Some of the many ways in which mathematical biology and statistics have been used in investigating the acquired immunodeficiency syndrome (AIDS) epidemic are reviewed. Aspects of the spread of the disease via social and sexual networks are discussed. The different kinds of data involved are critically compared. Some studies of the incubation period are briefly reviewed and some comments made on the role of adherence to therapy.
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References
1.
Centers for Disease Control . Update on acquired immune deficiency syndrome (AIDS) - United States . Morbidity and Mortality Weekly Reports1982; 31: 507-514 .
2.
Boily MC , Anderson RM.Sexual contact patterns between men and women and the spread of HIV-1 in urban centers in Africa . IMA Journal of Mathematics Applied in Medicine and Biology1991; 8: 221-247 .
3.
Ghani AC , Swinton J , Garnett GP.The role of sexual partnership networks in the epidemiology of gonorrhoea . Sexually Transmitted Diseases1991; 24: 45-56 .
4.
Neaigus A , Friedman SR , Curtis Ret al.The relevance of drug injectors’ social and risk networks for understanding and preventing HIV infection . Social Science and Medicine1991; 38: 67-78 .
5.
Friedman SR , Curtis R , Neaigus A , Jose B , Des Jarlais DC.Social networks, drug injectors’ lives and HIV/AIDS. New York: Kluwer Academic/Plenum , 1999.
6.
Anderson RM , May RM.Infectious diseases of humans. Oxford: Oxford University Press , 1991.
7.
Rand D.Correlation equations and pair approximations for spatial ecologies. In: McGlade J , ed. Advanced ecological theory: principles and applications. Oxford: Blackwell Science , 1999: 100-142.
8.
Watts CH , May RM.The influence of concurrent partnerships on the dynamics of HIV/AIDS . Mathematical Biosciences1992; 108: 89-104 .
9.
Morris M , Kretzschmar M.Concurrent partnerships and the spread of HIV . AIDS1997; 11: 641-648 .
10.
Ghani AC , Donnelly CA , Garnett GP.Sampling biases and missing data in explorations of sexual partner networks for the spread of sexually transmitted diseases . Statistics in Medicine1998; 17: 2079-2097 .
11.
Woodhouse DE , Rothenberg RB , Potterat JJet al.Mapping a social network of heterosexuals at high risk for HIV infection . AIDS1994; 8: 133-136 .
12.
Ghani AC , Ison CA , Ward Het al.Sexual partner networks in the transmission of sexually transmitted diseases: an analysis of gonorrhea cases in Sheffield, UK . Sexually Transmitted Diseases1996; 23: 498-503 .
13.
Bauch C , Rand DA.A moment closure model for sexually transmitted disease transmission through a concurrent partnership network . Proceedings of the Royal Society, London B2000; 267: 2019-2027 .
14.
Ferguson NM , Garnett GP.More realistic models of bacterial STD transmission dynamics: sexual partnership networks, pair models and moment closure . Sexually Transmitted Diseases2000; 27: 600-609 .
15.
Wasserman S , Faust K.Social network analysis: methods and applications. Cambridge: Cambridge University Press , 1994.
16.
Donnelly CA , Leisenring W , Kanki P , Awerbuch T , Sandberg S.Comparison of transmission rates of HIV-1 and HIV-2 in a cohort of prostitutes in Senegal . Bulletin of Mathematical Biology1993; 55: 731-743 .
17.
Jewell NP , Shiboski SC.Statistical analysis of HIV infectivity based on partner studies . Biometrics1990; 46: 1133-1150 .
18.
Shiboski SC , Padian NS.Epidemiologic evidence for time variation in HIV infectivity . Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology1998; 19: 527-535 .
19.
Quinn TC , Wawer MJ , Sewankambo Net al.Viral load and heterosexual transmission of human immunodeficiency virus type 1 . New England Journal of Medicine2000; 342: 921-929 .
20.
Ho DD , Moudgil T , Alam M.Quantitation of human immunodeficiency virus type 1 in the blood of infected persons . New England Journal of Medicine1989; 321: 1621-1625 .
21.
Coombs RW , Collier AC , Allain JP.Plasma viremia in human immunodeficiency virus infection . New England Journal of Medicine1989; 321: 1626-1631 .
22.
Notermans DW , De Wolf F , Oudshoorn Pet al.Evaluation of a second-generation nucleic acid sequence-based amplification assay for quantification of HIV type 1 RNA and the use of ultrasensitive protocol adaptions . AIDS Research and Human Retroviruses2000; 16: 1507-1517 .
23.
Jaffar S , Wilkins A , Ngom PTet al.Rate of decline of percentage CD4(+) cells is faster in HIV-1 than in HIV-2 infection . Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology1997; 16: 327-332 .
24.
Just JJ , Abrams E , Louie LGet al.Influence of host genotype on progression to acquired immunodeficiency syndrome among children infected with human immunodeficiency virus type 1 . Journal of Pediatrics1995; 127: 544-549 .
25.
Wei XP , Ghosh SK , Taylor MEet al.Viral dynamics in human immunodeficiency virus type 1 infection . Nature1995; 373: 117-122 .
26.
Ho DD , Neumann AU , Perelson AS , Chen W , Leonard JM , Markowitz M.Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection . Nature1995; 373: 123-126 .
27.
Wu HL , Ding AA , De Gruttola V.Estimation of HIV dynamic parameters . Statistics in Medicine1998; 17: 2463-2485 .
28.
Ding AA , Wu HL.A comparison study of models and fitting procedures for biphasic viral dynamics in HIV-1 infected patients treated with antiviral therapies . Biometrics2000; 56: 293-300 .
29.
Haber M , Watelet L , Halloran ME.On individual and population effectiveness of vaccination . International Journal of Epidemiology1995; 24: 1249-1260 .
30.
Black N.High-quality clinical databases: breaking down barriers . Lancet1999; 353: 1205-1206 .
31.
Phillips AN , Grabar S , Tassie JM , Costagliola D , Lundgren JD , Egger M.Use of observational databases to evaluate the effectiveness of antiretroviral therapy for HIV infection: comparison of cohort studies with randomized trials . AIDS1999; 13: 2075-2082 .
32.
Kleijnen J , Gotzsche P , Kunz RA , Oxman AD , Chalmers I.So what’s so special about randomisation? In: Maynard A , Chalmers I , eds. Non-random reflections on health services research. London: BMJ Publications , 1997: 93-106.
33.
Duley L.Commentary: sources of bias must be controlled . British Medical Journal1997; 315: 220-220 .
34.
Sacks H , Chalmers TC , Smith H.Randomized versus historical controls for clinical trials . American Journal of Medicine1982; 72: 233-240 .
35.
Chalmers TC , Celano P , Sacks HS , Smith H.Bias in treatment assignment in controlled clinical trials . New England Journal of Medicine1983; 309: 1358-1361 .
36.
Kunz R , Oxman AD.The unpredictability paradox: review of empirical comparisons of randomised and non-randomised clinical trials . British Medical Journal1998; 317: 1185-1190 .
37.
Concato J , Shah N , Horwitz RI.Randomized, controlled trials, observational studies, and the hierarchy of research designs . New England Journal of Medicine2000; 342: 1887-1892 .
38.
Benson K , Hartz AJ.A comparison of observational and randomized controlled trials . New England Journal of Medicine2000; 342: 1878-1886 .
39.
Britton A , McKee M , Black N , McPherson K , Sanderson C , Bain C.Choosing between randomised and non-randomised studies: a systematic review . Health Technology Assessment1998; 2(13).
40.
Pocock SJ , Elbourne DR.Randomized trials or observational tribulations?New England Journal of Medicine2000; 342: 1907-1909 .
41.
Barton S.Which clinical studies provide the best evidence? The best RCT still trumps the best observational study . British Medical Journal2000; 321: 255-256 .
42.
Solomon PJ , Isham VS.Disease surveillance and datacollectionissuesinepidemicmodeling . Statistical Methods in Medical Research2000; 9: 259-277 .
43.
Von Bargen J , Moorman A , Holmberg S.How many pills do patients with HIV infection take?Journal of the American Medical Association1998; 280: 29-29 .
44.
Nowak MA , Anderson RM , McLean AR , Wolfs TFW , Goudsmit J , May RM.Antigenic diversity thresholds and the development of AIDS . Science1991; 254: 963-969 .
45.
Frost SDW , McLean AR.Quasi-species dynamics and the emergence of drug-resistance during zidovudine therapy of HIV-infection . AIDS1994; 8: 323-332 .
46.
deJong MD , Veenstra J , Stilianakis NIet al.Host-parasite dynamics and outgrowth of virus containing a single K70R amino acid change in reverse transcriptase are responsible for the loss of human immunodeficiency virus type 1 RNA load suppression by zidovudine . Proceedings of the National Academy of Sciences of the United States of America1996; 93: 5501-5506 .
47.
Austin DJ , White NJ , Anderson RM.The dynamics of drug action on the within-host population growth of infectious agents: melding pharmacokinetics with pathogen population dynamics . Journal of Theoretical Biology1998; 194: 313-339 .
48.
Wahl LM , Nowak MA.Adherence and drug resistance: predictions for therapy outcome . Proceedings of the Royal Society, London B2000; 267: 835-843 .
49.
Ribeiro RM , Bonhoeffer S.Production of resistant HIV mutants during antiretroviral therapy . Proceedings of the National Academy of Sciences of the United States of America2000; 97: 7681-7686 .
50.
Hogg RS , Yip B , Chan Ket al.Nonadherence to triple combination therapy is predictive of AIDS progression and death in HIV-positive men and women . Seventh conference on retroviruses and opportunistic infections, 2000, Abstract 73: 89-89 .
51.
Bangsberg DR , Hecht FM , Charlebois EDet al.Adherence to protease inhibitors, HIV-1 viral load, and development of drug resistance in an indigent population . AIDS2000; 14: 357-366 .
52.
Descamps D , Flandre P , Calvez Vet al.Mechanisms of virologic failure in previously untreated HIV-infected patients from a trial of induction-maintenance therapy . Journal of the American Medical Association2000; 283: 205-211 .
53.
Eron JJ , Yetzer ES , Ruane PJet al.Efficacy, safety, and adherence with a twice-daily combination lamivudine/zidovudine tablet formulation, plus a protease inhibitor, in HIV infection . AIDS2000; 14: 671-681 .
54.
Powderly WG , Saag MS , Chapman Set al.Predictors of optimal virological response to potent antiretroviral therapy . AIDS1999; 13: 1873-1880 .
55.
Paterson DL , Swindells S , Mohr Jet al.Adherence to protease inhibitor therapy and outcomes in patients with HIV infection . Annals of Internal Medicine2000; 133: 21-30 .
56.
Kaufmann D , Pantaleo G , Sudre P , Telenti A for the Swiss HIV Cohort Study . CD4-cell count in HIV-1-infected individuals remaining viraemic with highly active antiretroviral therapy (HAART) . Lancet1998; 351: 723-724 .
57.
Haubrich RH , Little SJ , Currier JSet al.The value of patient-reported adherence to antiretroviral therapy in predicting virologic and immunologic response. California Collaborative Treatment Group . AIDS1999; 13: 1099-1107 .
58.
Singh N , Berman SM , Swindells Set al.Adherence of human immunodeficiency virus-infected patients to antiretroviral therapy . Clinical Infectious Diseases1999; 29: 824-830 .
59.
Watson DC , Farley JJ.Efficacy of and adherence to highly active antiretroviral therapy in children infected with human immunodeficiency virus type 1 . Pediatric Infectious Disease Journal1999; 18: 682-689 .
60.
Singh N , Squier C , Sivek C , Wagener M , Nguyen MH , Yu VL.Determinants of compliance with antiretroviral therapy in patients with human immunodeficiency virus: prospective assessment with implications for enhancing compliance . AIDS Care1996; 8: 261-269 .
61.
Gordillo V , del Amo J , Soriano V , Gonzalez-Lahoz J.Sociodemographic and psychological variables influencing adherence to antiretroviral therapy . AIDS1999; 13: 1763-1769 .
62.
Moatti JP , Spire B , Duran S.A review of sociobehavioural studies on adherence to antiretroviral treatments: beyond biomedical models?Revue d’Epidemiologie et de Sante Publique2000; 48: 182-197 .
63.
Catz SL , Kelly JA , Bogart LM , Benotsch EG , McAuliffe TL.Patterns, correlates, and barriers to medication adherence among persons prescribed new treatments for HIV disease . Health Psychology2000; 19: 124-133 .
64.
Bailey WC , Richards Jr JM , Brooks CM , Soong SJ , Windsor RA , Manzella BA.A randomized trial to improve self-management practices of adults with asthma . Archives of Internal Medicine1990; 150: 1664-1668 .
65.
Wilson SR , Scamagas P , German DFet al.A controlled trial of two forms of self-management education for adults with asthma . American Journal of Medicine1993; 94: 564-576 .
66.
Newell SA , Bowman JA , Cockburn JD.A critical review of interventions to increase compliance with medication-taking, obtaining medication refills, and appointment-keeping in the treatment of cardiovascular disease . Preventive Medicine1999; 29: 535-548 .
67.
Morrison A , Wertheimer AI , Berger ML.Interventions to improve antihypertensive drug adherence: a quantitative review of trials . Formulary2000; 35: 234-234 .
68.
Goetghebeur E, van Houwelingen HC, eds. Special issue on noncompliance in clinical trials . Statistics in Medicine1998; 17: 247-390 .
69.
Robins JM.Marginal structural models versus structural nested models as tools for causal inference. In: Halloran ME , Berry D eds. Statistical models in epidemiology: the environment and clinical trials. New York: Springer , 1999: 95-134.
70.
Hern-Eln M , Brumback B , Robins JM.Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men . Epidemiology2000; 11: 561-570 .
71.
Nagelkerke N , Fidler V , Bernsen R , Borgdorff M.Estimating treatment effects in randomized clinical trials in the presence of non-compliance . Statistics in Medicine2000; 19: 1849-1864 .
72.
Horne AD , Lachenbruch PA , Goldenthal KL.Intent-to-treat analysis and preventive vaccine efficacy . Vaccine2000; 19: 319-326 .
73.
Medley GF , Anderson RM , Cox DR , Billard L.Incubation period of AIDS in patients affected by blood transfusion . Nature1987; 328: 719-721 .
74.
Medley GF , Billard L , Cox DR , Anderson RM.The distribution of the incubation period for the acquired immunodeficiency syndrome (AIDS) . Proceedings of the Royal Society, London B1988; 233: 367-377 .
75.
Babiker A , Darby SC , De Angelis D , Ewart D , Porter K.Time from HIV-1 seroconversion to AIDS and death before widespread use of highly active antiretroviral therapy: a collaborative reanalysis . Lancet2000; 355: 1131-1137 .
76.
Cox DR , Oakes D.Analysis of survival data. London: Chapman and Hall , 1984: 177-178.
77.
Darby SC , Ewart DW , Giagrande PLF , Dolin PJ , Spooner RJD , Rizza CR.Mortality before and after HIV infection in the complete UK population of haemophiliacs . Nature1995; 377: 79-82 .
78.
Bunting C. Trends in HIV-1 prevalence and analysis of survival times of infected adults in rural Uganda. Masters dissertation, Department of Statistics, Oxford, 1999.
79.
Reeves GK.The overall distribution of survival times for UK AIDS patients . Philosophical Transactions of the Royal Society, London B1989; 325: 147-151 .
80.
Rothenberg R , Woelfe M , Stoneburner R , Milberg J , Parker R , Truman B.Survival with the acquired immunodeficiency syndrome - experience with 5833 cases in New York City . New England Journal of Medicine1987; 317: 1297-1302 .
81.
Isham V.Estimation of the incidence of HIV infection - the back projection method. Appendix 4. In: Short-term projection of HIV infection and AIDS. Report of a Working Group. London: HMSO , 1988.
82.
Isham V.Estimation of the incidence of HIV infection . Philosophical Transactions of the Royal Society, London B1989; 325: 113-121 .
83.
Brookmeyer R , Gail M.A method for obtaining short term projections and lower bounds on the size of the AIDS epidemic . Journal of the American Statistical Association1988; 83: 301-308 .
84.
Bailey NTJ.The mathematical theory of infections diseases and its applications, 2nd edn.London: Griffin , 1975.
85.
Prentice RL.Surrogate endpoints in clinical trials: definition and operational criteria . Statistics in Medicine1989; 8: 431-440 .
86.
Fleming TR , DeGruttola V , DeMets D.Surrogate endpoints. In: Armitage P , Colton T , eds. Encyclopedia of biostatistics 6. Chichester: Wiley , 1998: 4425-4431.
87.
Fleming TR , Prentice RL , Pepe MS , Glidden D.Surrogate and auxiliary endpoints in clinical trials, with potential applications in cancer and AIDS research . Statistics in Medicine1994; 13: 955-968 .
88.
Cox DR.Some remarks on failure-times, surrogate markers, degradation, wear and the quality of life . Lifetime Data Analysis1999; 5: 307-314 .
89.
Gilbert PB.Some statistical issues in the design of HIV-1 vaccine and treatment trials . Statistical Methods in Medical Research2000; 9: 207-229 .
90.
Brookmeyer R , Gail M.Biases in prevalent cohorts . Biometrics1987; 43: 739-749 .
91.
Brookmeyer R , Gail M.AIDS epidemiology. New York: Oxford University Press , 1994.
92.
Johnson AM , Wadsworth J , Wellings K , Anderson RM.Surveying sexual attitudes . Nature1990; 343: 109-109 .
93.
Gore SM , Bird AG.Drugs in British prisons . British Medical Journal1998; 316: 1256-1257 .
94.
Krebs JR , Anderson RM , Clutton-Brock Ret al.Badgers and bovine TB: conflicts between conservation and health . Science1998; 279: 817-818 .
95.
Bourne J , Donnelly C , Cox Det al.An epidemiological investigation into bovine tuberculosis: second report of the independent scientific group on cattle TB. London: Ministry of Agriculture, Fisheries and Food , 2000.
96.
Williams BG , Gouws E , Karim SSA.Where are we now? Where are we going? The demographic impact of HIV/AIDS in South Africa . South African Journal of Science2000; 96: 297-300 .
97.
Ramsay S.HIV-1c vaccine trial planned for South Africa . Lancet2000; 356: 1336-1336 .
