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Year 2022, , 656 - 664, 27.12.2022

Mehmet Merdan , Şeyma Şişman

https://doi.org/10.17776/csj.1127855

Abstract

Keywords

SDC model Fuzzy difference equation Generalized hukuhara.

References

  • [1] Segre, G., Turco, G.L., and Vercellone, G.V., Modelling Blood Glucose and Insuline Kinetics Normal, Diabetic and obsese Subjects, 22(2) (1973) 94-103.
  • [2] Rosado, Y.C., Mathematical model for Detecting Diabetes, Proceedings of The National Conference On Undergraduate Research (NCUR), University of Wisconsin LaCrosse La-Crosse, Wisconsin, (2009) 16 – 18.
  • [3] Ackerman, E., Gatewood, I., Rosevear, J., and Molnar, G., Blood glucose regulation and diabetes,Concepts and Models of Biomathematics, F. Heinmets, Ed., Marcel Decker, (1969) 131-156.
  • [4] Stahl, F., and Johansson , R., Diabetes mellitus modeling and short-term prediction based on blood glucose measurements, Mathematical Biosciences 217 (2009) 101–117.
  • [5] Boutayeb, A., Twizell, E. H., Achouayb, K., Chetouani, A., A mathematical model for the burden of diabetes and its complications, BioMedical Engineering Online 3 ( 2004) 20.
  • [6] Hill, J., Nielsen, M., Fox, M. H., Understanding the Social Factors That Contribute to Diabetes: A Means to Informing Health Care and Social Policies for the Chronically Ill,The Permanente Journal, 17(2) ( 2013) 67–72.
  • [7] Şişman, Ş., Merdan, M., Global stability of Susceptible Diabetes Complication (SDC) model in discrete time, Sigma J Eng Nat Sci, 39 (3) (2021) 290–312.
  • [8] Widyaningsih,P.,Affan,R.C.,Saputro,D.R.S. A Mathematical Model for The Epidemiology of Diabetes Mellitus with Lifestyle and Genetic Factors, IOP Conf. Series: Journal of Physics: Conf. Series, (2018) 1028-012110.
  • [9] Zadeh, L., Toward a generalized theory of uncertainty (GTU) – an outline, Information Sciences 172 (2005) 1–40.
  • [10] Dubois, D. , Prade, H., Towards fuzzy differential calculus: Part 3, Differentiation, Fuzzy Sets and Systems 8 (1982) 225–233.
  • [11] Puri, M.L., Ralescu, D.A., Differentials of fuzzy functions, Journal of Mathematical Analysis and Application 91 (1983) 552–558.
  • [12] Goetschel, R., Voxman, W., Elementary fuzzy calculus, Fuzzy Sets and Systems, 31 (1986) 18–43.
  • [13] Kandel, A., Byatt, W. J., Fuzzy differential equations, Proc. Int. Conf. Cybernatics and Society, Tokyo, Novenber, (1978) 1213-1216.
  • [14] Kandel, A., Byatt, W. J., Fuzzy sets, fuzzy algebra, and fuzzy statisticsProc. IEEE, 66 (1978) 1619-1639.
  • [15] Kaleva, O., Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987) 301-317.
  • [16] Zhang, Q., Yang, L., Liao, D.,On the fuzzy difference equation x_(n+1)=A+∑_(i=0)^k▒B/x_(n-i) , Engineering and Technology, 75 (2011).
  • [17] Chrysafis, K.A., Papadopoulos, B.K., and Papaschinopoulos, G., On the fuzzy difference equations of finance, Fuzzy Sets and Systems, 159 (2008) 3259-3270.
  • [18] Bede, B., Mathematics of Fuzzy Sets and Fuzzy Logic, Springer, (2013)
  • [19] Papaschinopoulos, G., Stefanidou, G., Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation, Fuzzy Sets and Systems 140 (2003) 523–539
  • [20] Zhang, Q., Zhang, W., Liu, J., Shao, Y.,(2014). On a Fuzzy Logistic Difference Equation, wseas transactions on mathematics, E-ISSN: 2224-2880, 282-290.
  • [21] Din, Q. Asymptotic Behavior of a Second-Order Fuzzy Rational Difference Equation, Hindawi Publishing Corporation Journal of Discrete Mathematics (2015) 7.
  • [22] Mahata, A., Mondal, S.P., Alam, S., and Roy, B., Mathematical model of glucose-insulin regulatory system on diabetes mellitus in fuzzy and crisp environment, Ecological Genetics and Genomics, (2016).
  • [23] Mahata, A., Roy, B., Mondal, S. P., Alam, S., Application of ordinary differential equation in glucose insulin regulatory system modeling in fuzzy environment, Ecological Genetics and Genomics 3(2017) 60-66.
  • [24] Salahshour, S., Aliahmadian, Mahata, A., Mondal, S. P., Alam, S., The Behavior of Logistic Equation with Alley Effect in Fuzzy Environment : Fuzzy Differential Equation Approach, International Journal of Applied and Computational Mathematics 4 (2) (2018) 1-20.
  • [25] Mahata, A., Mondal, S. P., Alam, S., Chakraborty, A., Dey, S.K., Goswami, A., Mathematical model for diabetes in fuzzy environment, Journal of Intelligent and Fuzzy Systems, 36 (3) (2018) 2923-2932.
  • [26] Mahata, A., Mondal, S. P., Ahmadian, A., Ismail, F., Alam, S., and Salahshour, S., Different Solution Strategies for Solving Epidemic Model in Imprecise Environment, , Complexity, (2018) 4902142.
  • [27] Khastan, A., Alijani, Z., On the new solutions to the fuzzy difference equation x_n+1=A+B/x_n , Fuzzy Sets and Systems 358, (2019), 64-83.
  • [28] Zhang, Q., Zhang, W., Lin, F., Li, D. On dynamic behavior of second-order exponential-type fuzzy difference equation, Fuzzy Sets and Systems,(2021), 169-187.
  • [29] Jia,L., Wang,C., Zhao, X., Wei,W. Dynamic Behavior of a Fractional-Type Fuzzy Difference System, Symmetry ,(2022), 14, 1337.
  • [30] Han, C., Li, L., Su, G. And Sun, T. Dynamical behaviors of a k-order fuzzy difference equation, Open Mathematics (2022), 20: 391–403.
  • [31] Darus, M., Wahab, A.F., Review on Fuzzy Difference Equation, Malaysian Journal of Fundamental & Applied Sciences, 8 (4) (2012) 176-180.
  • [32] Atanasov,K.,T., Intuitionistic Fuzzy Set. Fuzy Set and Systems, 20(1) (1986) 87-96.
  • [33] Diamond, P., Kloeden, P., Metric topology of fuzzy numbers and fuzzy analysis, In Fundamentals of Fuzzy Sets, Springer, (2000) 583-641.
  • [34] Goetschel Jr, R., Voxman, W., Elementary fuzzy calculus, Fuzzy sets and systems, 18(1) (1986) 31-43.
  • [35] Stefanini, L., A generalization of Hukuhara difference for interval and fuzzy arithmetic, Working Papers Series in Economics, Mathematics and Statistics, 48 (2008) 1-13.
  • [36] Samuelson, P.A., Conditions that a root of a polynomial be less than unity in absolute value, Ann. Math. Stat.,12 (1941) 360-364.

A Mathematical Model of Susceptible Diabetes Complication (SDC) Model in Discrete Time Fuzzy and Crisp Environment

Year 2022, , 656 - 664, 27.12.2022

Mehmet Merdan , Şeyma Şişman

https://doi.org/10.17776/csj.1127855

Abstract

In this study, we examined the mathematical model of the discrete-time equation system with susceptible diabetes complication (SDC), which is known to be caused by environmental and genetic factors in a fuzzy environment. From the diabetes complication (DC) model, the susceptible diabetes complication (SDC) model is being developed. It was obtained using definitions of how the behavior of this model changes in a fuzzy environment. A nonlinear differential equation system transforms the sensitive diabetes complication (SDC) model into a discrete time equation system. Stability analysis of the model with jury criterion was examined. In addition, numerical solutions and graphics of the analysis of the discrete model in fuzzy environment are obtained by using the MATLAB package program.

Keywords

SDC model Fuzzy difference equation Generalized hukuhara.

References

  • [1] Segre, G., Turco, G.L., and Vercellone, G.V., Modelling Blood Glucose and Insuline Kinetics Normal, Diabetic and obsese Subjects, 22(2) (1973) 94-103.
  • [2] Rosado, Y.C., Mathematical model for Detecting Diabetes, Proceedings of The National Conference On Undergraduate Research (NCUR), University of Wisconsin LaCrosse La-Crosse, Wisconsin, (2009) 16 – 18.
  • [3] Ackerman, E., Gatewood, I., Rosevear, J., and Molnar, G., Blood glucose regulation and diabetes,Concepts and Models of Biomathematics, F. Heinmets, Ed., Marcel Decker, (1969) 131-156.
  • [4] Stahl, F., and Johansson , R., Diabetes mellitus modeling and short-term prediction based on blood glucose measurements, Mathematical Biosciences 217 (2009) 101–117.
  • [5] Boutayeb, A., Twizell, E. H., Achouayb, K., Chetouani, A., A mathematical model for the burden of diabetes and its complications, BioMedical Engineering Online 3 ( 2004) 20.
  • [6] Hill, J., Nielsen, M., Fox, M. H., Understanding the Social Factors That Contribute to Diabetes: A Means to Informing Health Care and Social Policies for the Chronically Ill,The Permanente Journal, 17(2) ( 2013) 67–72.
  • [7] Şişman, Ş., Merdan, M., Global stability of Susceptible Diabetes Complication (SDC) model in discrete time, Sigma J Eng Nat Sci, 39 (3) (2021) 290–312.
  • [8] Widyaningsih,P.,Affan,R.C.,Saputro,D.R.S. A Mathematical Model for The Epidemiology of Diabetes Mellitus with Lifestyle and Genetic Factors, IOP Conf. Series: Journal of Physics: Conf. Series, (2018) 1028-012110.
  • [9] Zadeh, L., Toward a generalized theory of uncertainty (GTU) – an outline, Information Sciences 172 (2005) 1–40.
  • [10] Dubois, D. , Prade, H., Towards fuzzy differential calculus: Part 3, Differentiation, Fuzzy Sets and Systems 8 (1982) 225–233.
  • [11] Puri, M.L., Ralescu, D.A., Differentials of fuzzy functions, Journal of Mathematical Analysis and Application 91 (1983) 552–558.
  • [12] Goetschel, R., Voxman, W., Elementary fuzzy calculus, Fuzzy Sets and Systems, 31 (1986) 18–43.
  • [13] Kandel, A., Byatt, W. J., Fuzzy differential equations, Proc. Int. Conf. Cybernatics and Society, Tokyo, Novenber, (1978) 1213-1216.
  • [14] Kandel, A., Byatt, W. J., Fuzzy sets, fuzzy algebra, and fuzzy statisticsProc. IEEE, 66 (1978) 1619-1639.
  • [15] Kaleva, O., Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987) 301-317.
  • [16] Zhang, Q., Yang, L., Liao, D.,On the fuzzy difference equation x_(n+1)=A+∑_(i=0)^k▒B/x_(n-i) , Engineering and Technology, 75 (2011).
  • [17] Chrysafis, K.A., Papadopoulos, B.K., and Papaschinopoulos, G., On the fuzzy difference equations of finance, Fuzzy Sets and Systems, 159 (2008) 3259-3270.
  • [18] Bede, B., Mathematics of Fuzzy Sets and Fuzzy Logic, Springer, (2013)
  • [19] Papaschinopoulos, G., Stefanidou, G., Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation, Fuzzy Sets and Systems 140 (2003) 523–539
  • [20] Zhang, Q., Zhang, W., Liu, J., Shao, Y.,(2014). On a Fuzzy Logistic Difference Equation, wseas transactions on mathematics, E-ISSN: 2224-2880, 282-290.
  • [21] Din, Q. Asymptotic Behavior of a Second-Order Fuzzy Rational Difference Equation, Hindawi Publishing Corporation Journal of Discrete Mathematics (2015) 7.
  • [22] Mahata, A., Mondal, S.P., Alam, S., and Roy, B., Mathematical model of glucose-insulin regulatory system on diabetes mellitus in fuzzy and crisp environment, Ecological Genetics and Genomics, (2016).
  • [23] Mahata, A., Roy, B., Mondal, S. P., Alam, S., Application of ordinary differential equation in glucose insulin regulatory system modeling in fuzzy environment, Ecological Genetics and Genomics 3(2017) 60-66.
  • [24] Salahshour, S., Aliahmadian, Mahata, A., Mondal, S. P., Alam, S., The Behavior of Logistic Equation with Alley Effect in Fuzzy Environment : Fuzzy Differential Equation Approach, International Journal of Applied and Computational Mathematics 4 (2) (2018) 1-20.
  • [25] Mahata, A., Mondal, S. P., Alam, S., Chakraborty, A., Dey, S.K., Goswami, A., Mathematical model for diabetes in fuzzy environment, Journal of Intelligent and Fuzzy Systems, 36 (3) (2018) 2923-2932.
  • [26] Mahata, A., Mondal, S. P., Ahmadian, A., Ismail, F., Alam, S., and Salahshour, S., Different Solution Strategies for Solving Epidemic Model in Imprecise Environment, , Complexity, (2018) 4902142.
  • [27] Khastan, A., Alijani, Z., On the new solutions to the fuzzy difference equation x_n+1=A+B/x_n , Fuzzy Sets and Systems 358, (2019), 64-83.
  • [28] Zhang, Q., Zhang, W., Lin, F., Li, D. On dynamic behavior of second-order exponential-type fuzzy difference equation, Fuzzy Sets and Systems,(2021), 169-187.
  • [29] Jia,L., Wang,C., Zhao, X., Wei,W. Dynamic Behavior of a Fractional-Type Fuzzy Difference System, Symmetry ,(2022), 14, 1337.
  • [30] Han, C., Li, L., Su, G. And Sun, T. Dynamical behaviors of a k-order fuzzy difference equation, Open Mathematics (2022), 20: 391–403.
  • [31] Darus, M., Wahab, A.F., Review on Fuzzy Difference Equation, Malaysian Journal of Fundamental & Applied Sciences, 8 (4) (2012) 176-180.
  • [32] Atanasov,K.,T., Intuitionistic Fuzzy Set. Fuzy Set and Systems, 20(1) (1986) 87-96.
  • [33] Diamond, P., Kloeden, P., Metric topology of fuzzy numbers and fuzzy analysis, In Fundamentals of Fuzzy Sets, Springer, (2000) 583-641.
  • [34] Goetschel Jr, R., Voxman, W., Elementary fuzzy calculus, Fuzzy sets and systems, 18(1) (1986) 31-43.
  • [35] Stefanini, L., A generalization of Hukuhara difference for interval and fuzzy arithmetic, Working Papers Series in Economics, Mathematics and Statistics, 48 (2008) 1-13.
  • [36] Samuelson, P.A., Conditions that a root of a polynomial be less than unity in absolute value, Ann. Math. Stat.,12 (1941) 360-364.
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Mehmet Merdan 0000-0002-8509-3044

Şeyma Şişman 0000-0003-0756-4872

Publication Date December 27, 2022
Submission Date June 8, 2022
Acceptance Date September 1, 2022
Published in Issue Year 2022

Cite

APA Merdan, M., & Şişman, Ş. (2022). A Mathematical Model of Susceptible Diabetes Complication (SDC) Model in Discrete Time Fuzzy and Crisp Environment. Cumhuriyet Science Journal, 43(4), 656-664. https://doi.org/10.17776/csj.1127855