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COVID-19 has become endemic, with dynamics that reflect the waning of immunity and re-exposure, by contrast to the epidemic phase driven by exposure in immunologically naïve populations. Endemic does not, however, mean constant. Further evolution of SARS-CoV-2, as well as changes in behaviour and public health policy, continue to play a major role in the endemic load of disease and mortality. In this paper, we analyse evolutionary models to explore the impact that newly arising variants can have on the short-term and longer-term endemic load, characterizing how these impacts depend on the transmission and immunological properties of variants. We describe how evolutionary changes in the virus will increase the endemic load most for persistently immune-escape variants, by an intermediate amount for more transmissible variants, and least for transiently immune-escape variants. Balancing the tendency for evolution to favour variants that increase the endemic load, we explore the impact of vaccination strategies and non-pharmaceutical interventions (NPIs) that can counter these increases in the impact of disease. We end with some open questions about the future of COVID-19 as an endemic disease.