Analysis of COVID-19 Infection Dynamics: Extended SIR Model Approach

Abstract: This paper presents a detailed mathematical investigation into the dynamics of COVID-19 infections through extended Susceptible-Infected-Recovered (SIR) and Susceptible-Exposed-Infected-Recovered (SEIR) epidemiological models. By incorporating demographic factors such as birth and death rates, we enhance the classical Kermack-McKendrick framework to realistically represent long-term disease progression. Using empirical data from four COVID-19 epidemic waves in Orange County, California, between January 2020 and March 2022, we estimate key parameters and perform stability and bifurcation analyses. Our results consistently indicate endemic states characterized by stable spiral equilibria due to reproduction numbers (R0) exceeding unity across all waves. Additionally, the inclusion of vaccination demonstrates the potential to reduce the effective reproduction number below one, shifting the system towards a stable disease-free equilibrium. Our analysis underscores the critical role of latency periods in shaping epidemic dynamics and highlights actionable insights for public health interventions aimed at COVID-19 control and eventual eradication.