The relationship between real-time and discrete-generation models of epidemic spread. Math Biosci 216(1): 63–70
Pellis L, Ferguson NM, Fraser C (2009) Threshold parameters for a model of epidemic spread among households and workplaces. J R Soc Interface 6(40): 979–987
Picard P, Lefèvre C (1990) A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes. Adv Appl Probab 22(2): 269
Riley S (2007) Large-scale spatial-transmission models of infectious disease. Science 316(5829): 1298–1301
Riley S, Fraser C, Donnelly CA, Ghani AC, Abu-Raddad LJ, Hedley AJ, Leung GM, Ho LM, Lam TH, Thach TQ, Chau P, Chan KP, Lo SV, Leung PY, Tsang T, Ho W, Lee KH, Lau EMC, Ferguson NM, Anderson RM (2003) Transmission dynamics of the etiological agent of sars in hong kong: Impact of public health interventions. Science 300(5627): 1961–1966
Roberts M, Heesterbeek J (2007) Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection. J Math Biol 55(5): 803–816
Ross JV, House T, Keeling MJ (2010) Calculation of disease dynamics in a population of households. PLoS ONE 5(3): e9666
Svensson Å, Scalia-Tomba G (2001) Competing epidemics in closed populations. Research report/Mathematical Statistics, Stockholm University, 2001:8. Mathem Stat SU, Stockholm
UK census data (2001) http://www.ons.gov.uk/census/
UK National Statistics (2005) http://www.statistics.gov.uk/STATBASE/ssdataset.asp?vlnk=9390
Wald A (1947) Sequential analysis. Wiley, New York
Wallinga J, Lipsitch M (2007) How generation intervals shape the relationship between growth rates and reproductive numbers. Proc R Soc B: Biol Sci 274(1609): 599–604
Wu JT, Riley S, Fraser C, Leung GM (2006) Reducing the impact of the next influenza pandemic using household-based public health interventions. PLoS Med 3(9): 1532–1540
Xia Y, Bjørnstad ON, Grenfell BT (2004) Measles metapopulation dynamics: a gravity model for epidemiological coupling and dynamics. Am Nat 164(2): 267–281
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pellis, L., Ferguson, N.M. & Fraser, C. Epidemic growth rate and household reproduction number in communities of households, schools and workplaces. J. Math. Biol. 63, 691–734 (2011). https://doi.org/10.1007/s00285-010-0386-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-010-0386-0
Keywords
- Stochastic epidemic models
- Real-time growth rate
- Household reproduction number
- Structured populations
- Two levels of mixing
- Epidemic dynamics