Math. Model. Nat. Phenom.
Volume 12, Number 5, 2017Mathematical models in physiology
|Page(s)||99 - 119|
|Published online||13 October 2017|
A Stochastic Optimal Control Model for BCG Immunotherapy in Superficial Bladder Cancer
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: firstname.lastname@example.org
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.
Mathematics Subject Classification: 92C50 / 49K45 / 60H10 / 93E15 / 93E20
Key words: BCG immunotherapy / Stochastic Stability / Stochastic maximum principle / Bladder cancer / Optimal control
© EDP Sciences, 2017
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