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Metapopulation and Non-proportional Vaccination Models Overview

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Advances in the Mathematical Sciences

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 6))

Abstract

Influenza viruses are a major cause of morbidity and mortality worldwide. The 2009 influenza pandemic not only brought to our attention the strengths and weaknesses of the public health system but also changed their priorities. Vaccination still is the most powerful allay for preventing or mitigating influenza outbreaks or other diseases. In this article, we summarize our findings that arose from two different published research articles (see Cruz-Aponte et al. BMC Infect. Dis. 11(1), 207, (2011), [22], Herrera-Valdez et al. Math. Biosci. Eng. (MBE) 8(1), 21–48, (2011), [36]) that were presented at the Association of Woman in Mathematics (AWM) 2015 symposium at the University of Maryland College Park. The first one is a metapopulation model we constructed using the data from México’s 2009 epidemic patterns characterized by three peaks. These peak patterns were theoretically investigated via models that incorporate México’s general trends of land transportation, public health measures, and the academic calendar trends of that year. After studying many mathematical models that incorporated vaccination into the modeling, we were not satisfied with the simplification approaches that usually took place. Vaccinating only the susceptible individuals or vaccinating a fraction of the population was not realistic when supplies or daily administration capacity was considered. Hence, in the second project we presented a SIR-like model that explicitly takes into account vaccine supply and the number of vaccines administered per day and places data-informed limits on these parameters. The model that we refer to as non-proportional vaccination model is a theoretical improvement that provides more accurate predictions of the mitigating effects of vaccination than the typical proportional model. For some parameter regimes, proportional and non-proportional models behave the same, especially when the vaccination supplies were depleted but for others there were significant changes that predicted earlier or longer epidemics as we discuss further. Both of our models can be easily modified to be used by government and medical officials to create preparedness plans based on specific constraints.

This article is dedicated to my beloved friend Dr. Lukasz Adam Koscielski (1985–2015) cheers to you for a brief but brilliantly well lived life. Kocham Cie i Tesknie za Toba bardzo.

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Acknowledgments

I would like to thank the organizers of the Mathematical Biology session at the AWM Research Symposium for inviting me to give a talk: Dr. Erika T. Camacho and Dr. Talitha Washington. Also, I would like to thank my collaborators in the work presented at the symposium and summarize in this article Dr. Marco A. Herrera-Valdez, Dr. Erin C. McKiernan and Dr. Carlos Castillo Chavez for their support and their valuable discussions and feedback. Last but not least, thanks to my Figure 1 model register nurse Iris Aldecoa from Scottsdale Healthcare.

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Authors and Affiliations

  1. Department of Mathematics and Physics, University of Puerto Rico at Cayey, 205 Ave Antonio R Barcelo, Cayey, 00736-9997, Puerto Rico

    Mayteé Cruz-Aponte

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  1. Mayteé Cruz-Aponte
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Correspondence to Mayteé Cruz-Aponte .

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Editors and Affiliations

  1. National Security Agency , Fort Meade, Maryland, USA

    Gail Letzter

  2. Microsoft Research, Redmond, Washington, USA

    Kristin Lauter

  3. Department of Mathematics & Comp. Sci., Saint Louis University Department of Mathematics & Comp. Sci., Saint Louis, Missouri, USA

    Erin Chambers

  4. Department of Statistics, University of Missouri–Columbia, Columbia, Missouri, USA

    Nancy Flournoy

  5. Department of Mathematics , Boston College Department of Mathematics, Chestnut Hill, Massachusetts, USA

    Julia Elisenda Grigsby

  6. National Security Agency , Fort Meade, Maryland, USA

    Carla Martin

  7. Department of Mathematics & Comp. Sci., DeSales University Department of Mathematics & Comp. Sci., Center Valley, Pennsylvania, USA

    Kathleen Ryan

  8. Department of Mathematics, University of Maryland Department of Mathematics, College Park, Maryland, USA

    Konstantina Trivisa

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Cruz-Aponte, M. (2016). Metapopulation and Non-proportional Vaccination Models Overview. In: Letzter, G., et al. Advances in the Mathematical Sciences. Association for Women in Mathematics Series, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-34139-2_8

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