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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:
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##### References:
 [1] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Science Publications, 1991. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] C. Chiyakia, Z. Mukandavire, P. Das, F. Nyabadza, S. D. Hove Musekwa and H. 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Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.  Google Scholar [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.  Google Scholar [29] A. J. Lotka, Contribution to the theory of periodic reaction, J. Phys. Chem., 14 (1910), 271-274. doi: 10.1021/j150111a004.  Google Scholar [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.  Google Scholar [31] A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, 1925. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] R. M. 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