The Equilibrium and Pandemic Waves of COVID-19 in the US

Importance Removing the epidemic waves and reducing the instability level of an endemic critical point of COVID-19 dynamics are fundamental to the control of COVID-19 in the US. Objective To develop new mathematic models and investigate when and how will the COVID-19 in the US be evolved to endemic. Design, Setting, and Participants To solve the problem of whether mass vaccination against SARS-CoV-2 will ultimately end the COVID-19 pandemic, we defined a set of nonlinear ordinary differential equations as a mathematical model of transmission dynamics of COVID-19 with vaccination. Multi-stability analysis was conducted on the data for the daily reported new cases of infection from January 12, 2021 to December 12, 2022 across 50 states in the US using the developed dynamic model of COVID-19 and limit cycle theory. Main Outcomes and Measures Eigenvalues and the reproduction number under the disease-free equilibrium point and endemic equilibrium point were used to assess the stability of the disease-free equilibrium point and endemic equilibrium point. Both analytic analysis and numerical methods were used to determine the instability level of new cases of COVID-19 in the US under the different types of equilibrium points and to investigate how the system moves back and forth between stable and unstable states of the system and how the pandemic COVD-19 will evolve to endemic in the US. Results Multi-stability analysis identified two types of critical equilibrium points, disease-free endemic equilibrium points in the COVID-19 transmission dynamic system. The transmissional, recovery, vaccination rates and vaccination effectiveness during the major transmission waves of COVID-19 across 50 states in the US were estimated. These parameters in the model varied over time and across the 50 states. The eigenvalues and the reproduction numbers R and ![Graphic][1] in the disease-free equilibrium point and endemic equilibrium point were estimated to assess stability and classify equilibrium points. They also varied from state to state. The impacts of the transmission and vaccination parameters on the stability of COVID-19 were simulated, and stability attractor regions of these parameters were found and ranked for all 50 states in the US. The US experienced five major epidemic waves, endemic equilibrium points of which across 50 states were all in unstable states. However, the combination of re-infection and vaccination (hybrid immunity) may provide strong protection against COVID-19 infection, and stability analysis showed that these unstable equilibrium points were toward stable points. Theoretical analysis and real data analysis showed that additional epidemic waves may be possible in the future, but COVID-19 across all 50 sates in the US is rapidly moving toward stable endemicity. Conclusions and Relevance Both stability analysis and observed epidemic waves in the US indicated that the pandemic might not end with the disappearance of the virus. However, after enough people gained immune protection from vaccination and from natural infection, COVID-19 would become an endemic disease, as the stability analysis showed. Educating the population about multiple epidemic waves of the transmission dynamics of COVID-19 and designing optimal vaccine rollout are crucial for controlling the pandemic of COVID-19 and its evolving to endemic. Questio The US has already experienced five waves of the epidemic. We urgently need to know when and how will COVID-19 be evolved into endemic. Findings To solve the problem, we developed a mathematical model of transmission dynamics of COVID-19 with vaccination and performed a multi-stability analysis of COVID-19 transmission dynamics in the US. We found that COVID-19 dynamics of all 50 states in the US were getting closer and closer to endemic and stable states. Meaning COVID-19 dynamics of all 50 states in the US are toward stable states and will be evolved to endemic in the near future. ### Competing Interest Statement The authors have declared no competing interest. ### Funding Statement Zixin Hu in this study was partially supported by funding from the National Natural Science Foundation of China (3210040426 to Z.H.), the Shanghai Rising-Star Program (21QB1400900 to Z.H.), and was also partly supported by a grant from the major project of Study on Pathogenesis and Epidemic Prevention Technology System (2021YFC2302500) by the Ministry of Science and Technology of China. ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. 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Yes I have followed all appropriate research reporting guidelines and uploaded the relevant EQUATOR Network research reporting checklist(s) and other pertinent material as supplementary files, if applicable. Yes All data produced are available online at . . [1]: /embed/inline-graphic-1.gif