Mathematical Models of HIV: Methodologies and Applications

  • Emine YaylaliEmail author
  • Zikriye Melisa Erdogan
Conference paper
Part of the Lecture Notes in Management and Industrial Engineering book series (LNMIE)


HIV is one of the significant public health threats globally, with approximately 36.9 million people living with HIV and 1.8 million people becoming newly infected in 2017 (WHO fact sheet). To prevent HIV, to decrease its impact and to eventually eliminate this infectious disease; clinical, medical, epidemiological, economic, and modeling studies have been conducted in the last 30 years. In this study, we explore the mathematical modeling studies where HIV has been examined to understand the dynamics and spread of the disease as well as to improve HIV prevention. We surveyed HIV modeling literature, summarized primary modeling methodologies, and briefly discussed relevant studies. For each study included in this paper, we presented their modeling method, interventions included, target populations, implementation process, key results, and insights. Two most widely used modeling methodologies for HIV are Bernoulli process models and dynamic compartmental models similar to other infectious diseases. These methodologies have been discussed in detail in this paper. Other modeling methodologies included Markov models, agent-based simulation models, and discrete-event simulation models. Many studies focused on risk populations such as heterosexual (HET), men who have sex with men (MSM), people who inject drugs (PWID) and jail inmates. We included the cost-effectiveness studies where HIV prevention and treatment interventions and strategies are compared concerning their costs and benefits. In this survey, we provided a summary of existing modeling literature as well as suggestions for future studies. We concluded that application of modeling tools for HIV presents excellent opportunities for both decision-makers and public health policymakers while predicting the future of this disease, establishing the most cost-effective prevention strategies and evaluating possibilities for the elimination of HIV.


Mathematical Modeling Infectious Disease HIV AIDS Bernoulli Model Compartmental Model Markov Model Agent-Based Simulation 


  1. Adams LM, Kendall S, Smith A, Quigley E, Stuewig JB, Tangney JP (2013) HIV risk behaviors of male and female jail inmates prior to incarceration and one year post-release. AIDS Behav 17(8):2685–2694CrossRefGoogle Scholar
  2. Akpinar H (2012) Bulaşici hastaliklarin yayiliminin tahmininde deterministik modellerin kullanilmasi. Dspace RepositoryGoogle Scholar
  3. Alistar SS, Long EF, Brandeau ML, Beck EJ (2014) HIV epidemic control—a model for optimal allocation of prevention and treatment resources. Health Care Manag Sci 17(2):162–181CrossRefGoogle Scholar
  4. Baggaley RF, Irvine MA, Leber W, Cambiano V, Figueroa J, McMullen H, Hollingsworth TD (2017) Cost-effectiveness of screening for HIV in primary care: a health economics modelling analysis. Lancet HIV 4(10):e465–e474CrossRefGoogle Scholar
  5. Beyrer C, Baral SD, Van Griensven F, Goodreau SM, Chariyalertsak S, Wirtz AL, Brookmeyer R (2012) Global epidemiology of HIV infection in men who have sex with men. Lancet 380(9839): 367–377CrossRefGoogle Scholar
  6. Bos JM, Fennema JS, Postma MJ (2001) Cost-effectiveness of HIV screening of patients attending clinics for sexually transmitted diseases in Amsterdam. Aids 15(15):2031–2036CrossRefGoogle Scholar
  7. Bristow CC, Larson E, Anderson LJ, Klausner JD (2016) Cost-effectiveness of HIV and syphilis antenatal screening: a modelling study. Sex Transm Infect 92(5):340–346CrossRefGoogle Scholar
  8. Caro JJ, Briggs AH, Siebert U, Kuntz KM (2012) Modeling good research practices—overview: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force–1. Med Decis Making 32(5):667–677CrossRefGoogle Scholar
  9. Cipriano LE, Zaric GS, Holodniy M, Bendavid E, Owens DK, Brandeau ML (2012) Cost effectiveness of screening strategies for early identification of HIV and HCV infection in injection drug users. PLoS ONE 7(9):e45176CrossRefGoogle Scholar
  10. Downs AM, De Vincenzi I (1996) Probability of heterosexual transmission of HIV: relationship to the number of unprotected sexual contacts. JAIDS J Acquir Immune Defic Syndr 11(4):388–395CrossRefGoogle Scholar
  11. Fox J, White PJ, Weber J, Garnett GP, Ward H, Fidler S (2011) Quantifying sexual exposure to HIV within an HIV-serodiscordant relationship: development of an algorithm. Aids 25(8):1065–1082CrossRefGoogle Scholar
  12. Gilmour S, Li J, Shibuya K (2012) Projecting HIV transmission in Japan. PLoS ONE 7(8):e43473CrossRefGoogle Scholar
  13. Gopalappa C, Farnham PG, Hutchinson AB, Sansom SL (2012) Cost effectiveness of the National HIV/AIDS Strategy goal of increasing linkage to care for HIV-infected persons. JAIDS J Acquir Immune Defic Syndr 61(1):99–105CrossRefGoogle Scholar
  14. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42(4):599–653MathSciNetCrossRefGoogle Scholar
  15. Joint United Nations Programme on HIV/AIDS (UNAIDS). (2018) UNAIDS Data 2018. Retrieved from UNAIDS website:
  16. Hontelez JA, Nagelkerke N, Bärnighausen T, Bakker R, Tanser F, Newell ML, de Vlas SJ (2011) The potential impact of RV144-like vaccines in rural South Africa: a study using the STDSIM microsimulation model. Vaccine 29(36):6100–6106CrossRefGoogle Scholar
  17. Joint United Nations Programme on HIV/AIDS (UNAIDS). (2012) UNAIDS Report on the Global AIDS Epidemic 2012. Retrieved from UNAIDS website:
  18. Joint United Nations Programme on HIV/AIDS (UNAIDS) (2018). Fact Sheet-World AIDS Day 2018. Retrieved from UNAIDS website:
  19. Joint United Nations Programme on HIV/AIDS (UNAIDS) (2017). UNAIDS Data 2017. Retrieved from UNAIDS website:
  20. Joint United Nations Programme on HIV/AIDS (UNAIDS).(2018). UNAIDS Data 2018. Retrieved from UNAIDS website:
  21. Juusola JL, Brandeau ML, Owens DK, Bendavid E (2012) The cost-effectiveness of preexposure prophylaxis for HIV prevention in the United States in men who have sex with men. Ann Intern Med 156(8):541–550CrossRefGoogle Scholar
  22. Keeling MJ, Rohani P (2011) Modeling infectious diseases in humans and animals. Princeton University Press, PrincetonCrossRefGoogle Scholar
  23. Kretzschmar M, Wallinga J (2009) Mathematical models in infectious disease epidemiology. In: Modern infectious disease epidemiology, pp. 209–221. Springer, New YorkCrossRefGoogle Scholar
  24. Lasry A, Sansom SL, Hicks KA, Uzunangelov V (2012) Allocating HIV prevention funds in the United States: recommendations from an optimization model. PLoS ONE 7(6):e37545CrossRefGoogle Scholar
  25. Lasry A, Sansom SL, Hicks KA, Uzunangelov V (2011) A model for allocating CDC’s HIV prevention resources in the United States. Health Care Manag Sci 14(1):115–124CrossRefGoogle Scholar
  26. Longini IM, Clark WS, Gardner LI, Brundage JF (1991) The dynamics of CD4+ T-lymphocyte decline in HIV-infected individuals: a Markov modeling approach. J Acquir Immune Defic Syndr 4(11):1141–1147Google Scholar
  27. Lin F, Lasry A, Sansom SL, Wolitski RJ (2013) Estimating the impact of state budget cuts and redirection of prevention resources on the HIV epidemic in 59 California local health departments. PLoS ONE 8(3):e55713CrossRefGoogle Scholar
  28. Lin F, Farnham PG, Shrestha RK, Mermin J, Sansom SL (2016) Cost effectiveness of HIV prevention interventions in the US. Am J Prev Med 50(6):699–708CrossRefGoogle Scholar
  29. Macal CM, North MJ (2005) Tutorial on agent-based modeling and simulation. In: Proceedings of the winter simulation conference, p 14. IEEE, December 2005Google Scholar
  30. Macal CM, North MJ (2010) Tutorial on agent-based modelling and simulation. J. Simul 4(3):151–162CrossRefGoogle Scholar
  31. Pearson CR, Kurth AE, Cassels S, Martin DP, Simoni JM, Hoff P, Gloyd S (2007) Modeling HIV transmission risk among Mozambicans prior to their initiating highly active antiretroviral therapy. AIDS Care 19(5):594–604CrossRefGoogle Scholar
  32. Piot P, Quinn TC (2013) Response to the AIDS pandemic—a global health model. N Engl J Med 368(23):2210–2218CrossRefGoogle Scholar
  33. Pinkerton SD, Abramson PR (1998) The Bernoulli-process model of HIV transmission. In: Handbook of economic evaluation of HIV prevention programs, pp. 13–32. Springer, BostonCrossRefGoogle Scholar
  34. Pinkerton SD, Abramson PR (1996) Implications of increased infectivity in early-stage HIV infection: application of a Bernoulli-process model of HIV transmission. Evaluation Review 20(5):516–540CrossRefGoogle Scholar
  35. Pinkerton SD, Holtgrave DR, Valdiserri RO (1997) Cost-effectiveness of HIV-prevention skills training for men who have sex with men. Aids 11(3):347–357CrossRefGoogle Scholar
  36. Ross SM (2007) Introduction to probability models. Academic press, CambridgeGoogle Scholar
  37. Sanders GD, Bayoumi AM, Sundaram V, Bilir SP, Neukermans CP, Rydzak CE, Owens DK (2005) Cost-effectiveness of screening for HIV in the era of highly active antiretroviral therapy. N Engl J Med 352(6):570–585CrossRefGoogle Scholar
  38. Sayan M, Hinçal E, Şanlidağ T, Kaymakamzade B, Sa’ad FT, Baba IA (2017) Dynamics of HIV/AIDS in Turkey from 1985 to 2016. Qual Quant 52:711–723CrossRefGoogle Scholar
  39. Sorensen SW, Sansom SL, Brooks JT, Marks G, Begier EM, Buchacz K, Kilmarx PH (2012) A mathematical model of comprehensive test-and-treat services and HIV incidence among men who have sex with men in the United States. PloS one 7(2):e29098CrossRefGoogle Scholar
  40. Wang S, Moss JR, Hiller JE (2011) The cost-effectiveness of HIV voluntary counseling and testing in China. Asia Pac. J. Public Health 23(4):620–633CrossRefGoogle Scholar
  41. Wilson SR, Lavori PW, Brown NL, Kao YM (2003) Correlates of sexual risk for HIV infection in female members of heterosexual California Latino couples: an application of a Bernoulli process model. AIDS Behav 7(3):273–290CrossRefGoogle Scholar
  42. Xuan H, Xu L, Li L (2009) A CA-based epidemic model for HIV/AIDS transmission with heterogeneity. Ann Oper Res 168(1):81MathSciNetCrossRefGoogle Scholar
  43. Yaylali E, Farnham PG, Schneider KL, Landers SJ, Kouzouian O, Lasry A, Sansom SL (2016) From theory to practice: implementation of a resource allocation model in health departments. J. Public Health Manag Pract: JPHMP 22(6):567CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Industrial Engineering Department, Management FacultyIstanbul Technical UniversityIstanbulTurkey

Personalised recommendations