A Stochastic Optimal Control Model for BCG Immunotherapy in Superficial Bladder Cancer

¹ LERMA, Mohammadia School of Engineering, Mohammed V University in Rabat B.P 765, Agdal, Rabat, Morocco
² SAEDD Laboratory, ESTE, Cadi Ayyad University, Route d'Agadir, Essaouira Aljadida, Morocco
³ MTI unit, ENSAS, Cadi Ayyad university, Route Sidi Bouzid, Safi, Morocco

^* Corresponding author. E-mail: aboulaich@emi.ac.ma

Abstract

Urologic studies have reported in several papers that the optimal dose of Bacillus Calmette-Guérin (BCG) immunotherapy in superficial bladder cancer, is still a subject of research. Our main goal from this paper, is to find treatment regimens that minimize the total number of tumors in the presence of a diffusion process. For this, we devise a stochastic model in the form of a nonlinear system of four stochastic differential equations (SDEs) that describe tumor-immune dynamics after BCG instillations. Therefore, we study the existence and the stability results. Then, we introduce a control function in the mathematical model, to represent the dose of BCG intravesical therapy, and we seek its optimal values through the application of a stochastic version of Pontryagin's maximum principle. Finally, we present some numerical simulations using iterative stochastic Runge-Kutta progressive-regressive schemes which we propose for solving the optimality system of the obtained stochastic two-point boundary value problem.