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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, United States, United States |
2. | Department of Mathematics, Howard University, Washington, DC 20059, United States |
References:
[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.
doi: 10.1007/978-3-662-10067-7_2. |
[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.
doi: 10.1046/j.1468-1293.2002.00117.x. |
[5] |
F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, $2^{nd}$ edition, Springer, New York, 2012.
doi: 10.1007/978-1-4757-3516-1. |
[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.
doi: 10.3934/mbe.2008.5.713. |
[7] |
C. Castillo-Chavez, Z. Feng and W. Huang, On the computation $\mathcalR_0$ 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.
doi: 10.1007/978-3-642-93454-4. |
[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.
doi: 10.1016/j.jtbi.2009.10.032. |
[10] |
C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect Dis., 1 (2001), p1.
doi: 10.1186/1471-2334-1-1. |
[11] |
C. T. Codeço and F. C. Coelho, Trends in cholera epidemiology, PLoS Med., 3 (2006), e42.
doi: 10.1371/journal.pmed.0030042. |
[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.
doi: 10.1016/S0002-9343(02)01169-5. |
[13] |
K. Cooke and J. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Mathematical Biosciences, 16 (1973), 75-101.
doi: 10.1016/0025-5564(73)90046-1. |
[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.
doi: 10.1097/01.qai.0000179459.31562.16. |
[15] |
C. Elton and M. Nicholson, The ten-year cycle in numbers of the lynx in Canada, J. Animal Ecology, 11 (1942), 215-244.
doi: 10.2307/1358. |
[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.
doi: 10.1371/journal.pmed.0030007. |
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P. Hartman and C. Olech, On global asymptomatic stability of solutions of differential equations, Trans. Amer. Math. Soc., 104 (1962), 154-178. |
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H. W. Hethcote and J. A. Yorke, Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomathematics, 56. Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-662-07544-9. |
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F. C. Hoppensteadt, Mathematical Theories Among Populations: Demographics, Genetics, and Epidemics, SIAM, 1975.
doi: 10.1137/1.9781611970487. |
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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.
doi: 10.1016/S0025-5564(98)10057-3. |
[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.
doi: 10.1016/S0025-5564(02)00128-1. |
[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.
doi: 10.1006/jtbi.2000.2214. |
[23] |
J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic, Mathematical Biosciences, 90 (1988), 415-473.
doi: 10.1016/0025-5564(88)90078-8. |
[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.
doi: 10.1098/rspa.1927.0118. |
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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. |
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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.
doi: 10.1016/0025-5564(76)90125-5. |
[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-Ch\'avez et al.), 126 (2002), 295-311.
doi: 10.1007/978-1-4613-0065-6_17. |
[29] |
A. J. Lotka, Contribution to the theory of periodic reaction, J. Phys. Chem., 14 (1910), 271-274.
doi: 10.1021/j150111a004. |
[30] |
A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Natl. Acad. Sci. U.S., 6 (1920), 410-415.
doi: 10.1073/pnas.6.7.410. |
[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.
doi: 10.1016/j.tpb.2006.01.007. |
[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.
doi: 10.1080/17486700903325167. |
[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.
doi: 10.1136/bmj.d6016. |
[35] |
R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.
doi: 10.1038/261459a0. |
[36] | |
[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.
doi: 10.1097/00002030-200208160-00012. |
[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.
doi: 10.1016/S0140-6736(03)13802-0. |
[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.
doi: 10.1016/j.mcm.2008.12.012. |
[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.
doi: 10.1007/s11538-007-9204-2. |
[41] |
J. D. Murray, Mathematical Biology I: An Introduction, $3^rd$ 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.
doi: 10.1097/QAD.0b013e32834dcec9. |
[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.
doi: 10.1097/QCO.0b013e32835ba6b1. |
[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.
doi: 10.1137/S003613990343882X. |
[45] |
E. Odum, Fundamentals of Ecology, Bulletin of the Torrey Botanical Club, 82 (1955), 400-401.
doi: 10.2307/2482488. |
[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.
doi: 10.1007/BFb0065573. |
[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.
doi: 10.1056/NEJM199803263381301. |
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show all references
References:
[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.
doi: 10.1007/978-3-662-10067-7_2. |
[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.
doi: 10.1046/j.1468-1293.2002.00117.x. |
[5] |
F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, $2^{nd}$ edition, Springer, New York, 2012.
doi: 10.1007/978-1-4757-3516-1. |
[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.
doi: 10.3934/mbe.2008.5.713. |
[7] |
C. Castillo-Chavez, Z. Feng and W. Huang, On the computation $\mathcalR_0$ 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.
doi: 10.1007/978-3-642-93454-4. |
[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.
doi: 10.1016/j.jtbi.2009.10.032. |
[10] |
C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect Dis., 1 (2001), p1.
doi: 10.1186/1471-2334-1-1. |
[11] |
C. T. Codeço and F. C. Coelho, Trends in cholera epidemiology, PLoS Med., 3 (2006), e42.
doi: 10.1371/journal.pmed.0030042. |
[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.
doi: 10.1016/S0002-9343(02)01169-5. |
[13] |
K. Cooke and J. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Mathematical Biosciences, 16 (1973), 75-101.
doi: 10.1016/0025-5564(73)90046-1. |
[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.
doi: 10.1097/01.qai.0000179459.31562.16. |
[15] |
C. Elton and M. Nicholson, The ten-year cycle in numbers of the lynx in Canada, J. Animal Ecology, 11 (1942), 215-244.
doi: 10.2307/1358. |
[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.
doi: 10.1371/journal.pmed.0030007. |
[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.
doi: 10.1007/978-3-662-07544-9. |
[19] |
F. C. Hoppensteadt, Mathematical Theories Among Populations: Demographics, Genetics, and Epidemics, SIAM, 1975.
doi: 10.1137/1.9781611970487. |
[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.
doi: 10.1016/S0025-5564(98)10057-3. |
[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.
doi: 10.1016/S0025-5564(02)00128-1. |
[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.
doi: 10.1006/jtbi.2000.2214. |
[23] |
J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic, Mathematical Biosciences, 90 (1988), 415-473.
doi: 10.1016/0025-5564(88)90078-8. |
[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.
doi: 10.1098/rspa.1927.0118. |
[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.
doi: 10.1016/0025-5564(76)90125-5. |
[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-Ch\'avez et al.), 126 (2002), 295-311.
doi: 10.1007/978-1-4613-0065-6_17. |
[29] |
A. J. Lotka, Contribution to the theory of periodic reaction, J. Phys. Chem., 14 (1910), 271-274.
doi: 10.1021/j150111a004. |
[30] |
A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Natl. Acad. Sci. U.S., 6 (1920), 410-415.
doi: 10.1073/pnas.6.7.410. |
[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.
doi: 10.1016/j.tpb.2006.01.007. |
[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.
doi: 10.1080/17486700903325167. |
[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.
doi: 10.1136/bmj.d6016. |
[35] |
R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.
doi: 10.1038/261459a0. |
[36] | |
[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.
doi: 10.1097/00002030-200208160-00012. |
[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.
doi: 10.1016/S0140-6736(03)13802-0. |
[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.
doi: 10.1016/j.mcm.2008.12.012. |
[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.
doi: 10.1007/s11538-007-9204-2. |
[41] |
J. D. Murray, Mathematical Biology I: An Introduction, $3^rd$ 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.
doi: 10.1097/QAD.0b013e32834dcec9. |
[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.
doi: 10.1097/QCO.0b013e32835ba6b1. |
[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.
doi: 10.1137/S003613990343882X. |
[45] |
E. Odum, Fundamentals of Ecology, Bulletin of the Torrey Botanical Club, 82 (1955), 400-401.
doi: 10.2307/2482488. |
[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.
doi: 10.1007/BFb0065573. |
[47] |
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