# American Institute of Mathematical Sciences

## System response of an alcoholism model under the effect of immigration via non-singular kernel derivative

 1 Department of Mathematics, Balıkesir University, Balıkesir, 10145, Turkey 2 Department of Mathematics, Faculty of Sciences and Techniques, Moulay Ismail University, Errachidia, 52000, Morocco

* Corresponding author: Fırat Evirgen

Received  April 2019 Revised  July 2019 Published  November 2019

In this study, we aim to comprehensively investigate a drinking model connected to immigration in terms of Atangana-Baleanu derivative in Caputo type. To do this, we firstly extend the model describing drinking model by changing the derivative with time fractional derivative having Mittag-Leffler kernel. The existence and uniqueness of the drinking model solutions together with the stability analysis is shown by the help of Banach fixed point theorem. The special solution of the model is investigated using the Sumudu transformation and then, we present some numerical simulations for the different fractional orders to emphasize the effectiveness of the used derivative.

Citation: Fırat Evirgen, Sümeyra Uçar, Necati Özdemir, Zakia Hammouch. System response of an alcoholism model under the effect of immigration via non-singular kernel derivative. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020145
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##### References:
System behavior of the fractional drinking model (\protect5) with order $\eta = 0.3$ in respect to time $t = 1$ and $t = 40$
System behavior of the fractional drinking model (\protect5) with order $\eta = 0.5$ in respect to time $t = 1$ and $t = 40$
System behavior of the fractional drinking model (\protect5) with order $\eta = 0.7$ in respect to time $t = 1$ and $t = 40$
System behavior of the fractional drinking model (\protect5) with order $\eta = 0.9$ in respect to time $t = 1$ and $t = 40$
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