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July  2021, 14(7): 2151-2161. doi: 10.3934/dcdss.2020144

## Mathematical model of diabetes and its complication involving fractional operator without singular kernal

 1 Departement of Mathematics, Amity School Of Applied Sciences, Amity University Rajasthan, Jaipur-303002, India 2 School of Liberal Studies, Ambedkar University Delhi, Delhi-110006, India

* Corresponding author: Pranay Goswami

Received  April 2019 Revised  July 2019 Published  July 2021 Early access  November 2019

Diabetes is one of the burning issues of the whole world. It effected the world population rapidly. According to the WHO approx 415 million people are living with diabetes in the world and this figure is expected to rise up to 642 million by 2040. World various organizations raise their voice against the dire facts about the increasing graph of diabetes and its complicated patients. In this paper authors define the fractional model of diabetes and its complications involving to fractional operator with exponential kernel. The authors discuss the existence of the solution by using fixed point theorem and also show the uniqueness of the solution. To validate the model's efficiency the authors provided numerical simulation by using HPM. To strengthen the model the results have been presented in the form of graphs.

Citation: Ravi Shanker Dubey, Pranay Goswami. Mathematical model of diabetes and its complication involving fractional operator without singular kernal. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2151-2161. doi: 10.3934/dcdss.2020144
##### References:
 [1] K. G. M. M. Alberti and Z. Z. Paul, Definition, diagnosis and classification of diabetes mellitus and its complications. Part 1: Diagnosis and classification of diabetes mellitus, Provisional Report of A WHO Consultation, Diabetic Medicine, 15 (1998), 539–553. [2] Diet, Nutrition and the Prevention of Chronic Diseases: Report of A Joint WHO/FAO Expert Consultation, Vol. 916, World Health Organization, 2003. [3] Global action plan for the prevention and control of noncommunicable diseases 2013-2020, World Health Organization, [J]. 2013. [4] Global recommendations on physical activity for health[M], World Health Organization, 2010. [5] Global report on diseases, Geneva, World Health Organization, 2016. [6] Global status report on non communicable diseases 2015, Geneva, World Health Organization, 2015. [7] DIAMOND Project Group, Incidence and trends of childhood type 1 diabetes worldwide 1990-1999[J], Diabetic Medicine, 23 (2006), 857–866. [8] NCD risk factor collaboration (NCD-RisC), Worldwide trends in diabetes since 1980: A pooled analysis of 751 population-based studies with 4*4 million participants, Lancet 2016, 387 (2016), 1513-1530. doi: 10.1016/S0140-6736(16)00618-8. [9] WHO Guideline: Sugars intake in adults and children, Geneva: World Health Organization, 2015. [10] B. S. Alkahtani, O. J. Alkahtani, R. S. Dubey and P. Goswami, Solution of fractional oxygen diffusion problem having without singular kernel, J. Nonlinear Sci. Appl., 10 (2017), 299-307.  doi: 10.22436/jnsa.010.01.28. [11] B. S. Alkahtani, O. J. Algahtani, R. S. Dubey and P. Goswami, The Solution of Modified Fractional Bergmans Minimal Blood Glucose-Insulin Model, Entropy, 19 (2017), 114. doi: 10.3390/e19050114. [12] A. Atangana and B. S. T. Alkahtani, Analysis of non- homogenous heat model with new trend of derivative with fractional order, Chaos Solitons Fractals, 89 (2016), 566-571.  doi: 10.1016/j.chaos.2016.03.027. [13] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814355216. [14] A. Boutayeb, E. H. Twizell, K. Achouayb and A. Chetouani, A mathematical model for the burden of diabetes and its complications, Biomed. Eng. Online, 2 (2004), Article ID 20. [15] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, part II, Geophy. J. Int., 13 (1967), 529–539. doi: 10.1111/j.1365-246X.1967.tb02303.x. [16] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular Kernel, Prog. Fract. Diff. Appl., 1 (2015), 73-85. [17] V. B. L. Chaurasia and R. S. Dubey, Analytical solution for the differential equation containing generalized fractional derivative operators and Mittag-Leffler-type function, ISRN Appl. Math., 2011 (2011), Art. ID 682381, 9 pp. doi: 10.5402/2011/682381. [18] V. B. L. Chaurasia, R. S. Dubey and F. B. M. Belgacem, Fractional radial diffusion equation analytical solution via Hankel and Sumudu transforms, Math. Eng., Sci. Aerospace, 3 (2012), 1-10. [19] A. Debbouche and D. F. M. Torres, Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions, Appl. Math. Comput., 243 (2014), 161-175.  doi: 10.1016/j.amc.2014.05.087. [20] R. S. Dubey, B. S. T. Alkahtani and A. Atangana, Analytical solution of space-time fractional fokker plank equation by homotopy perturbation sumudu Transform method, Math. Prob. Eng., 2015, Art. ID 780929, 7 pp. doi: 10.1155/2015/780929. [21] R. S. Dubey and P. Goswami, Analytical solution of the nonlinear diffusion equation, European Phy. J. Plus, 133 (2018). doi: 10.1140/epjp/i2018-12010-6. [22] R. S. Dubey, F. B. M. Belgacem and P. Goswami, Homotopy perturbation approximate solutions for Bergmans minimal blood glucose-insulin model, Fractal Geo. and Nonlinear Anal. Medicine Biology, 2 (2016), 1-6. [23] R. S. Dubey, P. Goswami and F. B. M. Belgacem, Generalized time-fractional telegraph equation analytical solution by Sumudu and Fourier transforms, J. Frac. Cal. Appl., 5 (2014), 52-58. [24] T. Harder, E. Rodekamp, K. Schellong, J. W. Dudenhausen and A. Plagemann, Birth weight and subsequent risk of type 2 diabetes: A meta-analysis, Amer. J. Epidemiology, 165 (2007), 849-857.  doi: 10.1093/aje/kwk071. [25] I. W. Johnsson, B. Haglund, F. Ahlsson and J. Gustafsson, A high birth weight is associated with increased risk of type 2 diabetes and obesity, Pediatric Obesity, 10 (2015), 77-83.  doi: 10.1111/ijpo.230. [26] A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. [27] J. Losada and J. J. Nieto, Properties of the new fractional derivative without singular Kernel, Prog. Fract. Diff. Appl., 1 (2015), 87-92. [28] J. Luo, Ro ssouw, E. Tong, G. A. Giovino, C. C. Lee and C. Chen, Smoking and diabetes: Does the increased risk ever go away?, Amer. J. Epidemiology, 178 (2013), 937-945. [29] K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. [30] L. Morenga Te, S. Mallard and J. Mann, Dietary sugars and body weight: Systematic review and meta analyses of randomised controlled trials and cohort studies, British Medical J., 2013, Article ID 346, e7492. doi: 10.1136/bmj.e7492. [31] I. Podlubny, Fractional Differential Equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. [32] H. M. Srivastava, R. S. Dubey and M. Jain, A study of the fractional-order mathematical model of diabetes and its resulting complications, Math. Methods Appl. Sci., 42 (2019), 4570-4583.  doi: 10.1002/mma.5681. [33] P. H. Whincup, S. J. Kaye and C. G. Owen, Birth weight and risk of type 2 diabetes: A systematic review, J. Amer. Medical Asso., 300 (2008), 2886-2897. [34] D. Whiting, U. Nigel and R. Gojka, Diabetes: Equity and Social Determinants, Equity, social determinants and public health programmes, 2010. [35] C. Willi, P. Bodenmann, W. A. Ghali, P. D. Faris and J. Cornuz, Active smoking and the risk of type 2 diabetes: A systematic review and meta-analysis, J. Amer. Medical Asso., 298 (2007), 2654-2664.  doi: 10.1001/jama.298.22.2654. [36] X. J. Yang, J. A. T. Machado, C. Cattani and F. Gao, On a fractal LC-electric circuit modeled by local fractional calculus, Comm. Nonlinear Sci. Numer. Sim., 47 (2017), 200-206.  doi: 10.1016/j.cnsns.2016.11.017. [37] A. M. Yang, Y. Z. Zhang, C. Cattani, G. N. Xie, M. M. Rashidi, Y. J. Zhou and X. J. yang, Application of local fractional series expansion method to solve KleinGordon equations on Cantor sets, Abstract Appl. Anal., 2014, Art. ID 372741, 6 pp. doi: 10.1155/2014/372741.

show all references

##### References:
 [1] K. G. M. M. Alberti and Z. Z. Paul, Definition, diagnosis and classification of diabetes mellitus and its complications. Part 1: Diagnosis and classification of diabetes mellitus, Provisional Report of A WHO Consultation, Diabetic Medicine, 15 (1998), 539–553. [2] Diet, Nutrition and the Prevention of Chronic Diseases: Report of A Joint WHO/FAO Expert Consultation, Vol. 916, World Health Organization, 2003. [3] Global action plan for the prevention and control of noncommunicable diseases 2013-2020, World Health Organization, [J]. 2013. [4] Global recommendations on physical activity for health[M], World Health Organization, 2010. [5] Global report on diseases, Geneva, World Health Organization, 2016. [6] Global status report on non communicable diseases 2015, Geneva, World Health Organization, 2015. [7] DIAMOND Project Group, Incidence and trends of childhood type 1 diabetes worldwide 1990-1999[J], Diabetic Medicine, 23 (2006), 857–866. [8] NCD risk factor collaboration (NCD-RisC), Worldwide trends in diabetes since 1980: A pooled analysis of 751 population-based studies with 4*4 million participants, Lancet 2016, 387 (2016), 1513-1530. doi: 10.1016/S0140-6736(16)00618-8. [9] WHO Guideline: Sugars intake in adults and children, Geneva: World Health Organization, 2015. [10] B. S. Alkahtani, O. J. Alkahtani, R. S. Dubey and P. Goswami, Solution of fractional oxygen diffusion problem having without singular kernel, J. Nonlinear Sci. Appl., 10 (2017), 299-307.  doi: 10.22436/jnsa.010.01.28. [11] B. S. Alkahtani, O. J. Algahtani, R. S. Dubey and P. Goswami, The Solution of Modified Fractional Bergmans Minimal Blood Glucose-Insulin Model, Entropy, 19 (2017), 114. doi: 10.3390/e19050114. [12] A. Atangana and B. S. T. Alkahtani, Analysis of non- homogenous heat model with new trend of derivative with fractional order, Chaos Solitons Fractals, 89 (2016), 566-571.  doi: 10.1016/j.chaos.2016.03.027. [13] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814355216. [14] A. Boutayeb, E. H. Twizell, K. Achouayb and A. Chetouani, A mathematical model for the burden of diabetes and its complications, Biomed. Eng. Online, 2 (2004), Article ID 20. [15] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, part II, Geophy. J. Int., 13 (1967), 529–539. doi: 10.1111/j.1365-246X.1967.tb02303.x. [16] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular Kernel, Prog. Fract. Diff. Appl., 1 (2015), 73-85. [17] V. B. L. Chaurasia and R. S. Dubey, Analytical solution for the differential equation containing generalized fractional derivative operators and Mittag-Leffler-type function, ISRN Appl. Math., 2011 (2011), Art. ID 682381, 9 pp. doi: 10.5402/2011/682381. [18] V. B. L. Chaurasia, R. S. Dubey and F. B. M. Belgacem, Fractional radial diffusion equation analytical solution via Hankel and Sumudu transforms, Math. Eng., Sci. Aerospace, 3 (2012), 1-10. [19] A. Debbouche and D. F. M. Torres, Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions, Appl. Math. Comput., 243 (2014), 161-175.  doi: 10.1016/j.amc.2014.05.087. [20] R. S. Dubey, B. S. T. Alkahtani and A. Atangana, Analytical solution of space-time fractional fokker plank equation by homotopy perturbation sumudu Transform method, Math. Prob. Eng., 2015, Art. ID 780929, 7 pp. doi: 10.1155/2015/780929. [21] R. S. Dubey and P. Goswami, Analytical solution of the nonlinear diffusion equation, European Phy. J. Plus, 133 (2018). doi: 10.1140/epjp/i2018-12010-6. [22] R. S. Dubey, F. B. M. Belgacem and P. Goswami, Homotopy perturbation approximate solutions for Bergmans minimal blood glucose-insulin model, Fractal Geo. and Nonlinear Anal. Medicine Biology, 2 (2016), 1-6. [23] R. S. Dubey, P. Goswami and F. B. M. Belgacem, Generalized time-fractional telegraph equation analytical solution by Sumudu and Fourier transforms, J. Frac. Cal. Appl., 5 (2014), 52-58. [24] T. Harder, E. Rodekamp, K. Schellong, J. W. Dudenhausen and A. Plagemann, Birth weight and subsequent risk of type 2 diabetes: A meta-analysis, Amer. J. Epidemiology, 165 (2007), 849-857.  doi: 10.1093/aje/kwk071. [25] I. W. Johnsson, B. Haglund, F. Ahlsson and J. Gustafsson, A high birth weight is associated with increased risk of type 2 diabetes and obesity, Pediatric Obesity, 10 (2015), 77-83.  doi: 10.1111/ijpo.230. [26] A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. [27] J. Losada and J. J. Nieto, Properties of the new fractional derivative without singular Kernel, Prog. Fract. Diff. Appl., 1 (2015), 87-92. [28] J. Luo, Ro ssouw, E. Tong, G. A. Giovino, C. C. Lee and C. Chen, Smoking and diabetes: Does the increased risk ever go away?, Amer. J. Epidemiology, 178 (2013), 937-945. [29] K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. [30] L. Morenga Te, S. Mallard and J. Mann, Dietary sugars and body weight: Systematic review and meta analyses of randomised controlled trials and cohort studies, British Medical J., 2013, Article ID 346, e7492. doi: 10.1136/bmj.e7492. [31] I. Podlubny, Fractional Differential Equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. [32] H. M. Srivastava, R. S. Dubey and M. Jain, A study of the fractional-order mathematical model of diabetes and its resulting complications, Math. Methods Appl. Sci., 42 (2019), 4570-4583.  doi: 10.1002/mma.5681. [33] P. H. Whincup, S. J. Kaye and C. G. Owen, Birth weight and risk of type 2 diabetes: A systematic review, J. Amer. Medical Asso., 300 (2008), 2886-2897. [34] D. Whiting, U. Nigel and R. Gojka, Diabetes: Equity and Social Determinants, Equity, social determinants and public health programmes, 2010. [35] C. Willi, P. Bodenmann, W. A. Ghali, P. D. Faris and J. Cornuz, Active smoking and the risk of type 2 diabetes: A systematic review and meta-analysis, J. Amer. Medical Asso., 298 (2007), 2654-2664.  doi: 10.1001/jama.298.22.2654. [36] X. J. Yang, J. A. T. Machado, C. Cattani and F. Gao, On a fractal LC-electric circuit modeled by local fractional calculus, Comm. Nonlinear Sci. Numer. Sim., 47 (2017), 200-206.  doi: 10.1016/j.cnsns.2016.11.017. [37] A. M. Yang, Y. Z. Zhang, C. Cattani, G. N. Xie, M. M. Rashidi, Y. J. Zhou and X. J. yang, Application of local fractional series expansion method to solve KleinGordon equations on Cantor sets, Abstract Appl. Anal., 2014, Art. ID 372741, 6 pp. doi: 10.1155/2014/372741.
Flow Chart
Represents for the behavior of the solution C(t), with respect to t for different values of other perimeter defined above
Represents for the behavior of the solution E(t), with respect to t for different values of other perimeter defined above
Table-1
 A(t) The incidence of Diabetes Mellitus B(t) Number of person having diabetics without complications C(t) Number of person having diabetics with complications E(t) Size of population of diabetics at time $t$ $\delta$ The probability of a person having diabetic and developing complications $\varepsilon$ Natural rate of mortality $\lambda$ Rate of complications are recovered $\upsilon$ Rate of diabetic patients having complication and become severely disabled $\mu$ Rate of mortality due to diabetic complications
 A(t) The incidence of Diabetes Mellitus B(t) Number of person having diabetics without complications C(t) Number of person having diabetics with complications E(t) Size of population of diabetics at time $t$ $\delta$ The probability of a person having diabetic and developing complications $\varepsilon$ Natural rate of mortality $\lambda$ Rate of complications are recovered $\upsilon$ Rate of diabetic patients having complication and become severely disabled $\mu$ Rate of mortality due to diabetic complications
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