Discretization and bifurcation analysis of tumor immune interaction in fractional form

Muhammad Taj , Mohamed A. Khadimallah , Muzamal Hussain , Yahya Rashid , Waqas Ishaque , S.R. Mahmoud , Qamar Din , Afaf S. Alwabli , Abdelouahed Tounsi

Abstract

A tumor immune interaction is a main topic of interest in the last couple of decades because majority of human population suffered by tumor, formed by the abnormal growth of cells and is continuously interacted with the immune system. Because of its wide range of applications, many researchers have modeled this tumor immune interaction in the form of ordinary, delay and fractional order differential equations as the majority of biological models have a long range temporal memory. So in the present work, tumor immune interaction in fractional form provides an excellent tool for the description of memory and hereditary properties of inter and intra cells. So the interaction between effector-cells, tumor cells and interleukin-2 (IL-2) are modeled by using the definition of Caputo fractional order derivative that provides the system with long-time memory and gives extra degree of freedom. Moreover, in order to achieve more efficient computational results of fractional-order system, a discretization process is performed to obtain its discrete counterpart. Furthermore, existence and local stability of fixed points are investigated for discrete model. Moreover, it is proved that two types of bifurcations such as Neimark-Sacker and flip bifurcations are studied. Finally, numerical examples are presented to support our analytical results.

Key Words

tumor immune interaction; stability; Neimark-Sacker bifurcation; period-doubling bifurcation

Address

(1) Muhammad Taj, Waqas Ishaque — Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, 1300, Azad Kashmir, Pakistan
(2) Mohamed A. Khadimallah, Yahya Rashid — Prince Sattam Bin Abdulaziz University, College of Engineering, Civil Engineering Department, BP 655, Al-Kharj, 16273, Saudi Arabia
(3) Mohamed A. Khadimallah — Laboratory of Systems and Applied Mechanics, Polytechnic School of Tunisia, University of Carthage, Tunis, Tunisia
(4) Muzamal Hussain — Department of Mathematics, Govt. College University Faisalabad, 38000, Faisalabad, Pakistan
(5) Waqas Ishaque, Qamar Din — Department of Mathematics University of Poonch Rawalakot 12350, Azad Kashmir, Pakistan
(6) S.R. Mahmoud — GRC Department, Faculty of Applied studies, King Abdulaziz University, Jeddah, Saudi Arabia
(7) Afaf S. Alwabli — Department of Biological Sciences, Rabigh-Faculty of Science & Arts, King Abdulaziz University, Jeddah, Saudi Arabia
(8) Abdelouahed Tounsi — YFL (Yonsei Frontier Lab), Yonsei University, Seoul, Korea
(9) Abdelouahed Tounsi — Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals, 31261 Dhahran, Eastern Province, Saudi Arabia.

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