Fig. 4

A Illustration for \(\Delta t_{10\%-50\%}\) [defined in eq. (9)] taking two examples: a less fitter new variant (orange, \(\Delta t_{1}\)) and stronger new variant (blue, \(\Delta t_{2}\)) curves, respectively. B \(\Delta t_{10\%-50\%}\) for different relative fitness \(\hat{R}(t=0)_{\textrm{omicron}}/\hat{R}(t=0)_{\textrm{delta}}\), preimmunization (a) and cross-immunity (\(\epsilon\)) values. We see that in the regime of \(\hat{R}(t=0)_{\textrm{omicron}}/\hat{R}(t=0)_{\textrm{delta}}> 1.5\), preimmunization and cross-immunity play a minor role in the steepness of variant replacement. On x-axis labels, “new” ans “established” stand for omicron and delta, respectively