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

## 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

Received  January 2014 Revised  November 2014 Published  December 2014

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