Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-07-07T18:40:56.519Z Has data issue: false hasContentIssue false
  • English
  • Français

A Generating Function Approach to HIV Transmission with Dynamic Contact Rates

Published online by Cambridge University Press:  24 April 2014

E.O. Romero-Severson ,
G.D. Meadors  and
E.M. Volz
E.O. Romero-Severson*
Affiliation:
Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM
G.D. Meadors
Affiliation:
Department of Physics, University of Michigan, Ann Arbor, MI
E.M. Volz
Affiliation:
Department of Epidemiology, University of Michigan, Ann Arbor, MI
*
⋆ Corresponding author. E-mail: [email protected]
Get access

Abstract

The basic reproduction number, R0, is often defined as the average number of infections generated by a newly infected individual in a fully susceptible population. The interpretation, meaning, and derivation of R0 are controversial. However, in the context of mean field models, R0 demarcates the epidemic threshold below which the infected population approaches zero in the limit of time. In this manner, R0 has been proposed as a method for understanding the relative impact of public health interventions with respect to disease eliminations from a theoretical perspective. The use of R0 is made more complex by both the strong dependency of R0 on the model form and the stochastic nature of transmission. A common assumption in models of HIV transmission that have closed form expressions for R0 is that a single individual’s behavior is constant over time. In this paper we derive expressions for both R0 and probability of an epidemic in a finite population under the assumption that people periodically change their sexual behavior over time. We illustrate the use of generating functions as a general framework to model the effects of potentially complex assumptions on the number of transmissions generated by a newly infected person in a susceptible population. We find that the relationship between the probability of an epidemic and R0 is not straightforward, but, that as the rate of change in sexual behavior increases both R0 and the probability of an epidemic also decrease.

Keywords

HIVtransmission modelR0generating functionsbranching process
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 2: Epidemics models on networks , 2014 , pp. 121 - 135
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, R.M.. The epidemiology of HIV infection: Variable incubation plus infectious periods and heterogeneity in sexual activity. J. Roy. Stat. Soc. A. Sta. 151 (1988), 66–93. CrossRefGoogle Scholar
R.M. Anderson, R.M. May, B. Anderson. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, USA, 1992.
K.B. Athreya, P.E. Ney. Branching processes, volume 28. Springer-Verlag Berlin, 1972.
Ball, F.. The threshold behaviour of epidemic models. J. Appl. Probab., 20 (1983), 227–241. CrossRefGoogle Scholar
Bezemer, D., de Wolf, F., Boerlijst, M.C., van Sighem, A., Hollingsworth, T.D., Fraser, C.. 27 years of the HIV epidemic amongst men having sex with men in the Netherlands: An in depth mathematical model-based analysis. Epidemics, 2 (2010), 66–79. CrossRefGoogle Scholar
Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.. 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. CrossRefGoogle Scholar
Halkitis, P.N., Brockwell, S., Siconolfi, D.E., Moeller, R.W., Sussman, R.D., Mourgues, P.J., Cutler, B., Sweeney, M.M.. Sexual behaviors of adolescent, emerging and young adult men who have sex with men ages 13–29 in New York City. JAIDS, 56 (2011), 285–291. Google Scholar
T.E. Harris. The theory of branching processes. Courier Dover Publications, 2002.
Heesterbeek, J.A.P., Dietz, K.. The concept of R0 in epidemic theory. Stat. Neerl., 50 (1996), 89–110. CrossRefGoogle Scholar
Hethcote, H.W., Yorke, J.A., Nold, A.. Gonorrhea modeling: a comparison of control methods. Math. Biosci., 58 (1982), 93–109. CrossRefGoogle Scholar
Hollingsworth, T.D., Anderson, R.M., Fraser, C.. HIV-1 transmission, by stage of infection. J. Infect. Dis., 198 (2008), 687–693. CrossRefGoogle Scholar
Kendall, D.G.. Branching processes since 1873. J. London Math. Soc., 1 (1966), 385–406. CrossRefGoogle Scholar
Kretzschmar, M., van Duynhoven, Y.T., Severijnen, A.J.. Modeling prevention strategies for gonorrhea and chlamydia using stochastic network simulations. Am. J. Epidemiol., 144 (1996), 306–317. CrossRefGoogle Scholar
Liljeros, F., Edling, C.R., Amaral, L.A., Stanley, H.E., Aberg, Y.. The web of human sexual contacts. Nature, 411 (2001): 907–908. CrossRefGoogle Scholar
Longini, I.M., Clark, W.S., Byers, R.H., Ward, J.W., Darrow, W.W., Lemp, G.F., Hethcote, H.W.. Statistical analysis of the stages of HIV infection using a Markov model. Stat. Med., 8 (1989), 831–843. CrossRefGoogle ScholarPubMed
May, R.M., Lloyd, A.L.. Infection dynamics on scale-free networks. Phys. Rev. E, 64 (2001), 066112. CrossRefGoogle ScholarPubMed
Meyers, L.. Contact network epidemiology: Bond percolation applied to infectious disease prediction and control. B. Am. Math. Soc., 44 (2007), 63–86. CrossRefGoogle Scholar
Miller, J.C., Davoudi, B., Meza, R., Slim, A.C., Pourbohloul, B.. Epidemics with general generation interval distributions. J. Theor. Biol., 262 (2010), 107–115. CrossRefGoogle ScholarPubMed
Miller, J.C., Slim, A.C., Volz, E.M.. Edge-based compartmental modelling for infectious disease spread. J. R. Soc. Interface, 9 (2012), 890–906. CrossRefGoogle Scholar
Newman, M.E.J.. Spread of epidemic disease on networks. Phys. Rev. E, 66 (2002), 016128. CrossRefGoogle Scholar
Pastor-Satorras, R., Vespignani, A.. Epidemic spreading in scale-free networks. Phys. Rev. Lett., 86 (2001), 3200–3203. CrossRefGoogle Scholar
Pilcher, C.D., Joaki, G., Hoffman, I.F., Martinson, F.E.A., Mapanje, C., Stewart, P.W., Powers, K.A., Galvin, S., Chilongozi, D., Gama, S., Price, M.A., Fiscus, S.A., Cohen, M.S.. Amplified transmission of HIV-1: comparison of HIV-1 concentrations in semen and blood during acute and chronic infection. AIDS, 21 (2007), 1723–1730. CrossRefGoogle Scholar
Pinkerton, S.D.. Probability of HIV transmission during acute infection in Rakai, Uganda. AIDS Behav., 12 (2007), 677–684. CrossRefGoogle ScholarPubMed
Romero-Severson, E.O., Alam, S.J., Volz, E.M., Koopman, J.S.. Heterogeneity in number and type of sexual contacts in a gay urban cohort. Stat. Comm. Infect. Dis., 4 (2012). Google Scholar
Vitinghoff, E., Douglas, J., Judon, F., McKiman, D., MacQueen, K., Buchinder, S.P.. Per-contact risk of human immunodificiency virus tramnsmision between male sexual partners. Am. J. Epidemiol., 150 (1999), 306-311. CrossRefGoogle Scholar
Volz, E.. SIR dynamics in random networks with heterogeneous connectivity. J. Math. Biol., 56 (2008), 293–310. CrossRefGoogle Scholar
Wawer, M.J., Gray, R.H., Sewankambo, N.K., Serwadda, D., Li, X., Laeyendecker, O., Kiwanuka, N., Kigozi, G., Kiddugavu, M., Lutalo, T.. Rates of HIV-1 transmission per coital act, by stage of HIV-1 infection, in Rakai, Uganda. J. Infect. Dis., 191 (2005), 1403–1409. CrossRefGoogle ScholarPubMed
Zhang, X., Zhong, L., Romero-Severson, E., Alam, S.J., Henry, C.J., Volz, E.M., Koopman, J.S.. Episodic HIV risk behavior can greatly amplify HIV prevalence and the fraction of transmissions from acute HIV infection. Stat. Comm. Infect. Dis., 4 (2012). Google ScholarPubMed