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BioMed Research International
Volume 2014, Article ID 375862, 23 pages
http://dx.doi.org/10.1155/2014/375862
Research Article

The Impact of Coinfections and Their Simultaneous Transmission on Antigenic Diversity and Epidemic Cycling of Infectious Diseases

1Department of Statistics, Modelling and Economics, Centre for Infectious Disease Surveillance and Control, Public Health England, 61 Colindale Avenue, London NW9 5EQ, UK
2Medical Research Council Centre for Outbreak Analysis and Modelling, Department of Infectious Disease Epidemiology, Imperial College Faculty of Medicine, Norfolk Place, London W2 1PG, UK
3Centre for Nonlinear Complex Systems, Department of Physics, Yunnan University, Kunming, Yunnan 650091, China

Received 6 February 2014; Revised 18 April 2014; Accepted 18 April 2014; Published 22 June 2014

Academic Editor: Laith Abu-Raddad

Copyright © 2014 Xu-Sheng Zhang and Ke-Fei Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, UK, 1991.
  2. N. J. Dimmock, A. J. Easton, and K. N. Leppard, Introduction to Modern Virology, Blackwell Publishing, Oxford, UK, 6th edition, 2007.
  3. C. J. E. Metcalf, O. N. Bjørnstad, B. T. Grenfell, and V. Andreasen, “Seasonality and comparative dynamics of six childhood infections in pre-vaccination Copenhagen,” Proceedings of the Royal Society B: Biological Sciences, vol. 276, no. 1676, pp. 4111–4118, 2009. View at Publisher · View at Google Scholar · View at Scopus
  4. S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual, and P. Rohani, “Seasonality and the dynamics of infectious diseases,” Ecology Letters, vol. 9, no. 4, pp. 467–484, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. V. E. Pitzer, C. Viboud, B. A. Lopman, M. M. Patel, U. D. Parashar, and B. T. Grenfell, “Influence of birth rates and transmission rates on the global seasonality of rotavirus incidence,” Journal of the Royal Society Interface, vol. 8, no. 64, pp. 1584–1593, 2011. View at Publisher · View at Google Scholar · View at Scopus
  6. E. Lofgren, N. H. Fefferman, Y. N. Naumov, J. Gorski, and E. N. Naumova, “Influenza seasonality: underlying causes and modeling theories,” Journal of Virology, vol. 81, no. 11, pp. 5429–5436, 2007. View at Publisher · View at Google Scholar · View at Scopus
  7. H. Broutin, J.-F. Guégan, E. Elguero, F. Simondon, and B. Cazelles, “Large-scale comparative analysis of pertussis population dynamics: periodicity, synchrony, and impact of vaccination,” The American Journal of Epidemiology, vol. 161, no. 12, pp. 1159–1167, 2005. View at Publisher · View at Google Scholar · View at Scopus
  8. N. C. Grassly and C. Fraser, “Seasonal infectious disease epidemiology,” Proceedings of the Royal Society B: Biological Sciences, vol. 273, no. 1600, pp. 2541–2550, 2006. View at Publisher · View at Google Scholar · View at Scopus
  9. Y. A. Kuznetsov and C. Piccardi, “Bifurcation analysis of periodic SEIR and SIR epidemic models,” Journal of Mathematical Biology, vol. 32, no. 2, pp. 109–121, 1994. View at Google Scholar · View at Scopus
  10. K. Koelle and M. Pascual, “Disentangling extrinsic from intrinsic factors in disease dynamics: a nonlinear time series approach with an application to cholera,” The American Naturalist, vol. 163, no. 6, pp. 901–913, 2004. View at Publisher · View at Google Scholar · View at Scopus
  11. M. J. Ferrari, R. F. Grais, N. Bharti et al., “The dynamics of measles in sub-Saharan Africa,” Nature, vol. 451, no. 7179, pp. 679–684, 2008. View at Publisher · View at Google Scholar · View at Scopus
  12. J. Truscott, C. Fraser, S. Cauchemez et al., “Essential epidemiological mechanisms underpinning the transmission dynamics of seasonal influenza,” Journal of the Royal Society Interface, vol. 9, no. 67, pp. 304–312, 2012. View at Publisher · View at Google Scholar · View at Scopus
  13. S. F. Dowell, “Seasonal variation in host susceptibility and cycles of certain infectious diseases,” Emerging Infectious Diseases, vol. 7, no. 3, pp. 369–374, 2001. View at Google Scholar · View at Scopus
  14. J. Dushoff, J. B. Plotkin, S. A. Levin, and D. J. D. Earn, “Dynamical resonance can account for seasonality of influenza epidemics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 101, no. 48, pp. 16915–16916, 2004. View at Publisher · View at Google Scholar · View at Scopus
  15. N. C. Grassly, C. Fraser, and G. P. Garnett, “Host immunity and synchronized epidemics of syphilis across the United States,” Nature, vol. 433, no. 7024, pp. 417–421, 2005. View at Publisher · View at Google Scholar · View at Scopus
  16. H. W. Hethcote, “Mathematics of infectious diseases,” Society for Industrial and Applied Mathematics Review, vol. 42, no. 4, pp. 599–653, 2000. View at Google Scholar · View at Scopus
  17. H. W. Hethcote and S. A. Levin, “Periodicity in epidemiological models,” in Applied Mathematical Ecology, S. A. Levin, T. G. Hallam, and L. J. Gross, Eds., vol. 18 of Biomathematics, pp. 193–211, Springer, Berlin, Germany, 1989. View at Google Scholar
  18. Z. Feng and H. R. Thieme, “Recurrent outbreaks of childhood diseases revisited: the impact of isolation,” Mathematical Biosciences, vol. 128, no. 1-2, pp. 93–130, 1995. View at Publisher · View at Google Scholar · View at Scopus
  19. J. Tamerius, M. I. Nelson, S. Z. Zhou, C. Viboud, M. A. Miller, and W. J. Alonso, “Global influenza seasonality: reconciling patterns across temperate and tropical regions,” Environmental Health Perspectives, vol. 119, no. 4, pp. 439–445, 2011. View at Publisher · View at Google Scholar · View at Scopus
  20. B. Grenfell and O. Bjørnstad, “Sexually transmitted diseases: epidemic cycling and immunity,” Nature, vol. 433, no. 7024, pp. 366–367, 2005. View at Publisher · View at Google Scholar · View at Scopus
  21. G. Snounou, S. Viriyakosol, W. Jarra, S. Thaithong, and K. N. Brown, “Identification of the four human malaria parasite species in field samples by the polymerase chain reaction and detection of a high prevalence of mixed infections,” Molecular and Biochemical Parasitology, vol. 58, no. 2, pp. 283–292, 1993. View at Publisher · View at Google Scholar · View at Scopus
  22. B. Briseño-García, H. Gómez-Dantés, E. Argott-Ramírez et al., “Potential risk for dengue hemorrhagic fever: the isolation of serotype dengue-3 in Mexico,” Emerging Infectious Diseases, vol. 2, no. 2, pp. 133–135, 1996. View at Google Scholar
  23. S. Toda, R. Okamoto, T. Nishida et al., “Isolation of influenza A/H3 and B viruses from an influenza patient: confirmation of co-infection by two influenza viruses,” Japanese Journal of Infectious Diseases, vol. 59, no. 2, pp. 142–143, 2006. View at Google Scholar · View at Scopus
  24. L. Denoeud, C. Turbelin, S. Ansart, A.-J. Valleron, A. Flahault, and F. Carrat, “Predicting pneumonia and influenza mortality from morbidity data,” PLoS ONE, vol. 2, no. 5, article e464, 2007. View at Publisher · View at Google Scholar · View at Scopus
  25. M. Lipsitch and J. J. O'Hagan, “Patterns of antigenic diversity and the mechanisms that maintain them,” Journal of the Royal Society Interface, vol. 4, no. 16, pp. 787–802, 2007. View at Publisher · View at Google Scholar · View at Scopus
  26. J. Parkhill and B. W. Wren, “Bacterial epidemiology and biology—lessons from genome sequencing,” Genome Biology, vol. 12, no. 10, article 230, 2011. View at Publisher · View at Google Scholar · View at Scopus
  27. L. O. Bakaletz, “Developing animal models for polymicrobial diseases,” Nature Reviews Microbiology, vol. 2, no. 7, pp. 552–568, 2004. View at Publisher · View at Google Scholar · View at Scopus
  28. M. Peacey, R. J. Hall, S. Sonnberg et al., “Pandemic (H1N1) 2009 and seasonal influenza A (H1N1) co-infection, New Zealand, 2009,” Emerging Infectious Diseases, vol. 16, no. 10, pp. 1618–1620, 2010. View at Publisher · View at Google Scholar · View at Scopus
  29. W. Liu, Z.-D. Li, F. Tang et al., “Mixed infections of pandemic H1N1 and seasonal H3N2 viruses in 1 outbreak,” Clinical Infectious Diseases, vol. 50, no. 10, pp. 1359–1365, 2010. View at Publisher · View at Google Scholar · View at Scopus
  30. S. Räsänen, S. Lappalainen, S. Kaikkonen, M. Hämäläinen, M. Salminen, and T. Vesikari, “Mixed viral infections causing acute gastroenteritis in children in a waterborne outbreak,” Epidemiology and Infection, vol. 138, no. 9, pp. 1227–1234, 2010. View at Publisher · View at Google Scholar · View at Scopus
  31. H. Tanner, E. Boxall, and H. Osman, “Respiratory viral infections during the 2009-2010 winter seasons in Central England, UK: incidence and patterns of multiple virus co-infections,” European Journal of Clinical Microbiology and Infectious Diseases, vol. 31, no. 11, pp. 3001–3006, 2012. View at Google Scholar
  32. O. Balmer and M. Tanner, “Prevalence and implications of multiple-strain infections,” The Lancet Infectious Diseases, vol. 11, no. 11, pp. 868–878, 2011. View at Publisher · View at Google Scholar · View at Scopus
  33. E. C. Griffiths, A. B. Pedersen, A. Fenton, and O. L. Petchey, “The nature and consequences of coinfection in humans,” Journal of Infection, vol. 63, no. 3, pp. 200–206, 2011. View at Publisher · View at Google Scholar · View at Scopus
  34. M. Lipsitch, C. Colijn, T. Cohen, W. P. Hanage, and C. Fraser, “No coexistence for free: neutral null models for multistrain pathogens,” Epidemics, vol. 1, no. 1, pp. 2–13, 2009. View at Publisher · View at Google Scholar · View at Scopus
  35. T. DaPalma, B. P. Doonan, N. M. Trager, and L. M. Kasman, “A systematic approach to virus-virus interactions,” Virus Research, vol. 149, no. 1, pp. 1–9, 2010. View at Publisher · View at Google Scholar · View at Scopus
  36. X.-S. Zhang, J. Holt, and J. Colvin, “Synergism between plant viruses: a mathematical analysis of the epidemiological implications,” Plant Pathology, vol. 50, no. 6, pp. 732–746, 2001. View at Publisher · View at Google Scholar · View at Scopus
  37. J. Saldaña, S. F. Elena, and R. V. Solé, “Coinfection and superinfection in RNA virus populations: a selection-mutation model,” Mathematical Biosciences, vol. 183, no. 2, pp. 135–160, 2003. View at Publisher · View at Google Scholar · View at Scopus
  38. N. Ferguson, R. Anderson, and S. Gupta, “The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens,” Proceedings of the National Academy of Sciences of the United States of America, vol. 96, no. 2, pp. 790–794, 1999. View at Publisher · View at Google Scholar · View at Scopus
  39. D. A. T. Cummings, I. B. Schwartz, L. Billings, L. B. Shaw, and D. S. Burke, “Dynamic effects of antibody-dependent enhancement on the fitness of viruses,” Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 42, pp. 15259–15264, 2005. View at Publisher · View at Google Scholar · View at Scopus
  40. A. L. Graham, I. M. Cattadori, J. O. Lloyd-Smith, M. J. Ferrari, and O. N. Bjørnstad, “Transmission consequences of coinfection: cytokines writ large?” Trends in Parasitology, vol. 23, no. 6, pp. 284–291, 2007. View at Publisher · View at Google Scholar · View at Scopus
  41. S. Gorman, N. L. Harvey, D. Moro et al., “Mixed infection with multiple strains of murine cytomegalovirus occurs following simultaneous or sequential infection of immunocompetent mice,” Journal of General Virology, vol. 87, no. 5, pp. 1123–1132, 2006. View at Publisher · View at Google Scholar · View at Scopus
  42. B. M. Davis, A. E. Aiello, S. Dawid, P. Rohani, S. Shrestha, and B. Foxman, “Influenza and community-acquired pneumonia interactions: the impact of order and time of infection on population patterns,” The American Journal of Epidemiology, vol. 175, no. 5, pp. 363–367, 2012. View at Publisher · View at Google Scholar · View at Scopus
  43. S. L. Epstein, “Prior H1N1 influenza infection and susceptibility of Cleveland Family Study participants during the H2N2 pandemic of 1957: an experiment of nature,” Journal of Infectious Diseases, vol. 193, no. 1, pp. 49–53, 2006. View at Publisher · View at Google Scholar · View at Scopus
  44. K. Hancock, V. Veguilla, X. Lu et al., “Cross-reactive antibody responses to the 2009 pandemic H1N1 influenza virus,” The New England Journal of Medicine, vol. 361, no. 20, pp. 1945–1952, 2009. View at Publisher · View at Google Scholar · View at Scopus
  45. M. L. B. Hillaire, S. E. van Trierum, J. H. C. M. Kreijtz et al., “Cross-protective immunity against influenza pH1N1 2009 viruses induced by seasonal influenza A (H3N2) virus is mediated by virus-specific T-cells,” Journal of General Virology, vol. 92, no. 10, pp. 2339–2349, 2011. View at Publisher · View at Google Scholar · View at Scopus
  46. R. A. P. M. Perera, S. Riley, S. K. Ma, H.-C. Zhu, Y. Guan, and J. S. M. Peiris, “Seroconversion to pandemic (H1N1) 2009 virus and cross-reactive immunity to other swine influenza viruses,” Emerging Infectious Diseases, vol. 17, no. 10, pp. 1897–1899, 2011. View at Publisher · View at Google Scholar · View at Scopus
  47. J. M. Barry, C. Viboud, and L. Simonsen, “Cross-protection between successive waves of the 1918-1919 influenza pandemic: epidemiological evidence from US Army camps and from Britain,” Journal of Infectious Diseases, vol. 198, no. 10, pp. 1427–1434, 2008. View at Publisher · View at Google Scholar · View at Scopus
  48. T. E. Walton, M. M. Jochim, T. L. Barber, and L. H. Thompson, “Cross-protective immunity between equine encephalomyelitis viruses in equids,” The American Journal of Veterinary Research, vol. 50, no. 9, pp. 1442–1446, 1989. View at Google Scholar · View at Scopus
  49. R. Porrozzi, A. Teva, V. F. Amaral, M. V. Santos da Costa, and G. Grimaldi Jr., “Cross-immunity experiments between different species or strains of Leishmania in rhesus macaques (Macaca mulatta),” The American Journal of Tropical Medicine and Hygiene, vol. 71, no. 3, pp. 297–305, 2004. View at Google Scholar · View at Scopus
  50. S. Gupta, K. Trenholme, R. M. Anderson, and K. P. Day, “Antigenic diversity and the transmission dynamics of Plasmodium falciparum,” Science, vol. 263, no. 5149, pp. 961–963, 1994. View at Google Scholar · View at Scopus
  51. L. J. Abu-Raddad, B. I. S. van der Ventel, and N. M. Ferguson, “Interactions of multiple strain pathogen diseases in the presence of coinfection, cross immunity, and arbitrary strain diversity,” Physical Review Letters, vol. 100, no. 16, Article ID 168102, 2008. View at Publisher · View at Google Scholar · View at Scopus
  52. K. Dietz, “Epidemiologic interference of virus populations,” Journal of Mathematical Biology, vol. 8, no. 3, pp. 291–300, 1979. View at Google Scholar · View at Scopus
  53. C. Castillo-Chavez, H. W. Hethcote, V. Andreasen, S. A. Levin, and W. M. Liu, “Epidemiological models with age structure, proportionate mixing, and cross-immunity,” Journal of Mathematical Biology, vol. 27, no. 3, pp. 233–258, 1989. View at Publisher · View at Google Scholar · View at Scopus
  54. M. A. Nowak and R. M. May, “Superinfection and the evolution of parasite virulence,” Proceedings of the Royal Society B: Biological Sciences, vol. 255, no. 1342, pp. 81–89, 1994. View at Google Scholar · View at Scopus
  55. R. M. May and M. A. Nowak, “Coinfection and the evolution of parasite virulence,” Proceedings of the Royal Society B: Biological Sciences, vol. 261, no. 1361, pp. 209–215, 1995. View at Publisher · View at Google Scholar · View at Scopus
  56. M. Iannelli, M. Martcheva, and X.-Z. Li, “Strain replacement in an epidemic model with super-infection and perfect vaccination,” Mathematical Biosciences, vol. 195, no. 1, pp. 23–46, 2005. View at Publisher · View at Google Scholar · View at Scopus
  57. M. Martcheva and S. S. Pilyugin, “The role of coinfection in multidisease dynamics,” SIAM Journal on Applied Mathematics, vol. 66, no. 3, pp. 843–872, 2006. View at Publisher · View at Google Scholar · View at Scopus
  58. E. Delgado-Eckert, S. Ojosnegros, and N. Beerenwinkel, “The evolution of virulence in RNA viruses under a competition-colonization trade-off,” Bulletin of Mathematical Biology, vol. 73, no. 8, pp. 1881–1908, 2011. View at Publisher · View at Google Scholar · View at Scopus
  59. S. Gupta, N. Ferguson, and R. Anderson, “Chaos, persistence, and evolution of strain structure in antigenically diverse infectious agents,” Science, vol. 280, no. 5365, pp. 912–915, 1998. View at Publisher · View at Google Scholar · View at Scopus
  60. P. Rohani, C. J. Green, N. B. Mantilla-Beniers, and B. T. Grenfell, “Ecological interference between fatal diseases,” Nature, vol. 422, no. 6934, pp. 885–888, 2003. View at Publisher · View at Google Scholar · View at Scopus
  61. D. A. Vasco, H. J. Wearing, and P. Rohani, “Tracking the dynamics of pathogen interactions: modeling ecological and immune-mediated processes in a two-pathogen single-host system,” Journal of Theoretical Biology, vol. 245, no. 1, pp. 9–25, 2007. View at Publisher · View at Google Scholar · View at Scopus
  62. M. Recker, K. B. Blyuss, C. P. Simmons et al., “Immunological serotype interactions and their effect on the epidemiological pattern of dengue,” Proceedings of the Royal Society B: Biological Sciences, vol. 276, no. 1667, pp. 2541–2548, 2009. View at Publisher · View at Google Scholar · View at Scopus
  63. V. Andreasen, J. Lin, and S. A. Levin, “The dynamics of cocirculating influenza strains conferring partial cross-immunity,” Journal of Mathematical Biology, vol. 35, no. 7, pp. 825–842, 1997. View at Google Scholar
  64. J. R. Gog and B. T. Grenfell, “Dynamics and selection of many-strain pathogens,” Proceedings of the National Academy of Sciences of the United States of America, vol. 99, no. 26, pp. 17209–17214, 2002. View at Publisher · View at Google Scholar · View at Scopus
  65. O. Restif and B. T. Grenfell, “Integrating life history and cross-immunity into the evolutionary dynamics of pathogens,” Proceedings of the Royal Society B: Biological Sciences, vol. 273, no. 1585, pp. 409–416, 2006. View at Publisher · View at Google Scholar · View at Scopus
  66. M. Nuño, Z. Feng, M. Martcheva, and C. Castillo-Chavez, “Dynamics of two-strain influenza with isolation and partial cross-immunity,” SIAM Journal on Applied Mathematics, vol. 65, no. 3, pp. 964–982, 2005. View at Publisher · View at Google Scholar · View at Scopus
  67. L. J. Abu-Raddad and N. M. Ferguson, “The impact of cross-immunity, mutation and stochastic extinction on pathogen diversity,” Proceedings of the Royal Society B: Biological Sciences, vol. 271, no. 1556, pp. 2431–2438, 2004. View at Publisher · View at Google Scholar · View at Scopus
  68. M. Kamo and A. Sasaki, “The effect of cross-immunity and seasonal forcing in a multi-strain epidemic model,” Physica D: Nonlinear Phenomena, vol. 165, no. 3-4, pp. 228–241, 2002. View at Publisher · View at Google Scholar · View at Scopus
  69. P. Bharaj, H. S. Chahar, A. Pandey et al., “Concurrent infections by all four dengue virus serotypes during an outbreak of dengue in 2006 in Delhi, India,” Virology Journal, vol. 5, article 1, 2008. View at Publisher · View at Google Scholar · View at Scopus
  70. S. Vaccarella, S. Franceschi, R. Herrero et al., “Clustering of multiple human papillomavirus infections in women from a population-based study in Guanacaste, Costa Rica,” Journal of Infectious Diseases, vol. 204, no. 3, pp. 385–390, 2011. View at Publisher · View at Google Scholar · View at Scopus
  71. J. F. Brundage, “Interactions between influenza and bacterial respiratory pathogens: implications for pandemic preparedness,” The Lancet Infectious Diseases, vol. 6, no. 5, pp. 303–312, 2006. View at Publisher · View at Google Scholar · View at Scopus
  72. X.-S. Zhang, D. De Angelis, P. J. White, A. Charlett, R. G. Pebody, and J. McCauley, “Co-circulation of influenza A virus strains and emergence of pandemic via reassortment: the role of cross-immunity,” Epidemics, vol. 5, no. 1, pp. 20–33, 2013. View at Google Scholar
  73. M. J. Keeling and P. Rohani, Modeling Infectious Disease in Humans and Animals, Princeton University Press, Princeton, NJ, USA, 2007.
  74. P. Minayev and N. Ferguson, “Incorporating demographic stochasticity into multi-strain epidemic models: application to influenza A,” Journal of the Royal Society Interface, vol. 6, no. 40, pp. 989–996, 2009. View at Publisher · View at Google Scholar · View at Scopus
  75. M. Aguiar, B. Kooi, and N. Stollenwerk, “Epidemiology of dengue fever: a model with temporary cross-immunity and possible secondary infection shows bifurcations and chaotic behaviour in wide parameter regions,” Mathematical Modelling of Natural Phenomena, vol. 3, no. 4, pp. 48–70, 2008. View at Publisher · View at Google Scholar · View at Scopus
  76. B. W. Kooi, M. Aguiar, and N. Stollenwerk, “Analysis of an asymmetric two-strain dengue model,” Mathematical Biosciences, vol. 248, pp. 128–139, 2014. View at Google Scholar
  77. L. Mier-y-Teran-Romero, I. B. Schwartz, and D. A. T. Cummings, “Breaking the symmetry: immune enhancement increases persistence of dengue viruses in the presence of asymmetric transmission rates,” Journal of Theoretical Biology, vol. 332, pp. 203–210, 2013. View at Google Scholar
  78. N. M. Ferguson, A. P. Galvani, and R. M. Bush, “Ecological and immunological determinants of influenza evolution,” Nature, vol. 422, no. 6930, pp. 428–433, 2003. View at Publisher · View at Google Scholar · View at Scopus
  79. K. B. Blyuss and Y. N. Kyrychko, “On a basic model of a two-disease epidemic,” Applied Mathematics and Computation, vol. 160, no. 1, pp. 177–187, 2005. View at Publisher · View at Google Scholar · View at Scopus
  80. J. R. Gog and J. Swinton, “A status-based approach to multiple strain dynamics,” Journal of Mathematical Biology, vol. 44, no. 2, pp. 169–184, 2002. View at Publisher · View at Google Scholar · View at Scopus
  81. Y. Xia, J. R. Gog, and B. T. Grenfell, “Semiparametric estimation of the duration of immunity from infectious disease time series: influenza as a case-study,” Journal of the Royal Statistical Society C: Applied Statistics, vol. 54, no. 3, pp. 659–672, 2005. View at Publisher · View at Google Scholar · View at Scopus
  82. P.-Y. Boëlle, S. Ansart, A. Cori, and A.-J. Valleron, “Transmission parameters of the A/H1N1 (2009) influenza virus pandemic: a review,” Influenza and other Respiratory Viruses, vol. 5, no. 5, pp. 306–316, 2011. View at Publisher · View at Google Scholar · View at Scopus
  83. H. J. Wearing and P. Rohani, “Estimating the duration of pertussis immunity using epidemiological signatures,” PLoS Pathogens, vol. 5, no. 10, Article ID e1000647, 2009. View at Publisher · View at Google Scholar · View at Scopus
  84. H. J. Wearing and P. Rohani, “Ecological and immunological determinants of dengue epidemics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 103, no. 31, pp. 11802–11807, 2006. View at Publisher · View at Google Scholar · View at Scopus
  85. S. Bianco, L. B. Shaw, and I. B. Schwartz, “Epidemics with multistrain interactions: the interplay between cross immunity and antibody-dependent enhancement,” Chaos, vol. 19, no. 4, Article ID 043123, 2009. View at Google Scholar
  86. O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis, and Interpretation, John Wiley and Sons, Chichester, UK, 2000.
  87. X.-S. Zhang, “Escherichia coli O157 infection on cattle farms: the formulation of the force of infection and its effect on control effectiveness,” Epidemiology and Infection, vol. 140, no. 7, pp. 1215–1226, 2012. View at Google Scholar
  88. R.-H. Wang, Z. Jin, Q.-X. Liu, J. van de Koppel, and D. Alonso, “A simple stochastic model with environmental transmission explains multi-year periodicity in outbreaks of avian flu,” PLoS ONE, vol. 7, no. 2, Article ID e28873, 2012. View at Publisher · View at Google Scholar · View at Scopus
  89. D. Rios-Doria and G. Chowell, “Qualitative analysis of the level of cross-protection between epidemic waves of the 1918-1919 influenza pandemic,” Journal of Theoretical Biology, vol. 261, no. 4, pp. 584–592, 2009. View at Publisher · View at Google Scholar · View at Scopus
  90. P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, no. 1-2, pp. 29–48, 2002. View at Publisher · View at Google Scholar · View at Scopus
  91. B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1981.