Mathematical Biosciences and Engineering, 2015, 12(1): 135-161. doi: 10.3934/mbe.2015.12.135.

Primary: 92D30; Secondary: 37C75.

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Analysis of SI models with multiple interacting populations using subpopulations

1. Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21250
2. Department of Mathematics, Howard University, Washington, DC 20059

Computing endemic equilibria and basic reproductive numbers for systems of differential equations describing epidemiological systems with multiple connections between subpopulations is often algebraically intractable. We present an alternative method which deconstructs the larger system into smaller subsystems and captures theinteractions between the smaller systems as external forces using an approximate model. We bound the basic reproductive numbers of the full system in terms of the basic reproductive numbers of the smaller systems and use the alternate model to provide approximations for the endemic equilibrium. In addition to creating algebraically tractable reproductive numbers and endemic equilibria, we can demonstrate the influence of the interactions between subpopulations on the basic reproductive number of the full system. The focus of this paper is to provide analytical tools to help guide public health decisions with limited intervention resources.
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Keywords SI model; Reproductive number; stability; compartmental model; spread of HIV.

Citation: Evelyn K. Thomas, Katharine F. Gurski, Kathleen A. Hoffman. Analysis of SI models with multiple interacting populations using subpopulations. Mathematical Biosciences and Engineering, 2015, 12(1): 135-161. doi: 10.3934/mbe.2015.12.135

References

  • 1. Oxford Science Publications, 1991.
  • 2. Lecture Notes in Biomathematics, 86 (1990), 14-20.
  • 3. in Sea Fisheries: Their Investigation in the United Kingdom (ed. M. Graham), Edward Arnold, London, (1956), 372-441.
  • 4. HIV Med., 3 (2002), 195-199.
  • 5. $2^{nd}$ edition, Springer, New York, 2012.
  • 6. Mathematical Biosciences and Engineering, 5 (2008), 713-727.
  • 7. in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, The IMA Volumes in Mathematics and its Applications, 125 (2002), 229-250.
  • 8. Lecture Notes in Biomathematics, 83, 1989.
  • 9. Journal of Theoretical Biology, 263 (2010), 169-178.
  • 10. BMC Infect Dis., 1 (2001), p1.
  • 11. PLoS Med., 3 (2006), e42.
  • 12. Am. J. Med., 113 (2002), 91-98.
  • 13. Mathematical Biosciences, 16 (1973), 75-101.
  • 14. J Acquir. Immune Dec. Syndr., 41 (2006), 194-200.
  • 15. J. Animal Ecology, 11 (1942), 215-244.
  • 16. PLoS Med., 3 (2005), e7.
  • 17. Trans. Amer. Math. Soc., 104 (1962), 154-178.
  • 18. Lecture Notes in Biomathematics, 56. Springer-Verlag, Berlin, 1984.
  • 19. SIAM, 1975.
  • 20. Mathematical Biosciences, 155 (1999), 77-109.
  • 21. Math. Biosci., 181 (2003), 17-54.
  • 22. Journal of Theoretical Biology, 208 (2001), 227-249.
  • 23. Mathematical Biosciences, 90 (1988), 415-473.
  • 24. Proc. Roy. Soc. London B Biol. Sci., 115 (1927), 700-721.
  • 25. Proc. Roy. Soc. London B Biol. Sci., 138 (1932), 55-83.
  • 26. Proc. Roy. Soc. London B Biol. Sci., 141 (1933), 94-112.
  • 27. Mathematical Biosciences, 28 (1976), 221-236.
  • 28. in IMA Volumes in Mathematics and its Applications (eds. C. Castillo-Ch\'avez et al.), 126 (2002), 295-311.
  • 29. J. Phys. Chem., 14 (1910), 271-274.
  • 30. Proc. Natl. Acad. Sci. U.S., 6 (1920), 410-415.
  • 31. Williams and Wilkins, 1925.
  • 32. Theoretical Population Biology, 70 (2006), 174-182.
  • 33. Computational and Mathematical Methods in Medicine, 11 (2010), 201-222.
  • 34. BMJ, 343 (2011), d6016.
  • 35. Nature, 261 (1976), 459-467.
  • 36. Doubleday, 1976.
  • 37. AIDS, 16 (2002), 1663-1671.
  • 38. Lancet, 362 (2003), 22-29.
  • 39. Mathematical and Computer Modelling, 49 (2009), 1869-1882.
  • 40. Bulletin of Mathematical Biology, 69 (2007), 2061-2092.
  • 41. $3^rd$ edition, Springer, 2002.
  • 42. AIDS, 26 (2012), 335-343.
  • 43. Curr. Opin. Infect. Dis., 26 (2013), 17-25.
  • 44. SIAM Journal of Applied Mathematics, 65 (2005), 964-982.
  • 45. Bulletin of the Torrey Botanical Club, 82 (1955), 400-401.
  • 46. in On Global Univalence Theorems, Lecture Notes in Mathematics, 977 (1983), 59-467.
  • 47. N. Engl. J. Med., 338 (1998), 853-860.
  • 48. Differential Equations and Dynamical Systems: International Journal for Theory, Real World Modelling and Simulations, 19 (2011), 283-302.
  • 49. Mathematical Population Studies, 8 (2000), 205-229.
  • 50. $5^{th}$ edition, Oxford University Press, 2008.
  • 51. J. Fisher. Res. Board Can., 5 (1954), 559-623.
  • 52. $2^{nd}$ edition, E.P Dutton and Co., 1910.
  • 53. J. Hepatol., 42 (2005), 799-805.
  • 54. Bulletin of the Inter-American Tropical Tuna Commission, 1 (1954), 27-56.
  • 55. Lancet, 372 (2008), 293-299.
  • 56. Princeton Series in Theoretical and Computational Biology, 2003.
  • 57. AIDS, 24 (2010), 1527-1535.
  • 58. Journal of Mathematical Biology, 32 (1994), 233-249.
  • 59. Mem. R. Acad. Naz. dei Lincei, 4 (1926), 31-113.
  • 60. J. Cons. int. Explor. Mer., 3 (1928), 3-51.
  • 61. SIAM J. Appl. Math, 68 (2008), 1495-1502.

 

This article has been cited by

  • 1. K.F. Gurski, K.A. Hoffman, Influence of concurrency, partner choice, and viral suppression on racial disparity in the prevalence of HIV infected women, Mathematical Biosciences, 2016, 282, 91, 10.1016/j.mbs.2016.09.009

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