DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020120 DYNAMICAL SYSTEMS SERIES S A FRACTIONAL-ORDER DELAY DIFFERENTIAL MODEL WITH OPTIMAL CONTROL FOR CANCER TREATMENT BASED ON SYNERGY BETWEEN ANTI-ANGIOGENIC AND IMMUNE CELL THERAPIES Nasser Sweilam * Department of Mathematics, Faculty of Science Cairo University, Egypt Fathalla Rihan Department of Mathematical Sciences United Arab Emirates University Al-Ain, 15551, UAE Seham AL-Mekhlafi Department of Mathematics, Faculty of Education Sana’a University, Yemen Abstract. In this paper, we present an optimal control problem of fractional- order delay-differential model for cancer treatment based on the synergy be- tween anti-angiogenic and immune cells therapies. The governed model consists of eighteen differential equations. A discrete time-delay is incorporated to rep- resent the time required for the immune system to interact with the cancer cells, and fractional-order derivative is considered to reflect the memory and hereditary properties in the process. Two control variables for immunotherapy and anti-angiogenic therapy are considered to reduce the load of cancer cells. Necessary conditions that guarantee the existence and the uniqueness of the solution for the control problem have been considered. We approximate numer- ically the solution of the optimal control problem by solving the state system forward and adjoint system backward in time. Some numerical simulations are provided to validate the theoretical results. 1. Introduction. Nowadays, mathematical models can be considered as a suc- cessfully powerful tool to test hypotheses, confirm experiments, and simulate the dynamics of complex systems. Moreover, it has become an essential tool that is used to simulate tumor-immune cell interactions and to predict the therapeutic effect of various cancer treatments ( [2], [13], [15], [19], [23], [30], [39], [45]). Mathematical models, based on ordinary differential equations, delay differential equations, partial differential equations, have proven to be useful tools in analysing and understanding the interactions with viral, bacterial infections and cancerous cells [31]-[33]. It is well-known that Cancer can be considered as one of the biggest killer of humans over all the world ([17], [18], [25] ). In spite of recent advancements in 2010 Mathematics Subject Classification. 92C50, 93A30, 37N25, 37N35, 34K28. Key words and phrases. Anti-angiogenic therapy, hamiltonian, cancer treatment, fractional- calculus, delay differential equations, immunotherapy, optimal control. * Corresponding author: Nasser Sweilam. 1