Fuzzy Logic Approach for Solving an Optimal Control Problem of an Uninfected Hepatitis B Virus Dynamics

We aimed in this paper to use fuzzy logic approach to solve a hepatitis B virus optimal control problem. The approach efficiency is tested through a numerical comparison with the direct method by taking the values of determinant parameters of this disease for people administrating the drugs. Final results of both numerical methods are in good agreement with experimental data.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Ntaganda, J. and Gahamanyi, M. (2015) Fuzzy Logic Approach for Solving an Optimal Control Problem of an Uninfected Hepatitis B Virus Dynamics. Applied Mathematics, 6, 1524-1537. doi: 10.4236/am.2015.69136.

 [1] Hassan, M.M., Li, D., El-Deeb, A.S., et al. (2008) Association between Hepatitis B Virus and Pancreatic Cancer. Journal of Clinical Oncology, 26, 4557-4562. http://dx.doi.org/10.1200/JCO.2008.17.3526 [2] WHO (2013) Hepatitis B Fact Sheet No. 204. The World Health Organisation, Geneva. http://www.who.int/mediacentre/factsheets/fs204/en/ [3] Lin, C., Kao, J. (2008) Hepatitis B Viral Factors and Clinical Outcomes of Chronic Hepatitis B. Journal of Biomedical Science, 15, 137-145. http://dx.doi.org/10.1007/s11373-007-9225-8 [4] Anderson, R.M. and May, R.M. (1991) Infectious Disease of Humans: Dynamics and Control. Oxford University Press, Oxford. [5] Mann, J. and Roberts, M. (2011) Modelling the Epidemiology of hepatitis B in New Zealand. Journal of Theoretical Biology, 269, 266-272. http://dx.doi.org/10.1016/j.jtbi.2010.10.028 [6] Thornley, S., Bullen, C. and Roberts, M. (2008) Hepatitis B in a High Prevalence New Zealand Population: A Mathematical Model Applied to Infection Control Policy. Journal of Theoretical Biology, 254, 599-603. http://dx.doi.org/10.1016/j.jtbi.2008.06.022 [7] Medley, G.F., Lindop, N.A., Edmunds, W.J. and Nokes, D.J. (2001) Hepatitis-B Virus Endemicity: Heterogeneity, Catastrophic Dynamics and Control. Nature Medicine, 7, 619-624. http://dx.doi.org/10.1038/87953 [8] Zhao, S.-J., Xu, Z.-Y. and Lu, Y. (2000) A Mathematical Model of Hepatitis B Virus Transmission and Its Application for Vaccination Strategy in China. International Journal of Epidemiology, 29, 744-752. http://dx.doi.org/10.1093/ije/29.4.744 [9] Pang, J., Cui, J.-A. and Zhou, X. (2010) Dynamical Behavior of a Hepatitis B Virus Transmission Model with Vaccination. Journal of Theoretical Biology, 265, 572-578. http://dx.doi.org/10.1016/j.jtbi.2010.05.038 [10] Bhattacharyya, S. and Ghosh, S. (2010) Optimal Control of Vertically Transmitted Disease. Computational and Mathematical Methods in Medicine, 11, 369-387. http://dx.doi.org/10.1155/2010/520830 [11] Kar, T.K. and Batabyal, A. (2011) Stability Analysis Andoptimal Control of an SIR Epidemic Model with Vaccination. Biosystems, 104, 127-135. http://dx.doi.org/10.1016/j.biosystems.2011.02.001 [12] Kar, T.K. and Jana, S. (2013) A Theoretical Study on Mathematical Modelling of an Infectious Disease with Application of Optimal Control. Biosystems, 111, 37-50. http://dx.doi.org/10.1016/j.biosystems.2012.10.003 [13] Sheikhan, M. and Ghoreishi, S.A. (2012) Application of Covariance Matrix Adaptation-Evolution Strategy to Optimal Control of Hepatitis B Infection. Neural Computing and Applications, 23, 881-894. http://dx.doi.org/10.1007/s00521-012-1013-3 [14] Lai, C.L. and Yuen, M.F. (2007) The Natural History and Treatment of Chronic Hepatitis B: A Critical Evaluation of Standard Treatment Criteria and End Points. Annals of Internal Medicine, 147, 58-61. http://dx.doi.org/10.7326/0003-4819-147-1-200707030-00010 [15] Lenhart, S. and Workman, J.T. (2007) Optimal Control Applied to Biological Models. Mathematical and Computational Biology Series, Chapman & Hall/CRC, London. [16] Sheikhan, M. and Ghoreishi, S.A. (2012) Antiviral Therapy Using a Fuzzy Controller Optimized by Modified Evolutionary Algorithms: A Comparative Study. Neural Computing and Applications, 23, 1801-1813. http://dx.doi.org/10.1007/s00521-012-1146-4 [17] Hattaf, K., Rachik, M., Saadi, S. and Yousfi, N. (2009) Optimal Control of Treatment in a Basic Virus Infection Model. Applied Mathematical Sciences, 3, 949-958. [18] Elaiw, A.M., Alghamdi, M.A. and Aly, S. (2013) Hepatitis B Virus Dynamics: Modeling, Analysis, and Optimal Treatment Scheduling. Discrete Dynamics in Nature and Society, 2013, Article ID: 712829. http://dx.doi.org/10.1155/2013/712829 [19] Ntaganda, J.M. (2013) Fuzzy Logic Strategy for Solving an Optimal Control Problem of Glucose and Insulin in Diabetic Human. Open Journal of Applied Sciences, 3, 421-429. http://www.scirp.org/journal/ojapps [20] Ntaganda, J.M., Ousséni, S., Barro, G. and Mampassi, B. (2010) Using Fuzzy Logic Strategy for Solving a Therapeutic Optimal Control Problem of an HIV Dynamical Infection. Journal of Pure and Applied Mathematics: Advances and Applications, 3, 205-234. [21] Ntaganda, J.M., Daoussa Haggar, M.S. and Mampassi, B. (2014) Fuzzy Logic for Solving an Optimal Control Problem of Hypoxemic Hypoxia Tissue Blood Carbon Dioxide Exchange during Physical Activity. Open Journal of Applied Sciences, 4, 501-514. http://www.scirp.org/journal/ojapps [22] Song, X. and Neumann, A.U. (2007) Global Stability and Periodic Solution of the Viral Dynamics. Journal of Mathematical Analysis and Applications, 329, 281-297. http://dx.doi.org/10.1016/j.jmaa.2006.06.064 [23] Yakowitz, S.J. (1986) The Stagewise Kuhn-Tucker Condition and Differential Dynamic Programming. IEEE Transactions on Automatic Control, 31, 25-30. http://dx.doi.org/10.1109/tac.1986.1104123 [24] Krichen Masmoudi, N. and Derbel, N. (2003) Optimal Control of Nonlinear Systems by Fuzzy Logic. ISCIII, Nabeul. [25] Sugeno, M. and Murakami, K. (1984) Fuzzy Parking Control of Model Car. 23rd IEEE Conference on Decision and Control, Las Vegas, 12-14 December 1984, 902-903. http://dx.doi.org/10.1109/CDC.1984.272144 [26] Takagi, T. and Sugeno, M. (1985) Fuzzy Identification of Systems and Its Applications to Modeling and Control. IEEE Trans. Systems Man and Cybernetics, 15, 116-132. http://dx.doi.org/10.1109/TSMC.1985.6313399 [27] Park, M.I., Kim, E., Ji, S. and Park, M. (1987) A New Approach to Fuzzy Modeling. IEEE Transactions on Fuzzy Systems, 5, 328-337. [28] Jacobson, D., Lele, D. and Speyer, J.L. (1971) New Necessary Conditions of Optimality for Control Problems with State-Variable Inequality Constraints. Journal of Mathematical Analysis and Applications, 35, 255-284. http://dx.doi.org/10.1016/0022-247X(71)90219-8 [29] Kwakernaak, H. and Sivan, R. (1972) Linear Optimal Control Systems. Wiley-Interscience, New York. [30] Trélat, E. (2005) Contrôle optimal: Théeorie et applications, Vuibert, Collection Mathématiques Concrètes.