Modeling Household Effects in Epidemics

Households play a significant role in the spread of diseases, with close and frequent interactions among members facilitating the transmission process. Although measures to control disease transmission primarily target interactions between households, the size of households can significantly influence transmission rates. To examine the impact of household size and size distribution, it is necessary to model disease progression within individual households and between households. Since typically only a fraction of household members are infected, the model must consider the division of households into susceptible and infected subgroups. By considering the infection of a household as a process that splits it into a new, fully infected sub-household and a remaining susceptible sub-household, a compartmental ordinary differential equation (ODE) model is developed to describe the dynamics of these sub-households. Using this framework, it becomes possible to analytically compute key epidemiological indicators such as the basic reproduction number, prevalence, and peak of an infection wave in a population with a given distribution of household sizes. The validity of these findings is supported by graph-theoretic considerations. Furthermore, the numerical simulation results of this household-ODE model are compared with results from an agent-based model that incorporates realistic household size distributions from different countries. Both models demonstrate the significant role played by larger households in influencing the overall dynamics of disease transmission.