The progression of the Covid-19 pandemic in South Africa, as measured by the number of deaths in excess of the seasonally-expected mortality, is inconsistent with infectivity and fatality rates estimated elsewhere in the world.A compartment model applied in prognostic and inverse mode was used to explore the combinations of key parameters that could account for the outbreak timing and magnitude of Covid-19 mortality in the Western Cape Province in South Africa, where information on reported cases and deaths are most reliable.If it is assumed that the entire population is susceptible, the total number of deaths by October 2020, after six months of propagation, is far lower than expected for fatality and infection rates in the plausible range. The unimodal infection peak, followed by a substantial decline coinciding with the relaxation of non-pharmacological interventions, suggests that herd immunity has been reached, but serological tests for exposure indicate that too few people have been infected for this to be the case. We show that the number of excess deaths, timing of the peak infection and proportion of the population infected can all be explained by assuming 30-40% pre-existing cross-immunity in the population.The dynamics of Covid-19 in South Africa cannot be explained under the assumption of a fully susceptible population. Rather than postulating, against the evidence, that the coronavirus exhibits fundamentally low infectivity or morbidity rates in South Africa, we point out that wide-spread, pre-existing cross-reactive immunity to SARS-CoV-2, could explain the observations. If this is the case, the likelihood of a large second wave is low if those infected do not rapidly lose immunity.Funding Statement: The salaries of Engelbrecht and Scholes are funded by the University of the Witwatersrand. Otherwise, no dedicated source of funding supported this research, and neither author received funding from any source to write the paper.Declaration of Interests: The authors have nothing to disclose.