AUT Journal of Modeling and Simulation

AUT Journal of Modeling and Simulation

Artificial Neural Network-Based Feedback Control Strategy for Epidemiological SIR and SEIR Models

Document Type : Research Article

Authors
1 Ben M'Sick Faculty of Science, Hassan II University of Casablanca, Casablanca, Morocco.
2 Faculty of Science and Techniques, Abdelmalek Essaadi University, Tangier, Morocco.
Abstract
We investigate a branched artificial neural network (ANN) feedback controller for mitigating outbreaks in compartmental epidemic models. The architecture couples a shared trunk with two specialized branches and employs Soboleva–modified hyperbolic tangent (SMHT) activations to approximate the shape and boundedness of analytic control laws, yielding smooth, non–bang–bang signals suited to implementable interventions. The network is trained offline in supervised fashion on synthetic SIR trajectories labelled by a control that steers the infected population toward a low terminal target over a fixed horizon. On unseen SIR scenarios, the learned policy lowers peak prevalence and shortens outbreak duration relative to uncontrolled dynamics. When compared against simple baselines; however, the ANN achieves these outcomes with markedly smoother profiles and reduced actuation effort (time integral of the control), a property desirable for practice. Without retraining, the controller transfers to SEIR and retains qualitative benefits consistent with partial observability induced by the latent exposed class. We evaluate our suggested controller against conventional neural network baselines through ablation studies and robustness tests incorporating multiplicative process noise. The results demonstrate that our branched architecture reduces the attack size and peak infection with a comparable control effort. Importantly, the controller exhibits smooth, bounded actuation signals even when subjected to significant uncertainty. We discuss limitations and outline extensions: identification from data, observer design for latent/noisy states, explicit resource and rate constraints, and online adaptation under distribution shift.
Keywords
Subjects

[1] Brauer F, Castillo-Chavez C. Mathematical Models in Population Biology and Epidemiology. Springer, New York; 2012. (Texts Appl Math; 40). doi:10.1007/978-1-4614-1686-9.
[2] Brauer F, van den Driessche P, Wu J, Morel J-M, Takens F, Teissier B, eds. Mathematical Epidemiology. Springer, Berlin; 2008. (Lecture Notes Math; 1945). doi:10.1007/978-3-540-78911-6.
[3] Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc Lond A. 1927;115(772):700–21. doi:10.1098/rspa.1927.0118.
[4] Busenberg SN, Hadeler KP. Demography and epidemics. Math Biosci. 1990;101(1):63–74.
[5] Busenberg S, van den Driessche P. Analysis of a disease transmission model in a population with varying size. J Math Biol. 1990;28(3):257–70. doi:10.1007/BF00178776.
[6] Derrick WR, van den Driessche P. A disease transmission model in a nonconstant population. J Math Biol. 1993;31(5):495–512. doi:10.1007/BF00173889.
[7] Martcheva M, Castillo-Chavez C. Diseases with chronic stage in a population with varying size. Math Biosci. 2003;182(1):1–25.
[8] Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford; 1991.
[9] Chayoukh O, Zakary O. Transferring control strategies in epidemiological models using τ-equivalences. Commun Math Biol Neurosci. 2025;2025:54.
[10] Sharomi O, Malik T. Optimal control in epidemiology. Ann Oper Res. 2017;251(1):55–71. doi:10.1007/s10479-015-1834-4.
[11] Klamka J. Controllability of dynamical systems: a survey. Bull Pol Acad Sci Tech Sci. 2013;61(2):335–42.
[12] Sussmann HJ, Jurdjevic V. Controllability of nonlinear systems. J Differ Equ. 1972;12(1):95–116.
[13] Jung E, Lenhart S, Feng Z. Optimal control of treatments in a two-strain tuberculosis model. Discrete Contin Dyn Syst B. 2002;2(4):473–82. doi:10.3934/dcdsb.2002.2.473.
[14] Moualeu DP, Weiser M, Ehrig R, Deuflhard P. Optimal control for a tuberculosis model with undetected cases in Cameroon. Commun Nonlinear Sci Numer Simul. 2015;20(3):986–1003.
[15] Silva CJ, Torres DFM. Optimal control for a tuberculosis model with reinfection and post-exposure interventions. Math Biosci. 2013;244(2):154–64.
[16] Agusto FB, Adekunle AI. Optimal control of a two-strain tuberculosis–HIV/AIDS co-infection model. Biosystems. 2014;119:20–44.
[17] Whang S, Choi S, Jung E. A dynamic model for tuberculosis transmission and optimal treatment strategies in South Korea. J Theor Biol. 2011;279(1):120–31.
[18] Kim BN, Nah K, Chu C, Ryu SU, Kang YH, Kim Y. Optimal control strategy of Plasmodium vivax malaria transmission in Korea. Osong Public Health Res Perspect. 2012;3(3):128.
[19] Prosper O, Ruktanonchai N, Martcheva M. Optimal vaccination and bednet maintenance for the control of malaria in a region with naturally acquired immunity. J Theor Biol. 2014;353:142–56.
[20] Joshi HR. Optimal control of an HIV immunology model. Optim Control Appl Methods. 2002;23(4):199–213. doi:10.1002/oca.710.
[21] Fister KR, Lenhart S, McNally JS. Optimizing chemotherapy in an HIV model. Electron J Differ Equ. 1998.
[22] Yang Y, Xiao Y, Wu J. Pulse HIV vaccination: feasibility for virus eradication and optimal vaccination schedule. Bull Math Biol. 2013;75(5):725–51.
[23] Kwon HD, Lee J, Yang SD. Optimal control of an age-structured model of HIV infection. Appl Math Comput. 2012;219(5):2766–79.
[24] Roshanfekr M, Farahi MH, Rahbarian R. A different approach of optimal control on an HIV immunology model. Ain Shams Eng J. 2014;5(1):213–19.
[25] Zhou Y, Liang Y, Wu J. An optimal strategy for HIV multitherapy. J Comput Appl Math. 2014;263:326–37.
[26] Adams BM, Banks HT, Davidian M, Kwon HD, Tran HT, Wynne SN, et al. HIV dynamics: modeling, data analysis, and optimal treatment protocols. J Comput Appl Math. 2005;184(1):10–49. doi:10.1016/j.cam.2005.02.002.
[27] Costanza V, Rivadeneira PS, Biafore FL, D’Attellis CE. Optimizing thymic recovery in HIV patients through multidrug therapies. Biomed Signal Process Control. 2013;8(1):90–97.
[28] Zakary O, Rachik M, Elmouki I. On the impact of awareness programs in HIV/AIDS prevention: an SIR model with optimal control. Int J Comput Appl. 2016;133(9):1–6.
[29] Rong L, Perelson AS. Treatment of hepatitis C virus infection with interferon and small-molecule direct antivirals: viral kinetics and modeling. Crit Rev Immunol. 2010;30(2):131–48.
[30] Zakary O, Rachik M, Elmouki I. On effectiveness of an optimal antiviral bitherapy in HBV-HDV coinfection model. Int J Comput Appl. 2015;127(12):1–10.
[31] Agusto FB. Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model. Biosystems. 2013;113(3):155–64.
[32] Yan X, Zou Y. Optimal and sub-optimal quarantine and isolation control in SARS epidemics. Math Comput Model. 2007;47(1–2):235–45.
[33] Li Y. Optimal control for an epidemic model of COVID-19 with time-varying parameters. Mathematics. 2024;12(10):1484.
[34] Seddighi Chaharborj S, Seddighi Chaharborj S, Hassanzadeh Asl J, Phang PS. Controlling of pandemic COVID-19 using optimal control theory. Results Phys. 2021;26:104311.
[35] Pontryagin LS. Mathematical Theory of Optimal Processes. CRC Press, Boca Raton; 1987.
[36] Bolzoni L, Bonacini E, Soresina C, Groppi M. Time-optimal control strategies in SIR epidemic models. Math Biosci. 2017;292:86–96.
[37] Grigorieva EV, Khailov EN, Korobeinikov A. Optimal control for a SIR epidemic model with nonlinear incidence rate. Math Model Nat Phenom. 2016;11(4):89–104.
[38] Bakare EA, Nwagwo A, Danso-Addo E. Optimal control analysis of a SIR epidemic model with constant recruitment. Int J Appl Math Res. 2014;3(3):273–85.
[39] Zakary O, Rachik M, Elmouki I. On the analysis of a multi-regions discrete SIR epidemic model: an optimal control approach. Int J Dyn Control. 2017;5(3):917–30. doi:10.1007/s40435-016-0233-2.
[40] Lenhart S, Workman JT. Optimal Control Applied to Biological Models. Chapman & Hall/CRC, Boca Raton; 2007.
[41] Goodfellow I, Bengio Y, Courville A. Deep Learning. MIT Press, Cambridge, MA; 2016.
[42] Sutton RS, Barto AG. Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA; 2018.
[43] LeCun Y, Bengio Y, Hinton G. Deep learning. Nature. 2015;521(7553):436–44.
[44] Silver D, Huang A, Maddison CJ, Guez A, Sifre L, van den Driessche G, et al. Mastering the game of Go with deep neural networks and tree search. Nature. 2016;529(7587):484–89.
[45] Chayoukh O, Zakary O. Application of artificial intelligence to control a nonlinear SIR model. In: Chakir A, Andry JF, Ullah A, Bansal R, Ghazouani M, eds. Engineering Applications of Artificial Intelligence. Springer, Cham; 2024. p. 23–39. doi:10.1007/978-3-031-50300-9_2.
[46] Hattaf K, Lashari A, Louartassi Y, Yousfi N. A delayed SIR epidemic model with a general incidence rate. Electron J Qual Theory Differ Equ. 2013;2013(3):1–9.
[47] Jing W, Jin Z, Zhang J. An SIR pairwise epidemic model with infection age and demography. J Biol Dyn. 2018;12(1):486–508.
[48] AlQadi H, Bani-Yaghoub M. Incorporating global dynamics to improve the accuracy of disease models: example of a COVID-19 SIR model. PLoS One. 2022;17(4):e0265815.
[49] Wang H, Wu D, Luo J, Zhang J. Integrating socio-psychological factors in the SEIR model optimized by a genetic algorithm for COVID-19 trend analysis. Sci Rep. 2024;14:15684.
[50] Sun T, Jin B, Wu Y, Bao J. A study of the attenuation stage of a global infectious disease. Front Public Health. 2024;12:1379481.
[51] Zakary O, Bidah S, Rachik M. Optimizing infection trajectories: innovation in controllability of nonlinear SIR model. Rev Mex Ing Biomed. 2024;45(2):151–71.