Analysis of SI models with multiple interacting populations using subpopulations

Mathematical Biosciences and Engineering

Volumn 12, Issue 1, 2015, Pages 135-161

  Thomas, Evelyn K ^a   Gurski, Katharine F ^b   Hoffman, Kathleen A ^a  

^a UNIVERSITY OF MARYLAND BALTIMORE COUNTY   (United States)

^b HOWARD UNIVERSITY   (United States)

Author keywords

Compartmental model; Reproductive number; SI model; Spread of HIV; Stability

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; ENGINEERING; INDUSTRIAL ENGINEERING; MATHEMATICAL TECHNIQUES; STATISTICS;

BASIC REPRODUCTIVE NUMBER; COMPARTMENTAL MODEL; INTERACTING POPULATION; MULTIPLE CONNECTIONS; REPRODUCTIVE NUMBER; SI MODELS; SPREAD OF HIV; SYSTEMS OF DIFFERENTIAL EQUATIONS;

DIFFERENTIAL EQUATIONS;

ALGORITHM; BASIC REPRODUCTION NUMBER; BIOLOGICAL MODEL; BISEXUALITY; CHRONIC DISEASE; ECOLOGY; FEMALE; HIV INFECTIONS; HOMOSEXUALITY; HUMAN; MALE; POPULATION DYNAMICS; TIME;

ALGORITHMS; BASIC REPRODUCTION NUMBER; BISEXUALITY; CHRONIC DISEASE; ECOLOGY; EPIDEMIOLOGY; FEMALE; HIV INFECTIONS; HOMOSEXUALITY; HUMANS; MALE; MODELS, BIOLOGICAL; POPULATION DYNAMICS; TIME FACTORS;

EID: 84918537028     PISSN: 15471063     EISSN: 15510018     Source Type: Journal    
DOI: 10.3934/mbe.2015.12.135     Document Type: Article

References (61)

1. R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Science Publications, 1991.
2. N. T. Bailey, Application of stochastic epidemic modelling in the public health control of HIV/AIDS, Lecture Notes in Biomathematics, 86(1990), 14-20.
3. R. J. Beverton and S. J. Holt, The theory of fishing, in Sea Fisheries: Their Investigation in the United Kingdom (ed. M. Graham), Edward Arnold, London, (1956), 372-441.
4. F. Bonnet, P. Morlat, G. Chene, P. Mercie, D. Neau, M. Chossat, I. Decoin, F. Djossou, J. Beylot, F. Dabis and Groupe d'Epidemiologie Clinique du SIDA en Aquitaine (GECSA), Causes of death among HIV-infected patients in the era of highly active antiretroviral therapy, Bordeaux, France, 1998-1999, HIV Med., 3(2002), 195-199.
5. nd edition, Springer, New York, 2012.
6. C. Castillo-Chavez and B. Li, Spatial spread of sexually transmitted diseases within susceptible populations at demographic steady state, Mathematical Biosciences and Engineering, 5(2008), 713-727.
7. C. Castillo-Chavez, Z. Feng and W. Huang, On the computation IZq and its role on global stability, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, The IMA Volumes in Mathematics and its Applications, 125(2002), 229-250.
8. C. Castillo-Chavez, Mathematical and Statistical Approaches to AIDS Epidemiology, Lecture Notes in Biomathematics, 83, 1989.
9. C. Chiyakia, Z. Mukandavire, P. Das, F. Nyabadza, S. D. Hove Musekwa and H. Mwambi, Theoretical analysis of mixed Plasmodium malariae and Plasmodium falciparum infections with partial cross-immunity, Journal of Theoretical Biology, 263(2010), 169-178.
10. C. T. Codeco, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect Dis., 1(2001), p1.
11. C. T. Codeco and F. C. Coelho, Trends in cholera epidemiology, PLoS Med., 3(2006), e42.
12. M. H. Cohen, A. L. French, L. Benning, A. Kovacs, K. Anastos, M. Young, H. Minko and N. A. Hessol, Causes of death among women with human immunodeficiency virus infection in the era of combination antiretroviral therapy, Am. J. Med., 113(2002), 91-98.
13. K. Cooke and J. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Mathematical Biosciences, 16(1973), 75-101.
14. N. F. Crum, R. H. Rienburgh, S. Wegner, B. K. Agan, S. A. Tasker, K. M. Spooner, A. W. Armstrong, S. Fraser and M. R. Wallace, Comparisons of causes of death and mortality rates among HIV-infected persons: Analysis of the pre-, early, and late HAART eras, J Acquir. Immune Dec. Syndr., 41(2006), 194-200.
15. C. Elton and M. Nicholson, The ten-year cycle in numbers of the lynx in Canada, J. Animal Ecology, 11(1942), 215-244.
16. D. M. Hartley, J. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. Cholerae to cause epidemics?, PLoS Med., 3(2005), e7.
17. P. Hartman and C. Olech, On global asymptomatic stability of solutions of differential equations, Trans. Amer. Math. Soc, 104(1962), 154-178.
18. H. W. Hethcote and J. A. Yorke, Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomathematics, 56. Springer-Verlag, Berlin, 1984.
19. F. C. Hoppensteadt, Mathematical Theories Among Populations: Demographics, Genetics, and Epidemics, SIAM, 1975.
20. J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Mathematical Biosciences, 155(1999), 77-109.
21. J. M. Hyman, J. Li and E. A. Stanley, Modeling the impact of random screening and contact tracing in reducing the spread of HIV, Math. Biosci., 181(2003), 17-54.
22. J. M. Hyman, J. Li and E. A. Stanley, The initialization and sensitivity of multigroup models for the transmission of HIV, Journal of Theoretical Biology, 208(2001), 227-249.
23. J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic, Mathematical Biosciences, 90(1988), 415-473.
24. W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London B Biol. Sci., 115(1927), 700-721.
25. W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, part II, Proc. Roy. Soc. London B Biol. Sci., 138(1932), 55-83.
26. W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, part III, Proc. Roy. Soc. London B Biol. Sci., 141(1933), 94-112.
27. A. Lajmanovich and J. C. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28(1976), 221-236.
28. M. Y. Li and L. Wang, Global stability in some SEIR epidemic models, in IMA Volumes in Mathematics and its Applications (eds. C. Castillo-Chávez et al.), 126(2002), 295-311.
29. A. J. Lotka, Contribution to the theory of periodic reaction, J. Phys. Chem., 14(1910), 271-274.
30. A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Natl. Acad. Sci. U. S., 6(1920), 410-415.
31. A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, 1925.
32. S. Maggi and S. Rinaldi, A second-order impact model for forest fire regimes, Theoretical Population Biology, 70(2006), 174-182.
33. N. Malunguzaa, S. Mushayabasaa, C. Chiyaka and Z. Mukandavire, Modelling the effects of condom use and antiretroviral therapy in controlling HIV/AIDS among heterosexuals, homosexuals and bisexuals, Computational and Mathematical Methods in Medicine, 11(2010), 201-222.
34. M. May, M. Gompels, V. Delpech, K. Porter, F. Poct, M. Johnson, D. Dinn, A. Palfreeman, R. Gilson, B. Gazzard, T. Hill, J. Walsh, M. Fisher, C. Orkin, J. Ainsworth, L. Bansi, A. Phillips, C. Leen, M. Nelson, J. Anderson and C. Sabin, Impact of late diagnosis and treatment on life expectancy in people with HIV-1: UK Collaborative HIV Cohort (UK CHIC) Study, BMJ, 343(2011), d6016.
35. R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261(1976), 459-467.a
36. W. H. McNeill, Plagues and Peoples, Doubleday, 1976.
37. A. Mocroft, R. Brettle, O. Kirk, A. Blaxhult, J. M. Parkin, F. Antunes, P. Francioli, A. D'Arminio Monforte, Z. Fox, J. D. Lundgren and EuroSIDA study group, Changes in the cause of death among HIV positive subjects across Europe: results from the EuroSIDA study, AIDS, 16(2002), 1663-1671.
38. A. Mocroft, B. Ledergerber, C. Katlama, O. Kirk, P. Reiss, A. D'Arminio Monforte, B. Knysz, M. Dietrich, A. N. Phillips, J. D. Lundgren and EuroSIDA study group, Decline in the AIDS and death rates in the EuroSIDA study: An observational study, Lancet, 362(2003), 22-29.
39. Z. Mukandavire, C. Chiyaka, G. Magombedzea, G. Musukab and N. J. Malunguzaa, Assessing the effects of homosexuals and bisexuals on the intrinsic dynamics of HIV/AIDS in heterosexual settings, Mathematical and Computer Modelling, 49(2009), 1869-1882.
40. Z. Mukandavire and W. Garira, Age and sex structured model for assessing the demographic impact of mother-to-child transmission of HIV/AIDS, Bulletin of Mathematical Biology, 69(2007), 2061-2092.
41. J. D. Murray, Mathematical Biology I: An Introduction, 3rd edition, Springer, 2002.
42. F. Nakagawa, R. K. Lodwick, C. J. Smith, R. Smith, V. Cambiano, J. D. Lundgren, V. Delpech and A. N. Phillips, Projected life expectancy of people with HIV according to timing of diagnosis, AIDS, 26(2012), 335-343.
43. F. Nakagawa, M. May and A. Phillips, Life expectancy living with HIV: Recent estimates and future implications, Curr. Opin. Infect. Dis., 26(2013), 17-25.
44. M. Nuño, Z. Feng, M. Martcheva and C. Castillo-Chavez, Dynamics of two-strain influenza with isolation and partial cross-immunity, SIAM Journal of Applied Mathematics, 65(2005), 964-982.
45. E. Odum, Fundamentals of Ecology, Bulletin of the Torrey Botanical Club, 82(1955), 400-401.
46. C. Olech, On the global stability of an autonomous system on the plane, in On Global Univalence Theorems, Lecture Notes in Mathematics, 977(1983), 59-467.
47. F. J. Palella, K. M. Delaney, A. C. Moorman, M. O. Loveless, J. Fuhrer, G. A. Satten, D. J. Aschman, S. D. Holmberg, and the HIV Outpatient Study Investigators, Declining morbidity and mortality among patients with advanced human immunodeficiency virus infection, N. Engl. J. Med., 338(1998), 853-860.
48. C. N. Podder, O. Sharomi, A. B. Gumel, B. Song and E. Strawbridge, Mathematical analysis of a model for assessing the impact of antiretroviral therapy, voluntary testing and condom use in curtailing the spread of HIV, Differential Equations and Dynamical Systems: International Journal for Theory, Real World Modelling and Simulations, 19(2011), 283-302.
49. T. C. Porco and S. M. Blower, HIV vaccines: The effect of the mode of action on the coexistence of HIV subtypes, Mathematical Population Studies, 8(2000), 205-229.
50. th edition, Oxford University Press, 2008.
51. W. E. Ricker, Stock and Recruitment, J. Fisher. Res. Board Can., 5(1954), 559-623.
52. nd edition, E. P Dutton and Co., 1910.
53. D. Salmon Ceron, C. Lewden, P. Morlat, S. Bévilacqua, E. Jougla, F. Bonnet, L. Héripret, D. Costagliola, T. May and G. Chêne, Liver disease as a major cause of death among HIV infected patients: Role of hepatitis C and B viruses and alcohol, J. Hepatol., 42(2005), 799-805.
54. M. B. Schaefer, Some aspects of the dynamics of populations important to the management of commercial marine species, Bulletin of the Inter-American Tropical Tuna Commission, 1(1954), 27-56.
55. The Antiretroviral Therapy Cohort Collaboration, Life expectancy of individuals on combination antiretroviral therapy in high-income countries: A collaborative analysis of 14 cohort studies, Lancet, 372(2008), 293-299.
56. H. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, 2003.
57. A. Van Sighem, L. Gras, P. Reiss, K. Brinkman and F. De Wolf, Life expectancy of recently diagnosed asymptomatic HIV-infected patients approaches that of uninfected individuals, AIDS, 24(2010), 1527-1535.
58. J. X. Velasco-Hernandez and Y. H. Hsieh, Modelling the effect of treatment and behavior change in HIV-transmission dynamics, Journal of Mathematical Biology, 32(1994), 233-249.
59. V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. R. Acad. Naz. dei Lincei, 4(1926), 31-113.
60. V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, J. Cons. int. Explor. Mer., 3(1928), 3-51.
61. L. Wang and G.-Z. Dai, Global stability of virus spreading in complex heterogeneous networks, SIAM J. Appl. Math, 68(2008), 1495-1502.