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Modeling and optimal control of HIV/AIDS prevention through PrEP
Center for Research & Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal |
Pre-exposure prophylaxis (PrEP) consists in the use of an antiretroviral medication to prevent the acquisition of HIV infection by uninfected individuals and has recently demonstrated to be highly efficacious for HIV prevention. We propose a new epidemiological model for HIV/AIDS transmission including PrEP. Existence, uniqueness and global stability of the disease free and endemic equilibriums are proved. The model with no PrEP is calibrated with the cumulative cases of infection by HIV and AIDS reported in Cape Verde from 1987 to 2014, showing that it predicts well such reality. An optimal control problem with a mixed state control constraint is then proposed and analyzed, where the control function represents the PrEP strategy and the mixed constraint models the fact that, due to PrEP costs, epidemic context and program coverage, the number of individuals under PrEP is limited at each instant of time. The objective is to determine the PrEP strategy that satisfies the mixed state control constraint and minimizes the number of individuals with pre-AIDS HIV-infection as well as the costs associated with PrEP. The optimal control problem is studied analytically. Through numerical simulations, we demonstrate that PrEP reduces HIV transmission significantly.
References:
[1] |
U. L. Abbas, R. M. Anderson and J. W. Mellors,
Potential impact of antiretroviral chemoprophylaxis on HIV-1 transmission in resource-limited settings, PLoS ONE, 2 (2007), e875, 1-11.
doi: 10.1371/journal.pone.0000875. |
[2] |
F. B. Agusto, S. Lenhart, A. B. Gumel and A. Odoi,
Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis, Math. Meth. Appl. Sci., 34 (2011), 1873-1887.
doi: 10.1002/mma.1486. |
[3] |
S. S. Alistar, P. M. Grant and E. Bendavid, Comparative effectiveness and cost-effectiveness of antiretroviral therapy and pre-exposure prophylaxis for HIV prevention in South Africa BMC Med. 12 (2014), p46.
doi: 10.1186/1741-7015-12-46. |
[4] |
E. J. Arts and D. J. Hazuda,
HIV-1 antiretroviral drug therapy, Cold Spring Harb. Perspect. Med., 2 (2012), 1-23.
doi: 10.1101/cshperspect.a007161. |
[5] |
M. H. A. Biswas, L. T. Paiva and M. R. de Pinho,
A SEIR model for control of infectious diseases with constraints, Math. Biosci. Eng., 11 (2014), 761-784.
doi: 10.3934/mbe.2014.11.761. |
[6] |
C. Celum, T. B. Hallett and J. M. Baeten,
HIV-1 prevention with ART and PrEP: Mathematical modeling insights into Resistance, effectiveness, and public health impact, J. Infect. Dis., 208 (2013), 189-191.
doi: 10.1093/infdis/jit154. |
[7] |
F. Clarke,
Functional Analysis, Calculus of Variations and Optimal Control Graduate Texts in Mathematics, 264, Springer, London, 2013.
doi: 10.1007/978-1-4471-4820-3. |
[8] |
F. Clarke and M. R. de Pinho,
Optimal control problems with mixed constraints, SIAM J. Control Optim., 48 (2010), 4500-4524.
doi: 10.1137/090757642. |
[9] |
M. S. Cohen, Y. Q. Chen and M. McCauley,
Prevention of HIV-1 infection with early antiretroviral therapy, New England Journal of Medicine, 365 (2011), 493-505.
doi: 10.1056/NEJMoa1105243. |
[10] |
S. G. Deeks, S. R. Lewin and D. V. Havlir,
The end of AIDS: HIV infection as a chronic disease, The Lancet, 382 (2013), 1525-1533.
doi: 10.1016/S0140-6736(13)61809-7. |
[11] |
J. Del Romero, M. B. Baza, I. Río, A. Jerónimo, M. Vera, V. Hernando, C. Rodríguez and J. Castilla, Natural conception in HIV-serodiscordant couples with the infected partner in suppressive antiretroviral therapy: A prospective cohort study Medicine (Baltimore) 95 (2016), e4398.
doi: 10.1097/MD.0000000000004398. |
[12] |
R. Denysiuk, C. J. Silva and D. F. M. Torres,
Multiobjective approach to optimal control for a tuberculosis model, Optim. Methods Softw., 30 (2015), 893-910.
doi: 10.1080/10556788.2014.994704. |
[13] |
E. F. Drabo, J. W. Hay, R. Vardavas, Z. R. Wagner and N. Sood,
A cost-effectiveness analysis of pre-exposure prophylaxis for the prevention of HIV among Los Angeles County: Men who have sex with men, Clin. Infect. Dis., 63 (2016), 1495-1504.
doi: 10.1093/cid/ciw578. |
[14] |
C. L. Gay and M. S. Cohen,
Antiretrovirals to prevent HIV infection: Pre-and postexposure prophylaxis, Curr. Infect. Dis. Rep., 10 (2008), 323-331.
doi: 10.1007/s11908-008-0052-5. |
[15] |
H. W. Hethcote,
The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[16] |
D. Hincapié-Palacio, J. Ospina and D. F. M. Torres,
Approximated analytical solution to an Ebola optimal control problem, Int. J. Comput. Methods Eng. Sci. Mech., 17 (2016), 382-390.
doi: 10.1080/15502287.2016.1231236. |
[17] |
S. B. Kim, M. Yoon, N. S. Ku, M. H. Kim, J. E. Song and J. Y. Ahn,
Mathematical modeling of HIV prevention measures including pre-exposure prophylaxis on HIV incidence in South Korea, PLoS ONE, 9 (2014), e90080, 1-9.
doi: 10.1371/journal.pone.0090080. |
[18] |
V. Lakshmikantham, S. Leela and A. A. Martynyuk,
Stability Analysis of Nonlinear Systems Marcel Dekker, Inc. , New York and Basel, 1989. |
[19] |
J. P. LaSalle,
The Stability of Dynamical Systems SIAM, Philadelphia, PA, 1976. |
[20] |
A. P. Lemos-Paião, C. J. Silva and D. F. M. Torres,
An epidemic model for cholera with optimal control treatment, J. Comput. Appl. Math., 318 (2017), 168-180.
doi: 10.1016/j.cam.2016.11.002. |
[21] |
J. Liu and T. Zhang,
Global stability for a tuberculosis model, Math. Comput. Modelling, 54 (2011), 836-845.
doi: 10.1016/j.mcm.2011.03.033. |
[22] |
M. R. Loutfy, W. Wu and M. Letchumanan,
Systematic review of HIV transmission between heterosexual serodiscordant couples where the HIV-positive partner is fully suppressed on antiretroviral therapy, PLoS ONE, 8 (2013), e55747, 1-12.
doi: 10.1371/journal.pone.0055747. |
[23] |
J. F. G. Monteiro, S. Galea, T. Flanigan, M. L. Monteiro, S. R. Friedman and B. D. L. Marshall,
Evaluating HIV prevention strategies for populations in key affected groups: The example of Cabo Verde, Int. J. Public Health, 60 (2015), 457-466.
doi: 10.1007/s00038-015-0676-9. |
[24] |
A. Rachah and D. F. M. Torres,
Dynamics and optimal control of Ebola transmission, Math. Comput. Sci., 10 (2016), 331-342.
doi: 10.1007/s11786-016-0268-y. |
[25] |
R. de Cabo Verde, Rapport de Progrès sur la Riposte au SIDA au Cabo Verde – 2015, Comité de Coordenação do Combate a Sida, 2015. Google Scholar |
[26] |
D. Rocha, C. J. Silva and D. F. M. Torres, Stability and optimal control of a delayed HIV model Math. Methods Appl. Sci. in press, 2016.
doi: 10.1002/mma.4207. |
[27] |
H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres,
Vaccination models and optimal control strategies to dengue, Math. Biosci., 247 (2014), 1-12.
doi: 10.1016/j.mbs.2013.10.006. |
[28] |
H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres,
Seasonality effects on dengue: Basic reproduction number, sensitivity analysis and optimal control, Math. Methods Appl. Sci., 39 (2016), 4671-4679.
doi: 10.1002/mma.3319. |
[29] |
C. J. Silva, H. Maurer and D. F. M. Torres,
Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321-337.
doi: 10.3934/mbe.2017021. |
[30] |
C. J. Silva and D. F. M. Torres,
A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639-4663.
doi: 10.3934/dcds.2015.35.4639. |
[31] |
C. J. Silva and D. F. M. Torres,
A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde, Ecological Complexity, 30 (2017), 70-75.
doi: 10.1016/j.ecocom.2016.12.001. |
[32] |
C. D. Spinner, C. Boesecke, A. Zink, H. Jessen, H-J. Stellbrink, J. K. Rockstroh and S. Esser,
HIV pre-exposure prophylaxis (PrEP): A review of current knowledge of oral systemic HIV PrEP in humans, Infection, 44 (2016), 151-158.
doi: 10.1007/s15010-015-0850-2. |
[33] |
UNAIDS, Global AIDS Update 2016 Joint United Nations Programme on HIV/AIDS, Geneva, 2016. http://www.unaids.org/en/resources/documents/2016/Global-AIDS-update-2016 Google Scholar |
[34] |
P. van den Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[35] |
R. A. Weiss,
How does HIV cause AIDS?, Science, 260 (1993), 1273-1279.
doi: 10.1126/science.8493571. |
[36] |
WHO, Policy Brief on Oral Pre-Exposure Prophylaxis of HIV Infection (PrEP) Geneva, 2015. http://www.who.int/hiv/pub/prep/policy-brief-prep-2015/en/ Google Scholar |
[37] |
D. P. Wilson, M. G. Law, A. E. Grulich, D. A. Cooper and J. M. Kaldor,
Relation between HIV viral load and infectiousness: A model-based analysis, The Lancet, 372 (2008), 314-320.
doi: 10.1016/S0140-6736(08)61115-0. |
[38] |
World Bank Data, Cabo Verde, World Development Indicators, http://data.worldbank.org/country/cape-verde Google Scholar |
[39] |
M. Zwahlen and M. Egger, Progression and mortality of untreated HIV-positive individuals living in resource-limited settings: Update of literature review and evidence synthesis Report on UNAIDS obligation no. HQ/05/422204,2006. Google Scholar |
[40] |
https://www.hiv.gov/hiv-basics/hiv-prevention/using-hiv-medication-to-reduce-risk/pre-exposure-prophylaxis Google Scholar |
[41] |
http://www.who.int/hiv/mediacentre/news/southafrican-strategy-sex-workers/en Google Scholar |
[42] |
http://data.worldbank.org/indicator/SP.POP.TOTL?locations=CV Google Scholar |
show all references
References:
[1] |
U. L. Abbas, R. M. Anderson and J. W. Mellors,
Potential impact of antiretroviral chemoprophylaxis on HIV-1 transmission in resource-limited settings, PLoS ONE, 2 (2007), e875, 1-11.
doi: 10.1371/journal.pone.0000875. |
[2] |
F. B. Agusto, S. Lenhart, A. B. Gumel and A. Odoi,
Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis, Math. Meth. Appl. Sci., 34 (2011), 1873-1887.
doi: 10.1002/mma.1486. |
[3] |
S. S. Alistar, P. M. Grant and E. Bendavid, Comparative effectiveness and cost-effectiveness of antiretroviral therapy and pre-exposure prophylaxis for HIV prevention in South Africa BMC Med. 12 (2014), p46.
doi: 10.1186/1741-7015-12-46. |
[4] |
E. J. Arts and D. J. Hazuda,
HIV-1 antiretroviral drug therapy, Cold Spring Harb. Perspect. Med., 2 (2012), 1-23.
doi: 10.1101/cshperspect.a007161. |
[5] |
M. H. A. Biswas, L. T. Paiva and M. R. de Pinho,
A SEIR model for control of infectious diseases with constraints, Math. Biosci. Eng., 11 (2014), 761-784.
doi: 10.3934/mbe.2014.11.761. |
[6] |
C. Celum, T. B. Hallett and J. M. Baeten,
HIV-1 prevention with ART and PrEP: Mathematical modeling insights into Resistance, effectiveness, and public health impact, J. Infect. Dis., 208 (2013), 189-191.
doi: 10.1093/infdis/jit154. |
[7] |
F. Clarke,
Functional Analysis, Calculus of Variations and Optimal Control Graduate Texts in Mathematics, 264, Springer, London, 2013.
doi: 10.1007/978-1-4471-4820-3. |
[8] |
F. Clarke and M. R. de Pinho,
Optimal control problems with mixed constraints, SIAM J. Control Optim., 48 (2010), 4500-4524.
doi: 10.1137/090757642. |
[9] |
M. S. Cohen, Y. Q. Chen and M. McCauley,
Prevention of HIV-1 infection with early antiretroviral therapy, New England Journal of Medicine, 365 (2011), 493-505.
doi: 10.1056/NEJMoa1105243. |
[10] |
S. G. Deeks, S. R. Lewin and D. V. Havlir,
The end of AIDS: HIV infection as a chronic disease, The Lancet, 382 (2013), 1525-1533.
doi: 10.1016/S0140-6736(13)61809-7. |
[11] |
J. Del Romero, M. B. Baza, I. Río, A. Jerónimo, M. Vera, V. Hernando, C. Rodríguez and J. Castilla, Natural conception in HIV-serodiscordant couples with the infected partner in suppressive antiretroviral therapy: A prospective cohort study Medicine (Baltimore) 95 (2016), e4398.
doi: 10.1097/MD.0000000000004398. |
[12] |
R. Denysiuk, C. J. Silva and D. F. M. Torres,
Multiobjective approach to optimal control for a tuberculosis model, Optim. Methods Softw., 30 (2015), 893-910.
doi: 10.1080/10556788.2014.994704. |
[13] |
E. F. Drabo, J. W. Hay, R. Vardavas, Z. R. Wagner and N. Sood,
A cost-effectiveness analysis of pre-exposure prophylaxis for the prevention of HIV among Los Angeles County: Men who have sex with men, Clin. Infect. Dis., 63 (2016), 1495-1504.
doi: 10.1093/cid/ciw578. |
[14] |
C. L. Gay and M. S. Cohen,
Antiretrovirals to prevent HIV infection: Pre-and postexposure prophylaxis, Curr. Infect. Dis. Rep., 10 (2008), 323-331.
doi: 10.1007/s11908-008-0052-5. |
[15] |
H. W. Hethcote,
The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[16] |
D. Hincapié-Palacio, J. Ospina and D. F. M. Torres,
Approximated analytical solution to an Ebola optimal control problem, Int. J. Comput. Methods Eng. Sci. Mech., 17 (2016), 382-390.
doi: 10.1080/15502287.2016.1231236. |
[17] |
S. B. Kim, M. Yoon, N. S. Ku, M. H. Kim, J. E. Song and J. Y. Ahn,
Mathematical modeling of HIV prevention measures including pre-exposure prophylaxis on HIV incidence in South Korea, PLoS ONE, 9 (2014), e90080, 1-9.
doi: 10.1371/journal.pone.0090080. |
[18] |
V. Lakshmikantham, S. Leela and A. A. Martynyuk,
Stability Analysis of Nonlinear Systems Marcel Dekker, Inc. , New York and Basel, 1989. |
[19] |
J. P. LaSalle,
The Stability of Dynamical Systems SIAM, Philadelphia, PA, 1976. |
[20] |
A. P. Lemos-Paião, C. J. Silva and D. F. M. Torres,
An epidemic model for cholera with optimal control treatment, J. Comput. Appl. Math., 318 (2017), 168-180.
doi: 10.1016/j.cam.2016.11.002. |
[21] |
J. Liu and T. Zhang,
Global stability for a tuberculosis model, Math. Comput. Modelling, 54 (2011), 836-845.
doi: 10.1016/j.mcm.2011.03.033. |
[22] |
M. R. Loutfy, W. Wu and M. Letchumanan,
Systematic review of HIV transmission between heterosexual serodiscordant couples where the HIV-positive partner is fully suppressed on antiretroviral therapy, PLoS ONE, 8 (2013), e55747, 1-12.
doi: 10.1371/journal.pone.0055747. |
[23] |
J. F. G. Monteiro, S. Galea, T. Flanigan, M. L. Monteiro, S. R. Friedman and B. D. L. Marshall,
Evaluating HIV prevention strategies for populations in key affected groups: The example of Cabo Verde, Int. J. Public Health, 60 (2015), 457-466.
doi: 10.1007/s00038-015-0676-9. |
[24] |
A. Rachah and D. F. M. Torres,
Dynamics and optimal control of Ebola transmission, Math. Comput. Sci., 10 (2016), 331-342.
doi: 10.1007/s11786-016-0268-y. |
[25] |
R. de Cabo Verde, Rapport de Progrès sur la Riposte au SIDA au Cabo Verde – 2015, Comité de Coordenação do Combate a Sida, 2015. Google Scholar |
[26] |
D. Rocha, C. J. Silva and D. F. M. Torres, Stability and optimal control of a delayed HIV model Math. Methods Appl. Sci. in press, 2016.
doi: 10.1002/mma.4207. |
[27] |
H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres,
Vaccination models and optimal control strategies to dengue, Math. Biosci., 247 (2014), 1-12.
doi: 10.1016/j.mbs.2013.10.006. |
[28] |
H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres,
Seasonality effects on dengue: Basic reproduction number, sensitivity analysis and optimal control, Math. Methods Appl. Sci., 39 (2016), 4671-4679.
doi: 10.1002/mma.3319. |
[29] |
C. J. Silva, H. Maurer and D. F. M. Torres,
Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321-337.
doi: 10.3934/mbe.2017021. |
[30] |
C. J. Silva and D. F. M. Torres,
A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639-4663.
doi: 10.3934/dcds.2015.35.4639. |
[31] |
C. J. Silva and D. F. M. Torres,
A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde, Ecological Complexity, 30 (2017), 70-75.
doi: 10.1016/j.ecocom.2016.12.001. |
[32] |
C. D. Spinner, C. Boesecke, A. Zink, H. Jessen, H-J. Stellbrink, J. K. Rockstroh and S. Esser,
HIV pre-exposure prophylaxis (PrEP): A review of current knowledge of oral systemic HIV PrEP in humans, Infection, 44 (2016), 151-158.
doi: 10.1007/s15010-015-0850-2. |
[33] |
UNAIDS, Global AIDS Update 2016 Joint United Nations Programme on HIV/AIDS, Geneva, 2016. http://www.unaids.org/en/resources/documents/2016/Global-AIDS-update-2016 Google Scholar |
[34] |
P. van den Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[35] |
R. A. Weiss,
How does HIV cause AIDS?, Science, 260 (1993), 1273-1279.
doi: 10.1126/science.8493571. |
[36] |
WHO, Policy Brief on Oral Pre-Exposure Prophylaxis of HIV Infection (PrEP) Geneva, 2015. http://www.who.int/hiv/pub/prep/policy-brief-prep-2015/en/ Google Scholar |
[37] |
D. P. Wilson, M. G. Law, A. E. Grulich, D. A. Cooper and J. M. Kaldor,
Relation between HIV viral load and infectiousness: A model-based analysis, The Lancet, 372 (2008), 314-320.
doi: 10.1016/S0140-6736(08)61115-0. |
[38] |
World Bank Data, Cabo Verde, World Development Indicators, http://data.worldbank.org/country/cape-verde Google Scholar |
[39] |
M. Zwahlen and M. Egger, Progression and mortality of untreated HIV-positive individuals living in resource-limited settings: Update of literature review and evidence synthesis Report on UNAIDS obligation no. HQ/05/422204,2006. Google Scholar |
[40] |
https://www.hiv.gov/hiv-basics/hiv-prevention/using-hiv-medication-to-reduce-risk/pre-exposure-prophylaxis Google Scholar |
[41] |
http://www.who.int/hiv/mediacentre/news/southafrican-strategy-sex-workers/en Google Scholar |
[42] |
http://data.worldbank.org/indicator/SP.POP.TOTL?locations=CV Google Scholar |








Year | 1987 | 1988 | 1989 | 1990 | 1991 | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 |
HIV/AIDS | 61 | 107 | 160 | 211 | 244 | 303 | 337 | 358 | 395 | 432 | 471 | 560 | 660 | 779 | 913 | 1064 | 1233 | 1493 | 1716 | 2015 | 2334 | 2610 | 2929 | 3340 | 3739 | 4090 | 4537 | 4946 |
Population | 323972 | 328861 | 334473 | 341256 | 349326 | 358473 | 368423 | 378763 | 389156 | 399508 | 409805 | 419884 | 429576 | 438737 | 447357 | 455396 | 462675 | 468985 | 474224 | 478265 | 481278 | 483824 | 486673 | 490379 | 495159 | 500870 | 507258 | 513906 |
Year | 1987 | 1988 | 1989 | 1990 | 1991 | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 |
HIV/AIDS | 61 | 107 | 160 | 211 | 244 | 303 | 337 | 358 | 395 | 432 | 471 | 560 | 660 | 779 | 913 | 1064 | 1233 | 1493 | 1716 | 2015 | 2334 | 2610 | 2929 | 3340 | 3739 | 4090 | 4537 | 4946 |
Population | 323972 | 328861 | 334473 | 341256 | 349326 | 358473 | 368423 | 378763 | 389156 | 399508 | 409805 | 419884 | 429576 | 438737 | 447357 | 455396 | 462675 | 468985 | 474224 | 478265 | 481278 | 483824 | 486673 | 490379 | 495159 | 500870 | 507258 | 513906 |
Symbol | Description | Value | Reference |
Initial population | [38] | ||
Recruitment rate | [38] | ||
Natural death rate | [38] | ||
HIV transmission rate | Estimated | ||
Modification parameter | Assumed | ||
Modification parameter | Assumed | ||
HIV treatment rate for |
[30] | ||
Default treatment rate for |
[30] | ||
AIDS treatment rate | [30] | ||
Default treatment rate for |
[30] | ||
AIDS induced death rate | [39] |
Symbol | Description | Value | Reference |
Initial population | [38] | ||
Recruitment rate | [38] | ||
Natural death rate | [38] | ||
HIV transmission rate | Estimated | ||
Modification parameter | Assumed | ||
Modification parameter | Assumed | ||
HIV treatment rate for |
[30] | ||
Default treatment rate for |
[30] | ||
AIDS treatment rate | [30] | ||
Default treatment rate for |
[30] | ||
AIDS induced death rate | [39] |
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