On a mathematical model of the influence of memory on cancer treatment using gene therapy

Cancer is one of the diseases that causes the most deaths in the world. Gene therapy is a recent alternative technique that has shown good clinical results in the treatment of gene changes and mutations caused by cancer. In this work, we will study the effects of gene therapy treatment by adding the hypothesis of the existence of immunological memory to the dynamics of cancer propagation through the introduction of multi-fractional order derivatives in the mathematical modeling for the problem. From a theoretical point of view, we will prove that the problem is well-posed in Hadamard's sense. We present numerical simulations for different scenarios, which demonstrate evidence that patients' previous contact with the virus involved in gene therapy can harm the effectiveness of the treatment.