Optimized Fixed-Time Synergetic Controller via a modified Salp Swarm Algorithm for Acute and Chronic HBV Transmission System

  • Saadi Achour Laboratory of Pure and Applied Mathematics, University Amar Telidji of Laghouat, Algeria
  • Khalil Mokhtari Department of Industrial Engineering, University Abbes Laghrour of Khenchela, Algeria
  • Abdelaziz Rahmoune Laboratory of Pure and Applied Mathematics, University Amar Telidji of Laghouat, Algeria
  • Fares Yazid Laboratory of Pure and Applied Mathematics, University Amar Telidji of Laghouat, Algeria
Keywords: Salp Swarm optimization Algorithm, Synergetic controller, Fixed-time control, Lyapunov stability, Hepatitis B Virus, Epidemic system


In this paper, we propose a Salp Swarm Algorithm (SSA) Optimized Fixed-Time Synergetic Control (FTSC) strategy for the spread of hepatitis B infection. The utilization of the SSA optimization algorithm for optimizing the Synergetic Control (SC) fraction parameters presents a non-trivial challenge due to the restriction that only odd numbers can be used for the fractional power. Therefore, an enhanced and adapted version of the SSA algorithm is proposed to effectively address this specific scenario. Our strategic approach centers on the reduction of susceptible, acutely infected, and chronically infected individuals by employing control parameters like isolation, treatment, and vaccination. The objective is to drive these target state variables to their smallest values in a fixed-time, thereby effectively controlling the epidemic. We support our proposal with numerical simulations to demonstrate the feasibility and effectiveness of the control strategy. A comparison is conducted between FTSC and SC in scenarios with and without optimization. The results indicated that FTSC holds a distinct advantage, consistently demonstrating significant progress, with up to 30\% reduction in the total convergence time to zero, outperforming SC in each case.


Abusnaina, A. A., Ahmad, S., Jarrar, R., and Mafarja, M. Training neural networks using salp swarm algorithm for pattern classification. In Proceedings of the 2nd international conference on future networks and distributed systems (2018), pp. 1–6.

Al-Hussein, A.-B. A., Tahir, F. R., and Pham, V.-T. Fixed-time synergetic control for chaos suppression in endocrine glucose–insulin regulatory system. Control Engineering Practice 108 (2021), 104723.

Al-Sadeq, D. W., Taleb, S. A., Zaied, R. E., Fahad, S. M., Smatti, M. K., Rizeq, B. R., Al Thani, A. A., Yassine, H. M., and Nasrallah, G. K. Hepatitis b virus molecular epidemiology, host-virus interaction, coinfection, and laboratory diagnosis in the mena region: An update. Pathogens 8, 2 (2019), 63.

Alter, M. J., Evatt, B. L., Margolis, H. S., Biswas, R., Epstein, J. S., Feinstone, S. M., Finlayson, J. S., Tankersley, D., Alter, H. J., and Hoofnagel, J. H. Public health service interagency guidelines for screening donors of blood, plasma, organs, tissues, and semen for evidence of hepatitis b and hepatitis c. American Journal of Infection Control 19, 5 (1991), 32A–41A.

Boonyaprapasorn, A., Choopojcharoen, T., Pengwang, E., Maneewarn, T., Natsupakpong, S., Sa-Ngiamsunthorn, P., Suechoei, A., and Wechsathol, W. Control of ebola epidemic system based on terminal synergetic controller design. In 2019 IEEE International Conference on Automatic Control and Intelligent Systems (I2CACIS) (2019), IEEE, pp. 204–209.

Boonyaprapasorn, A., Natsupakpong, S., Ngiamsunthorn, P. S., and Thung-od, K. An application of finite time synergetic control for vaccination in epidemic systems. In 2017 IEEE Conference on Systems, Process and Control (ICSPC) (2017), IEEE, pp. 30–35.

Camacho-Villalon, C. L., Dorigo, M., and Stutzle, T. An analysis of why cuckoo search does not bring any novel ideas to optimization. Computers & Operations Research 142 (2022), 105747.

Camacho-Villalon, C. L., Dorigo, M., and Stutzle, T. Exposing the grey wolf, moth-flame, whale, firefly, bat, and antlion algorithms: six misleading optimization techniques inspired by bestial metaphors. International Transactions in Operational Research 30, 6 (2023), 2945–2971.

Castelli, M., Manzoni, L., Mariot, L., Nobile, M. S., and Tangherloni, A. Salp swarm optimization: a critical review. Expert Systems with Applications 189 (2022), 116029.

Dontwi, I., Obeng-Denteh, W., Obiri- Apraku, L., and Andam, E. Modelling hepatitis b in a high prevalence district in ghana.

Huang, S., Xiong, L., Wang, J., Li, P., Wang, Z., and Ma, M. Fixed-time synergetic controller for stabilization of hydraulic turbine regulating system. Renewable Energy 157 (2020), 1233–1242.

Kamyad, A. V., Akbari, R., Heydari, A. A., Heydari, A., et al. Mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis b virus. Computational and mathematical methods in medicine 2014 (2014).

Kanagaraj, G., Masthan, S. S., and Vincent, F. Y. Meta-heuristics based inverse kinematics of robot manipulator’s path tracking capability under joint limits. In Mendel (2022), vol. 28, pp. 41–54.

Kanchanaharuthai, A. Nonlinear controller design for hydraulic turbine regulating systems via immersion and invariance. International Journal of Control (2023), 1–15.

Khan, T., Zaman, G., and Algahtani, O. Transmission dynamic and vaccination of hepatitis b epidemic model. WULFENIA J 22, 2 (2015), 230–241.

Khan, T., Zaman, G., and Chohan, M. I. The transmission dynamic and optimal control of acute and chronic hepatitis b. Journal of biological dynamics 11, 1 (2017), 172–189.

Kheireddine, C., Yassine, A., Fawzi, S., and Khalil, M. A robust synergetic controller for quadrotor obstacle avoidance using b´ezier curve versus b-spline trajectory generation. Intelligent Service Robotics 15, 1 (2022), 143–152.

Kim, G., and Kim, A. Scan context: Egocentric spatial descriptor for place recognition within 3d point cloud map. In 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (2018), IEEE, pp. 4802–4809.

Kolesnikov, A. A. Introduction of synergetic control. In 2014 American control conference (2014), IEEE, pp. 3013–3016.

Kondratiev, I., Santi, E., and Dougal, R. Robust nonlinear synergetic control for m-parallelconnected dc-dc boost converters. In 2008 IEEE power electronics specialists conference (2008), IEEE, pp. 2222–2228.

Krupovic, M., Blomberg, J., Coffin, J. M., Dasgupta, I., Fan, H., Geering, A. D., Gifford, R., Harrach, B., Hull, R., Johnson, W., et al. Ortervirales: new virus order unifying five families of reverse-transcribing viruses. Journal of virology 92, 12 (2018), 10–1128.

Kudela, J. A critical problem in benchmarking and analysis of evolutionary computation methods. Nature Machine Intelligence 4, 12 (2022), 1238–1245.

Li, D., Xi, W., Zhang, Z., Ren, L., Deng, C., Chen, J., Sun, C., Zhang, N., and Xu, J. Oral microbial community analysis of the patients in the progression of liver cancer. Microbial Pathogenesis 149 (2020), 104479.

Lu, S., Tian, C., and Yan, P. Adaptive extended state observer-based synergetic control for a long-stroke compliant microstage with stress stiffening. IEEE/ASME Transactions on Mechatronics 25, 1 (2019), 259–270.

Magnius, L., Mason, W. S., Taylor, J., Kann, M., Glebe, D., D´eny, P., Sureau, C., Norder, H., and Consortium, I. R. Ictv virus taxonomy profile: Hepadnaviridae. Journal of General Virology 101, 6 (2020), 571–572.

Merta, J., and Brandejsk`y, T. Threedimensional genetic algorithm for the kirkman´ s schoolgirl problem. In Mendel (2018), Vysok´e uˇcen´ı technick´e v Brnˇe.

Mirjalili, S., Gandomi, A. H., Mirjalili, S. Z., Saremi, S., Faris, H., and Mirjalili, S. M. Salp swarm algorithm: A bio-inspired optimizer for engineering design problems. Advances in engineering software 114 (2017), 163–191.

Nana-Kyere, S., Ackora-Prah, J., Okyere, E., Marmah, S., and Afram, T. Hepatitis b optimal control model with vertical transmission. Appl. Math 7, 1 (2017), 5–13.

Nusawardhana, Zak, S., and Crossley, W. Nonlinear synergetic optimal controllers. Journal of guidance, control, and dynamics 30, 4 (2007), 1134–1147.

Oussama, M., Abdelghani, C., and Lakhdar, C. Efficiency and robustness of type-2 fractional fuzzy pid design using salps swarm algorithm for a wind turbine control under uncertainty. ISA transactions 125 (2022), 72–84.

Santi, E., Monti, A., Li, D., Proddutur, K., and Dougal, R. Synergetic control for power electronics applications: A comparison with the sliding mode approach. Journal of Circuits, Systems, and Computers 13, 04 (2004), 737–760.

Sayiner, M., Golabi, P., and Younossi, Z. M. Disease burden of hepatocellular carcinoma: a global perspective. Digestive diseases and sciences 64 (2019), 910–917.

Sharifi, A., and Salarieh, H. An adaptive synergetic controller applied to heavy-duty gas turbine unit. Applied Energy 333 (2023), 120535.

Solanki, P., and Deep, K. Quadratic approximation salp swarm algorithm for function optimization. OPSEARCH (2023), 1–33.

Sun, Y., Demagny, H., Faure, A., Pontanari, F., Jalil, A., Bresciani, N., Yildiz, E., Korbelius, M., Perino, A., Schoonjans, K., et al. Asparagine protects pericentral hepatocytes during acute liver injury. The Journal of Clinical Investigation 133, 7 (2023).

Veselov, G. E., Sklyarov, A. A., and Sklyarov, S. A. Synergetic approach to quadrotor helicopter control with attractor-repeller strategy of nondeterministic obstacles avoidance. In 2014 6th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT) (2014), IEEE, pp. 228–235.

Wang, J., Pang, J., and Liu, X. Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model. Journal of biological dynamics 8, 1 (2014), 99–116.

Wang, J., Zhang, R., and Kuniya, T. The stability analysis of an sveir model with continuous age-structure in the exposed and infectious classes. Journal of biological dynamics 9, 1 (2015), 73–101.

Williams, R. Global challenges in liver disease. Hepatology 44, 3 (2006), 521–526.

Zaman, G., Kang, Y. H., and Jung, I. H. Stability analysis and optimal vaccination of an sir epidemic model. BioSystems 93, 3 (2008), 240–249.

Zaman, G., Kang, Y. H., and Jung, I. H. Optimal treatment of an sir epidemic model with time delay. BioSystems 98, 1 (2009), 43–50.

Zioui, N., Tadjine, M., and Boucherit, M. S. Super-twisting algorithm approach to control tumors growth. In 3rd International Conference on Systems and Control (2013), IEEE, pp. 457–461.

How to Cite
Achour, S., Mokhtari, K., Rahmoune, A. and Yazid, F. 2023. Optimized Fixed-Time Synergetic Controller via a modified Salp Swarm Algorithm for Acute and Chronic HBV Transmission System. MENDEL. 29, 2 (Dec. 2023), 191-201. DOI:https://doi.org/10.13164/mendel.2023.2.191.
Research articles