A stochastic SIR network epidemic model with preventive dropping of edges

Ball, Frank; Britton, Tom; Yin Leung, Ka; Sirl, David

Abstract

A Markovian SIR (Susceptible – Infectious - Recovered) model is considered for the spread of an epidemic on a configuration model network, in which susceptible individuals may take preventive measures by dropping edges to infectious neighbours. An effective degree formulation of the model is used in conjunction with the theory of density dependent population processes to obtain a law of large numbers and a functional central limit theorem for the epidemic as the population size N - ¥, assuming that the degrees of individuals are bounded. A central limit theorem is conjectured for the final size of the epidemic. The results are obtained for both the Molloy–Reed (in which the degrees of individuals are deterministic) and Newman–Strogatz–Watts (in which the degrees of individuals are independent and identically distributed) versions of the configuration model. The two versions yield the same limiting deterministic model but the asymptotic variances in the central limit theorems are greater in the Newman–Strogatz–Watts version. The basic reproduction number R0 and the process of susceptible individuals in the limiting deterministic model, for the model with dropping of edges, are the same as for a corresponding SIR model without dropping of edges but an increased recovery rate, though, when R0 > 1, the probability of a major outbreak is greater in the model with dropping of edges. The results are specialised to the model without dropping of edges to yield conjectured central limit theorems for the final size of Markovian SIR epidemics on configuration-model networks, and for the giant components of those networks. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations are good, even for moderate N.

Citation

Ball, F., Britton, T., Yin Leung, K., & Sirl, D. (2019). A stochastic SIR network epidemic model with preventive dropping of edges. Journal of Mathematical Biology, 78(6), 1875-1951. https://doi.org/10.1007/s00285-019-01329-4

• Journal Article Type: Article
• Acceptance Date: Jan 18, 2019
• Online Publication Date: Mar 13, 2019
• Publication Date: May 1, 2019
• Deposit Date: Feb 6, 2019
• Publicly Available Date: Mar 14, 2020
• Journal: Journal of Mathematical Biology
• Print ISSN: 0303-6812
• Electronic ISSN: 1432-1416
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 78
• Issue: 6
• Pages: 1875-1951
• DOI: https://doi.org/10.1007/s00285-019-01329-4
• Keywords: Agricultural and Biological Sciences (miscellaneous); Modelling and Simulation; Applied Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/1522863
• Publisher URL: https://link.springer.com/article/10.1007%2Fs00285-019-01329-4
• Additional Information: Received: 4 June 2018; Revised: 18 January 2019; First Online: 13 March 2019
• Contract Date: Feb 6, 2019