A minimal model for household effects in epidemics

Abstract

<jats:p>Shelter-in-place and other confinement strategies implemented in the current COVID-19 pandemic have created stratified patterns of contacts between people: close contacts within households and more distant contacts between the households. The epidemic transmission dynamics is significantly modified as a consequence. We introduce a minimal model that incorporates these household effects in the framework of mean-field theory and numerical simulations. We show that the reproduction number <jats:italic>R</jats:italic><jats:sub>0</jats:sub> depends on the household size in a surprising way: linearly for relatively small households, and as a square root of size for larger households. We discuss the implications of the findings for the lockdown, test, tracing, and isolation policies.</jats:p>

A minimal model for household-based testing and tracing in epidemics

“…The assumptions made are close to those in [1]. An individual can be in one of the following states (we assume no natural immunity): susceptible s, infected i, recovered r and quarantined q, as shown in Figure 1.…”

“…A household state can therefore be described by a four-dimensional vector of integers, (s, i, q, r). We make the same assumption as in [1]: the intra-household rate of infection is fast. Then the simplified state diagram for a household is shown in Figure 2.…”

“…In a previous work [1] we proposed a minimal model describing epidemics in such a compartmentalized population. The key feature of the model was the separation of infection into a fast intra-household mode and a slow inter-household mode.…”

A random-walk-based epidemiological model

“…Classic epidemiological models 1,2 often assume panmictic populations: every infected person has an equal chance to affect any other person in the population. While these models have been successful in describing an ideal dynamics of epidemic spread, they do not account for inhomogeneity: an infected person has higher probability to transmit the disease to a member of their household 3,4 or to a person in their locale. The overall reduction of long-distance travel due to the COVID-19 pandemic makes the latter cause of inhomogeneity especially relevant: many infected persons (but not all) spread the infection only in their immediate vicinity.…”

“…There are two approaches to geographical inhomogeneity: detailed analyses based on contact-tracing and mobility data, 5,6 or analyses based on stylized models. 4,7,8 The first approach, while potentially highly accurate, requires a large number of parameters and is not robust with respect to unforeseen changes in mobility patterns. The second approach, in contrast, requires a small number of well-defined parameters and provides a sound intuition about their effects on the outcome.…”