The Annals of Applied Probability

Epidemics with two levels of mixing

Frank Ball, Denis Mollison, and Gianpaolo Scalia-Tomba

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We consider epidemics with removal (SIR epidemics) in populations that mix at two levels: global and local. We develop a general modelling framework for such processes, which allows us to analyze the conditions under which a large outbreak is possible, the size of such outbreaks when they can occur and the implications for vaccination strategies, in each case comparing our results with the simpler homogeneous mixing case.

More precisely, we consider models in which each infectious individual i has a global probability $p_G$ for infecting each other individual in the population and a local probability $p_L$, typically much larger, of infecting each other individual among a set of neighbors $\mathscr{N}(i)$. Our main concern is the case where the population is partitioned into local groups or households, but our approach also applies to cases where neighborhoods do not form a partition, for instance, to spatial models with a mixture of local (e.g., nearest-neighbor) and global contacts.

We use a variety of theoretical approaches: a random graph framework for the initial exposition of the simple case where an individual's contacts are independent; branching process approximations for the general threshold result; and an embedding representation for rigorous results on the final size of outbreaks.

From the applied viewpoint the key result is that, compared with the homogeneous mixing model in which individuals make contacts simply with probability $p_G$, the local infectious contacts have an "amplification" effect. The basic reproductive ratio of the epidemic is increased from its individual-to-individual value $R_G$ in the absence of local infections to a group-to-group value $R_* = \mu R_G$, where $\mu$ is the mean size of an outbreak, started by a randomly chosen individual, in which only local infections count. Where the groups are large and the within-group epidemics are above threshold, this amplification can permit an outbreak in the whole population at very low levels of $p_G$, for instance, for $p_G = O(1/Nn)$ in a population of N divided into groups of size n.

The implication of these results for control strategies is that vaccination should be directed preferentially toward reducing $\mu$; we discuss the conditions under which the equalizing strategy, aimed at leaving unvaccinated sets of neighbors of equal sizes, is optimal. We also discuss the estimation of our threshold parameter $R_*$ from data on epidemics among households.

Article information

Ann. Appl. Probab. Volume 7, Number 1 (1997), 46-89.

First available in Project Euclid: 14 October 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92D30: Epidemiology 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 05C80: Random graphs [See also 60B20] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Epidemic household vaccination estimation final outcome random graphs Reed-Frost branching processes Gontcharoff polynomials threshold parameters asymptotic distribution embedding $R_0$


Ball, Frank; Mollison, Denis; Scalia-Tomba, Gianpaolo. Epidemics with two levels of mixing. Ann. Appl. Probab. 7 (1997), no. 1, 46--89. doi:10.1214/aoap/1034625252.

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