Optimal Control Model for Dual Treatment of Delayed Type-II Diabetes Infection in Human Population
Bassey Echeng Bassey, Department of Mathematics/Statistics, Cross River University of Technology, Calabar, Nigeria.
Following the seeming insurmountable medical cure for the dreaded type-II diabetes, several and concurrent notable scientific research works for the most appropriate approach for the treatment and management of the aforementioned disease have been on the increase. In this paper, using ordinary differential equations, we formulated a set of pent-linear mathematical type-II diabetes dynamic model. The novelty of investigation was primed by a tri-linear optimal maximization of model predominant state variables following methodological application of designated bilinear control functions in the presence of incorporated time delay lag. With the derived model, the system invariant and boundedness of solutions as well as stability analysis was scientifically investigated. To achieve study set goal, the model was transformed to an optimal control problem and analysis performed using classical Pontryagin’s maximum principle. The system optimal characterization, existence of an optimal control pair and optimality system were comprehensively established. Numerical illustrative examples were then conducted. The result that follows conspicuously indicated a pragmatic flow of the model as evidenced by highly tri-linear maximization and sequence reversion of type-II diabetes’ early infection stages. Moreso, the near zero reduction of chronic type-II diabetes infection was a further affirmation of model ingenuity, which is a step towards achieving bioscientific and biotechnological height needed for this 21st century. Suggested therefore, is a more chemotherapy inclusive and possible extensively articulated method for a possible eradication of this dreaded type-II diabetes.
Hyperglycemia, Hypoglycemia, Multifactorial-Infection, Optimal-Control-Function, Penalty-Multiplier, Tri-Linear-Maximization, Type-II-Diabetes
International Diabetes Federation [IDF], IDF Report 2003. Retrieved date: [12, November, 2018], online available at www.idf.org/home/index.cfm
Derouich M., Boutayeb A., Boutayeb W. and Lamlili M. (2004) Optimal control approach to the dynamics of a population of diabetics. Applied Mathematical Sciences, 8 (56): 2773-2782.
Mayo Foundation for Medical Education and Research (MFMER), 1998-2018. Retrieved date: [12, November, 2018], online available at https: //www.mayoclinic.org/diseases-conditions/type-2-diabetes/symptoms-causes/syc-20351193
World Health Organization (2006) Definition and Diagnosis of Diabetes Mellitus and Intermediate Hyperglycemia, WHO, Geneva. Retrieved date: [12, November, 2018], online available at http: //www.who.int/diabetes/publications/Definition%20and%20diagnosis%20of%20diabetes_new.pdf
American Diabetes Association (2001) Fasting Versus Postload Glucose Levels. Diabetes Care, 24, 1855. Retrieved date: [12, November, 2018], online available at http://care.diabetesjournals.org/content/24/11/1855.short
Makroglou A., Li J. and Kuang Y. (2006) Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview. Applied Numerical Mathematics, 56 (2006): 559–573.
Cellgevity and Diabetes (2018). Retrieved date: [12, November, 2018], online available at https: //www.npennington.com/cellgevity-and-diabetes.html
Ajmera I., Swat M., Laibe C., Le Novère N. and Chelliah V. (2013) The impact of mathematical modeling on the understanding of diabetes and related complications. CPT: Pharmacometrics & Systems Pharmacology (2013) 2: e54, 1-14.
Boutayeb A., Chetouani A., Achouyab A. and E. H. Twizell E. H. (2006) A Non-linear Population Model of Diabetes Mellitus. J. Appl. Math.& Computing, l. 21 (1-2): 127 – 139.
Permatasari A. H., Tjahjana R. H. and Udjiani T. (2018) Global stability for linear system and controllability for nonlinear system in the dynamics model of diabetics population. IOP Conf. Series: Journal of Physics: Conf. Series 1025, (2018) 012086, doi: 10.1088/1742-6596/1025/1/012086.
Kenchaiah S., Evans J. C., Levy D., Benjamine E. I. Larson M. G., Kannel W. B. and Vasan R. S. (2002) Obesity and the Risk of Heart Failure. New England Journal of Medicine, 347, 305-313.
Boutayeb W., Lamlili M., Boutayeb A. and Derouich M. (2015) A simulation model for the dynamics of a population of diabetics with and without complications using optimal control. Bioinformatics and Biomedical Engineering, 9043, 589-98.
Boutayeb W., Lamlili M., Boutayeb A. and Derouich M. (2016) The dynamics of a population of healthy people, pre-diabetics, and diabetics with and without complications with optimal control Proceedings of the Mediterranean. Conference on Information and Communication Technologies, 380, 463-71.
Permatasari A. H., Tjahjana R. H. and Udjiani T. (2017) Existence and characterization of optimal control in mathematics model of diabetics’ population. IOP Conf. Series: Journal of Physics: Conf. Series 983, (2018) 012069, doi: 10.1088/1742-6596/983/1/012069.
Boutayeb A. and M. Derouich M. M. (2002) Age structured models for diabetes in East Morocco. Mathematics and Computers in Simulation, 58 (2002), 215- 229.
Yusuf T. T. (2015) Optimal Control of Incidence of Medical Complications in a Diabetic Patients 'population. FUTA Journal of Research in Sciences, 2015 (1): 180-189.
Boutayeb A. and Chetouani A. (2007) A population model of diabetes and prediabetes. International Journal of Computer Mathematics, 84 (1) (2007), 57 – 66.
Boutayeb A., Boutayeb W., Lamlili M. E. N. and Boutayeb S. (2013) Indirect cost of Diabetes in the Arab region. International Journal of Diabetology and Vascular Disease Research, 1 (7): 1-6.
Hattaf K. and Yousfi, N. (2012) Optimal Control of a Delayed HIV Infection Model with Immune Response Using an Efficient Numerical Method. Biomathematics, 2012, 1-7.
Zhu H. and Zou X. (2009) Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete and continuous dynamical systems, series B, 12 (2): 511–524.
Bassey E. B. (2017) Optimal control model for immune effectors response and multiple chemotherapy treatment (MCT) of dual delayed HIV - pathogen infections. SDRP Journal of Infectious Diseases Treatment & Therapy, 1 (1): 1-18.
Bassey E. B. (2018) Dynamic optimal control model for dual-pair treatment functions of dual delayed HIV-pathogen infections. Journal of Mathematical Sciences: Advances and Applications, 51 (1): 1-50.
Wang K., Wang W., Pang H. and Liu X. (2004) Complex dynamic behavior in a viral model with delayed immune response. Physica D., 226, 197-208.
Osman S., Makinde O. D. and Theuri D. M. (2018) Stability analysis and modeling of listeriosis dynamics in human and animal populations. Global Journal of Pure and Applied Mathematics, 14 (1): 115-138.
Hattaf K. and Yousfi N. (2012) Two optimal treatments of HIV infection model. world Journal of modeling and Simulation, 8 (1): 27-35.
Culshaw R., Ruan S. and Spiteri R. J. (2004) Optimal HIV Treatment by Maximizing Immune Response. Journal of Mathematical Biology, 48 (5): 545-562.
Joshi H. R. (2002) Optimal Control of an HIV Immunology Model. Optimal Control Applications and Methods, 23: 199-213.
Butler S., Kirschner, D. and Lenhart S. (1997) Optimal control of the chemotherapy affecting the infectivity of HIV, Editors: O. Arino, D. Axelrod and M. Kimmel. Advances in Mathematical Population Dynamics- Molecules, Cells and Man, 557-569.
Perelson S. A., Kirschner E. D. and De Boer R. (1993) Dynamics of HIV-infection of CD4+ T cells. Mathematical Biosciences, 114, 81-125.
Fleming W. and Rishel R. (1975) Deterministic and Stochastic Optimal Control. Springer Verlag, New York.
Lukes D. L. (1982) Differential Equations: Classical to Controlled, vol. 162 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA.
Bassey B. E. and Bassey D. R. (2018) Optimal Control Dynamics: Control Interventions for Eradication of Ebola Virus Infection. International Journal of Mathematical Sciences and Computing (IJMSC), 4 (3): 42-65, 2018. DOI: 10.5815/ijmsc.2018.03.04.
Fister K. R., Lenhart S. and McNally J. S. (1998) Optimizing chemotherapy in an HIV Model. Electr. J. Diff. Eq., 32, 1-12.
Kirschner D. and Webb G. F. (1996) A Model for Treatment Strategy in the Chemotherapy of AIDS. Bull. Math. Biol., 58: 367-390.
Bassey B. E. (2017) Dynamic optimal control model for period multiple chemotherapy (PMC) treatment of dual HIV-pathogen infections. J Anal Pharm Res., 6 (3): 00176, 1-22. DOI: 10.15406/japlr.2017.06.00176.