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

TL;DR

The paper extends classical SIR/SEIR epidemic models by incorporating demographic turnover to analyze long-term COVID-19 dynamics and endemism. It uses four regional waves to estimate wave-specific transmission rates and to examine fixed points through linearization, identifying a transcritical bifurcation at $R_0=1$ and stable endemic spirals when $R_0>1$. The study shows that vaccination can reduce the effective reproduction number to $R_v=R_0(1-p)$, potentially eliminating the endemic state if $R_v<1$, and highlights the latency period in shaping outbreak timing via the SEIR extension. Phase-portrait and bifurcation analyses provide mechanistic insights for policy: achieving sufficient vaccination coverage and timely interventions can transition the system from endemicity to disease elimination, albeit with caveats about population structure and immunity dynamics.

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.

Figures (3)

• Figure 1: COVID-19 infection waves in Orange County (2020-2022) showing the four distinct surges analyzed in this study: the initial summer 2020 surge, winter 2020-2021 surge, Delta variant surge (mid-2021), and Omicron variant surge (early 2022). Data shown as 7-day rolling sums normalized by population.
• Figure 2: Bifurcation diagram for the extended SIR model illustrating the transition in equilibrium stability as the basic reproduction number $R_0$ varies. Stability regimes indicated are: stable disease-free equilibrium ($R_0 < 1$), transcritical bifurcation at $R_0 = 1$, and stable endemic equilibrium ($R_0 > 1$). This diagram clearly illustrates the critical threshold that defines epidemic outbreak dynamics.
• Figure 3: Phase portrait illustrating the dynamics of Wave 1 of the COVID-19 epidemic using the extended SIR model. Blue curves represent trajectories of epidemic progression starting from various initial conditions, while red vectors indicate the direction and relative speed of change within the system. The yellow diagonal line denotes the boundary of biologically feasible states. Equilibrium points are visible as convergence points along trajectories, highlighting the stable spiral behavior observed in this wave.