Critical immune and vaccination thresholds for determining multiple influenza epidemic waves.

Previous influenza pandemics (1918, 1957, and 1968) have all had multiple waves. The 2009 pandemic influenza A (H1N1) (pandemic H1N1) started in April 2009 and was followed, in the United States (US) and temperate Northern Hemisphere, by a second wave during the fall of 2009. The ratio of susceptible and immune individuals in a population at the end of a wave determines the potential and magnitude of a subsequent wave. As influenza vaccines are not completely protective, there was a combined immunity in the population at the beginning of 2010 (due to vaccination and due to previous natural infection), and it was uncertain if this mixture of herd immunity was enough to prevent a third wave of pandemic influenza during the winter of 2010. Motivated by this problem, we developed a mathematical deterministic two-group epidemic model with vaccination and calibrated it for the 2009 pandemic H1N1. Then, applying methods from mathematical epidemiology we developed a scheme that allowed us to determine critical thresholds for vaccine-induced and natural immunity that would prevent the spread of influenza. Finally, we estimated the level of combined immunity in the US during winter 2010. Our results suggest that a third wave was unlikely if the basic reproduction number R(0) were below 1.6, plausible if the original R(0) was 1.6, and likely if the original R(0) was 1.8 or higher. Given that the estimates for the basic reproduction number for pandemic influenza place it in the range between 1.4 and 1.6 (Bacaer and Ait Dads, 2011; Fraser et al., 2009; Munayco et al., 2009; Pourbohloul et al., 2009; Tuite et al., 2010; White et al., 2009; Yang et al., 2009), our approach accurately predicted the absence of a third wave of influenza in the US during the winter of 2010. We also used this scheme to accurately predict the second wave of pandemic influenza in London and the West Midlands, UK during the fall of 2009.