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J Theor Biol. 2009 Dec 21;261(4):584-92. doi: 10.1016/j.jtbi.2009.08.020. Epub 2009 Aug 22.

Qualitative analysis of the level of cross-protection between epidemic waves of the 1918-1919 influenza pandemic.

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Mathematical, Computational and Modeling Sciences Center, School of Human Evolution and Social Change, Arizona State University, Tempe, AZ 85287, USA.


The 1918-1919 influenza pandemic was composed of multiple waves within a period of nine months in several regions of the world. Increasing our understanding of the mechanisms responsible for this multi-wave profile has important public health implications. We model the transmission dynamics of two strains of influenza interacting via cross-immunity to simulate two temporal waves of influenza and explore the impact of the basic reproduction number, as a measure of transmissibility associated to each influenza strain, cross-immunity and the timing of the onset of the second influenza epidemic on the pandemic profile. We use time series of case notifications during the 1918 influenza pandemic in Geneva, Switzerland, for illustration. We calibrate our mathematical model to the initial wave of infection to estimate the basic reproduction number of the first wave and the corresponding timing of onset of the second influenza variant. We use this information to explore the impact of cross-immunity levels on the dynamics of the second wave of influenza. Our results for the 1918 pandemic in Geneva, Switzerland, indicate that a second wave can occur whenever R 01 < 1.5 or when cross-immunity levels are less than 0.58 for our estimated R 02 of 2.4. We also explore qualitatively profiles of two-wave pandemics and compare them with real temporal profiles of the 1918 influenza pandemic in other regions of the world including several Scandinavian cities, New York City, England and Wales, and Sydney, Australia. Pandemic profiles are classified into three broad categories namely "right-handed", "left-handed", and "M-shape". Our results indicate that avoiding a second influenza epidemic is plausible given sufficient levels of cross-protection are attained via natural infection during an early (herald) wave of infection or vaccination campaigns prior to a second wave. Furthermore, interventions aimed at mitigating the first pandemic wave may be counterproductive by increasing the chances of a second wave of infection that could potentially be more virulent than the first.

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