Abstract
A nonlinear mathematical model for the spread of influenza A (H1N1) infectious diseases including the role of vaccination is proposed and analyzed. It is assumed that the susceptibles become infected by direct contact with infectives and exposed population. We take under consideration that only a susceptible person can be vaccinated and that the vaccine is not 100% efficient. The model is analyzed using stability theory of differential equations and numerical simulation. We have found a threshold condition, in terms of vaccination reproduction number \({\mathcal{R}_V}\) which is, if less than one, the disease dies out provided the vaccine efficacy is high enough, and otherwise the infection is maintained in the population. It is also shown that the spread of an infectious disease increases as the infective rate increases.
![](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40065-012-0013-6/MediaObjects/40065_2012_13_Article_Figa_HTML.gif)
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bowman C.S., Arino J., Moghadas S.M.: Evaluation of vaccination strategies during pandemic outbreaks. Math. Biosci. Eng. 8(1), 113–122 (2011)
Collins, G.E.; Akritas, A.G.: Polynomial real root isolation using Descartes rule of signs. In: Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computations, pp. 272–275. Yorktown Heights, NY (1976)
Flu Home. http://www.flu.gov/prevention-vaccination/index.html
Fonda A.: Uniformly persistent semidynamical systems. Proc. Am. Math. Soc. 104(1), 111–116 (1988)
Gumel A.B., Connell McCluskey C., Watmough J.: An SVEIR model for assessing potential impact of an imperfect anti-SARS vaccine. Math. Biosci. Eng. 3(3), 485–512 (2006)
Hethcote, H.W.: Three basic epidemiological models. In: Gross, L.; Hallam, T.G.; Levin, S.A. (eds.) Applied Mathematical Ecology. Springer, Berlin (1989)
Hsieh, Y.H.; Fisman, D.N.; Wu, J.H.: On epidemic modeling in real time: an application to the 2009 Novel A (H1N1) influenza outbreak in Canada. BMC Res. Notes 3, 283 (2010)
Kolata, G.: Flu: The Story of the Great Influenza Pandemic of 1918 and the Search for the Virus that Caused it. Farrar, Straus and Giroux, New York (1999)
Lakshmikantham, V.; Leela, S.: Differential and Integral Inequalities, vols. I and II. Academic Press, New York (1969)
Leekha, S.; Zitterkopf, N.L.; Espy, M.J.; Smith, T.F.; Thompson, R.L.; et al.: Duration of influenza A virus shedding in hospitalized patients and implications for infection control. Infect. Control Hosp. Epidemiol. 28, 1071–1076 (2007)
Miller M.A., Viboud C., Balinska M., Simonsen L.: The signature features of influenza pandemics—implications for policy. N. Engl. J. Med. 360, 2595–2598 (2009)
Mukandavire Z., Garira W.: Sex-structured HIV/AIDS model to analyse the effects of condom use with application to Zimbabwe. J. Math. Biol. 54, 669–699 (2007)
Patterson K.D., Pyle G.F.: The geography and mortality of the 1918 influenza pandemic. Bull. Hist. Med. 65, 4–21 (1991)
Swine influenza A (H1N1) infection in two children—Southern California, March-April 2009. MMWR Morb Mortal Wkly Rep 58, 400–402 (2009)
Thieme R.H.: Persistence under relaxed point-dissipativity (with application to an endemic model). SIAM J. Math. Anal. 24, 407–435 (1993)
Tracht, S.M.; Del Valle, S.Y.; Hyman, J.M.: Mathematical modeling of the effectiveness of facemasks in reducing the spread of novel influenza A (H1N1). PLoS One 5(2), e9018(2010)
Tuite A.R., Greer A.L., Whelan M. et al.: Estimated epidemiologic parameters and morbidity associated with pandemic H1N1 influenza. CMAJ 182(2), 131–136 (2009)
Varga, R.S.: Matrix Iterative Analysis. Prentice-Hall, Inc., Englewood Cliffs (1962)
van den Driessche P., Watmough J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)
World Health Organisation: Global Alert and Response. Pandemic (H1N1) 2009—update 69. http://www.who.int/csr/don/2009_10_09/en/index.html
World Health Organisation: Global Alertand Response (GAR) WHO Recommendations on Pandemic (H1N1) Vaccines. http://www.who.int/csr/disease/swineflu/notes/h1n1_vaccine_20090713/en/ (2009)
Acknowledgments
We would like to thank the anonymous referees for their careful reading of the original manuscript and their many valuable comments and suggestions that greatly improved the presentation of this work. This work is supported by the National Natural Science Foundation of China (No. 11071011) and Natural Science Foundation of the Education Department of Henan Province (Nos. 2010A110017 and 2011B110028).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Zhou, X., Guo, Z. Analysis of an influenza A (H1N1) epidemic model with vaccination. Arab. J. Math. 1, 267–282 (2012). https://doi.org/10.1007/s40065-012-0013-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-012-0013-6