HARİCİ REENFEKSİYONLARA BAĞLI TÜBERKÜLOZUN UYARLAMALI KALMAN FİLTRE TEMELLİ OPTİMAL KONTROLÜ

Öz

Tüberküloz gibi epidemiyolojik hastalıkları matematiksel modeller üzerinden incelemek hastalığın gelecekteki dinamiklerini yorumlamak için faydalıdır. Bu modellerin varlığında, hastalığı ortadan kaldırabilecek stratejileri hesaplamak mümkün olmaktadır. Bu çalışmada, aktif tüberkülozlu, evde ya da hastanede tedavi edilen bireylerde ölçülemeyen tüberküloz dinamiklerini kontrol etmek için uyarlanabilir kokusuz bir Kalman filtre (UKKF) tabanlı optimal denetleyici tasarlanmıştır. Harici reenfeksiyona bağlı tüberküloz hastalarının evde ve hastanede tedavi görmeleri durumunun incelenmesi, optimal tedavi seçenekleri araştırılmasına olanak sağlamaktadır. Bu sayede küçük bir bulaşıcı insan grubu varlığında bile hastalığın uzun vadede kalıcı olmasının önüne geçileceği düşünülmektedir. Tüberküloz modelinden elde edilen tahmin ve kontrol sonuçlarına göre, tasarlanan uyarlanabilir optimal denetleyicinin, bulaşıcı virüs taşıyan kişilerin eve ya da hastaneye geçişi için doğru kararlar verebilecek nitelikte olduğu kanısına varılmıştır.

Anahtar Kelimeler

• Optimal kontrol
• Uyarlamalı kontrol
• Kalman filtre
• Tüberkuloz
• Harici reenfeksiyon.

ADAPTIVE KALMAN FILTERING BASED OPTIMAL CONTROL OF TUBERCULOSIS DYNAMICS WITH EXOGENOUS REINFECTIONS

Öz

Examining epidemiological diseases such as tuberculosis through mathematical models is useful for interpreting the future dynamics of the disease. In the presence of these models, it is possible to calculate strategies that can exterminate the disease. In this study, an adaptive unscented Kalman filter (AUKF) -based optimal controller has been designed to control unknown tuberculosis dynamics in individuals treated with active tuberculosis, at home or in hospital. The investigation of the treatment of tuberculosis patients at home and in hospital due to exogenous reinfections helps to search for optimal treatment options. In this way, even in the presence of a small group of infectious people, the long-term persistence of the disease is thought to be prevented. According to the estimation and control results obtained from the tuberculosis model, it was concluded that the designed adaptive optimal controller was able to make the right decision about the transfer of infected persons to the home or to the hospital.

Anahtar Kelimeler

• Optimal control
• Adaptive control
• Kalman filter
• Tuberculosis
• Exogenous reinfection

Kaynakça

1. Castillo-Chavez, C., and Feng, Z., 1997. To treat or not to treat: the case of tuberculosis. Journal of mathematical biology, 35(6):629–656.
2. Cetin, M., and Beyhan, S., 2019. Adaptive stabilization of uncertain cortex dynamics under joint estimates and input constraints. IEEE Transactions on Circuits and Systems II: Express Briefs, 66(4):627–631.
3. Das, M., Dey, A., Sadhu, S., and Ghoshal T., 2015. Adaptive central difference filter for non-linear state estimation. IET Science, Measurement & Technology, 9(6):728–733.
4. Earn, DJ., Brauer, F., Driessche, P., and Wu, J., 2008. Mathematical epidemiology. Springer.
5. Gao, D., and Huang, N., 2018. Optimal control analysis of a tuberculosis model. Applied Mathematical Modelling, 58:47–64.
6. Hajiyev, C., and Soken, H. E., 2014. Robust adaptive unscented Kalman filter for attitude estimation of pico satellites. International Journal of Adaptive Control and Signal Processing, 28(2):107–120.
7. Hethcote, H., 2000. The mathematics of infectious diseases. SIAM review, 42(4):599–653.
8. Huo, H., and Zou, M., 2016. Modelling effects of treatment at home on tuberculosis transmission dynamics. Applied Mathematical Modelling, 40(21-22):9474–9484.

9. Itik, M., 2016. Optimal control of nonlinear systems with input constraints using linear time varying approximations. Nonlinear Analysis: Modelling and Control, 21(3):400–412.
10. Julier, S., and Uhlmann, J., 2004. Unscented filtering and nonlinear estimation. Proceedings of the IEEE, 92(3):401–422.
11. Khajanchi, S., Das, D., and Kar, T., 2018. Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation. Physica A: Statistical Mechanics and its Applications, 497:52–71.
12. Liang, X., Xu, J., and Zhang H., 2017. Optimal control and stabilization for networked control systems with packet dropout and input delay. IEEE Transactions on Circuits and Systems II: Express Briefs, 64(9):1087–1091.
13. Silva, C. J., and Torres, D., 2015. Optimal control of tuberculosis: A review. In Dynamics, Games and Science, pages 701–722.
14. Yıldız, T., and Karaoğlu, E., 2019. Optimal control strategies for tuberculosis dynamics with exogenous reinfections in case of treatment at home and treatment in hospital. Nonlinear Dynamics, 97(4):2643–2659.
15. Waaler, H., Geser, A., and Andersen, S., 1962. The use of mathematical models in the study of the epidemiology of tuberculosis. American Journal of Public Health and the Nations Health, 52(6):1002–1013.
16. Wan, E., and Merwe, R., 2000. The unscented Kalman filter for nonlinear estimation. In Adaptive Systems for Signal Processing, Communications, and Control Symposium, IEEE, pages 153–158.
17. World Health Organization et al. Global tuberculosis report 2017. World Health Organization.