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February  2018, 11(1): 119-141. doi: 10.3934/dcdss.2018008

Modeling and optimal control of HIV/AIDS prevention through PrEP

Cristiana J. Silva , and  Delfim F. M. Torres

Center for Research & Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal

* Corresponding author: Cristiana J. Silva

Received  September 2016 Revised  February 2017 Published  January 2018

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.

Keywords: PrEP, HIV/AIDS model, global stability, optimal control, Cape Verde.
Mathematics Subject Classification: Primary: 34C60, 92D30; Secondary: 34D23, 49K15.
Citation: Cristiana J. Silva, Delfim F. M. Torres. Modeling and optimal control of HIV/AIDS prevention through PrEP. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 119-141. doi: 10.3934/dcdss.2018008
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems Marcel Dekker, Inc. , New York and Basel, 1989.  Google Scholar

[19]

J. P. LaSalle, The Stability of Dynamical Systems SIAM, Philadelphia, PA, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

R. A. Weiss, How does HIV cause AIDS?, Science, 260 (1993), 1273-1279.  doi: 10.1126/science.8493571.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems Marcel Dekker, Inc. , New York and Basel, 1989.  Google Scholar

[19]

J. P. LaSalle, The Stability of Dynamical Systems SIAM, Philadelphia, PA, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

R. A. Weiss, How does HIV cause AIDS?, Science, 260 (1993), 1273-1279.  doi: 10.1126/science.8493571.  Google Scholar

[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.  Google Scholar

[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

25,42]. The $l_2$ norm of the difference between the real total population of Cape Verde and our prediction gives an error of $1.9\%$ of individuals per year with respect to the total population of Cape Verde in 2014">Figure 1.  Model (1) fitting the total population of Cape Verde between 1987 and 2014 [25,42]. The $l_2$ norm of the difference between the real total population of Cape Verde and our prediction gives an error of $1.9\%$ of individuals per year with respect to the total population of Cape Verde in 2014
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25]. The $l_2$ norm of the difference between the real data and the cumulative cases of infection by HIV/AIDS given by model (1) gives, in both cases, an error of $0.03\%$ of individuals per year with respect to the total population of Cape Verde in 2014">Figure 2.  Model (1) fitting the data of cumulative cases of HIV and AIDS infection in Cape Verde between 1987 and 2014 [25]. The $l_2$ norm of the difference between the real data and the cumulative cases of infection by HIV/AIDS given by model (1) gives, in both cases, an error of $0.03\%$ of individuals per year with respect to the total population of Cape Verde in 2014
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with PrEP" refers to the case $(\psi, \theta) = (0.1, 0.001)$ and "no PrEP" refers to the case $(\psi, \theta) = (0, 0)$">Figure 3.  Top left: cumulative HIV and AIDS cases. Top right: pre-AIDS HIV infected individuals $I$. Bottom left: HIV-infected individuals under ART treatment $C$. Bottom right: HIV-infected individuals with AIDS symptoms $A$. Expression "with PrEP" refers to the case $(\psi, \theta) = (0.1, 0.001)$ and "no PrEP" refers to the case $(\psi, \theta) = (0, 0)$
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$-\, -$" is the solution of the optimal control problem with no mixed state control constraint and "$\cdot \, -$" is the solution of model (10) for $\psi = 0.9$">Figure 4.  Top left: Individuals under PrEP, $E$. Top right: pre-AIDS HIV infected individuals, $I$. Bottom left: HIV-infected individuals under ART treatment, $C$. Bottom right: HIV-infected individuals with AIDS symptoms, $A$. The continuous line is the solution of model (10) for $\psi = 0.1$, the dashed line "$-\, -$" is the solution of the optimal control problem with no mixed state control constraint and "$\cdot \, -$" is the solution of model (10) for $\psi = 0.9$
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Figure 5.  Solutions of the optimal control problem with no mixed state control constraint. (a) Optimal control. (b) Total number of individuals that take PrEP at each instant of time
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$-\, -$" is the solution of the optimal control problem with the mixed state control constraint (21) and "$\cdot \, -$" is the solution of model (10) for $\psi = 0.9$">Figure 6.  Top left: Individuals under PrEP, $E$. Top right: pre-AIDS HIV infected individuals, $I$. Bottom left: HIV-infected individuals under ART treatment, $C$. Bottom right: HIV-infected individuals with AIDS symptoms, $A$. The continuous line is the solution of model (10) for $\psi = 0.1$, the dashed line "$-\, -$" is the solution of the optimal control problem with the mixed state control constraint (21) and "$\cdot \, -$" is the solution of model (10) for $\psi = 0.9$
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Figure 7.  (a) Optimal control $\tilde{u}$ considering the mixed state control constraint (21). (b) Total number of individuals under PrEP at each instant of time for $t \in [0, 25]$ associated with the optimal control $\tilde{u}$. (c) Total number of individuals under PrEP at each instant of time for $t \in [0, 25]$ associated with $\psi = 0.61$
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Figure 8.  Extremals of the optimal control problem (17)–(20) with $\theta = 0.001$
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Table 1.  Cumulative cases of infection by HIV/AIDS and total population in Cape Verde in the period 1987–2014 [25,42]
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
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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
Table 2.  Parameters of the HIV/AIDS model (1) for Cape Verde
Symbol Description Value Reference
$N(0)$ Initial population $323 972$ [38]
$\Lambda$ Recruitment rate $13045$ [38]
$\mu$ Natural death rate $1/69.54$ [38]
$\beta$ HIV transmission rate $0.752$ Estimated
$\eta_C$ Modification parameter $0.015$, $0.04$ Assumed
$\eta_A$ Modification parameter $1.3$, $1.35$ Assumed
$\phi$ HIV treatment rate for $I$ individuals $1$ [30]
$\rho$ Default treatment rate for $I$ individuals $0.1 $ [30]
$\alpha$ AIDS treatment rate $0.33 $ [30]
$\omega$ Default treatment rate for $C$ individuals $0.09$ [30]
$d$ AIDS induced death rate $1$ [39]
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Symbol Description Value Reference
$N(0)$ Initial population $323 972$ [38]
$\Lambda$ Recruitment rate $13045$ [38]
$\mu$ Natural death rate $1/69.54$ [38]
$\beta$ HIV transmission rate $0.752$ Estimated
$\eta_C$ Modification parameter $0.015$, $0.04$ Assumed
$\eta_A$ Modification parameter $1.3$, $1.35$ Assumed
$\phi$ HIV treatment rate for $I$ individuals $1$ [30]
$\rho$ Default treatment rate for $I$ individuals $0.1 $ [30]
$\alpha$ AIDS treatment rate $0.33 $ [30]
$\omega$ Default treatment rate for $C$ individuals $0.09$ [30]
$d$ AIDS induced death rate $1$ [39]
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