Modeling Hybrid Crossover Dynamics of Immuno-Chemotherapy and Gene Therapy: A Numerical Approach

A piecewise fractional differential equation (deterministic–stochastic differential equation or vice versa) has appeared in recent literature. Piecewise operators are used to study crossover real data effectively. By using stochastic–deterministic piecewise hybrid fractional derivatives, with hybrid fractional-order and variable-order fractional operators, this paper extends the deterministic model of immuno-chemotherapy with gene therapy and time delay. The immuno-chemotherapy with gene therapy combines traditional chemotherapy and immunotherapy with genetic engineering techniques to enhance the immune system's ability to target and destroy cancer cells. This offers a multifaceted approach to cancer treatment, potentially enhancing effectiveness while minimizing side effects. Two approximation techniques are used to solve the proposed model numerically. In the hybrid fractional deterministic models, we use the nonstandard Caputo proportional constant finite difference method, and in the stochastic models, we use the nonstandard Milstein technique. We examine the stability analysis of these methods to ensure their efficiency. Both the theoretical results and the efficiency of the methods are confirmed by numerical tests. New piecewise operators are illustrated by the curves presented. Studying the introduced piecewise fractional differential immuno-chemotherapy model with gene therapy and time delay along with a stochastic case greatly explained the dynamics of immuno-chemotherapy interaction.