Analysis and dynamical behavior of fractional-order cancer model with vaccine strategy

In recent year, the world has witnessed the arrival of deadly diseases like cancer over all the global levels. To fight back this disease or control the spread, mankind relies on modeling and medicine to control, cure, and behavior of the cancer diseases. We developed the fractional-order immunotherapy bladder cancer model and used the BCG vaccine for treatment by using the Caputo fractional derivative operator φɛ(0,1] . A mathematical model has four variables B,E, T_i, T_u which represent the vaccine for the immune system, effector cells, total population of affected, and unaffected cells, respectively. In this model, we have two cases according to the growth rate of cells. The fractional-order system is stable in both cases and gives the solution infeasible region for uniqueness, positivity, and boundedness to illustrate the treatment of cancer. The effect of fractional parameter on our obtained solutions is presented, and a comparison is made with the classical ordinary derivative operator. It is worthy to observe that fractional derivatives show significant changes and memory effects as compared with ordinary derivatives to control the disease at the initial stage to overcome the risk of living with cancer.