Mathematical analysis of an age- and socio-economic-stratified transmission model

Abstract

The COVID-19 pandemic underscored the need for models that capture heterogeneity in disease transmission and outcomes. We present a deterministic epidemic model that integrates age and socio-economic stratification to examine how population structure influences epidemic thresholds and stability. The framework extends the classic susceptible–exposed–infectious–recovered paradigm to eight compartments: susceptible, exposed, asymptomatic, undetected symptomatic, detected symptomatic, hospitalized, recovered and deceased. Individuals are divided into 21 age groups and 10 deprivation deciles, enabling joint assessment of demographic and socio-economic disparities in transmission. A block-structured age-deprivation contact matrix and a time-dependent testing function reflect evolving test capacity. Analytical results establish positivity and boundedness of solutions, with local and global stability of the disease-free equilibrium (DFE) when the basic reproduction number R0<1. In the absence of demographic processes, no strictly positive endemic equilibrium exists, though a quasi-endemic equilibrium emerges when R0>1. Using the next-generation matrix (NGM) approach, we derive an explicit expression for R0 and identify key parameters influencing transmission. Contact rates, recovery and hospitalization most strongly affect R0, while testing and isolation alone cannot reduce transmission below threshold levels. The model highlights earlier and higher peaks among deprived populations, informing equitable and targeted control strategies.