Journal of Mathematical Biology

, Volume 73, Issue 4, pp 919–946 | Cite as

Stochastic modelling of the eradication of the HIV-1 infection by stimulation of latently infected cells in patients under highly active anti-retroviral therapy

  • Daniel Sánchez-TaltavullEmail author
  • Arturo Vieiro
  • Tomás Alarcón


HIV-1 infected patients are effectively treated with highly active anti-retroviral therapy (HAART). Whilst HAART is successful in keeping the disease at bay with average levels of viral load well below the detection threshold of standard clinical assays, it fails to completely eradicate the infection, which persists due to the emergence of a latent reservoir with a half-life time of years and is immune to HAART. This implies that life-long administration of HAART is, at the moment, necessary for HIV-1-infected patients, which is prone to drug resistance and cumulative side effects as well as imposing a considerable financial burden on developing countries, those more afflicted by HIV, and public health systems. The development of therapies which specifically aim at the removal of this latent reservoir has become a focus of much research. A proposal for such therapy consists of elevating the rate of activation of the latently infected cells: by transferring cells from the latently infected reservoir to the active infected compartment, more cells are exposed to the anti-retroviral drugs thus increasing their effectiveness. In this paper, we present a stochastic model of the dynamics of the HIV-1 infection and study the effect of the rate of latently infected cell activation on the average extinction time of the infection. By analysing the model by means of an asymptotic approximation using the semi-classical quasi steady state approximation (QSS), we ascertain that this therapy reduces the average life-time of the infection by many orders of magnitudes. We test the accuracy of our asymptotic results by means of direct simulation of the stochastic process using a hybrid multi-scale Monte Carlo scheme.


HIV-1 HAART Latently infection Stochastic modelling Antigen stimulation 

Mathematics Subject Classification

34E20 Singular perturbations turning point theory WKB methods 34E13 Multiple scale methods 92B05 General biology and biomathematics 



D.S.T. and T.A. acknowledge the spanish Ministry for Science and Innovation (MICINN) for funding under Grant MTM2011-29342 and Generalitat de Catalunya for financial support under Grant 2014SGR1307. A.V. has been supported by Grants MTM2010-16425 and MTM2013-41168-P (Spain) and 2014 SGR 1145 (Catalonia). This work was done during the stay of A.V. at the Centre de Recerca Matemàtica (CRM, Catalunya), he warmly thanks the CRM for the facilities and support. The computing facilities of the UB Dynamical Systems Group have been largely used, the authors thank J. Timoneda for the technical support. T.A. acknowledges support from the Ministry of Economy and Competitivity (MINECO) for funding awarded to the Barcelona Graduate School of Mathematics under the “María de Maeztu” programme, Grant Number MDM-2014-0445.


  1. Alarcón T (2014) Stochastic quasi-steady state approximations for asymptotic solutions of the chemical master equation. J Chem Phys 140:184109CrossRefGoogle Scholar
  2. Alarcón T, Jensen HJ (2011) Quiescence: a mechanism for escaping the effects of drug on cell populations. J R Soc Interface 8(54):99–106CrossRefGoogle Scholar
  3. Alarcón T, Page KM (2007) Mathematical models of the vegf receptor and its role in cancer therapy. J R Soc Interface 4:283–304CrossRefGoogle Scholar
  4. Arnol’d VI (1989) Mathematical methods of classical mechanics, vol 60. Springer, New YorkCrossRefzbMATHGoogle Scholar
  5. Assaf M, Meerson B (2006) Spectral theory of metastability and extinction in birth-death systems. Phys Rev Lett 97(20):200602CrossRefGoogle Scholar
  6. Assaf M, Meerson B (2007) Spectral theory of metastability and extinction in a branching-annihilation reaction. Phys Rev E 75(3):031122CrossRefGoogle Scholar
  7. Assaf M, Meerson B (2010) Extinction of metastable stochastic populations. Phys Rev E 81:021116CrossRefGoogle Scholar
  8. Assaf M, Meerson B, Sasorov PV (2010) Large fluctuations in stochastic population dynamics: momentum-space calculations. J Stat Mech:P07018Google Scholar
  9. Batut C, Belabas K, Bernardi D, Cohen H, Olivier M (1995) Users’ guide to PARI/GP.
  10. Bohil AB, Robertson BW, Cheney RE (2006) Myosin-x is a molecular motor that functions in filopodia formation. Proc Nat Acad Sci 103(33):12411–12416CrossRefGoogle Scholar
  11. Cabré X, Fontich E, De La Llave R (2005) The parameterization method for invariant manifolds iii: overview and applications. J Differ Equations 218(2):444–515MathSciNetCrossRefzbMATHGoogle Scholar
  12. Cao Y, Gillespie D, Petzold L (2005) Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. J Comput Phys 206(2):395–411MathSciNetCrossRefzbMATHGoogle Scholar
  13. Chomont N, El-Far M, Ancuta P, Trautmann L, Procopio FA, Yassine-Diab B, Boucher G, Boulassel M-R, Brenchley JM, Schacker TW, Hill BJ, Ghattas DCDG, Routy J-P, Haddad EK, Sekaly R-P (2009) HIV reservoir size and persistence are driven by T cell survival and homeostatic proliferation. Nature Med 15:893–901CrossRefGoogle Scholar
  14. Chun T-W, Carruth L, Finzi D, Shen X, DiGiuseppe JA, Taylor H, Hermankova M, Chadwick K, Margolick J, Quinn TC, Kuo Y-H, Brookmeyer R, Zeiger MA, Barditch-Crovo P, Siliciano RF (1997) Quantification of latent tissue reservoirs and total body viral load in HIV-1 infection. Nature 387:183–188CrossRefGoogle Scholar
  15. Chun TW, Finzi D, Margolick J, Chadwick K, Schwartz D, Siliciano RF (1995) In vivo fate of HIV-1-infected T cells: quantitative analysis of the transition to stable latency. Nat Med 1:1284–1290CrossRefGoogle Scholar
  16. Conway JM, Coombs D (2011) A stochastic model of latently infected cell reactivation and viral blip generation in treated HIV patients. PLoS Comput Biol 7:e1002033Google Scholar
  17. de la Cruz R, Guerrero P, Spill F, Alarcón T (2014) The effects of intrinsic noise on the behaviour of bistable systems in quasi-steady state conditions (submitted)Google Scholar
  18. Doering CR, Sargsyan KV, Sander LM (2005) Extinction times for birth-death processes: exact results, continuum asymptotics, and the failure of the fokker-planck approximation. Multiscale Model Simul 3:283MathSciNetCrossRefzbMATHGoogle Scholar
  19. Dykman MI, Mori E, Ross J, Hunt PM (1994) Large fluctuations and optimal paths in chemical kinetics. J Chem Phys 100:5735–5750CrossRefGoogle Scholar
  20. Dykman MI, Schwartz IB, Landsman AS (2008) Disease extinction in the presence of random vaccination. Phys Rev Lett 101:078101CrossRefGoogle Scholar
  21. Elgart V, Kamenev A (2004) Rare event statistics in reaction-diffusion systems. Phys Rev E 70:041106MathSciNetCrossRefGoogle Scholar
  22. Freidlin MI, Wentzell AD (1998) Random perturbations of dynamical systems, vol. 260 of grundlehren der mathematischen wissenschaften (fundamental principles of mathematical sciences)Google Scholar
  23. Gardiner CW (2009) Stochatic methods. Springer, BerlinGoogle Scholar
  24. Gelfreich V, Simó C, Vieiro A (2013) Dynamics of 4D symplectic maps near a double resonance. Phys D 243(1):92–110MathSciNetCrossRefzbMATHGoogle Scholar
  25. Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22(4):403–434MathSciNetCrossRefGoogle Scholar
  26. Goldstein H (1980) Classical mechanicsGoogle Scholar
  27. Gottesman O, Meerson B (2012) Multiple extinction routes in stochastic population models. Phys Rev E 85:021140CrossRefGoogle Scholar
  28. Gray C, Karl G, Novikov V (2004) Progress in classical and quantum variational principles. Reports Progress Phys 67(2):159MathSciNetCrossRefGoogle Scholar
  29. Gray C, Taylor EF (2007) When action is not least. Am J Phys 75(5):434–458CrossRefGoogle Scholar
  30. Haro A, Canadell M, Figueras J, Luque A, Mondelo J (2016) The parameterization method for invariant manifolds: from rigorous results to effective computations. Springer, New YorkCrossRefzbMATHGoogle Scholar
  31. Herz AVM, Bonhoefer S, Anderson RM, May RM, Nowak MA (1996) Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proc Natl Acad Sci 93:7247–7251CrossRefGoogle Scholar
  32. Ho DD, Neumann AU, Perelson AS, Chen W, Leonard JM, Markowitz M (1995) Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 373:123–126CrossRefGoogle Scholar
  33. Ho DD, Rota TR, Hirsch MS (1986) Infection of monocyte/macrophages by human T lymphotropic virus type III. J Clin Invest 77:1712–1715CrossRefGoogle Scholar
  34. Jones LE, Perelson AS (2007) Transient viremia, plasma viral load and reservoir replenishment in HIV-infected patients on antiretroviral therapy. J Acquir Immune Defic Syndr 45:483–493CrossRefGoogle Scholar
  35. Jorba À, Zou M (2005) A software package for the numerical integration of ODEs by means of high-order Taylor methods. Exp Math 14(1):99–117MathSciNetCrossRefzbMATHGoogle Scholar
  36. Kamenev A, Meerson B (2008) Extinction of an infectious disease: a large fluctuation in a non-equilibrium system. Phys Rev E 77:061107MathSciNetCrossRefGoogle Scholar
  37. Katlama C, Deeks SG, Autran B, Martinez-Picado J, van Luzen J, Rouzioux C, Miller M, Vella S, Schmitz JE, Ahlers J, Richman DD, Sekaly RP (2013) Barriers to a cure for HIV: new ways to target and eradicate HIV-1 reservoirs. Lancet 381:2109–2117CrossRefGoogle Scholar
  38. Kent SJ, Reece JC, Petravic J, Martyushev A, Kramski M, Rose RD, Cooper DA, Kelleher AD, Emery S, Cameron PU, Lewin SR, Davenport MP (2013) The search for an HIV cure: tackling latent infection. Lancet Infect Dis 13:614–621CrossRefGoogle Scholar
  39. Kepler TB, Perelson AS (1998) Drug concentration heterogeneityof drug resistance facilitates the evolution of drug resistance. Proc Natl Acad Sci 95:11514–11519CrossRefzbMATHGoogle Scholar
  40. Khasin M, Dykman MI (2009) Extinction rate fragility in population dynamics. Phys Rev Lett 103:068101CrossRefGoogle Scholar
  41. Khasin M, Dykman MI, Meerson B (2010) Speeding up disease extinction with a limited amount of vaccine. Phys Rev E 81:051925MathSciNetCrossRefGoogle Scholar
  42. Kim H, Perelson AS (2006) Viral and latent reservoir persistence in HIV-1-infected patients on therapy. PLoS Comp Biol 2:e135CrossRefGoogle Scholar
  43. Kubo R, Matsou K, Kitahara K (1973) Fluctuations and relaxation of macrovariables. J Stat Phys 9:51–96CrossRefGoogle Scholar
  44. Landau L, Lifshitz E (1976) Course of theoretical physics. Vol. 1. Mechanics (trans: Russian by Skyes JB, Bell JS). Pergamon Press, Oxford, New York, Toronto, ONGoogle Scholar
  45. Markowitz M, Louie M, Hurley A, Sun E, Mascio MD, Perelson AS, Ho DD (2003) A novel antiviral intervention results in more accurate assessment of human immunodeficiency virus type 1 replication dynamics and t-cell decay in vivo. J Virol 77:5037–5038CrossRefGoogle Scholar
  46. Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  47. Ovaskainen O, Meerson B (2010) Stochastic models of population extinction. Trends Ecol. Evol. 25:646–652CrossRefGoogle Scholar
  48. Perelson AS, Essunger P, Cao Y, Vesanen M, Hurley A, Saksela K, Markowitz M, Ho DD (1997) Decay characteristics of HIV-1-infected compartments during combination therapy. Nature 387:188–191CrossRefGoogle Scholar
  49. Perelson AS, Neumann AU, Markowitz M, Leonard JM, Ho DD (1996) HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271:1582–1586CrossRefGoogle Scholar
  50. Pierson T, McArthur J, Siciliano RF (2000) Reservoirs for HIV-1: mechanisms for viral persistence in the presence of antiviral immune response and antiretroviral therapy. Annu Rev Immunol 18:665–708CrossRefGoogle Scholar
  51. Rong L, Perelson AS (2009) Modeling latently infected cell activation: viral and latent reservoir persistence, and viral blips in HIV-infected patients on potent therapy. PLoS Comput Biol 5:e1000533Google Scholar
  52. Rong L, Perelson AS (2009) Modelling HIV persistence, the latent reservoir, and viral blips. J Theor Biol 260:308–331MathSciNetCrossRefGoogle Scholar
  53. Schwartz IB, Billings L, Dykman MI, Landsman AS (2009) Predicting extinction rates in stochastic epidemic models. J Stat Mech:P01005Google Scholar
  54. Shan L, Deng K, Shroff NS, Durand CM, Rabi SA, Yang H-C, Zhang H, Margolick JB, Blankson JN, Siciliano RF (2012) Stimulation of HIV-1-specific cytolityc T lymphocytes facilitates elimination of latent viral reservoir after virus reactivation. Immunity 36:491–501CrossRefGoogle Scholar
  55. Siliciano JD, Kajdas J, Finzi D, Quinn TC, Chadwick K, Margolick JB, Kovacs C, Gange SJ, Siliciano RF (2003) Long-term follow-up studies confirm the stability of the latent reservoir for hiv-1 in resting CD4+ T cells. Nat Med 9:727–728CrossRefGoogle Scholar
  56. Simó C (1990) On the analytical and numerical approximation of invariant manifolds. In: Les Méthodes Modernes de la Mécanique Céleste. Modern methods in celestial mechanics vol. 1, pp 285-329Google Scholar
  57. Trono D, Lint CV, Rouziux C, Verdin E, Barre-Sinoussi F, Chun T-W, Chomont N (2010) HIV persistence and the prospect of long-term drug-free remissions for HIV-infected individuals. Science 329:174–180CrossRefGoogle Scholar
  58. Van Kampen NG (2007) Stochastic processes in physics and chemistry. Elsevier, The NetherlandszbMATHGoogle Scholar
  59. Ward MJ (1998) Exponential asymptotics and convection-diffusion reactions models. In: Analysing multiscale phenomena using singular perturbation methods, Proceedings of Symposia in Applied Mathematics, vol. 56, pp 151-184. AMS Short CoursesGoogle Scholar
  60. Wei X, Ghosh SK, Taylor ME, Johnson VA, Emini EA, Deutsch P, Lifson JD, Bonhoeffer S, Nowak MA, Hahn BH, Saag MS, Shaw GM (1995) Viral dynamics in human immunodeficiency virus type 1 infection. Nature 373:117–122CrossRefGoogle Scholar
  61. Zhang Z-Q, Schuler T, Zupancic M, Wietgrefe S, Staskus KA, Reimann KA, Reinhart TA, Rogan M, Cavert W, Miller CJ, Veazey RS, Notermans D, Little S, Danner SA, Richman DD, Havlir D, Wong J, Jordan HL, Schacker TW, Racz P, Tenner-Racz K, Letvin NL, Wolinsky S, Haase AT (1999) Sexual transmission and propagation of SIV and HIV in resting and activated CD4+ T Cells. Science 286:1712–1715CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Daniel Sánchez-Taltavull
    • 1
    • 4
    Email author
  • Arturo Vieiro
    • 2
  • Tomás Alarcón
    • 3
    • 4
    • 5
    • 6
  1. 1.Regenerative Medicine ProgramOttawa Hospital Research InstituteOttawaCanada
  2. 2.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain
  3. 3.ICREA (Institució Catalana de Recerca i Estudis Avançats)BarcelonaSpain
  4. 4.Centre de Recerca Matemàtica, Edifici CBarcelonaSpain
  5. 5.Departament de MatemàtiquesUniversitat Autònoma de Barcelona, BellaterraBarcelonaSpain
  6. 6.Barcelona Graduate School of Mathematics (BGSMath)BarcelonaSpain

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