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.
Highlights
The basic reproduction number R0 is arguably the most important epidemiological parameter because of its clear biological interpretation and its properties: in the simplest epidemic models, where individuals are all identical, mix homogeneously, the population is large and the initial number of infectives is small, (i) a large epidemic is possible if and only if R0 > 1, (ii) when R0 > 1, vaccinating a fraction 1 − 1/R0 of individuals chosen uniformly at random – or, equivalently, isolating the same fraction of infected individuals before they have the chance to transmit further – is sufficient to prevent a large outbreak and (iii) the fraction of the population infected by a large epidemic depends only on R0
A notable exception is if the underlying epidemic model is Markovian and most of our numerical examples are for such models
We know that both R0 and RI are strictly greater than 1, and we have proved above that RI ≥ RV ≥ R0, so we need consider only a declining epidemic
Summary
The basic reproduction number R0 is arguably the most important epidemiological parameter because of its clear biological interpretation and its properties: in the simplest epidemic. We show that R0 for these models may be obtained more from the discrete-time Lotka-Euler equation (cf Equation (5)) that describes the asymptotic (Malthusian) geometric growth rate of the mean population size of an associated branching process, which approximates the early phase of the epidemic. Variant of R2 from [7] Perfect vaccine-associated reproduction number Leaky vaccine-associated reproduction number Exponential-growth-associated reproduction number
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