Semi-Random Mixing Epidemic Model: Integrating Explicit Household and Non-household Interactions

Understanding how infectious diseases spread through populations requires models that capture real human interactions more realistically. Many classical epidemic models assume that everyone mixes randomly, overlooking the structured and clustered nature of daily contacts within households, workplaces, and schools. These simplifications can limit their predictive capability and the design of effective control strategies. To improve on this limitation, we develop an epidemic modelling framework that integrates explicitly both household and non-household interactions. Building on the semi-random mixing (SeRaMix) concept in the literature, the model captures how people move between different major contact settings (household and work/school) each day, interacting within and across clusters of contacts. We introduce a novel formulation linking contact duration and proximity to infection risk, enabling a more realistic representation of disease-specific transmission factors in an equation-based model. Analytical derivation of the basic reproduction number (R0) demonstrates how epidemic potential distinctly depends on social behaviour, mobility, and biological parameters. Validation against an individual-based model confirms that this equation-based framework reproduces epidemic dynamics within this structured network of interaction. Sensitivity analyses identify the number of contacts at home and work/school, mobility, and inter-household connections as key non-pathogen-dependent drivers of epidemic growth. This framework offers a more realistic approach to analysing the impacts of non-pharmaceutical interventions--such as reduced mobility, hybrid working, and household bubbles--on outbreak trajectories.