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This paper introduces a mathematical framework for determining second surge behavior of COVID-19 cases in the United States. Within this framework, a flexible algorithmic approach selects a set of turning points for each state, computes distances between them, and determines whether each state is in (or over) a first or second surge. Then, appropriate distances between normalized time series are used to further analyze the relationships between case trajectories on a month-by-month basis. Our algorithm shows that 31 states are experiencing second surges, while four of the 10 largest states are still in their first surge, with case counts that have never decreased. This analysis can aid in highlighting the most and least successful state responses to COVID-19. [ABSTRACT FROM AUTHOR]