March  2022, 15(3): 639-654. doi: 10.3934/dcdss.2021156

Stability and optimal control of a delayed HIV/AIDS-PrEP model

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

Received  February 2020 Revised  March 2021 Published  March 2022 Early access  December 2021

In this paper, we propose a time-delayed HIV/AIDS-PrEP model which takes into account the delay on pre-exposure prophylaxis (PrEP) distribution and adherence by uninfected persons that are in high risk of HIV infection, and analyze the impact of this delay on the number of individuals with HIV infection. We prove the existence and stability of two equilibrium points, for any positive time delay. After, an optimal control problem with state and control delays is proposed and analyzed, where the aim is to find the optimal strategy for PrEP implementation that minimizes the number of individuals with HIV infection, with minimal costs. Different scenarios are studied, for which the solutions derived from the Minimum Principle for Multiple Delayed Optimal Control Problems change depending on the values of the time delays and the weights constants associated with the number of HIV infected individuals and PrEP. We observe that changes on the weights constants can lead to a passage from bang-singular-bang to bang-bang extremal controls.

Citation: Cristiana J. Silva. Stability and optimal control of a delayed HIV/AIDS-PrEP model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 639-654. doi: 10.3934/dcdss.2021156
References:
[1]

S. S. AlistarP. 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), 1-11.  doi: 10.1186/1741-7015-12-46.

[2]

C. F. Cáceres, The promises and challenges of pre-exposure prophylaxis as part of the emerging paradigm of combination HIV prevention, Journal of the International AIDS Society, 18 (2015), 19949.  doi: 10.7448/IAS.18.4.19949.

[3]

W. Chen, Dynamics and control of a financial system with time-delayed feedbacks, Chaos, Solitons and Fractals, 37 (2008), 1198-1207.  doi: 10.1016/j.chaos.2006.10.016.

[4]

S. G. DeeksS. 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.

[5]

U. Foryś and B. Zduniak, Two-stage model of carcinogenic mutations with the influence of delays, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2501-2519.  doi: 10.3934/dcdsb.2014.19.2501.

[6]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A modeling language for mathematicalprogramming, The Scientific Press, South San Francisco, California, 1993.

[7]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, J. Ind. Manag. Optim., 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413.

[8]

L. Göllmann and H. Maurer, Optimal control problems with time delays: Two case studies in biomedicine, Math. Biosci. Eng., 15 (2018), 1137-1154.  doi: 10.3934/mbe.2018051.

[9]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[10]

R. Heffron, et al., Efficacy of Oral PrEP for HIV prevention among women with abnormal vaginal microbiota: A randomized, placebo controlled comparison, Lancet HIV, 4 (2017), e449–e456. doi: 10.1016/S2352-3018(17)30110-8.

[11]

E. KaraoǧluE. Yılmaz and H. Merdan, Stability and bifurcation analysis of two-neuron network with discrete and distributed delays, Neurocomputing, 182 (2016), 102-110. 

[12] Y. Kuang, Delay Differential Equations With Applications in Population Dynamics, Academic Press, Boston, MA, 1993. 
[13]

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

[14]

A. P. Lemos-PaiaoC. J. Silva and D. F. M. Torres, A sufficient optimality condition for non-linear delayed optimal control problems, Pure Appl. Funct. Anal., 4 (2019), 345-361. 

[15]

C. Liu and M. Han, Time-delay optimal control of a fed-batch production involving multiple feeds, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1697-1709.  doi: 10.3934/dcdss.2020099.

[16]

C. Liu, X. Ma, B. Liu, C. Chen and H. Zhang, HIV-1 functional cure: Will the dream come true?, BMC Med., (2015), Art. 284. doi: 10.1186/s12916-015-0517-y.

[17]

S. NicaiseJ. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[18]

M. J. PiotrowskaM. BodnarJ. Poleszczuk and U. Foryś, Mathematical modelling of immune reaction against gliomas: Sensitivity analysis and influence of delays, Nonlinear Anal. Real World Appl., 14 (2013), 1601-1620.  doi: 10.1016/j.nonrwa.2012.10.020.

[19]

D. RochaC. J. Silva and D. F. M. Torres, Stability and optimal control of a delayed HIV model, Math. Methods Appl. Sci., 41 (2018), 2251-2260.  doi: 10.1002/mma.4207.

[20]

F. RodriguesC. J. SilvaD. F. M. Torres and H. Maurer, Optimal control of a delayed HIV model, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 443-458.  doi: 10.3934/dcdsb.2018030.

[21]

C. J. Silva and H. Maurer, Optimal control of HIV treatment and immunotherapy combination with state and control delays, Optimal Control Appl. Methods, 41 (2019), 537-554.  doi: 10.1002/oca.2558.

[22]

C. J. SilvaH. 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.

[23]

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.

[24]

C. J. Silva and D. F. M. Torres, Modeling and optimal control of HIV/AIDS prevention through PrEP, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 119-141.  doi: 10.3934/dcdss.2018008.

[25]

C. J. Silva and D. F. M. Torres, Errata to - Modeling and optimal control of HIV/AIDS prevention through PrEP, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1619-1621.  doi: 10.3934/dcdss.2020343.

[26]

A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.

[27]

D. P. WilsonM. G. LawA. E. GrulichD. 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.

[28]

L. XuH. Chen and B. Zhang, The Challenge of a "Functional Cure" for AIDS by gene modified HSCT therapy, Curr. Stem. Cell. Res. Ther., 10 (2015), 492-498.  doi: 10.2174/1574888X10666150519094026.

[29]

X. YangL. Chen and J. Chen, Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Comput. Math. Appl., 32 (1996), 109-116.  doi: 10.1016/0898-1221(96)00129-0.

[30]

https://www.who.int/hiv/topics/prep/en/, Accessed on 31 January 2020.

[31]

https://www.who.int/news-room/detail/11-12-2019-study-links-prep-use-against-hiv-with-high-sti-risk, Accessed on 31 January 2020.

[32]

https://www.unaids.org/en/pre-exposure-prophylaxis, Accessed on 31 January 2020.

show all references

References:
[1]

S. S. AlistarP. 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), 1-11.  doi: 10.1186/1741-7015-12-46.

[2]

C. F. Cáceres, The promises and challenges of pre-exposure prophylaxis as part of the emerging paradigm of combination HIV prevention, Journal of the International AIDS Society, 18 (2015), 19949.  doi: 10.7448/IAS.18.4.19949.

[3]

W. Chen, Dynamics and control of a financial system with time-delayed feedbacks, Chaos, Solitons and Fractals, 37 (2008), 1198-1207.  doi: 10.1016/j.chaos.2006.10.016.

[4]

S. G. DeeksS. 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.

[5]

U. Foryś and B. Zduniak, Two-stage model of carcinogenic mutations with the influence of delays, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2501-2519.  doi: 10.3934/dcdsb.2014.19.2501.

[6]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A modeling language for mathematicalprogramming, The Scientific Press, South San Francisco, California, 1993.

[7]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, J. Ind. Manag. Optim., 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413.

[8]

L. Göllmann and H. Maurer, Optimal control problems with time delays: Two case studies in biomedicine, Math. Biosci. Eng., 15 (2018), 1137-1154.  doi: 10.3934/mbe.2018051.

[9]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[10]

R. Heffron, et al., Efficacy of Oral PrEP for HIV prevention among women with abnormal vaginal microbiota: A randomized, placebo controlled comparison, Lancet HIV, 4 (2017), e449–e456. doi: 10.1016/S2352-3018(17)30110-8.

[11]

E. KaraoǧluE. Yılmaz and H. Merdan, Stability and bifurcation analysis of two-neuron network with discrete and distributed delays, Neurocomputing, 182 (2016), 102-110. 

[12] Y. Kuang, Delay Differential Equations With Applications in Population Dynamics, Academic Press, Boston, MA, 1993. 
[13]

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

[14]

A. P. Lemos-PaiaoC. J. Silva and D. F. M. Torres, A sufficient optimality condition for non-linear delayed optimal control problems, Pure Appl. Funct. Anal., 4 (2019), 345-361. 

[15]

C. Liu and M. Han, Time-delay optimal control of a fed-batch production involving multiple feeds, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1697-1709.  doi: 10.3934/dcdss.2020099.

[16]

C. Liu, X. Ma, B. Liu, C. Chen and H. Zhang, HIV-1 functional cure: Will the dream come true?, BMC Med., (2015), Art. 284. doi: 10.1186/s12916-015-0517-y.

[17]

S. NicaiseJ. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[18]

M. J. PiotrowskaM. BodnarJ. Poleszczuk and U. Foryś, Mathematical modelling of immune reaction against gliomas: Sensitivity analysis and influence of delays, Nonlinear Anal. Real World Appl., 14 (2013), 1601-1620.  doi: 10.1016/j.nonrwa.2012.10.020.

[19]

D. RochaC. J. Silva and D. F. M. Torres, Stability and optimal control of a delayed HIV model, Math. Methods Appl. Sci., 41 (2018), 2251-2260.  doi: 10.1002/mma.4207.

[20]

F. RodriguesC. J. SilvaD. F. M. Torres and H. Maurer, Optimal control of a delayed HIV model, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 443-458.  doi: 10.3934/dcdsb.2018030.

[21]

C. J. Silva and H. Maurer, Optimal control of HIV treatment and immunotherapy combination with state and control delays, Optimal Control Appl. Methods, 41 (2019), 537-554.  doi: 10.1002/oca.2558.

[22]

C. J. SilvaH. 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.

[23]

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.

[24]

C. J. Silva and D. F. M. Torres, Modeling and optimal control of HIV/AIDS prevention through PrEP, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 119-141.  doi: 10.3934/dcdss.2018008.

[25]

C. J. Silva and D. F. M. Torres, Errata to - Modeling and optimal control of HIV/AIDS prevention through PrEP, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1619-1621.  doi: 10.3934/dcdss.2020343.

[26]

A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.

[27]

D. P. WilsonM. G. LawA. E. GrulichD. 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.

[28]

L. XuH. Chen and B. Zhang, The Challenge of a "Functional Cure" for AIDS by gene modified HSCT therapy, Curr. Stem. Cell. Res. Ther., 10 (2015), 492-498.  doi: 10.2174/1574888X10666150519094026.

[29]

X. YangL. Chen and J. Chen, Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Comput. Math. Appl., 32 (1996), 109-116.  doi: 10.1016/0898-1221(96)00129-0.

[30]

https://www.who.int/hiv/topics/prep/en/, Accessed on 31 January 2020.

[31]

https://www.who.int/news-room/detail/11-12-2019-study-links-prep-use-against-hiv-with-high-sti-risk, Accessed on 31 January 2020.

[32]

https://www.unaids.org/en/pre-exposure-prophylaxis, Accessed on 31 January 2020.

Figure 1.  Partial derivative of $ R_0(\tau) $ with respect to $ \tau $ (with logarithmic scale in the $ Y $ axis): (a) $ \beta = 0.0752 $ corresponding to $ R_0(\tau) < 1 $; (b) $ \beta = 0.752 $ corresponding to $ R_0(\tau) > 1 $
Figure 2.  Influence of the time delay on $ R_0(\tau) $, for $ \tau \in [0, 50] $ and $ \beta \in [0.16, 1.72] $
Figure 3.  Stability of the disease free equilibrium $ \Sigma_0 $ for $ d = 1 $, $ \Lambda = 10724 $, $ \beta = 0.0752 $ and the other parameter values from Table 2, with $ t \in [0, 1000] $ and $ \tau \in \{0, 10,100 \} $
Figure 4.  Stability of the endemic equilibrium $ \Sigma_+ $ for $ d = 1 $, $ \Lambda = 10724 $, $ \beta = 0.752 $ and the other parameter values from Table 2, with $ t \in [0, 1000] $ and $ \tau \in \{0, 10,100 \} $
Figure 5.  Extremal solutions $ \tilde{S} $, $ \tilde{I} $ and $ \tilde{C} $ associated to the extremal control $ \tilde{u} $, given by (18)
Figure 6.  Extremal solutions $ \tilde{A} $, $ \tilde{E} $ and extremal control $ \tilde{u} $
Figure 7.  Extremal control $ \tilde{u} $ and associated state trajectory $ \tilde{I} $, for different weight constants: $ w_1 = w_2 = 1 $; $ w_1 = 1 $, $ w_2 = 50 $; $ w_1 = 5 $, $ w_2 = 1 $
Table 1.  Description of the parameters of the HIV/AIDS-PrEP model (2)
Symbol Description
$ \Lambda $ Recruitment rate
$ \mu $ Natural death rate
$ \lambda $ Infection rate for $ S $ individuals
$ \beta $ Transmission coefficient for HIV transmission
$ \eta_C $ Modification parameter
$ \eta_A $ Modification parameter
$ \phi $ HIV treatment rate for $ I $ individuals
$ \rho $ Default treatment rate for $ I $ individuals
$ \alpha $ AIDS treatment rate
$ \omega $ Default treatment rate for $ C $ individuals
$ d $ AIDS induced death rate
$ \psi $ Proportion of susceptible individuals that takes PrEP
$ \theta $ Proportion of susceptible individuals who default PrEP
Symbol Description
$ \Lambda $ Recruitment rate
$ \mu $ Natural death rate
$ \lambda $ Infection rate for $ S $ individuals
$ \beta $ Transmission coefficient for HIV transmission
$ \eta_C $ Modification parameter
$ \eta_A $ Modification parameter
$ \phi $ HIV treatment rate for $ I $ individuals
$ \rho $ Default treatment rate for $ I $ individuals
$ \alpha $ AIDS treatment rate
$ \omega $ Default treatment rate for $ C $ individuals
$ d $ AIDS induced death rate
$ \psi $ Proportion of susceptible individuals that takes PrEP
$ \theta $ Proportion of susceptible individuals who default PrEP
Table 2.  Parameters values of models (3) and (11), taken from [24]
Symbol Value Symbol Value
$ \mu $ $ 1/69.54 $ $ \rho $ $ 0.1 $
$ \Lambda $ $ 10724 $ $ \alpha $ $ 0.33 $
$ \beta $ $ 0.582 $ $ \omega $ $ 0.09 $
$ \eta_C $ $ 0.04 $ $ d $ $ 0 $
$ \eta_A $ $ 1.35 $ $ \psi $ $ 0.1 $
$ \phi $ $ 1 $ $ \theta $ $ 0.01 $
Symbol Value Symbol Value
$ \mu $ $ 1/69.54 $ $ \rho $ $ 0.1 $
$ \Lambda $ $ 10724 $ $ \alpha $ $ 0.33 $
$ \beta $ $ 0.582 $ $ \omega $ $ 0.09 $
$ \eta_C $ $ 0.04 $ $ d $ $ 0 $
$ \eta_A $ $ 1.35 $ $ \psi $ $ 0.1 $
$ \phi $ $ 1 $ $ \theta $ $ 0.01 $
Table 3.  Cost functional and switching times for different weight constant values
Weight constant values Cost functional $ J(\tilde{u}) $ Switching time $ t_1 $ Switching time $ t_2 $
$ w_1 = w_2 = 1 $ $ J(\tilde{u}) \simeq 1818.64 $ $ t_1 \simeq 13.30 $ $ t_2 \simeq 18.02 $
$ w_1 = 1, \, w_2 = 50 $ $ J(\tilde{u}) \simeq 2125.28 $ $ t_1 \simeq 2.93 $ $ t_2 \simeq 11.28 $
$ w_1 = 5, \, w_2 = 1 $ $ J(\tilde{u}) \simeq 9020.09 $ $ t_1 \simeq 19.10 $
Weight constant values Cost functional $ J(\tilde{u}) $ Switching time $ t_1 $ Switching time $ t_2 $
$ w_1 = w_2 = 1 $ $ J(\tilde{u}) \simeq 1818.64 $ $ t_1 \simeq 13.30 $ $ t_2 \simeq 18.02 $
$ w_1 = 1, \, w_2 = 50 $ $ J(\tilde{u}) \simeq 2125.28 $ $ t_1 \simeq 2.93 $ $ t_2 \simeq 11.28 $
$ w_1 = 5, \, w_2 = 1 $ $ J(\tilde{u}) \simeq 9020.09 $ $ t_1 \simeq 19.10 $
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