# American Institute of Mathematical Sciences

2015, 12(4): 803-840. doi: 10.3934/mbe.2015.12.803

## An age-structured model for the coupled dynamics of HIV and HSV-2

 1 Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, United States, United States, United States, United States

Received  March 2014 Revised  October 2014 Published  April 2015

Evidence suggests a strong correlation between the prevalence of HSV-2 (genital herpes) and the perseverance of the HIV epidemic. HSV-2 is an incurable viral infection, characterized by periodic reactivation. We construct a model of the co-infection dynamics between the two diseases by incorporating a time-since-infection variable to track the alternating periods of infectiousness of HSV-2. The model considers only heterosexual relationships and distinguishes three population groups: males, general population females, and female sex workers. We calculate the basic reproduction numbers for each disease that provide threshold conditions, which determine whether a disease dies out or becomes endemic in the absence of the other disease. We also derive the invasion reproduction numbers that determine whether or not a disease can invade into a population in which the other disease is endemic. The calculations of the invasion reproduction numbers suggest a new aspect in their interpretation - the class from which the initial disease carrier arises is important for understanding the invasion dynamics and biological interpretation of the expressions of the reproduction numbers. Sensitivity analysis is conducted to examine the role of model parameters in influencing the model outcomes. The results are discussed in the last section.
Citation: Georgi Kapitanov, Christina Alvey, Katia Vogt-Geisse, Zhilan Feng. An age-structured model for the coupled dynamics of HIV and HSV-2. Mathematical Biosciences & Engineering, 2015, 12 (4) : 803-840. doi: 10.3934/mbe.2015.12.803
##### References:
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##### References:
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Koss, Transactional sex with casual and main partners among young south african men in the rural eastern cape: Prevalence, predictors, and associations with gender-based violence, Social Science & Medicine, 65 (2007), 1235-1248, URL http://www.sciencedirect.com/science/article/pii/S0277953607002328. doi: 10.1016/j.socscimed.2007.04.029. [17] K. L. Dunkle, R. K. Jewkes, H. C. Brown, G. E. Gray, J. A. McIntryre and S. D. Harlow, Transactional sex among women in soweto, south africa: Prevalence, risk factors and association with HIV infection, Social Science & Medicine, 59 (2004), 1581-1592, URL http://www.sciencedirect.com/science/article/pii/S0277953604000504. doi: 10.1016/j.socscimed.2004.02.003. [18] Z. Feng, Z. Qiu, Z. Sang, C. Lorenzo and J. Glasser, Modeling the synergy between hsv-2 and hiv and potential impact of hsv-2 therapy, Mathematical Biosciences, 245 (2013), 171-187. doi: 10.1016/j.mbs.2013.07.003. [19] A. Foss, P. Vickerman, Z. Chalabi, P. Mayaud, M. Alary and C. Watts, Dynamic modeling of herpes simplex virus type-2 (hsv-2) transmission: Issues in structural uncertainty, Bulletin of Mathematical Biology, 71 (2009), 720-749. doi: 10.1007/s11538-008-9379-1. [20] E. Freeman, H. Weiss, J. Glynn, P. Cross, J. Whitworth and R. Hayes, Herpes simplex virus 2 infection increases hiv acquisition in men and women: systematic review and meta-analysis of longitudinal studies, AIDS, 20 (2006), 73-83. doi: 10.1097/01.aids.0000198081.09337.a7. [21] R. Granich, C. F. Gilks, C. Dye, K. M. De Cock and B. G. Williams, Universal voluntary hiv testing with immediate antiretroviral therapy as a strategy for elimination of hiv transmission: a mathematical model, The Lancet, 373 (2009), 48-57. doi: 10.1016/S0140-6736(08)61697-9. [22] R. H. Gray and M. J. Wawer, Probability of heterosexual hiv-1 transmission per coital act in sub-saharan africa, Journal of Infectious Diseases, 205 (2012), 351-352, URL http://jid.oxfordjournals.org/content/205/3/351.short. doi: 10.1093/infdis/jir751. [23] R. H. Gray, M. J. Wawer, R. Brookmeyer, N. K. Sewankambo, D. Serwadda, F. Wabwire-Mangen, T. Lutalo, X. Li, T. vanCott and T. C. Quinn, Probability of hiv-1 transmission per coital act in monogamous, heterosexual, hiv-1-discordant couples in rakai, uganda, The Lancet, 357 (2001), 1149-1153. doi: 10.1016/S0140-6736(00)04331-2. [24] R. Gupta, T. Warren and A. Wald, Genital herpes, The Lancet, 370 (2007), 2127-2137. doi: 10.1016/S0140-6736(07)61908-4. [25] D. Hamby, A review of techniques for parameter sensitivity analysis of environmental models, Environmental Monitoring and Assessment, 32 (1994), 135-154. doi: 10.1007/BF00547132. [26] J. Helton, J. Johnson, C. Sallaberry and C. 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