## Annals of Applied Probability

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

### Epidemics with two levels of mixing

Frank Ball, Denis Mollison, and Gianpaolo Scalia-Tomba

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 14 October 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1034625252

**Digital Object Identifier**

doi:10.1214/aoap/1034625252

**Mathematical Reviews number (MathSciNet)**

MR1428749

**Zentralblatt MATH identifier**

0909.92028

**Subjects**

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.)

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aoap/1034625252