# American Institute of Mathematical Sciences

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