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Journal of Mathematical Biology

, Volume 57, Issue 3, pp 311–331 | Cite as

Deterministic epidemiological models at the individual level

  • Kieran J. SharkeyEmail author
Open Access
Article

Abstract

In many fields of science including population dynamics, the vast state spaces inhabited by all but the very simplest of systems can preclude a deterministic analysis. Here, a class of approximate deterministic models is introduced into the field of epidemiology that reduces this state space to one that is numerically feasible. However, these reduced state space master equations do not in general form a closed set. To resolve this, the equations are approximated using closure approximations. This process results in a method for constructing deterministic differential equation models with a potentially large scope of application including dynamic directed contact networks and heterogeneous systems using time dependent parameters. The method is exemplified in the case of an SIR (susceptible-infectious-removed) epidemiological model and is numerically evaluated on a range of networks from spatially local to random. In the context of epidemics propagated on contact networks, this work assists in clarifying the link between stochastic simulation and traditional population level deterministic models.

Keywords

Master equations ODE Heterogeneous contact networks Population dynamics Individual based models Deterministic models Epidemic Pair approximations 

Mathematics Subject Classification (2000)

92D25 92D30 92D40 00A71 00A72 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematical SciencesThe University of LiverpoolLiverpoolUK
  2. 2.Epidemiology Research Group, Department of Veterinary Clinical ScienceThe University of LiverpoolLeahurstUK
  3. 3.Manchester Interdisciplinary BiocentreThe University of ManchesterManchesterUK

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