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Long- and short-term effects of cross-immunity in epidemic dynamics

The vertebrate immune system is capable of strong, focused adaptive responses that depend on T-cell specificity in recognizing antigenic sequences of a pathogen. Recognition tolerance and antigenic convergence cause cross-immune reactions that extend prompt, specific responses to rather similar pathogens. This suggests that reaching herd-immunity might be facilitated during successive epidemic outbreaks (e.g., SARS-CoV-2 waves with different variants). Qualitative studies play down this possibility because cross-immune protection is seldom sterilizing. We use minimal quantitative models to study how cross-immunity affects epidemic dynamics over short and long timescales. In the short scale, we investigate models of sterilizing and attenuating immunity, finding equivalences between both mechanisms---thus suggesting a key role of attenuating protection in achieving herd immunity. Our models render maps in epidemic-parameter space that discern threatening variants depending on acquired cross-immunity levels. We illustrate this application with SARS-CoV-2 data, including protection due to vaccination rates across countries. In the long-time scale, we model sterilizing cross-immunity between rolling pathogens to characterize statistical properties of successful strains. We find that sustained cross-immune protection alters the regions of epidemic-parameter space where large outbreaks happen. Our results suggest an optimistic revision concerning prospects for herd protection based on cross-immunity, including for the SARS-CoV-2 pandemics.