Equilibrium points and their stability of COVID-19 in US

Abstract

This study aims to develop an advanced mathematic model and investigate when and how will the COVID-19 in the US be evolved to endemic. We employed a nonlinear ordinary differential equations-based model to simulate COVID-19 transmission dynamics, factoring in vaccination efforts. Multi-stability analysis was performed on daily new infection data from January 12, 2021 to December 12, 2022 across 50 states in the US. Key indices such as eigenvalues and the basic reproduction number were utilized to evaluate stability and investigate how the pandemic COVD-19 will evolve to endemic in the US. The transmissional, recovery, vaccination rates, vaccination effectiveness, eigenvalues and reproduction numbers ($R_0$ and $R_0^{\mathit{end}}$) in the endemic equilibrium point were estimated. The stability attractor regions for these parameters were identified and ranked. Our multi-stability analysis revealed that while the endemic equilibrium points in the 50 states remain unstable, there is a significant trend towards stable endemicity in the US. The study's stability analysis, coupled with observed epidemiological waves in the US, suggested that the COVID-19 pandemic may not conclude with the virus's eradication. Nevertheless, the virus is gradually becoming endemic. Effectively strategizing vaccine distribution is pivotal for this transition.