Modelling the natural history of HIV infection in individuals and its epidemiological implications F. A. B. Coutinho L. F. Lopez M. N. Burattini E. Massad Article

Received: 22 August 1999 Accepted: 04 July 2001 DOI :
10.1006/bulm.2001.0257

Cite this article as: Coutinho, F.A.B., Lopez, L.F., Burattini, M.N. et al. Bull. Math. Biol. (2001) 63: 1041. doi:10.1006/bulm.2001.0257
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Abstract The variation of viraemia in the natural course of HIV infection is expected to have major influence on the probability of transmission and, consequently, on the epidemiology of HIV/AIDS.

In this paper we propose a model which takes into account the time evolution of HIV viraemia (measured as HIV-RNA copies per ml of blood) in an infected individual and its impact on the threshold for the establishment of an endemic level, and mainly on the relative contribution of each of the clinical phases of the infection to the total transmission of HIV per infected individual.

We consider that an infected individual passes through three phases of viraemia. The first phase, which lasts for 6–7 weeks, is characterized by very high viraemia. In the second phase, which lasts about 10 years, the viraemia is much lower, increasing again in the last phase, which lasts up to two years, and ends in full-blown AIDS.

We show that the relative contribution of each phase to the total transmission of HIV is very sensitive to the model we assume for the dependence of the transmissibility of HIV on the viral load. For instance, if we assume that transmissibility is proportional to the decimal logarithm of viraemia, then the second phase predominates always. Due to the epidemiological importance of this fact, it is clear that further improvement on virological research to better understand the dependence of HIV transmissibility on the viral concentration in biological fluids is necessary.

References Ahlgreen, D. J., M. K. Corny and A. C. Stein (1990). Model-based optmization of infectivity parameters: a study of the early epidemics in San Francisco.

J. Acquir. Immune Defic. Syndr.
3 , 631–642.

Google Scholar Anderson, R. M. and R. M. May (1991).

Infectious Diseases of Humans , Oxford: Oxford Science Publications.

Google Scholar Aoki-Sei, S., R. Yarchoan, S. Kageyama, D. T. Hoekzema, J. M. Pluda, K. M. Wyvill, S. Broder and H. Mitsuya (1992). Plasma HIV-1 viraemia in HIV-1 infected individuals assessed by polymerase chain reaction.

AIDS Res. Hum. Retroviruses
8 , 1263–1270.

CrossRef Google Scholar Arfken, G. (1970).

Mathematical Methods for Physicists , 2nd edn, New York: Academic.

Google Scholar Bagasra, O., S. P. Hauptman, H. W. Lischner, M. Sachs and R. J. Pomerantz (1992). Detection of human immunodeficiency virus type 1 provirus in mononuclear cells by in situ polymerase chain reaction.

N. Engl. J. Med.
326 , 1385–1391.

CrossRef Google Scholar Bagnarelli, P., S. Menzo, A. Manzin, M. Giacca, P. E. Varaldo and M. Clementi (1991). Detection of human immunodeficiency virus type 1 genomic RNA in plasma samples by reverse-transcription polymerase chain reaction.

J. Med. Virol.
34 , 89–95.

Google Scholar Blower, S. M., H. B. Gershengorn and R. M. Grant (2000). A tale of two futures: HIV and antiretroviral therapy in San Francisco.

Science
287 , 650–654.

CrossRef Google Scholar Bravo, R.

et al. (1995). The quantification of human immunodeficiency virus viraemia in patients with rapid and slow progression of the disease.

Med. Clin. (Barc.)
104 , 530–534.

Google Scholar Brookmeyer, R. and M. H. Gail (1994).

AIDS Epidemiology: A Quantitative Approach , Oxford: Oxford University Press.

Google Scholar Cameron, D. W., J. N. Simonsen, L. J. D’Costa, A. R. Ronald, G. M. Maitha, M. N. Gakinya, M. Cheang, J. O. Ndinya-Achola, P. Piot and R. C. Brunham (1989). Female to male transmission of human immunodeficiency virus type 1: risk factors for seroconversion in men.

Lancet
2 , 403–407.

CrossRef Google Scholar Connor, R. I., H. Mohri, Y. Cao and D. D. Ho (1993). Increased viral burden and cytopathicity correlate temporally with CD4

^{+} T-lymphocyte decline and clinical progression in human immunodeficiency virus type-1 infected individuals.

J. Virol.
67 , 1772–1777.

Google Scholar Coombs, R. W., A. C. Collier, J. P. Allain, B. Nikora, M. Leuther, G. F. Gjerset and L. Corey (1989). Plasma viraemia in human immunodeficiency virus infection.

N. Engl. J. Med.
321 , 1626–1631.

CrossRef Google Scholar Coutinho, F. A. B., E. Massad, M. N. Burattini and R. X. Menezes (1999). A theoretical model for the evolution of virulence in sexually transmitted HIV/AIDS.

Rev. Saúde Pública
33 , 329–333.

CrossRef Google Scholar Coutinho, F. A. B., E. Massad, M. N. Burattini, H. M. Yang and R. S. Azevedo-Neto (1993). Effects of vaccination programs on transmission rates of infections and related threshold conditions for control.

IMA J. Math. Appl. Med. Biol.
10 , 187–206.

MATH Google Scholar Coutinho, F. A. B., E. Massad, L. F. Lopez, M. N. Burattini, C. J. Struchiner and R. S. Azevedo (1999). Modelling heterogeneitis in individual frailties in epidemic models.

Math. Comput. Model.
30 , 97–115.

CrossRef MATH Google Scholar Dickover, R. E., E. M. Garratty, S. A. Herman, M. S. Sim, S. Plaeger, P. J. Boyer, M. Keller, A. Deveikis, E. R. Stiehm and Y. J. Bryson (1996). Identification of levels of maternal HIV-1 RNA associated with risk of perinatal transmission. Effect of maternal zydovudine treatment on viral load.

JAMA
275 , 599–605.

CrossRef Google Scholar Easterbrook, P. J. (1994). Non-progression in HIV infection.

AIDS
8 , 1179–1182.

Google Scholar Erb, P., S. Krauchi, D. Burgin, K. Biedermann, C. Camli, C. H. Rudin and the Swiss HIV and pregnancy colaborative study group (1994). Quantitative anti-p24 determinations can predict the risk of vertical transmission.

J. Acquir. Immune Defic. Syndr.
7 , 261–264.

Google Scholar Garcia, P. M.

et al. (1999). Maternal levels of plasma human immunodeficiency virus type 1 RNA and the risk of transmission.

N. Engl. J. Med.
341 , 394–402.

CrossRef Google Scholar Graziosi, C., G. Pantaleo, L. Butini, J. F. Demareste, M. S. Saag, G. M. Shaw and A. S. Faucci (1993). Kinetic of human immunodeficiency virus type 1 (HIV-1) DNA and RNA synthesis during primary HIV-1 infection.

Proc. Natl Acad. Sci. USA
90 , 6405–6409.

CrossRef Google Scholar Griensven, van, G. J. P., E. M. M. de Vroome, F. de Wolf, J. Goudsmit, M. Roos and R. A. Coutinho (1990). Risk factors for progression of human immunodeficiency virus (HIV) infection among seroconverted and seropositive homosexual men.

Am. J. Epidemiol.
132 , 203–210.

Google Scholar Griffel, D. H. (1981).

Applied Functional Analysis , Chichester: Ellis Horwood.

MATH Google Scholar Henrard, D. R., J. Phillips, S. Burchett, J. B. Jackson, P. Kataaha, F. Mmiro and C. Ndugwa (1992). Plasma viral load in symptom free women and vertical transmission of HIV.

Lancet
340 , 1470–1471.

CrossRef Google Scholar Hethcote, H. W. and J. W. Van Ark (1992).

Modelling HIV transmission and AIDS in the United States , Berlin: Springer-Verlag.

Google Scholar Ho, D. D., T. Moudgil and M. Alan (1989). Quantitation of human immunodeficiency virus type 1 in the blood of infected persons.

N. Engl. J. Med.
321 , 1621–1625.

CrossRef Google Scholar Holodniy, M., D. A. Katzenstein, S. Sengupta, A. M. Wang, C. Casipit, D. H. Schwarts, M. Konrad, E. Groves and T. C. Merigan (1991). Detection and quantification of human immunodeficiency virus RNA in patients serum by use of the polymerase chain reaction.

J. Infect. Dis.
163 , 862–866.

Google Scholar Houn, H. Y., A. A. Pappas, and E. M. Walker Jr (1987). Status of current clinical tests for human immunodeficiency virus (HIV): applications and limitations.

Ann. Clin. Lab. Sci.
17 , 279–285.

Google Scholar Hyman, J. M. and F. A. Stanley (1988). Using mathematical models to understand the Aids epidemics.

Math. Biosci.
90 , 415–473.

MathSciNet CrossRef MATH Google Scholar Jackson, J. B. and H. H. Balfour Jr (1988). Practical diagnostic testing for human immunodeficiency virus.

Clin. Microbiol. Rev.
1 , 124–138.

Google Scholar Jacquez, J. A., J. S. Koopman, C. P. Simon and I. M. Longini Jr (1994). Role of the primary infection in epidemics of HIV infection in gay cohorts.

J. Acquir. Immune Defic. Syndr.
7 , 1169–1184.

Google Scholar Jurriaans, S. (1995). Virus load in HIV-1 Infection, PhD thesis, Univerrsity of Amsterdam.

Keet, I. P. M., P. Krijnen, N. Koot, J. M. A. Lange, F. Miedema, J. Goudsmit and R. A. Coutinho (1993). Predictors of rapid progression to AIDS in HIV-1 seroconverters.

AIDS
7 , 51–57.

Google Scholar Kievits, T., B. van Gemen, D. van Strijp, R. Schukkink, M. Dircks, H. Adriaanse, L. Malek, R. Sooknanan and P. Lens (1991). NASBA isothermal enzymatic in vitru nucleic acid amplification optimized for the diagnosis of HIV-1 infection.

J. Virol. Methods
35 , 273–286.

CrossRef Google Scholar Koopman, S. J., J. A. Jacquez, W. W. Gavin, C. P. Simon, B. Foxman, M. P. Stephen, D. Barth-Jones, L. A. Andrew and K. Lange (1997). The role of early HIV infection in the spread of HIV through populations.

J. Acquir. Immune Defic. Syndr. Hum. Retrovirol.
14 , 249–258.

Google Scholar Layne, S. P., M. J. Merges, M. Dembo, J. L. Spouge, S. R. Conley, J. P. Moor, J. L. Raina, H. Renz, H. R. Gelderblom and P. L. Nara (1992). Factors underlying spontaneous inactivation and susceptibility to neutralization of human immunodeficiency virus.

Virology
189 , 695–714.

CrossRef Google Scholar Levin, B. R., J. J. Bull and F. M. Stewart (1996). The intrinsic rate of increase of HIV/AIDS: epidemiological and evolutionary implications.

Math. Biosci.
132 , 69–96.

CrossRef MATH Google Scholar Lifson, A. R., S. P. Buchbinder, H. W. Sheppard, A. C. Mawle, J. C. Wilber, N. Stanley, C. E. Hart, N. A. Hessol and S. D. Holmberg (1991). Long-term human immunodeficiency virus infection in asymptomathic homosexual and bisexual men with normal CD4

^{+} lymphocyte counts: immunologic and virologic characteristics.

J. Infect. Dis.
163 , 959–965.

Google Scholar Lopez, L. F. and F. A. B. Coutinho (2000). On the existence and uniqueness of a solution of an integral equation which appear in epidemiology.

J. Math. Biol.
40 , 199–228.

MathSciNet CrossRef MATH Google Scholar Massad, E. (1989). A homogeneously mixing population model for the AIDS epidemic.

Math. Comput. Model.
12 , 89–96.

MATH CrossRef Google Scholar May, R. M. and R. M. Anderson (1990). Parasite-host coevolution.

Parasitology
100 , S89–S101.

CrossRef Google Scholar Mellors, J. W., C. R. Rinaldo Jr, P. Gupta, R. M. White, J. A. Todd and L. A. Kingsley (1996). Prognosis in HIV-1 infection predicted by the quantity of virus in plasma.

Science
272 , 1167–1170.

Google Scholar Osmond, D., P. Bacchetti, R. E. Chaisson, T. Kelly, R. Stempel, J. Carlson and A. R. Moss (1988). Time of exposure and risk of HIV infection in homosexual partners of men with AIDS.

Am. J. Public Health
78 , 944–948.

CrossRef Google Scholar Piatak, M. J. Jr, M. S. Saag, L. C. Yang, S. J. Clark, J. C. Kappes, K. C. Luk, P. H. Hahn, G. M. Shaw and J. D. Lifson (1993). High levels of HIV-1 in plasma during all stages of infection determined by competitive PCR.

Science
259 , 1749–1759.

Google Scholar Plummer, F. A., J. N. Simonsen and D. W. Cameron

et al. (1991). Cofactors in male-female sexual transmission of human immunodeficiency virus type 1.

J. Infect. Dis.
163 , 233.

Google Scholar Quinn, T. C., M. J. Wawer, N. Sewankambo, D. Serwardda, C. J. Li, F. Wabwire-Mangen, M. O. Meehan, T. Lutalo and R. H. Gray (2000). Viral load and heterosexual transmission of human immunodeficiency virus type 1.

N. Engl. J. Med.
342 , 921–929.

CrossRef Google Scholar Sano, K. and M. H. Lee (1987). Antibody that inhibits human immunodeficiency virus reverse transnriptase and association with inability to isolate virus.

J. Clin. Microbiol.
25 , 2415–2417.

Google Scholar Simmonds, P. (1990). Variation in HIV virus load of individuals at different stages in infection: possible relationship with risk of transmission.

AIDS
4(Suppl. 1) , S77–S83.

Google Scholar Simonsen, J. N., D. W. Cameron, M. N. Gakinya, J. O. Ndinya-Achola, L. J. D’Costa, P. Karasira, M. Cheang, A. R. Ronald, P. Piot and F. A. Plummer (1988). Human immunodeficiency virus infection among men with sexually transmitted disease. Experience from a center in Africa.

N. Engl. J. Med.
319 , 274–278.

CrossRef Google Scholar Vella, S.

et al. (2000). Plasma HIV-1 copy number and in vitro infectivity of plasma prior to and during combination antiretroviral treatment.

Antiviral Res.
47 , 189–198.

CrossRef Google Scholar © Society for Mathematical Biology 2001

Authors and Affiliations F. A. B. Coutinho L. F. Lopez M. N. Burattini E. Massad 1. School of Medicine The University of São Paulo, and LIM01/HCFMUSP. Brazil