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:
[1]

L. J. Abu-Raddad, J. T. Schiffer, R. Ashley, G. Mumtaz, R. A. Alsallaq, F. A. Akala, I. Semini, G. Riedner and D. Wilson, Hsv-2 serology can be predictive of hiv epidemic potential and hidden sexual risk behavior in the middle east and north africa,, Epidemics, 2 (2010), 173.

[2]

J. Baeten, B. Richardson, L. Lavreys, J. Rakwar, K. Mandaliya, J. Bwayo and J. Kreiss, Female-to-male infectivity of hiv-1 among circumcised and uncircumcised kenyan men,, The journal of infectious diseases, 191 (2005), 546. doi: 10.1086/427656.

[3]

J. Benedetti, L. Corey and R. Ashley, Recurrence rates in genital herpes after symptomatic first-episode infection,, Annals of Internal Medicine, 121 (1994), 847. doi: 10.7326/0003-4819-121-11-199412010-00004.

[4]

J. K. Benedetti, J. Zeh and L. Corey, Clinical reactivation of genital herpes simplex virus infection decreases in frequency over time,, Annals of Internal Medicine, 131 (1999), 14. doi: 10.7326/0003-4819-131-1-199907060-00004.

[5]

S. Blower and L. Ma, Calculating the contribution of herpes simplex virus type 2 epidemics to increasing hiv incidence:treatment implications,, Clin Infect Dis, 39 (2004). doi: 10.1086/422361.

[6]

S. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An hiv model, as an example,, International Statistical Review/Revue Internationale de Statistique, 62 (1994), 229. doi: 10.2307/1403510.

[7]

S. Blower, T. Porco and G. Darby, Predicting and preventing the emergence of antiviral drug resistance in hsv-2,, Nature Medicine, 4 (1998), 673. doi: 10.1038/nm0698-673.

[8]

D. D. Brewer, J. J. Potterat, S. B. Garrett, S. Q. Muth, J. M. Roberts, D. Kasprzyk, D. E. Montano and W. W. Darrow, Prostitution and the sex discrepancy in reported number of sexual partners,, Proceedings of the National Academy of Sciences, 97 (2000), 12385. doi: 10.1073/pnas.210392097.

[9]

A. Brewis and M. Meyer, Marital coitus across the life course,, Journal of Biosocial Science, 37 (2005), 499. doi: 10.1017/S002193200400690X.

[10]

M. Carael, E. Slaymaker, R. Lyerla and S. Sarkar, Clients of sex workers in different regions of the world: Hard to count,, Sexually Transmitted Infections, 82 (2006). doi: 10.1136/sti.2006.021196.

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C. Celum, A. Wald, J. Hughes, J. Sanchez, S. Reid, S. Delany-Moretlwe, F. Cowan, M. Casapia, A. Ortiz, J. Fuchs, S. Buchbinder, B. Koblin, S. Zwerski, S. Rose, J. Wang and L. Corey, Effect of aciclovir on hiv-1 acquisition in herpes simplex virus 2 seropositive women and men who have sex with men: a randomised, double-blind, placebo-controlled trial,, The Lancet, 371 (2008), 2109. doi: 10.1016/S0140-6736(08)60920-4.

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S. Chibaya, M. Kgosimore and E. S. Massawe, Mathematical analysis of drug resistance in vertical transmission of hiv/aids,, Open Journal of Epidemiology, 3 (2013), 139.

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L. Corey, H. G. Adams, Z. A. Brown and K. K. Holmes, Genital herpes simplex virus infections: Clinical manifestations, course, and complications,, Annals of Internal Medicine, 98 (1983), 958. doi: 10.7326/0003-4819-98-6-958.

[14]

L. Corey, A. Wald, C. Celum and T. Quinn, The effects of herpes simplex virus-2 on hiv-1 acquisition and transmission: a review of two overlapping epidemics,, J Acquir Immune Defic Syndr, 35 (2004), 435. doi: 10.1097/00126334-200404150-00001.

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D. C. Des Jarlais, K. Arasteh, C. McKnight, D. C. Perlman, H. L. F. Cooper and H. Hagan, Hsv-2 infection as a cause of female/male and racial/ethnic disparities in hiv infection,, PLoS ONE, 8 (2013).

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K. L. Dunkle, R. Jewkes, M. Nduna, N. Jama, J. Levin, Y. Sikweyiya and M. P. 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. doi: 10.1016/j.socscimed.2007.04.029.

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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. 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. 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. doi: 10.1007/s11538-008-9379-1.

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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. doi: 10.1097/01.aids.0000198081.09337.a7.

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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. 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. 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. doi: 10.1016/S0140-6736(00)04331-2.

[24]

R. Gupta, T. Warren and A. Wald, Genital herpes,, The Lancet, 370 (2007), 2127. 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. doi: 10.1007/BF00547132.

[26]

J. Helton, J. Johnson, C. Sallaberry and C. Storlie, Survey of sampling-based methods for uncertainty and sensitivity analysis,, Reliability Engineering & System Safety, 91 (2006), 1175. doi: 10.1016/j.ress.2005.11.017.

[27]

T. D. Hollingsworth, R. M. Anderson and C. Fraser, Hiv-1 transmission, by stage of infection,, The Journal of Infectious Diseases, 198 (2008), 687. doi: 10.1086/590501.

[28]

M. Hunter, The materiality of everyday sex: Thinking beyond 'prostitution', African Studies, 61 (2002), 99. doi: 10.1080/00020180220140091.

[29]

R. K. Jewkes, M. Nduna, P. Jama, K. L. Dunkle and J. B. Levin, Steadys, roll-ons and hit and runs: using indigenous typology to measure number of sexual partners, 2002,, Poster Presentation: The XIV International AIDS Conference., ().

[30]

L. F. Johnson, R. E. Dorrington, D. Bradshaw and T. M. Pillay-Van Wyk Victoria Rehle, Sexual behaviour patterns in south africa and their association with the spread of hiv: Insights from a mathematical model,, Demographic Research, 21 (2009), 289. doi: 10.4054/DemRes.2009.21.11.

[31]

Q. A. Karim, S. S. Karim, K. Soldan and M. Zondi, Reducing the risk of hiv infection among south african sex workers: Socioeconomic and gender barriers,, American Journal of Public Health, 85 (1995), 1521. doi: 10.2105/AJPH.85.11.1521.

[32]

D. E. Kirschner and G. F. Webb, A mathematical model of combined drug therapy of hiv infection,, Journal of Theoretical Medicine, 1 (1997), 25. doi: 10.1080/10273669708833004.

[33]

S. P. Kouyoumjian, G. R. Mumtaz, P. Vickerman and L. J. Abu-Raddad, P3.235 global ecological study of hiv and hsv-2 prevalence,, Sexually Transmitted Infections, 89 (2013). doi: 10.1136/sextrans-2013-051184.0692.

[34]

E. Lagarde, C. Enel and G. Pison, Reliability of reports of sexual behavior: A study of married couples in rural west africa,, American Journal of Epidemiology, 141 (1995), 1194.

[35]

A. G. Langenberg, L. Corey, R. L. Ashley, W. P. Leong and S. E. Straus, A prospective study of new infections with herpes simplex virus type 1 and type 2,, New England Journal of Medicine, 341 (1999), 1432. doi: 10.1056/NEJM199911043411904.

[36]

S.-G. Mahiane, C. Legeai, D. Taljaard, A. Latouche, A. Puren, A. Peillon, J. Bretagnolle, P. Lissouba, E.-P. N. Nguéma, E. Gassiat and B. Auvert, Transmission probabilities of hiv and herpes simplex virus type 2, effect of male circumcision and interaction: a longitudinal study in a township of south africa,, AIDS, 23 (2009), 377. doi: 10.1097/QAD.0b013e32831c5497.

[37]

G. Mertz, J. Benedetti, R. Ashley, S. Selke and L. Corey, Risk factors for the sexual transmission of genital herpes,, Annals of Internal Medicine, 116 (1992), 197. doi: 10.7326/0003-4819-116-3-197.

[38]

L. Morison, H. A. Weiss, A. Buvé, M. Caraël, S.-C. Abega, F. Kaona, J. Kanhonou L.and Chege and R. J. Hayes, Commercial sex and the spread of hiv in four cities in sub-saharan africa,, AIDS, 15 (2001). doi: 10.1097/00002030-200108004-00007.

[39]

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[40]

National Department of Health, The National Antenatal Sentinel Hiv and Syphilis Prevalence Survey,, south africa, (2010).

[41]

G. Ramjee and N. S. Weber Amy E.and Morar, Recording sexual behavior: Comparison of recall questionnaires with a coital diary,, Sexually Transmitted Diseases, 26 (1999), 374. doi: 10.1097/00007435-199908000-00002.

[42]

L. Rong, Z. Feng and A. Perelson, Mathematical analysis of age-structured hiv-1 dynamics with combination antiretroviral therapy,, SIAM Journal on Applied Mathematics, 67 (2007), 731. doi: 10.1137/060663945.

[43]

P. K. Roy, A. N. Chatterjee, D. Greenhalgh and Q. J. Khan, Long term dynamics in a mathematical model of hiv-1 infection with delay in different variants of the basic drug therapy model,, Nonlinear Analysis: Real World Applications, 14 (2013), 1621. doi: 10.1016/j.nonrwa.2012.10.021.

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M. J. Wawer, R. H. Gray, N. K. Sewankambo, D. Serwadda, X. Li, O. Laeyendecker, N. Kiwanuka, G. Kigozi, M. Kiddugavu, T. Lutalo, F. Nalugoda, F. Wabwire-Mangen, M. P. Meehan and T. C. Quinn, Rates of hiv-1 transmission per coital act, by stage of hiv-1 infection, in rakai, uganda,, Journal of Infectious Diseases, 191 (2005), 1403. doi: 10.1086/429411.

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show all references

References:
[1]

L. J. Abu-Raddad, J. T. Schiffer, R. Ashley, G. Mumtaz, R. A. Alsallaq, F. A. Akala, I. Semini, G. Riedner and D. Wilson, Hsv-2 serology can be predictive of hiv epidemic potential and hidden sexual risk behavior in the middle east and north africa,, Epidemics, 2 (2010), 173.

[2]

J. Baeten, B. Richardson, L. Lavreys, J. Rakwar, K. Mandaliya, J. Bwayo and J. Kreiss, Female-to-male infectivity of hiv-1 among circumcised and uncircumcised kenyan men,, The journal of infectious diseases, 191 (2005), 546. doi: 10.1086/427656.

[3]

J. Benedetti, L. Corey and R. Ashley, Recurrence rates in genital herpes after symptomatic first-episode infection,, Annals of Internal Medicine, 121 (1994), 847. doi: 10.7326/0003-4819-121-11-199412010-00004.

[4]

J. K. Benedetti, J. Zeh and L. Corey, Clinical reactivation of genital herpes simplex virus infection decreases in frequency over time,, Annals of Internal Medicine, 131 (1999), 14. doi: 10.7326/0003-4819-131-1-199907060-00004.

[5]

S. Blower and L. Ma, Calculating the contribution of herpes simplex virus type 2 epidemics to increasing hiv incidence:treatment implications,, Clin Infect Dis, 39 (2004). doi: 10.1086/422361.

[6]

S. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An hiv model, as an example,, International Statistical Review/Revue Internationale de Statistique, 62 (1994), 229. doi: 10.2307/1403510.

[7]

S. Blower, T. Porco and G. Darby, Predicting and preventing the emergence of antiviral drug resistance in hsv-2,, Nature Medicine, 4 (1998), 673. doi: 10.1038/nm0698-673.

[8]

D. D. Brewer, J. J. Potterat, S. B. Garrett, S. Q. Muth, J. M. Roberts, D. Kasprzyk, D. E. Montano and W. W. Darrow, Prostitution and the sex discrepancy in reported number of sexual partners,, Proceedings of the National Academy of Sciences, 97 (2000), 12385. doi: 10.1073/pnas.210392097.

[9]

A. Brewis and M. Meyer, Marital coitus across the life course,, Journal of Biosocial Science, 37 (2005), 499. doi: 10.1017/S002193200400690X.

[10]

M. Carael, E. Slaymaker, R. Lyerla and S. Sarkar, Clients of sex workers in different regions of the world: Hard to count,, Sexually Transmitted Infections, 82 (2006). doi: 10.1136/sti.2006.021196.

[11]

C. Celum, A. Wald, J. Hughes, J. Sanchez, S. Reid, S. Delany-Moretlwe, F. Cowan, M. Casapia, A. Ortiz, J. Fuchs, S. Buchbinder, B. Koblin, S. Zwerski, S. Rose, J. Wang and L. Corey, Effect of aciclovir on hiv-1 acquisition in herpes simplex virus 2 seropositive women and men who have sex with men: a randomised, double-blind, placebo-controlled trial,, The Lancet, 371 (2008), 2109. doi: 10.1016/S0140-6736(08)60920-4.

[12]

S. Chibaya, M. Kgosimore and E. S. Massawe, Mathematical analysis of drug resistance in vertical transmission of hiv/aids,, Open Journal of Epidemiology, 3 (2013), 139.

[13]

L. Corey, H. G. Adams, Z. A. Brown and K. K. Holmes, Genital herpes simplex virus infections: Clinical manifestations, course, and complications,, Annals of Internal Medicine, 98 (1983), 958. doi: 10.7326/0003-4819-98-6-958.

[14]

L. Corey, A. Wald, C. Celum and T. Quinn, The effects of herpes simplex virus-2 on hiv-1 acquisition and transmission: a review of two overlapping epidemics,, J Acquir Immune Defic Syndr, 35 (2004), 435. doi: 10.1097/00126334-200404150-00001.

[15]

D. C. Des Jarlais, K. Arasteh, C. McKnight, D. C. Perlman, H. L. F. Cooper and H. Hagan, Hsv-2 infection as a cause of female/male and racial/ethnic disparities in hiv infection,, PLoS ONE, 8 (2013).

[16]

K. L. Dunkle, R. Jewkes, M. Nduna, N. Jama, J. Levin, Y. Sikweyiya and M. P. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0140-6736(00)04331-2.

[24]

R. Gupta, T. Warren and A. Wald, Genital herpes,, The Lancet, 370 (2007), 2127. 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. doi: 10.1007/BF00547132.

[26]

J. Helton, J. Johnson, C. Sallaberry and C. Storlie, Survey of sampling-based methods for uncertainty and sensitivity analysis,, Reliability Engineering & System Safety, 91 (2006), 1175. doi: 10.1016/j.ress.2005.11.017.

[27]

T. D. Hollingsworth, R. M. Anderson and C. Fraser, Hiv-1 transmission, by stage of infection,, The Journal of Infectious Diseases, 198 (2008), 687. doi: 10.1086/590501.

[28]

M. Hunter, The materiality of everyday sex: Thinking beyond 'prostitution', African Studies, 61 (2002), 99. doi: 10.1080/00020180220140091.

[29]

R. K. Jewkes, M. Nduna, P. Jama, K. L. Dunkle and J. B. Levin, Steadys, roll-ons and hit and runs: using indigenous typology to measure number of sexual partners, 2002,, Poster Presentation: The XIV International AIDS Conference., ().

[30]

L. F. Johnson, R. E. Dorrington, D. Bradshaw and T. M. Pillay-Van Wyk Victoria Rehle, Sexual behaviour patterns in south africa and their association with the spread of hiv: Insights from a mathematical model,, Demographic Research, 21 (2009), 289. doi: 10.4054/DemRes.2009.21.11.

[31]

Q. A. Karim, S. S. Karim, K. Soldan and M. Zondi, Reducing the risk of hiv infection among south african sex workers: Socioeconomic and gender barriers,, American Journal of Public Health, 85 (1995), 1521. doi: 10.2105/AJPH.85.11.1521.

[32]

D. E. Kirschner and G. F. Webb, A mathematical model of combined drug therapy of hiv infection,, Journal of Theoretical Medicine, 1 (1997), 25. doi: 10.1080/10273669708833004.

[33]

S. P. Kouyoumjian, G. R. Mumtaz, P. Vickerman and L. J. Abu-Raddad, P3.235 global ecological study of hiv and hsv-2 prevalence,, Sexually Transmitted Infections, 89 (2013). doi: 10.1136/sextrans-2013-051184.0692.

[34]

E. Lagarde, C. Enel and G. Pison, Reliability of reports of sexual behavior: A study of married couples in rural west africa,, American Journal of Epidemiology, 141 (1995), 1194.

[35]

A. G. Langenberg, L. Corey, R. L. Ashley, W. P. Leong and S. E. Straus, A prospective study of new infections with herpes simplex virus type 1 and type 2,, New England Journal of Medicine, 341 (1999), 1432. doi: 10.1056/NEJM199911043411904.

[36]

S.-G. Mahiane, C. Legeai, D. Taljaard, A. Latouche, A. Puren, A. Peillon, J. Bretagnolle, P. Lissouba, E.-P. N. Nguéma, E. Gassiat and B. Auvert, Transmission probabilities of hiv and herpes simplex virus type 2, effect of male circumcision and interaction: a longitudinal study in a township of south africa,, AIDS, 23 (2009), 377. doi: 10.1097/QAD.0b013e32831c5497.

[37]

G. Mertz, J. Benedetti, R. Ashley, S. Selke and L. Corey, Risk factors for the sexual transmission of genital herpes,, Annals of Internal Medicine, 116 (1992), 197. doi: 10.7326/0003-4819-116-3-197.

[38]

L. Morison, H. A. Weiss, A. Buvé, M. Caraël, S.-C. Abega, F. Kaona, J. Kanhonou L.and Chege and R. J. Hayes, Commercial sex and the spread of hiv in four cities in sub-saharan africa,, AIDS, 15 (2001). doi: 10.1097/00002030-200108004-00007.

[39]

National Center for HIV/AIDS, Viral Hepatitis, STD & TB Prevention, CDC, Hiv surveillance - epidemiology of hiv infection (through 2011), 2011,, URL , ().

[40]

National Department of Health, The National Antenatal Sentinel Hiv and Syphilis Prevalence Survey,, south africa, (2010).

[41]

G. Ramjee and N. S. Weber Amy E.and Morar, Recording sexual behavior: Comparison of recall questionnaires with a coital diary,, Sexually Transmitted Diseases, 26 (1999), 374. doi: 10.1097/00007435-199908000-00002.

[42]

L. Rong, Z. Feng and A. Perelson, Mathematical analysis of age-structured hiv-1 dynamics with combination antiretroviral therapy,, SIAM Journal on Applied Mathematics, 67 (2007), 731. doi: 10.1137/060663945.

[43]

P. K. Roy, A. N. Chatterjee, D. Greenhalgh and Q. J. Khan, Long term dynamics in a mathematical model of hiv-1 infection with delay in different variants of the basic drug therapy model,, Nonlinear Analysis: Real World Applications, 14 (2013), 1621. doi: 10.1016/j.nonrwa.2012.10.021.

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