Skip to main content
Article thumbnail
Location of Repository

Calculation of disease dynamics in a population of households

By Joshua V. Ross, Thomas A. House and Matthew James Keeling


Early mathematical representations of infectious disease dynamics assumed a single, large, homogeneously mixing population. Over the past decade there has been growing interest in models consisting of multiple smaller subpopulations (households, workplaces, schools, communities), with the natural assumption of strong homogeneous mixing within each subpopulation, and weaker transmission between subpopulations. Here we consider a model of SIRS (susceptible-infectious-recovered-suscep​tible) infection dynamics in a very large (assumed infinite) population of households, with the simplifying assumption that each household is of the same size (although all methods may be extended to a population with a heterogeneous distribution of household sizes). For this households model we present efficient methods for studying several quantities of epidemiological interest: (i) the threshold for invasion; (ii) the early growth rate; (iii) the household offspring distribution; (iv) the endemic prevalence of infection; and (v) the transient dynamics of the process. We utilize these methods to explore a wide region of parameter space appropriate for human infectious diseases. We then extend these results to consider the effects of more realistic gamma-distributed infectious periods. We discuss how all these results differ from standard homogeneous-mixing models and assess the implications for the invasion, transmission and persistence of infection. The computational efficiency of the methodology presented here will hopefully aid in the parameterisation of structured models and in the evaluation of appropriate responses for future disease outbreaks

Topics: RA
Publisher: Public Library of Science
Year: 2010
OAI identifier:

Suggested articles


  1. (2002). A general model for stochastic sir epidemics with two levels of mixing. doi
  2. (1991). A generalized stochastic model for the analysis of infectious disease final size data. doi
  3. (1986). A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemics models. doi
  4. (2007). An event-based model of superspreading in epidemics. doi
  5. (1998). Collective dynamics of ‘‘small-world.
  6. (2002). Containing bioterrorist smallpox. doi
  7. (2006). Demographic structure and pathogen dynamics on the network of livestock movements in great britain. doi
  8. (1956). Deterministic and stochastic models for recurrent epidemics.
  9. (2008). Deterministic epidemic models with explicit household structure. doi
  10. (2005). Duelling timescales of host movement and disease recovery determine invasion of disease in structured populations. doi
  11. (1997). Epidemics with two levels of mixing. doi
  12. (1982). Estimating household and community transmission parameters for influenza.
  13. (2007). Estimating individual and household reproduction numbers in an emerging epidemic. doi
  14. (2004). Forecast and control of epidemics in a globalized world. doi
  15. (2008). Household structure and infectious disease transmission. doi
  16. (1992). Infectious diseases of humans. Oxford: doi
  17. (2003). Integrals for continuous-time markov chains. doi
  18. (2009). Integrating stochasticity and network structure into an epidemic model. doi
  19. (1997). Markov chains. Cambridge: doi
  20. (2000). Mathematical Epidemiology of Infectious Diseases Model Building, Analysis and Interpretation. doi
  21. (1927). McKendrick A doi
  22. (2008). Modeling infectious diseases in humans and animals. doi
  23. (1999). Network structural dynamics and infectious disease propagation. doi
  24. (2005). Networks and epidemic models. doi
  25. (2001). Networks and pathogens. doi
  26. (2008). On methods for studying stochastic disease dynamics. doi
  27. (2009). On parameter estimation in population models ii: multi-dimensional processes and transient dynamics. Theor Pop doi
  28. (2002). Path integrals for continuous-time markov chains. doi
  29. (2007). Recent network evolution increases the potential for large epidemics in the british cattle population. doi
  30. (2006). Reducing the impact of the next influenza pandemic using household-based public health interventions. doi
  31. (2009). Reproductive numbers, epidemic spread and control in a community of households. doi
  32. (2004). Sis epidemics with household structure: the self-consistent field method. doi
  33. (1970). Solutions of ordinary differential equations as limits of pure jump markov processes. doi
  34. (1999). Stochastic and deterministic models for sis epidemics among a population partitioned into households. doi
  35. (2002). Stochastic effects on endemic infection levels of disseminating versus local contacts. doi
  36. (2000). Stochastic Epidemic Models and Their Statistical Analysis. doi
  37. (2001). Stochastic multitype sir epidemics among a population partitioned into households. doi
  38. (2006). Strategies for mitigating an influenza pandemic. doi
  39. (2005). Superspreading and the effect of individual variation on disease emergence. doi
  40. (1995). Susceptible-infectious-recovered epidemic models with dynamic partnerships. doi
  41. (1967). The distribution of the total size of an epidemic.
  42. (1998). The effect of community structure on the immunity coverage required to prevent epidemics. doi
  43. (1995). The effect of household distribution on transmission and control of highly infectious diseases. doi
  44. (1999). The effects of local spatial structure on epidemiological invasions. doi
  45. (1957). The mathematical theory of epidemics. doi
  46. (2009). Threshold behaviour and final outcome of an epidemic on a random network with household structure. doi
  47. (2009). Threshold parameters for a model of epidemic spread among households and workplaces. doi
  48. (2002). Understanding the persistence of measles: reconciling theory, simulation and observation. doi
  49. (2007). Utility of r0 as a predictor of disease invasion in structured populations. doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.