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A general model for stochastic SIR epidemics with two levels of mixing
1School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. frank.ball@nottingham.ac.uk
Abstract:
This paper is concerned with a general stochastic model for susceptible-->infective-->removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution 'centred' on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R(*) governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective's household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.
Insights
This study introduces a general stochastic model for epidemics, analyzing how local and global contacts influence disease spread in large populations. It identifies a key threshold parameter determining epidemic occurrence and impact.
Area of Science:
- Epidemiology
- Mathematical Biology
- Stochastic Processes
Background:
- Understanding epidemic dynamics in finite populations is crucial for public health.
- Previous models often simplified contact structures, limiting applicability.
- The interplay of local and global contacts in disease transmission requires further investigation.
Purpose of the Study:
- To develop a general stochastic model for susceptible-infective-removed (SIR) epidemics in finite populations.
- To analyze the impact of both local and global contacts on epidemic threshold behavior.
- To unify the study of epidemic models by introducing concepts of local infectious clumps and susceptibility sets.
Main Methods:
- Development of a general stochastic SIR model with local and global contacts.
- Asymptotic analysis for large populations with fixed local contact distributions.
- Introduction of 'local infectious clump' and 'local susceptibility set' concepts.
- Derivation of a threshold parameter R(*) for global epidemic occurrence.
Main Results:
- A unified approach to threshold behavior in epidemic models is established.
- The threshold parameter R(*) is determined, governing the potential for global epidemics.
- Formulas are derived for the probability of a global epidemic and the proportion of infected susceptibles.
- The model is specialized to specific contact structures: households, overlapping groups, and the great circle model.
Conclusions:
- The developed model provides a robust framework for analyzing epidemic dynamics with heterogeneous contact patterns.
- The threshold parameter R(*) is a critical determinant of epidemic potential, applicable across various contact structures.
- The findings offer insights into disease spread mechanisms and can inform public health interventions.