A study on fractional tumour–immune–vitamins model for intervention of vitamins

Abstract

Cancer is a very challenging issue in terms of people’s health and lives around the world. Cancer refers to a collection of illnesses distinguished by the formation of tumour cells, malignant cells, or cancer cells that have the potential to become cancerous. Millions of people across the globe are suffering from this illness. Several studies on cancer and related diseases have been made in recent years. The main goal of this article is to computationally examine the tumour–immune–vitamins (TIV) system dynamics, which incorporates the effect of vitamin intervention on enhancing the immune system and its aspects on squeezing and delaying tumour cell production and division. We analyse the model with different fractional approaches to learn more about its dynamics. Initially, the TIV model is proposed with Caputo–Fabrizio (CF) derivative. We investigate the existence and uniqueness of the solutions of the suggested fractional model using fixed-point results. Furthermore, we modify the TIV system with conformable and β-conformable fractional operators in the Liouville–Caputo (LC) sense. For numerical solution, we present the two-step Adams–Bashforth numerical approach for the CF operator and the Adams–Moulton numerical method for fractional conformable operators in the LC sense. Lastly, numerical outcomes are demonstrated graphically for various choices of the fractional-order parameters.

Keywords

• CF derivative
• Fractional conformable and β-conformable derivatives
• Existence and uniqueness
• Numerical simulation