Traditional epidemiological models rely heavily on Ordinary Differential Equations (ODEs) to track temporal changes in disease states. However, ODEs often struggle to capture the continuous interaction density across age cohorts in a stationary state. This article proposes a Fredholm Integral Equation (FIE) approach to model the steady-state distribution of an infectious disease within a population. By utilizing a transmission kernel that represents the aggregate contact patterns between various age groups over a fixed period, we derive a model that focuses on equilibrium rather than evolution. We apply the Nystrom method for numerical simulation. Results indicate that age-specific contact intensity is the primary driver of infection "hotspots," independent of initial infection time. This approach provides a robust framework for long-term public health planning and targeted vaccination strategies.

Keywords: Fredholm Integral Equations, Age-structured Modeling, Steady-state Distribution, Epidemiology, Numerical Simulation, Nyström Method.