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2016, 13(1): 171-191. doi: 10.3934/mbe.2016.13.171

## Modelling HIV superinfection among men who have sex with men

 1 Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049, China 2 Department of Epidemiology and Biostatistics, Nanjing Medical University, Nanjing 210029, China

Received  October 2014 Revised  July 2015 Published  October 2015

Superinfection, a phenomenon that an individual infected by one HIV strain is re-infected by the second heterologous HIV strain, occurs in HIV infection. A mathematical model is formulated to examine how superinfection affects transmission dynamics of drug sensitive/resistant strains. Three reproduction numbers are defined: reproduction numbers $R_r$ and $R_s$ for drug-resistant and drug-sensitive strains, respectively, and the invasion reproduction number $R_s^r$. The disease-free equilibrium always exists and is locally stable when the larger of $R_s$ and $R_r$ is less than one. The drug resistant strain-only equilibrium is locally stable when $R_r>1$ and $R_s^r<1$. Numerical studies show that as the superinfection coefficient of the sensitive strain increases the system may (1) change to bistable states of disease-free equilibrium and the coexistence state from the stable disease-free equilibrium under no superinfection; (2) experience the stable resistant-strain only equilibrium, the bistable states of resistant-strain only equilibrium and the coexistence state, and the stable coexistence state in turn. This implies that superinfection of the sensitive strain is beneficial for two strains to coexist. While superinfection of the resistant strain makes resistant strain more likely to be sustained. The findings suggest that superinfection induces the complicated dynamics, and brings more difficulties in antiretroviral therapy.
Citation: Xiaodan Sun, Yanni Xiao, Zhihang Peng. Modelling HIV superinfection among men who have sex with men. Mathematical Biosciences & Engineering, 2016, 13 (1) : 171-191. doi: 10.3934/mbe.2016.13.171
##### References:
 [1] T. M. Allen and M. Altfeld, HIV-1 superinfection, J. Allergy. Clin. Immun., 112 (2003), 829-835. doi: 10.1016/j.jaci.2003.08.037. [2] M. Amaku, M. N. Burattini, F. A. B. Coutinho and E. Massad, Modeling the dynamics of viral evolution considering competition within individuals hosts and at population level: the effects of treatment, B. Math. Biol., 72 (2010), 1294-1314. doi: 10.1007/s11538-009-9495-6. [3] R. F. Baggaley, G. P. Garnett and N. M. Ferguson, Modelling the impact of antiretroviral use in resource-poor settings, Plos Med., 3 (2006), e124. doi: 10.1371/journal.pmed.0030124. [4] D. R. Bangsberg, S. Perry, E. D. Charlebois, R. A. Clark, M. Roberston and et al., Non-adherence to highly active antiretroviral therapy predicts progression to AIDS, AIDS, 15 (2001), 1181-1183. doi: 10.1097/00002030-200106150-00015. [5] S. M. Blower, H. B. Gershengorn and R. M. Grant, A tale of two futures: HIV and antiretroviral therapy in San Francisco, Science, 287 (2000), 650-654. doi: 10.1126/science.287.5453.650. [6] C. Chiyaka, Z. Mukandavire, P. Das, F. Nyabadza and et al., Theoretical analysis of mixed Plasmodium malariae and Plasmodium falciparum infections with partial cross-immunity, J. Theor. Biol., 263 (2010), 169-178. doi: 10.1016/j.jtbi.2009.10.032. [7] B. Chohan, L. Lavrey, S. M. Rainwaer and J. Overbaugh, Evidence for frequent reinfection with human immunodeficiency virus type 1 of a different subtype, J. Virol., 79 (2005), 10701-10708. doi: 10.1128/JVI.79.16.10701-10708.2005. [8] M. S. Cohen, I. F. Hoffman, R. A. Royce, P. Kazebe, J. R. Dyer and et al., Reduction of concentraion of HIV-in in semen after treatment of urethritis: implications for prevention of sexual transmission of HIV-1, Lancet, 349 (1996), 1868-1873. [9] O. Diekmann, J. A. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [10] G. H. Friedland and A. Williams, Attaining higher goals in HIV treatment: The central importance of adherence, AIDS, 13 (1999), S61-S72. [11] F. Gao, D. L. Robertson, S. G. Morrison, H. Hui, S. Craig and et al., The heterosexual human immunodeficiency virus type 1 epidemic in Thailand is caused by an intersubtype (A/E) recombinant of African origin, J. Virol., 70 (1996), 7013-7029. [12] M. J. Gonzales, E. Delwart, S. Y. Rhee, R. Tsui, A. R. Zolopa and et al., Lack of detectable human immunodeficiency virus type 1 superinfection during 1072 person-years of observation, J. Infect. Dis., 188 (2003), 397-405. doi: 10.1086/376534. [13] G. S. Gottlieb, D. C. Nickle, M. A. Jensen, K. G. Wong, J. Grobler and et al., Dual HIV-1 infection associated with rapid disease progression, Lancet, 363 (2004), 619-622. doi: 10.1016/S0140-6736(04)15596-7. [14] P. Hartman, Ordinary Differential Equation, Bikh$\brevea$user, Boston, 1982. [15] A. E. Jetzt, H. Yu, G. J. Klarmann, Y. Ron, B. D. Preston and et al., High rate of recombination throughout the human immunodeficiency virus type 1 genome, J. Virol., 74 (2000), 1234-1240. doi: 10.1128/JVI.74.3.1234-1240.2000. [16] S. Jost, M.-C. Bernard, L. Kaiser, B. Hirschel, B. Autran and et al., A patient with HIV-superinfection, New. Engl. J. Med., 347 (2002), 731-736. doi: 10.1056/NEJMoa020263. [17] K. K. Koelsch, D. M. Smith, S. J. Little, C. C. Ignacio, T. R. Macaranas and et al., Clade B HIV superinfection with wild-type virus after primary infection with drug resistant clade B virus, AIDS, 17 (2003), F11-F16. doi: 10.1097/00002030-200305020-00001. [18] T. Kuwata, Y. Miyazaki, T. Igarashi, J. Takehisa and M. Hayami, The rapid spread of recombinants during a natural in vitro infection with two human immunodeficiency virus type 1 strains, J. Virol. 71 (1997), 7088-7091. [19] Q. Li, S. Cao, X. Chen, G. Sun, Y. Liu Y and Z. Jia, Stability analysis of an HIV/AIDS dynamics model with drug resistance, Discrete. Dyn. Nat. Soc., (2012), 162527, 13 pages. doi: 10.1155/2012/162527. [20] J. Lou, J. Wu, L. Chen, Y. Ruan and Y. Shao, A sex-role-preference model for HIV transmission among men who have sex with men in China, BMC Public Health, 9 (2009), S1-S10. doi: 10.1186/1471-2458-9-S1-S10. [21] F. Mammano, C. Petit and F. Clavel, Resistance-associated loss of viral fitness in human immunodeficiency virus type 1: Phenotypic analysis of protease and gag coevolution in protease inhibitor-treated patients, J. Virol., 72 (1998), 7632-7637. [22] M. Martcheva, B. M. Bolker and R. D. Holt, Vaccine-induced pathogen strain replacement: What are the mechanisms? J. Roy. Soc. Interface, 5 (2008), 3-13. doi: 10.1098/rsif.2007.0236. [23] M. P. Pernas, C. Casado, R. Fuentes, M. J. Pérez-Elías and C. López-Galíndez, A dual superinfection and recombination within HIV-1 subtype B 12 years after primoinfection, J. Acq. immun. Def. Synd., 42 (2006), 12-18. doi: 10.1097/01.qai.0000214810.65292.73. [24] A. Piantadosi, B. Chohan, V. Chohan, R. S. McClelland and J. Overbaugh, Chronic HIV-1 infection frequently fails to protect against superinfection, Plos Pathog., 3 (2007), e177. doi: 10.1371/journal.ppat.0030177. [25] A. Piantadosi, M. O. Ngayo, B. Chohan and J. Overbaugh, Examination of a second region of the HIV type 1 genome reveals additional cases of superinfection, AIDS Res. Hum. Retrov., 24 (2008), 1221-1224. doi: 10.1089/aid.2008.0100. [26] T. C. Porco and S. M. Blower, HIV vaccines: the effect of the mode of action on the coexistence of HIV subtypes, Math. Popul. Stud., 8 (2000), 205-229. doi: 10.1080/08898480009525481. [27] A. Rachinger, I. G. Stolte, T. D. Van de Ven, J. A. Burger, M. Prins and et al., Absence of HIV-1 superinfection 1 year after infection between 1985-1987 coincides with a reduction in sexual risk behavior in the seroincident Amsterdam cohort of homosexual men, Clin. Infect. Dis., 50 (2010), 1309-1315. doi: 10.1086/651687. [28] S. M. Raimundo, H. M. Yang, E. Venturino and E. Massad, Modeling the emergence of HIV-1 drug resistance resulting from antiretroviral therapy: Insights from theoretical and numerical studies, BioSystems, 108 (2012), 1-13. doi: 10.1016/j.biosystems.2011.11.009. [29] A. Ramos, D. J. Hu, L. Nguyen, K-O. Phan, S. Vanichseni and et al., Intersubtype human immunodeficiency virus type 1 superinfection following seroconversion to primary infection in two injection drug users, J. Virol., 76 (2002), 7444-7452. doi: 10.1128/JVI.76.15.7444-7452.2002. [30] A. D. Redd, C. E. Mullis, D. Serwadda, X. Kong, C. Martens and et al., The rates of HIV superinfection and primary HIV incidence in a general population in Rakai, Uganda, J. Infect. Dis, 206 (2012), 267-274. doi: 10.1093/infdis/jis325. [31] A. K. Sethi, D. D. Celentano, S. J. Gange, R. D. Moore and J. E. Gallant, Association between adherence to antiretroviral therapy and human immunodeficiency virus drug resistance, Clin. Infect. Dis., 37 (2003), 1112-1118. doi: 10.1086/378301. [32] O. Sharomi and A. B. Gumel, Dynamical analysis of a multi-strain model of HIV in the presence of antiretroviral drugs, J. Biol. Dyn., 2 (2008), 323-345. doi: 10.1080/17513750701775599. [33] D. M. Smith, D. D. Richman and S. J. Little, HIV superinfection, J. Infect. Dis., 192 (2005), 438-444. doi: 10.1086/431682. [34] D. M. Smith, J. K. Wong, G. K. Hightoer, C. C. Ignacio, K. K. Koelsch and et al., Incidence of HIV superinfection following primary infection, J. Amer. Med. Assoc., 292 (2004), 1177-1178. doi: 10.1001/jama.292.10.1177. [35] X. Sun, Y. Xiao, Z. Peng and N. Wang, Modelling HIV/AIDS epidemic among men who have sex with men in China, BioMed Res. Int., 2013 (2013), Article ID 413260, 18 pages. doi: 10.1155/2013/413260. [36] J. Takehisa, L. Zekeng, E. Ido, Y. Yamaguchi-Kabata, I. Mboudjeka and et al., Human immunodeficiency virus type 1 intergroup(M/O) recombination in Cameroon, J. Virol., 73 (1999), 6810-6820. [37] R. Tsui, B. L. Herring, J. D. Barbour, R. M. Grant, P. Bacchetti and et al., Human immuodeficiency virus type 1 superinfection was not detected following 215 years of injection drug user exposure, J. Virol., 78 (2004), 94-103. doi: 10.1128/JVI.78.1.94-103.2004. [38] R. Vardavas and S. Blower, The emergence of HIV transmitted resistance in Botswana: "When will the WHO detection threshold be exceeded?" Plos One, 2 (2007), e152. doi: 10.1371/journal.pone.0000152. [39] J. X. Velasco-Hernandez, A model for chagas disease involving transmission by vectors and blood transfusion, Theor. Popul. Biol., 46 (1994), 1-31. doi: 10.1006/tpbi.1994.1017. [40] P. Venden Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [41] P. L. Vernazza, B. L. Gilliam, M. Flepp, J. R. Dyer, A. C. Frank and et al., Effect of antiviral treatment on the shedding of HIV-1 in semen, AIDS, 11 (1997), 1249-1254. doi: 10.1097/00002030-199710000-00008. [42] W. Walter, Differential and Integral Inequalities, Springer, Berlin, 1970. [43] X. Wang, L. Yang, H. Li and et al., Factors associated with HIV virologic failure among patients on HAART for one year at three sentinel surveillance sities in China, Curr. HIV. Res., 9 (2011), 103-111. doi: 10.2174/157016211795569122. [44] D. P. Wilson, J. Kahn and S. M. Blower, Predicting the epidemiological impact of antiretroviral allocation strategies in KwaZulu-Natal: The effect of the urban-rural divide, PNAS, 103 (2006), 14228-14233. doi: 10.1073/pnas.0509689103. [45] D. P. Wilson, M. G. Law, A. E. Grulich, D. A. Cooper and J. M. Kaldor, Relation between HIV viral load and infectiousness: a model-based analysis, Lancet, 372 (2008), 314-320. doi: 10.1016/S0140-6736(08)61115-0. [46] H. Xing, Y. Ruan, J. Li and et al., HIV Drug resistance and its impact on antiretroviral therapy in chinese HIV-infected patients, Plos one, 8 (2013), e54917. doi: 10.1371/journal.pone.0054917. [47] M. Xiridou, M. Kretzschmar and R. Geskus, Competition of pathogen strains leading to infection with variable infectivity and the effect of treatment, Math. Biosci., 197 (2005), 153-172. doi: 10.1016/j.mbs.2005.04.007. [48] X. Xu, Y. Xiao and N. Wang, Modeling sexual transmission of HIV/AIDS in Jiangsu province, China, Math. Method. Appl. Sci., 36 (2012), 234-248. doi: 10.1002/mma.2599.

show all references

##### References:
 [1] T. M. Allen and M. Altfeld, HIV-1 superinfection, J. Allergy. Clin. Immun., 112 (2003), 829-835. doi: 10.1016/j.jaci.2003.08.037. [2] M. Amaku, M. N. Burattini, F. A. B. Coutinho and E. Massad, Modeling the dynamics of viral evolution considering competition within individuals hosts and at population level: the effects of treatment, B. Math. Biol., 72 (2010), 1294-1314. doi: 10.1007/s11538-009-9495-6. [3] R. F. Baggaley, G. P. Garnett and N. M. Ferguson, Modelling the impact of antiretroviral use in resource-poor settings, Plos Med., 3 (2006), e124. doi: 10.1371/journal.pmed.0030124. [4] D. R. Bangsberg, S. Perry, E. D. Charlebois, R. A. Clark, M. Roberston and et al., Non-adherence to highly active antiretroviral therapy predicts progression to AIDS, AIDS, 15 (2001), 1181-1183. doi: 10.1097/00002030-200106150-00015. [5] S. M. Blower, H. B. Gershengorn and R. M. Grant, A tale of two futures: HIV and antiretroviral therapy in San Francisco, Science, 287 (2000), 650-654. doi: 10.1126/science.287.5453.650. [6] C. Chiyaka, Z. Mukandavire, P. Das, F. Nyabadza and et al., Theoretical analysis of mixed Plasmodium malariae and Plasmodium falciparum infections with partial cross-immunity, J. Theor. Biol., 263 (2010), 169-178. doi: 10.1016/j.jtbi.2009.10.032. [7] B. Chohan, L. Lavrey, S. M. Rainwaer and J. Overbaugh, Evidence for frequent reinfection with human immunodeficiency virus type 1 of a different subtype, J. Virol., 79 (2005), 10701-10708. doi: 10.1128/JVI.79.16.10701-10708.2005. [8] M. S. Cohen, I. F. Hoffman, R. A. Royce, P. Kazebe, J. R. Dyer and et al., Reduction of concentraion of HIV-in in semen after treatment of urethritis: implications for prevention of sexual transmission of HIV-1, Lancet, 349 (1996), 1868-1873. [9] O. Diekmann, J. A. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [10] G. H. Friedland and A. Williams, Attaining higher goals in HIV treatment: The central importance of adherence, AIDS, 13 (1999), S61-S72. [11] F. Gao, D. L. Robertson, S. G. Morrison, H. Hui, S. Craig and et al., The heterosexual human immunodeficiency virus type 1 epidemic in Thailand is caused by an intersubtype (A/E) recombinant of African origin, J. Virol., 70 (1996), 7013-7029. [12] M. J. Gonzales, E. Delwart, S. Y. Rhee, R. Tsui, A. R. Zolopa and et al., Lack of detectable human immunodeficiency virus type 1 superinfection during 1072 person-years of observation, J. Infect. Dis., 188 (2003), 397-405. doi: 10.1086/376534. [13] G. S. Gottlieb, D. C. Nickle, M. A. Jensen, K. G. Wong, J. Grobler and et al., Dual HIV-1 infection associated with rapid disease progression, Lancet, 363 (2004), 619-622. doi: 10.1016/S0140-6736(04)15596-7. [14] P. Hartman, Ordinary Differential Equation, Bikh$\brevea$user, Boston, 1982. [15] A. E. Jetzt, H. Yu, G. J. Klarmann, Y. Ron, B. D. Preston and et al., High rate of recombination throughout the human immunodeficiency virus type 1 genome, J. Virol., 74 (2000), 1234-1240. doi: 10.1128/JVI.74.3.1234-1240.2000. [16] S. Jost, M.-C. Bernard, L. Kaiser, B. Hirschel, B. Autran and et al., A patient with HIV-superinfection, New. Engl. J. Med., 347 (2002), 731-736. doi: 10.1056/NEJMoa020263. [17] K. K. Koelsch, D. M. Smith, S. J. Little, C. C. Ignacio, T. R. Macaranas and et al., Clade B HIV superinfection with wild-type virus after primary infection with drug resistant clade B virus, AIDS, 17 (2003), F11-F16. doi: 10.1097/00002030-200305020-00001. [18] T. Kuwata, Y. Miyazaki, T. Igarashi, J. Takehisa and M. Hayami, The rapid spread of recombinants during a natural in vitro infection with two human immunodeficiency virus type 1 strains, J. Virol. 71 (1997), 7088-7091. [19] Q. Li, S. Cao, X. Chen, G. Sun, Y. Liu Y and Z. Jia, Stability analysis of an HIV/AIDS dynamics model with drug resistance, Discrete. Dyn. Nat. Soc., (2012), 162527, 13 pages. doi: 10.1155/2012/162527. [20] J. Lou, J. Wu, L. Chen, Y. Ruan and Y. Shao, A sex-role-preference model for HIV transmission among men who have sex with men in China, BMC Public Health, 9 (2009), S1-S10. doi: 10.1186/1471-2458-9-S1-S10. [21] F. Mammano, C. Petit and F. Clavel, Resistance-associated loss of viral fitness in human immunodeficiency virus type 1: Phenotypic analysis of protease and gag coevolution in protease inhibitor-treated patients, J. Virol., 72 (1998), 7632-7637. [22] M. Martcheva, B. M. Bolker and R. D. Holt, Vaccine-induced pathogen strain replacement: What are the mechanisms? J. Roy. Soc. Interface, 5 (2008), 3-13. doi: 10.1098/rsif.2007.0236. [23] M. P. Pernas, C. Casado, R. Fuentes, M. J. Pérez-Elías and C. López-Galíndez, A dual superinfection and recombination within HIV-1 subtype B 12 years after primoinfection, J. Acq. immun. Def. Synd., 42 (2006), 12-18. doi: 10.1097/01.qai.0000214810.65292.73. [24] A. Piantadosi, B. Chohan, V. Chohan, R. S. McClelland and J. Overbaugh, Chronic HIV-1 infection frequently fails to protect against superinfection, Plos Pathog., 3 (2007), e177. doi: 10.1371/journal.ppat.0030177. [25] A. Piantadosi, M. O. Ngayo, B. Chohan and J. Overbaugh, Examination of a second region of the HIV type 1 genome reveals additional cases of superinfection, AIDS Res. Hum. Retrov., 24 (2008), 1221-1224. doi: 10.1089/aid.2008.0100. [26] T. C. Porco and S. M. Blower, HIV vaccines: the effect of the mode of action on the coexistence of HIV subtypes, Math. Popul. Stud., 8 (2000), 205-229. doi: 10.1080/08898480009525481. [27] A. Rachinger, I. G. Stolte, T. D. Van de Ven, J. A. Burger, M. Prins and et al., Absence of HIV-1 superinfection 1 year after infection between 1985-1987 coincides with a reduction in sexual risk behavior in the seroincident Amsterdam cohort of homosexual men, Clin. Infect. Dis., 50 (2010), 1309-1315. doi: 10.1086/651687. [28] S. M. Raimundo, H. M. Yang, E. Venturino and E. Massad, Modeling the emergence of HIV-1 drug resistance resulting from antiretroviral therapy: Insights from theoretical and numerical studies, BioSystems, 108 (2012), 1-13. doi: 10.1016/j.biosystems.2011.11.009. [29] A. Ramos, D. J. Hu, L. Nguyen, K-O. Phan, S. Vanichseni and et al., Intersubtype human immunodeficiency virus type 1 superinfection following seroconversion to primary infection in two injection drug users, J. Virol., 76 (2002), 7444-7452. doi: 10.1128/JVI.76.15.7444-7452.2002. [30] A. D. Redd, C. E. Mullis, D. Serwadda, X. Kong, C. Martens and et al., The rates of HIV superinfection and primary HIV incidence in a general population in Rakai, Uganda, J. Infect. Dis, 206 (2012), 267-274. doi: 10.1093/infdis/jis325. [31] A. K. Sethi, D. D. Celentano, S. J. Gange, R. D. Moore and J. E. Gallant, Association between adherence to antiretroviral therapy and human immunodeficiency virus drug resistance, Clin. Infect. Dis., 37 (2003), 1112-1118. doi: 10.1086/378301. [32] O. Sharomi and A. B. Gumel, Dynamical analysis of a multi-strain model of HIV in the presence of antiretroviral drugs, J. Biol. Dyn., 2 (2008), 323-345. doi: 10.1080/17513750701775599. [33] D. M. Smith, D. D. Richman and S. J. Little, HIV superinfection, J. Infect. Dis., 192 (2005), 438-444. doi: 10.1086/431682. [34] D. M. Smith, J. K. Wong, G. K. Hightoer, C. C. Ignacio, K. K. Koelsch and et al., Incidence of HIV superinfection following primary infection, J. Amer. Med. Assoc., 292 (2004), 1177-1178. doi: 10.1001/jama.292.10.1177. [35] X. Sun, Y. Xiao, Z. Peng and N. Wang, Modelling HIV/AIDS epidemic among men who have sex with men in China, BioMed Res. Int., 2013 (2013), Article ID 413260, 18 pages. doi: 10.1155/2013/413260. [36] J. Takehisa, L. Zekeng, E. Ido, Y. Yamaguchi-Kabata, I. Mboudjeka and et al., Human immunodeficiency virus type 1 intergroup(M/O) recombination in Cameroon, J. Virol., 73 (1999), 6810-6820. [37] R. Tsui, B. L. Herring, J. D. Barbour, R. M. Grant, P. Bacchetti and et al., Human immuodeficiency virus type 1 superinfection was not detected following 215 years of injection drug user exposure, J. Virol., 78 (2004), 94-103. doi: 10.1128/JVI.78.1.94-103.2004. [38] R. Vardavas and S. Blower, The emergence of HIV transmitted resistance in Botswana: "When will the WHO detection threshold be exceeded?" Plos One, 2 (2007), e152. doi: 10.1371/journal.pone.0000152. [39] J. X. Velasco-Hernandez, A model for chagas disease involving transmission by vectors and blood transfusion, Theor. Popul. Biol., 46 (1994), 1-31. doi: 10.1006/tpbi.1994.1017. [40] P. Venden Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [41] P. L. Vernazza, B. L. Gilliam, M. Flepp, J. R. Dyer, A. C. Frank and et al., Effect of antiviral treatment on the shedding of HIV-1 in semen, AIDS, 11 (1997), 1249-1254. doi: 10.1097/00002030-199710000-00008. [42] W. Walter, Differential and Integral Inequalities, Springer, Berlin, 1970. [43] X. Wang, L. Yang, H. Li and et al., Factors associated with HIV virologic failure among patients on HAART for one year at three sentinel surveillance sities in China, Curr. HIV. Res., 9 (2011), 103-111. doi: 10.2174/157016211795569122. [44] D. P. Wilson, J. Kahn and S. M. Blower, Predicting the epidemiological impact of antiretroviral allocation strategies in KwaZulu-Natal: The effect of the urban-rural divide, PNAS, 103 (2006), 14228-14233. doi: 10.1073/pnas.0509689103. [45] D. P. Wilson, M. G. Law, A. E. Grulich, D. A. Cooper and J. M. Kaldor, Relation between HIV viral load and infectiousness: a model-based analysis, Lancet, 372 (2008), 314-320. doi: 10.1016/S0140-6736(08)61115-0. [46] H. Xing, Y. Ruan, J. Li and et al., HIV Drug resistance and its impact on antiretroviral therapy in chinese HIV-infected patients, Plos one, 8 (2013), e54917. doi: 10.1371/journal.pone.0054917. [47] M. Xiridou, M. Kretzschmar and R. Geskus, Competition of pathogen strains leading to infection with variable infectivity and the effect of treatment, Math. Biosci., 197 (2005), 153-172. doi: 10.1016/j.mbs.2005.04.007. [48] X. Xu, Y. Xiao and N. Wang, Modeling sexual transmission of HIV/AIDS in Jiangsu province, China, Math. Method. Appl. Sci., 36 (2012), 234-248. doi: 10.1002/mma.2599.
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