Reproduction numbers for epidemic models with households and other social structures II: comparisons and implications for vaccination

Ball, Frank; Pellis, Lorenzo; Trapman, Pieter

Abstract

In this paper we consider epidemic models of directly transmissible SIR (susceptible - infective - recovered) and SEIR (with an additional latent class) infections in fully-susceptible populations with a social structure, consisting either of households or of households and workplaces. We review most reproduction numbers defined in the literature for these models, including the basic reproduction number R0 introduced in the companion paper of this, for which we provide a simpler, more elegant derivation. Extending previous work, we provide a complete overview of the inequalities among these reproduction numbers and resolve some open questions. Special focus is put on the exponential-growth-associated reproduction number Rr, which is loosely defined as the estimate of R0 based on the observed exponential growth of an emerging epidemic obtained when the social structure is ignored. We show that for the vast majority of the models considered in the literature Rr >= R0 when R0 >=1 and Rr <= R0 when R0 <= 1. We show that, in contrast to models without social structure, vaccination of a fraction 1-1/R0 of the population, chosen uniformly at random, with a perfect vaccine is usually insufficient to prevent large epidemics. In addition, we provide significantly sharper bounds than the existing ones for bracketing the critical vaccination coverage between two analytically tractable quantities, which we illustrate by means of extensive numerical examples.

Citation

Ball, F., Pellis, L., & Trapman, P. (2016). Reproduction numbers for epidemic models with households and other social structures II: comparisons and implications for vaccination. Mathematical Biosciences, 274, https://doi.org/10.1016/j.mbs.2016.01.006

• Journal Article Type: Article
• Acceptance Date: Jan 7, 2016
• Online Publication Date: Feb 2, 2016
• Publication Date: Apr 1, 2016
• Deposit Date: Jun 22, 2016
• Publicly Available Date: Jun 22, 2016
• Journal: Mathematical Biosciences
• Print ISSN: 0025-5564
• Electronic ISSN: 1879-3134
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 274
• DOI: https://doi.org/10.1016/j.mbs.2016.01.006
• Keywords: SIR epidemic; Household; Social structure; Basic reproduction number; Vaccination; Exponential growth rate
• Public URL: https://nottingham-repository.worktribe.com/output/977230
• Publisher URL: http://www.sciencedirect.com/science/article/pii/S0025556416000171
• Contract Date: Jun 22, 2016