# American Institute of Mathematical Sciences

• Previous Article
New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques
• DCDS-S Home
• This Issue
• Next Article
Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions

## 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 & Continuous Dynamical Systems - S, 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. Baskonus, T. Mekkaoui, H. 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. Bhrawy, S. S. Ezz-Eldien, E. H. Abdelkawy, M. 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. Bonyah, K. O. Okosun, O. 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. Paparao, V. 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. Singh, D. Kumar, M. 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. TAlkahtani, I. 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. Baskonus, T. Mekkaoui, H. 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. Bhrawy, S. S. Ezz-Eldien, E. H. Abdelkawy, M. 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. Bonyah, K. O. Okosun, O. 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. Paparao, V. 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. Singh, D. Kumar, M. 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. TAlkahtani, I. 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.
Approximate solution for $\alpha = 0.3$
Approximate solution for $\alpha = 0.5$
Approximate solution for $\alpha = 0.65$
Approximate solution for $\alpha = 0.75$
Approximate solution for $\alpha = 0.95$
 [1] Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 867-880. doi: 10.3934/dcdss.2020050 [2] Francesco Mainardi. On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2267-2278. doi: 10.3934/dcdsb.2014.19.2267 [3] Mehmet Yavuz, Necati Özdemir. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 995-1006. doi: 10.3934/dcdss.2020058 [4] Jean Daniel Djida, Juan J. Nieto, Iván Area. Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 609-627. doi: 10.3934/dcdss.2020033 [5] Antonio Coronel-Escamilla, José Francisco Gómez-Aguilar. A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 561-574. doi: 10.3934/dcdss.2020031 [6] Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 [7] Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775 [8] Genady Ya. Grabarnik, Misha Guysinsky. Livšic theorem for banach rings. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4379-4390. doi: 10.3934/dcds.2017187 [9] Hari Bercovici, Viorel Niţică. A Banach algebra version of the Livsic theorem. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 523-534. doi: 10.3934/dcds.1998.4.523 [10] Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297 [11] Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237 [12] Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017 [13] Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709 [14] Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979 [15] Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381 [16] Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377 [17] Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621 [18] Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure & Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645 [19] Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467 [20] Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248

2017 Impact Factor: 0.561

## Tools

Article outline

Figures and Tables