doi: 10.3934/dcdss.2020278

Numerical analysis of polio model: A new approach to epidemiological model using derivative with Mittag-Leffler Kernel

1. 

Department of Mathematics Education, Adıyaman University, Adıyaman, Turkey

2. 

Faculty of Mathematics and Statistics Ton Duc Thang University, Ho Chi Minh City, Vietnam

* Corresponding author: koladematthewowolabi@tdtu.edu.vn (K. M. Owolabi)

Received  July 2019 Revised  October 2019 Published  February 2020

The goal of this study is to analyze and obtaining a new numerical approach to an important mathematical model called polio, which is one of the highly infectious and dangerous diseases challenging many lives in most developing nations, most especially in Africa, Latin America and Asia. A number of research outputs has proven beyond doubt that modelling with non-integer order derivative is much more accurate and reliable when compared with the integer order counterparts. In the present case, an extension is given to the polio model by replacing the classical time derivative with the newly defined operator known as the Atangana-Baleanu fractional derivative which has its formulation based on the noble Mittag-Leffler kernel. This derivative has been tested and applied in number of ways to model a range of physical phenomena in science and engineering. The Picard Lindelof theorem is applied to determine the condition under which the proposed model has a solution, also to show that the solution exists and unique. Local stability analysis of the disease free equilibrium and endemic state is also discussed. A novel approximation based on Adams-Bashforth method is formulated to numerically approximate the fractional derivative operator. To justify the theoretical findings, some numerical results obtained for different instances of fractional order are presented.

Citation: Berat Karaagac, Kolade M. Owolabi. Numerical analysis of polio model: A new approach to epidemiological model using derivative with Mittag-Leffler Kernel. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020278
References:
[1]

M. Agarwal and A. S. Bhadauria, Modeling spread of polio with the role of vaccination, Appl. Appl. Math., 6 (2011), 552-571.   Google Scholar

[2] A. Atangana, Fractional Operators With Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018.   Google Scholar
[3]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm Sci., 20 (2016), 763-769.   Google Scholar

[4]

A. Atangana and J. F. Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos, Solitons and Fract., 114 (2018), 516-535.  doi: 10.1016/j.chaos.2018.07.033.  Google Scholar

[5]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.  Google Scholar

[6]

A. Atangana, Blind in a commutative world: Simple illustrations with functions and chaotic attractors, Chaos, Solitons and Fract., 114 (2018), 347-363.  doi: 10.1016/j.chaos.2018.07.022.  Google Scholar

[7]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), Art. 3, 21 pp. doi: 10.1051/mmnp/2018010.  Google Scholar

[8]

G. Bencskó and T. Ferenci, Effective case/infection ratio of poliomyelitis in vaccinated populations, Epidemiol Infect, 144 (2016), 1933-1942.   Google Scholar

[9]

C. J. BrowneR. J. Smith and L. Bourouiba, From regional pulse vaccination to global disease eradication: insights from a mathematical model of poliomyelitis, J. Math. Biol., 71 (2015), 215-253.  doi: 10.1007/s00285-014-0810-y.  Google Scholar

[10]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11. doi: 10.18576/pfda/020101.  Google Scholar

[11]

A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, Verlag, New York, 1997. Google Scholar

[12]

A. Dénes and L. Székely, Global dynamics of a mathematical model for the possible re-emergence of polio, Math. Biosci., 293 (2017), 64-74.  doi: 10.1016/j.mbs.2017.08.010.  Google Scholar

[13]

A. EsenAl aattinB. Karaagac and O. Tasbozan, Finite difference methods for fractional gas dynamics equation, Appl. Math. Inf. Sci., 4 (2016), 1-4.   Google Scholar

[14]

B. Karaagac, Analysis of the cable equation with non-local and non-singular kernel fractional derivative, Eur. Phys. J. Plus, 54 (2018), 133. doi: 10.1140/epjp/i2018-11916-1.  Google Scholar

[15]

B. Karaagac, A study on fractional Klein Gordon equation with non-local and non-singular kernel, Chaos, Solitons and Fract, 126 (2019), 218-229.  doi: 10.1016/j.chaos.2019.06.010.  Google Scholar

[16]

B. Karaagac, New exact solutions for some fractional order differential equations via improved sub-equation method, Discrete Cont Dyn -S, 12 (2019), 447-454.   Google Scholar

[17]

B. KaraagacY. UcarN. M. Yagmurlu and A. Esen, A New Perspective on The Numerical Solution for Fractional Klein Gordon Equation, Politeknik, 22 (2019), 443-451.  doi: 10.2339/politeknik.428986.  Google Scholar

[18]

M. M. Khader and K. M. Saad, A numerical study by using the Chebyshev collocation method for a problem of biological invasion: Fractional Fisher equation, Int J Biomath, 11 (2018), 1850099, 15pp. doi: 10.1142/S1793524518500997.  Google Scholar

[19]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, 2006.  Google Scholar

[20]

A. MedvinskyS. PetrovskiiI. TikhonovaH. Malchow and B. L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.  Google Scholar

[21]

V. F. Morales-DelgadoJ. F. Gómez-AguilarK. M.SaadM. A.Khan and P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach., Physica A Stat. Mech. Appl., 523 (2019), 48-65.  doi: 10.1016/j.physa.2019.02.018.  Google Scholar

[22]

V. F. Morales-DelgadoJ. F. Gómez-AguilarK. Saad and R. F. Escobar Jiménez, Application of the Caputo-Fabrizio and Atangana-Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect, Math Method Appl Sci., 42 (2019), 1167-1193.  doi: 10.1002/mma.5421.  Google Scholar

[23]

K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Math. Model. Nat. Phenom., 13 (2018), Art. 7, 17 pp. doi: 10.1051/mmnp/2018006.  Google Scholar

[24]

K. M. Owolabi and A. Atangana, Modelling and formation of spatiotemporal patterns of fractional predation system in subdiffusion and superdiffusion scenarios, Eur. Phys. J. Plus, 133 (2018), 43. doi: 10.1140/epjp/i2018-11886-2.  Google Scholar

[25]

K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus, 133 (2018), 15. doi: 10.1140/epjp/i2018-11863-9.  Google Scholar

[26]

K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, Eur. Phys. J. Plus, 133 (2018), 98. doi: 10.1140/epjp/i2018-11951-x.  Google Scholar

[27]

K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos, Solitons and Fract, 111 (2018), 119-127.  doi: 10.1016/j.chaos.2018.04.019.  Google Scholar

[28]

K. M. Owolabi and A. Atangana, Chaotic behaviour in system of noninteger-order ordinary differential equations, Chaos, Solitons and Fract, 115 (2018), 362-370.  doi: 10.1016/j.chaos.2018.07.034.  Google Scholar

[29]

K. M. Owolabi, Analysis and numerical simulation of multicomponent system with Atangana-Baleanu fractional derivative, Chaos, Solitons and Fract., 115 (2018), 127-134.  doi: 10.1016/j.chaos.2018.08.022.  Google Scholar

[30]

K. M. Owolabi, Numerical patterns in system of integer and non-integer order derivatives, Chaos, Solitons and Fract., 115 (2018), 143-153.  doi: 10.1016/j.chaos.2018.08.010.  Google Scholar

[31]

K. M. Owolabi, Numerical patterns in reaction-diffusion system with the Caputo and Atangana-Baleanu fractional derivatives, Chaos, Solitons and Fract., 115 (2018), 160-169.  doi: 10.1016/j.chaos.2018.08.025.  Google Scholar

[32]

K. M. Owolabi and J. F. Gómez-Aguilar, Numerical simulations of multilingual competition dynamics with nonlocal derivative, Chaos, Solitons and Fract., 117 (2018), 175-182.  doi: 10.1016/j.chaos.2018.10.020.  Google Scholar

[33]

K. M. Owolabi, Computational study of noninteger order system of predation, Chaos, 29 (2019), 013120, 14pp. doi: 10.1063/1.5079616.  Google Scholar

[34]

K. M. Owolabi and Z. Hammouch, Mathematical modeling and analysis of two-variable system with noninteger-order derivative, Chaos, 29 (2019), 013145, 15pp. doi: 10.1063/1.5086909.  Google Scholar

[35]

K. M. Owolabi and A. Atangana, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems, Chaos, 29 (2019), 023111, 12pp. doi: 10.1063/1.5085490.  Google Scholar

[36] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, CA, 1999.   Google Scholar
[37]

Polio Global Eradication Initiative, Circulating Vaccine-derived Poliovirus, 2016. Google Scholar

[38]

K. M. Saad, S. Deniz and D. Baleanu, On a new modified fractional analysis of Nagumo equation, Int J Biomath, 12 (2019), 1950034, 15pp. doi: 10.1142/S1793524519500347.  Google Scholar

[39]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[40]

K. M. Thompson and R. J. Duintjer Tebbens, Modeling the dynamics of oral poliovirus vaccine cessation, J Infect Dis, 210 (2014), S475-S484. doi: 10.1093/infdis/jit845.  Google Scholar

[41]

M. Toufik and A. Atangana, New numerical approximation of fractional derivative with nonlocal and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132 (2017), 444. Google Scholar

[42]

World Health Organization, Polio vaccines: WHO position paper - March, 2016, 2016. Google Scholar

show all references

References:
[1]

M. Agarwal and A. S. Bhadauria, Modeling spread of polio with the role of vaccination, Appl. Appl. Math., 6 (2011), 552-571.   Google Scholar

[2] A. Atangana, Fractional Operators With Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018.   Google Scholar
[3]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm Sci., 20 (2016), 763-769.   Google Scholar

[4]

A. Atangana and J. F. Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos, Solitons and Fract., 114 (2018), 516-535.  doi: 10.1016/j.chaos.2018.07.033.  Google Scholar

[5]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.  Google Scholar

[6]

A. Atangana, Blind in a commutative world: Simple illustrations with functions and chaotic attractors, Chaos, Solitons and Fract., 114 (2018), 347-363.  doi: 10.1016/j.chaos.2018.07.022.  Google Scholar

[7]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), Art. 3, 21 pp. doi: 10.1051/mmnp/2018010.  Google Scholar

[8]

G. Bencskó and T. Ferenci, Effective case/infection ratio of poliomyelitis in vaccinated populations, Epidemiol Infect, 144 (2016), 1933-1942.   Google Scholar

[9]

C. J. BrowneR. J. Smith and L. Bourouiba, From regional pulse vaccination to global disease eradication: insights from a mathematical model of poliomyelitis, J. Math. Biol., 71 (2015), 215-253.  doi: 10.1007/s00285-014-0810-y.  Google Scholar

[10]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11. doi: 10.18576/pfda/020101.  Google Scholar

[11]

A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, Verlag, New York, 1997. Google Scholar

[12]

A. Dénes and L. Székely, Global dynamics of a mathematical model for the possible re-emergence of polio, Math. Biosci., 293 (2017), 64-74.  doi: 10.1016/j.mbs.2017.08.010.  Google Scholar

[13]

A. EsenAl aattinB. Karaagac and O. Tasbozan, Finite difference methods for fractional gas dynamics equation, Appl. Math. Inf. Sci., 4 (2016), 1-4.   Google Scholar

[14]

B. Karaagac, Analysis of the cable equation with non-local and non-singular kernel fractional derivative, Eur. Phys. J. Plus, 54 (2018), 133. doi: 10.1140/epjp/i2018-11916-1.  Google Scholar

[15]

B. Karaagac, A study on fractional Klein Gordon equation with non-local and non-singular kernel, Chaos, Solitons and Fract, 126 (2019), 218-229.  doi: 10.1016/j.chaos.2019.06.010.  Google Scholar

[16]

B. Karaagac, New exact solutions for some fractional order differential equations via improved sub-equation method, Discrete Cont Dyn -S, 12 (2019), 447-454.   Google Scholar

[17]

B. KaraagacY. UcarN. M. Yagmurlu and A. Esen, A New Perspective on The Numerical Solution for Fractional Klein Gordon Equation, Politeknik, 22 (2019), 443-451.  doi: 10.2339/politeknik.428986.  Google Scholar

[18]

M. M. Khader and K. M. Saad, A numerical study by using the Chebyshev collocation method for a problem of biological invasion: Fractional Fisher equation, Int J Biomath, 11 (2018), 1850099, 15pp. doi: 10.1142/S1793524518500997.  Google Scholar

[19]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, 2006.  Google Scholar

[20]

A. MedvinskyS. PetrovskiiI. TikhonovaH. Malchow and B. L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.  Google Scholar

[21]

V. F. Morales-DelgadoJ. F. Gómez-AguilarK. M.SaadM. A.Khan and P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach., Physica A Stat. Mech. Appl., 523 (2019), 48-65.  doi: 10.1016/j.physa.2019.02.018.  Google Scholar

[22]

V. F. Morales-DelgadoJ. F. Gómez-AguilarK. Saad and R. F. Escobar Jiménez, Application of the Caputo-Fabrizio and Atangana-Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect, Math Method Appl Sci., 42 (2019), 1167-1193.  doi: 10.1002/mma.5421.  Google Scholar

[23]

K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Math. Model. Nat. Phenom., 13 (2018), Art. 7, 17 pp. doi: 10.1051/mmnp/2018006.  Google Scholar

[24]

K. M. Owolabi and A. Atangana, Modelling and formation of spatiotemporal patterns of fractional predation system in subdiffusion and superdiffusion scenarios, Eur. Phys. J. Plus, 133 (2018), 43. doi: 10.1140/epjp/i2018-11886-2.  Google Scholar

[25]

K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus, 133 (2018), 15. doi: 10.1140/epjp/i2018-11863-9.  Google Scholar

[26]

K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, Eur. Phys. J. Plus, 133 (2018), 98. doi: 10.1140/epjp/i2018-11951-x.  Google Scholar

[27]

K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos, Solitons and Fract, 111 (2018), 119-127.  doi: 10.1016/j.chaos.2018.04.019.  Google Scholar

[28]

K. M. Owolabi and A. Atangana, Chaotic behaviour in system of noninteger-order ordinary differential equations, Chaos, Solitons and Fract, 115 (2018), 362-370.  doi: 10.1016/j.chaos.2018.07.034.  Google Scholar

[29]

K. M. Owolabi, Analysis and numerical simulation of multicomponent system with Atangana-Baleanu fractional derivative, Chaos, Solitons and Fract., 115 (2018), 127-134.  doi: 10.1016/j.chaos.2018.08.022.  Google Scholar

[30]

K. M. Owolabi, Numerical patterns in system of integer and non-integer order derivatives, Chaos, Solitons and Fract., 115 (2018), 143-153.  doi: 10.1016/j.chaos.2018.08.010.  Google Scholar

[31]

K. M. Owolabi, Numerical patterns in reaction-diffusion system with the Caputo and Atangana-Baleanu fractional derivatives, Chaos, Solitons and Fract., 115 (2018), 160-169.  doi: 10.1016/j.chaos.2018.08.025.  Google Scholar

[32]

K. M. Owolabi and J. F. Gómez-Aguilar, Numerical simulations of multilingual competition dynamics with nonlocal derivative, Chaos, Solitons and Fract., 117 (2018), 175-182.  doi: 10.1016/j.chaos.2018.10.020.  Google Scholar

[33]

K. M. Owolabi, Computational study of noninteger order system of predation, Chaos, 29 (2019), 013120, 14pp. doi: 10.1063/1.5079616.  Google Scholar

[34]

K. M. Owolabi and Z. Hammouch, Mathematical modeling and analysis of two-variable system with noninteger-order derivative, Chaos, 29 (2019), 013145, 15pp. doi: 10.1063/1.5086909.  Google Scholar

[35]

K. M. Owolabi and A. Atangana, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems, Chaos, 29 (2019), 023111, 12pp. doi: 10.1063/1.5085490.  Google Scholar

[36] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, CA, 1999.   Google Scholar
[37]

Polio Global Eradication Initiative, Circulating Vaccine-derived Poliovirus, 2016. Google Scholar

[38]

K. M. Saad, S. Deniz and D. Baleanu, On a new modified fractional analysis of Nagumo equation, Int J Biomath, 12 (2019), 1950034, 15pp. doi: 10.1142/S1793524519500347.  Google Scholar

[39]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[40]

K. M. Thompson and R. J. Duintjer Tebbens, Modeling the dynamics of oral poliovirus vaccine cessation, J Infect Dis, 210 (2014), S475-S484. doi: 10.1093/infdis/jit845.  Google Scholar

[41]

M. Toufik and A. Atangana, New numerical approximation of fractional derivative with nonlocal and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132 (2017), 444. Google Scholar

[42]

World Health Organization, Polio vaccines: WHO position paper - March, 2016, 2016. Google Scholar

Figure 1.  Fractional evolution of non-diffusive form of model (6) using initial condition (4) for different $ \alpha $ at time $ t = 10 $
Figure 2.  Numerical solution of (6) with $ \kappa = 0.5 $ for different $ \alpha $
Figure 3.  Effect of $ \alpha $ with $ \kappa = 0.05 $
Figure 4.  Behaviour of species with $ \alpha = 0.89 $ for different instances of $ \gamma $ at $ t = 20 $
Figure 5.  Effect of $ \beta = 0.01, 0.02, 0.03, 0.04 $ for $ \alpha = 0.90 $
Figure 6.  Diffusive evolution of fractional model (6) for $ \alpha = 0.88 $ and time $ t = 10 $
Figure 7.  Diffusive evolution of fractional model (6) for $ \alpha = 0.96 $ and time $ t = 10 $
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