
-
Previous Article
Fractional Laplacians : A short survey
- DCDS-S Home
- This Issue
-
Next Article
Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $
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 |
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.
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. Google Scholar |
[2] |
Diet, Nutrition and the Prevention of Chronic Diseases: Report of A Joint WHO/FAO Expert Consultation, Vol. 916, World Health Organization, 2003. Google Scholar |
[3] |
Global action plan for the prevention and control of noncommunicable diseases 2013-2020, World Health Organization, [J]. 2013. Google Scholar |
[4] |
Global recommendations on physical activity for health[M], World Health Organization, 2010. Google Scholar |
[5] |
Global report on diseases, Geneva, World Health Organization, 2016. Google Scholar |
[6] |
Global status report on non communicable diseases 2015, Geneva, World Health Organization, 2015. Google Scholar |
[7] |
DIAMOND Project Group, Incidence and trends of childhood type 1 diabetes worldwide 1990-1999[J], Diabetic Medicine, 23 (2006), 857–866. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[34] |
D. Whiting, U. Nigel and R. Gojka, Diabetes: Equity and Social Determinants, Equity, social determinants and public health programmes, 2010. Google Scholar |
[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. Google Scholar |
[2] |
Diet, Nutrition and the Prevention of Chronic Diseases: Report of A Joint WHO/FAO Expert Consultation, Vol. 916, World Health Organization, 2003. Google Scholar |
[3] |
Global action plan for the prevention and control of noncommunicable diseases 2013-2020, World Health Organization, [J]. 2013. Google Scholar |
[4] |
Global recommendations on physical activity for health[M], World Health Organization, 2010. Google Scholar |
[5] |
Global report on diseases, Geneva, World Health Organization, 2016. Google Scholar |
[6] |
Global status report on non communicable diseases 2015, Geneva, World Health Organization, 2015. Google Scholar |
[7] |
DIAMOND Project Group, Incidence and trends of childhood type 1 diabetes worldwide 1990-1999[J], Diabetic Medicine, 23 (2006), 857–866. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[34] |
D. Whiting, U. Nigel and R. Gojka, Diabetes: Equity and Social Determinants, Equity, social determinants and public health programmes, 2010. Google Scholar |
[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. |



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 |
The probability of a person having diabetic and developing complications | |
Natural rate of mortality | |
Rate of complications are recovered | |
Rate of diabetic patients having complication and become severely disabled | |
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 |
The probability of a person having diabetic and developing complications | |
Natural rate of mortality | |
Rate of complications are recovered | |
Rate of diabetic patients having complication and become severely disabled | |
Rate of mortality due to diabetic complications |
[1] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003 |
[2] |
Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021023 |
[3] |
Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021019 |
[4] |
Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021017 |
[5] |
Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021020 |
[6] |
Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021014 |
[7] |
Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021062 |
[8] |
Maha Daoud, El Haj Laamri. Fractional Laplacians : A short survey. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021027 |
[9] |
George A. Anastassiou. Iyengar-Hilfer fractional inequalities. Mathematical Foundations of Computing, 2021 doi: 10.3934/mfc.2021004 |
[10] |
María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 |
[11] |
Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021022 |
[12] |
Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2993-3020. doi: 10.3934/dcds.2020394 |
[13] |
Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39. |
[14] |
Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021021 |
[15] |
Zhimin Chen, Kaihui Liu, Xiuxiang Liu. Evaluating vaccination effectiveness of group-specific fractional-dose strategies. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021062 |
[16] |
Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021004 |
[17] |
Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021068 |
[18] |
Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021033 |
[19] |
Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021038 |
[20] |
Flank D. M. Bezerra, Rodiak N. Figueroa-López, Marcelo J. D. Nascimento. Fractional oscillon equations; solvability and connection with classical oscillon equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021067 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]