Model-free based control of a HIV/AIDS prevention model

Loïc Michel; Cristiana J. Silva; Delfim F. M. Torres; Loïc Michel; Cristiana J. Silva; Delfim F. M. Torres

doi:10.3934/mbe.2022034

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 1: 759-774. doi: 10.3934/mbe.2022034

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Model-free based control of a HIV/AIDS prevention model

1.
École Centrale de Nantes-LS2N, UMR 6004 CNRS, Nantes 44300, France
2.
Univ Lyon, INSA Lyon, Université Claude Bernard Lyon 1, École Centrale de Lyon, CNRS, Ampère, UMR 5005, Villeurbanne 69621, France
3.
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal

Academic Editor: Baojun Song

Received: 14 September 2021 Accepted: 02 November 2021 Published: 22 November 2021

Controlling an epidemiological model is often performed using optimal control theory techniques for which the solution depends on the equations of the controlled system, objective functional and possible state and/or control constraints. In this paper, we propose a model-free control approach based on an algorithm that operates in 'real-time' and drives the state solution according to a direct feedback on the state solution that is aimed to be minimized, and without knowing explicitly the equations of the controlled system. We consider a concrete epidemic problem of minimizing the number of HIV infected individuals, through the preventive measure pre-exposure prophylaxis (PrEP) given to susceptible individuals. The solutions must satisfy control and mixed state-control constraints that represent the limitations on PrEP implementation. Our model-free based control algorithm allows to close the loop between the number of infected individuals with HIV and the supply of PrEP medication 'in real time', in such a manner that the number of infected individuals is asymptotically reduced and the number of individuals under PrEP medication remains below a fixed constant value. We prove the efficiency of our approach and compare the model-free control solutions with the ones obtained using a classical optimal control approach via Pontryagin maximum principle. The performed numerical simulations allow us to conclude that the model-free based control strategy highlights new and interesting performances compared with the classical optimal control approach.
- HIV/AIDS model,
- model-free control,
- real-time control,
- intelligent PID control,
- tracking
Citation: Loïc Michel, Cristiana J. Silva, Delfim F. M. Torres. Model-free based control of a HIV/AIDS prevention model[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 759-774. doi: 10.3934/mbe.2022034

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Abstract

Controlling an epidemiological model is often performed using optimal control theory techniques for which the solution depends on the equations of the controlled system, objective functional and possible state and/or control constraints. In this paper, we propose a model-free control approach based on an algorithm that operates in 'real-time' and drives the state solution according to a direct feedback on the state solution that is aimed to be minimized, and without knowing explicitly the equations of the controlled system. We consider a concrete epidemic problem of minimizing the number of HIV infected individuals, through the preventive measure pre-exposure prophylaxis (PrEP) given to susceptible individuals. The solutions must satisfy control and mixed state-control constraints that represent the limitations on PrEP implementation. Our model-free based control algorithm allows to close the loop between the number of infected individuals with HIV and the supply of PrEP medication 'in real time', in such a manner that the number of infected individuals is asymptotically reduced and the number of individuals under PrEP medication remains below a fixed constant value. We prove the efficiency of our approach and compare the model-free control solutions with the ones obtained using a classical optimal control approach via Pontryagin maximum principle. The performed numerical simulations allow us to conclude that the model-free based control strategy highlights new and interesting performances compared with the classical optimal control approach.

References

[1]	C. J. Silva, 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. doi: 10.3934/dcds.2015.35.4639
[2]	J. Djordjevic, C. J. Silva, D. F. M. Torres, A stochastic SICA epidemic model for HIV transmission, Appl. Math. Lett., 84 (2018), 168–175. doi: 10.1016/j.aml.2018.05.005. doi: 10.1016/j.aml.2018.05.005
[3]	X. Wang, C. Wang, K. Wang, Extinction and persistence of a stochastic SICA epidemic model with standard incidence rate for HIV transmission, Adv. Differ. Equations, 2021 (2021). doi: 10.1186/s13662-021-03392-y. doi: 10.1186/s13662-021-03392-y
[4]	C. J. Silva, D. F. M. Torres, Stability of a fractional HIV/AIDS model, Math. Comput. Simul., 164 (2019), 180–190. doi: 10.1016/j.matcom.2019.03.016. doi: 10.1016/j.matcom.2019.03.016
[5]	M. Z. Ullah, D. Baleanu, A new fractional SICA model and numerical method for the transmission of HIV/AIDS, Math. Methods Appl. Sci., 44 (2021), 8648–8659. doi: 10.1002/mma.7292. doi: 10.1002/mma.7292
[6]	S. Vaz, D. F. M. Torres, A dynamically-consistent nonstandard finite difference scheme for the SICA model, Math. Biosci. Eng., 18 (2021), 4552–4571. doi: 10.3934/mbe.2021231. doi: 10.3934/mbe.2021231
[7]	C. J. Silva, D. F. M. Torres, A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde, Ecol. Complexity, 30 (2017), 70–75. doi: 10.1016/j.ecocom.2016.12.001. doi: 10.1016/j.ecocom.2016.12.001
[8]	C. J. Silva, 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. doi: 10.3934/dcdss.2018008
[9]	E. M. Lotfi, M. Mahrouf, M. Maziane, C. J. Silva, D. F. M. Torres, N. Yousfi, A minimal HIV-AIDS infection model with general incidence rate and application to Morocco data, Stat. Optim. Inf. Comput., 7 (2019), 588–603. doi: 10.19139/soic.v7i3.834. doi: 10.19139/soic.v7i3.834
[10]	C. J. Silva, D. F. M. Torres, On SICA models for HIV transmission, in Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control, 302 (2020), 155–179. doi: 10.1007/978-3-030-49896-2_6.
[11]	Centers for Disease Control and Prevention, Pre-Exposure Prophylaxis (PrEP), 2021. Available from: https://www.cdc.gov/hiv/risk/prep/index.html.
[12]	M. Fliess, C. Join, Model-free control, Int. J. Control, 86 (2013), 2228–2252. doi: 10.1080/00207179.2013.810345. doi: 10.1080/00207179.2013.810345
[13]	K. J. Aström, P. R. Kumar, Control: a perspective, Automatica, 50 (2014), 3–43. doi: 10.1016/j.automatica.2013.10.012. doi: 10.1016/j.automatica.2013.10.012
[14]	M. Fliess, C. Join, An alternative to PIs and PIDs: intelligent proportional-derivative regulators, Int. J. Robust Nonlin., (2021). doi: 10.1002/rnc.5657. doi: 10.1002/rnc.5657
[15]	O. Bara, M. Fliess, C. Join, J. Day, S. M. Djouadi, Toward a model-free feedback control synthesis for treating acute inflammation, J. Theor. Biol., 448 (2018), 26–37. doi: 10.1016/j.jtbi.2018.04.003. doi: 10.1016/j.jtbi.2018.04.003
[16]	K. Hamiche, M. Fliess, C. Join, H. Abouaïssa, Bullwhip effect attenuation in supply chain management via control-theoretic tools and short-term forecasts: a preliminary study with an application to perishable inventories, in 6th International Conference on Control, Decision and Information Technologies (CoDIT), (2019), 1492–1497. doi: 10.1109/CoDIT.2019.8820297.
[17]	O. Bara, M. Fliess, C. Join, J. Day, S. Djouadi, Model-free immune therapy: a control approach to acute inflammation, in 15th European Control Conference (ECC), 2016. arXiv: 1607.07259.
[18]	C. Join, J. Bernier, S. Mottelet, M. Fliess, S. Rechdaoui-Guérin, S. Azimi, et al., A simple and efficient feedback control strategy for wastewater denitrification, IFAC-PapersOnLine, 50 (2017), 7657–7662. doi: 10.1016/j.ifacol.2017.08.1167. doi: 10.1016/j.ifacol.2017.08.1167
[19]	T. MohammadRidha, C. Moog, E. Delaleau, M. Fliess, C. Join, A variable reference trajectory for model-free glycemia regulation, in SIAM Conference on Control & its Applications (SIAM CT15), (2015). doi: 10.1137/1.9781611974072.9.
[20]	S. Tebbani, M. Titica, C. Join, M. Fliess, D. Dumur, Model-based versus model-free control designs for improving microalgae growth in a closed photobioreactor: some preliminary comparaisons, in 24th Mediterranean Conference on Control and Automation (MED), IEEE, (2016), 683–688. doi: 10.1109/MED.2016.7535870.
[21]	T. MohammadRidha, M. Aït-Ahmed, L. Chaillous, M. Krempf, I. Guilhem, J. Y. Poirier, et al., Model free iPID control for glycemia regulation of type-1 diabetes, IEEE Trans. Biomed. Eng., 65 (2018), 199–206. doi: 10.1109/TBME.2017.2698036. doi: 10.1109/TBME.2017.2698036
[22]	L. Michel, A para-model agent for dynamical systems, preprint, 2018. arXiv: 1202.4707.
[23]	M. Fliess, C. Join, Commande sans modèle et commande à modèle restreint, in e-STA Sciences et Technologies de l'Automatique, SEE-Société de l'Electricité, de l'Electronique et des Technologies de l'Information et de la Communication, 5 (2008), 1–23. Available from: https://hal.inria.fr/inria-00288107v3/document.
[24]	M. Fliess, C. Join, Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control? IFAC Proc. Vol., 42 (2009), 1531–1550. doi: 10.3182/20090706-3-FR-2004.00256. doi: 10.3182/20090706-3-FR-2004.00256
[25]	T. P. Nascimento, M. Saska, Position and attitude control of multi-rotor aerial vehicles: a survey, Annu. Rev. Control, 48 (2019), 129–146. doi: 10.1016/j.arcontrol.2019.08.004. doi: 10.1016/j.arcontrol.2019.08.004
[26]	M. Porcelli, P. L. Toint, BFO, a trainable derivative-free brute force optimizer for nonlinear bound-constrained optimization and equilibrium computations with continuous and discrete variables, ACM Trans. Math. Software, 44 (2017), 1–25. doi: 10.1145/3085592. doi: 10.1145/3085592
[27]	D. P. Wilson, M. G. Law, A. E. Grulich, D. A. Cooper, J. M. Kaldor, Relation between HIV viral load and infectiousness: a model-based analysis, Lancet, 372 (2008), 314–320. doi: 10.1016/S0140-6736(08)61115-0. doi: 10.1016/S0140-6736(08)61115-0
[28]	S. G. Deeks, S. R. Lewin, D. V. Havlir, The end of AIDS: HIV infection as a chronic disease, Lancet, 382 (2013), 1525–1533. doi: 10.1016/S0140-6736(13)61809-7. doi: 10.1016/S0140-6736(13)61809-7
[29]	M. Fazlyab, A. Ribeiro, M. Morari, V. M. Preciado, Analysis of optimization algorithms via integral quadratic constraints: nonstrongly convex problems, SIAM J. Optim., 28 (2018), 2654–2689. doi: 10.1137/17M1136845. doi: 10.1137/17M1136845
[30]	J. M. Sanz-Serna, K. C. Zygalakis, The connections between Lyapunov functions for some optimization algorithms and differential equations, SIAM J. Numer. Anal., 59 (2021), 1542–1565. doi: 10.1137/20M1364138. doi: 10.1137/20M1364138

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