Chaos, Solitons and Fractals 133 (2020) 109619 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative Behzad Ghanbari a,b , Sunil Kumar c,∗ , Ranbir Kumar c a Department of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran b Department of Mathematics, Faculty of Engineering and Natural Sciences,Bahçe ¸ sehir University, Istanbul 34349, Turkey c Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand 831014, India a r t i c l e i n f o Article history: Received 10 December 2019 Revised 2 January 2020 Accepted 8 January 2020 Keywords: Modelling Fractional Immunogenetic tumours model Immune cells Non-singular kernel Tumor cells Atangana - Baleanu (AB) derivative Adam Bashforth’s Moulton method a b s t r a c t Mathematical biology is one of the interesting research area of applied mathematics that describes the accurate description of phenomena in biology and related health issues. The use of new mathematical tools and definitions in this area of research will have a great impact on improving community health by controlling some diseases. This is the best reason for doing new research using the latest tools available to us. In this work, we will make novel numerical approaches to the immunogenetic tumour model to using differential and integral operators with Mittag-Leffler law. To be more precise, the fractional Atangana- Baleanu derivative has been utilized in the structure of proposed model. This paper proceeds by examin- ing and proving the convergence and uniqueness of the solution of these equations. The Adam Bashforth’s Moulton method will then be used to solve proposed fractional immunogenetic tumour model. Numeri- cal simulations for the model are obtained to verify the applicability and computational efficiency of the considered process. Similar models in this field can also be explored similarly to what has been done in this article. © 2020 Elsevier Ltd. All rights reserved. 1. Introduction Nowadays, mathematical modeling on infectious disease model of biological science using the fractional-order system of differ- ential equations has gained much attention over the past few years. The biology is the fundamental in the most literal sense of the word such as about how things live, breathe, and die. How- ever, several disease models like fractional immunogenetic tumour model of biological and engineering science have been successful being formulated and analysed the behaviours of its solution due to exciting nature of fractional derivatives and integrals. The non- integer derivatives have important characteristics called memory effect, and such exclusive property does not exist in the classical derivatives. These derivatives are nonlocal opposed to the local be- haviour of integer derivatives. The tumor is the most dangerous life-threatening disease for the human body, which is a mass or lump of tissue that may resemble swelling or morbid enlargement. Differently, a tumor is an abnormal growth of cells that serves no purpose. Typically, the cells in our body grow, die, replaced and di- vide to produce new cells in a controlled and orderly manner, but ∗ Corresponding author. E-mail addresses: b.ghanbary@yahoo.com (B. Ghanbari), skumar.math@nitjsr. ac.in (S. Kumar). sometimes cells reproduce, grow, and division too much quickly, then tumor can develop. Mainly, tumors are divided into three cat- egories benign, premalignant, and malignant. The benign tumors are not cancerous and the cells of premalignant tumors are not yet cancerous, but they have the potential to become malignant. While the malignant are most cancerous and dangerous tumors for the human body. The cells can grow and spread into other parts of the body [1–7]. A mathematical model is a description of a system using math- ematical concepts and language which may help to explain a sys- tem and to study the effects of different components, and to make predictions about behaviour. The process of developing mathemat- ical models is named as mathematical modeling which has impor- tance in the field of natural sciences and engineering disciplines as well as in the social sciences. Several mathematical models of disease are presented through a system of nonlinear ordinary dif- ferential equations without time delay. Let us consider a mathe- matical model to describe the interactions between immune and tumor cells as [8–10]:  dx(t ) dt =  x (x) +  x (x, y) −  x (x, y) −  x (x), t ≥ 0, dy(t ) dt = yφ (y) −  y (x, y), t ≥ 0, (1.1) where x(t) indicates the density of immune cells and y(t) repre- sents the density of tumor cells. https://doi.org/10.1016/j.chaos.2020.109619 0960-0779/© 2020 Elsevier Ltd. All rights reserved.