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March  2020, 13(3): 519-537. doi: 10.3934/dcdss.2020029

Analysis of a Lymphatic filariasis-schistosomiasis coinfection with public health dynamics: Model obtained through Mittag-Leffler function

1. 

Department of Mathematics Education Kumasi Campus, University of Education Winneba, Ghana, Kumasi Ashanti Region, Box 1277, Ghana

2. 

Department of Mathematics Education, University of Education Winneba, Winneba, Central region, Box 25, Ghana

* Corresponding author: ebbonya@gmail.com

Received  April 2018 Revised  June 2018 Published  March 2019

In this paper, Lymphatic filariasis-schistosomiasis coinfected model is studied within the context of fractional derivative order based on Mittag-Leffler function of ABC in the Caputo sense. The existence and uniqueness of system model solution is derived by employing a well- known Banach fixed point theorem. The numerical solution based on the Mittag-Leffler function suggests that the dynamics of the coinfected model is well explored using fractional derivative order because of non-singularity.

Citation: Ebenezer Bonyah, Samuel Kwesi Asiedu. Analysis of a Lymphatic filariasis-schistosomiasis coinfection with public health dynamics: Model obtained through Mittag-Leffler function. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 519-537. doi: 10.3934/dcdss.2020029
References:
[1]

A. Atangana and I. Koca, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlin. Sci. Appl., 9 (2016), 2467-2480.  doi: 10.22436/jnsa.009.05.46.

[2]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 89 (2016), 763-769. 

[3]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fract., 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.

[4]

A. Atangana and J. F. Gomez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus., 133 (2018), 166.  doi: 10.1140/epjp/i2018-12021-3.

[5]

H. M. BaskonusT. MekkaouiH. Hammouch and H. Bulut, Active control of a Chaotic fractional order economic system, Abstr. Appl. Anal., 17 (2015), 5771-5783.  doi: 10.3390/e17085771.

[6]

A. H. BhrawyS. S. Ezz-EldienE. H. AbdelkawyM. A. Doha and D. Baleanu, Solving fractional optimal control problems within a Chebyshev- Legendre operational technique, Int. J. Cont., 90 (2017), 1230-1244.  doi: 10.1080/00207179.2016.1278267.

[7]

E. BonyahK. O. OkosunO. O. Okosun and L. Ossei, Mathematical modeling of Lymphatic filariasis-schistosomiasis co-infection dynamics:Insight through public education, Int. Jour. Eco. Devel., 33 (2017). 

[8]

H. Bulut, H. M. Baskonus and F. B. M. Belgacem, The analytical solutions of some fractional ordinary differential equations by Sumudu transform method, Abstr. Appl. Anal., 2013 (2013), Art. ID 203875, 6 pp. doi: 10.1155/2013/203875.

[9]

K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, Spr. Plus., 5 (2016), 1643.  doi: 10.1186/s40064-016-3295-x.

[10]

K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo abrizio derivative in Riemann Liouville sense, Spr. Plus., 99 (2017), 171-179.  doi: 10.1016/j.chaos.2017.04.008.

[11]

K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Spr. Plus., 103 (2017), 544-554.  doi: 10.1016/j.chaos.2017.07.013.

[12]

N. Ozalp and I. Koca, A fractional order nonlinear dynamical model of interpersonal relationships, Adv. Diff. Equ., 189 (2012), 1-7.  doi: 10.1186/1687-1847-2012-189.

[13]

A. Paparao and K. L. Narayan, Solving fractional optimal control problems within a Chebyshev- Legendre operational technique, Int. J. Cont., 32 (2017), 75-86. 

[14]

A. V. PaparaoV. S. Kalesha and A. Paparao, Dynamics of directly transmitted viral micro parasite model, Int. J. Ecol. Devel., 32 (2017), 88-97. 

[15]

C. M. A. Pinto and A. R. M. Carvalho, New findings on the dynamics of HIV and TB coinfection models, Appl. math. comp., 242 (2014), 36-46.  doi: 10.1016/j.amc.2014.05.061.

[16]

J. SinghD. KumarM. A. Qurashi and D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Diff. Equ., 88 (2017), 1-16.  doi: 10.1186/s13662-017-1139-9.

[17]

B. S. TAlkahtaniI. Koca and A. Atangana, Analysis of a new model of H1N1 spread: Model obtained via Mittag-Leffler function, Adv. Mech. Eng., 9 (2017), 1-8. 

show all references

References:
[1]

A. Atangana and I. Koca, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlin. Sci. Appl., 9 (2016), 2467-2480.  doi: 10.22436/jnsa.009.05.46.

[2]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 89 (2016), 763-769. 

[3]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fract., 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.

[4]

A. Atangana and J. F. Gomez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus., 133 (2018), 166.  doi: 10.1140/epjp/i2018-12021-3.

[5]

H. M. BaskonusT. MekkaouiH. Hammouch and H. Bulut, Active control of a Chaotic fractional order economic system, Abstr. Appl. Anal., 17 (2015), 5771-5783.  doi: 10.3390/e17085771.

[6]

A. H. BhrawyS. S. Ezz-EldienE. H. AbdelkawyM. A. Doha and D. Baleanu, Solving fractional optimal control problems within a Chebyshev- Legendre operational technique, Int. J. Cont., 90 (2017), 1230-1244.  doi: 10.1080/00207179.2016.1278267.

[7]

E. BonyahK. O. OkosunO. O. Okosun and L. Ossei, Mathematical modeling of Lymphatic filariasis-schistosomiasis co-infection dynamics:Insight through public education, Int. Jour. Eco. Devel., 33 (2017). 

[8]

H. Bulut, H. M. Baskonus and F. B. M. Belgacem, The analytical solutions of some fractional ordinary differential equations by Sumudu transform method, Abstr. Appl. Anal., 2013 (2013), Art. ID 203875, 6 pp. doi: 10.1155/2013/203875.

[9]

K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, Spr. Plus., 5 (2016), 1643.  doi: 10.1186/s40064-016-3295-x.

[10]

K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo abrizio derivative in Riemann Liouville sense, Spr. Plus., 99 (2017), 171-179.  doi: 10.1016/j.chaos.2017.04.008.

[11]

K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Spr. Plus., 103 (2017), 544-554.  doi: 10.1016/j.chaos.2017.07.013.

[12]

N. Ozalp and I. Koca, A fractional order nonlinear dynamical model of interpersonal relationships, Adv. Diff. Equ., 189 (2012), 1-7.  doi: 10.1186/1687-1847-2012-189.

[13]

A. Paparao and K. L. Narayan, Solving fractional optimal control problems within a Chebyshev- Legendre operational technique, Int. J. Cont., 32 (2017), 75-86. 

[14]

A. V. PaparaoV. S. Kalesha and A. Paparao, Dynamics of directly transmitted viral micro parasite model, Int. J. Ecol. Devel., 32 (2017), 88-97. 

[15]

C. M. A. Pinto and A. R. M. Carvalho, New findings on the dynamics of HIV and TB coinfection models, Appl. math. comp., 242 (2014), 36-46.  doi: 10.1016/j.amc.2014.05.061.

[16]

J. SinghD. KumarM. A. Qurashi and D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Diff. Equ., 88 (2017), 1-16.  doi: 10.1186/s13662-017-1139-9.

[17]

B. S. TAlkahtaniI. Koca and A. Atangana, Analysis of a new model of H1N1 spread: Model obtained via Mittag-Leffler function, Adv. Mech. Eng., 9 (2017), 1-8. 

Figure 1.  Approximate solution for $ \alpha = 0.3 $
Figure 2.  Approximate solution for $\alpha = 0.5$
Figure 3.  Approximate solution for $\alpha = 0.65$
Figure 4.  Approximate solution for $\alpha = 0.75$
Figure 5.  Approximate solution for $ \alpha = 0.95 $
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