Stationary distribution and persistence of a stochastic mathematical model for prostate cancer with pulsed therapy

Abstract

Intermittent androgen deprivation therapy is one of the most commonly used therapeutic regimens for treating prostate cancer. Immunotherapy with dendritic cells, which act as the most robust antigen-presenting cells, are regarded as an effective method for the treatment of advanced prostate cancer. This paper utilizes impulsive differential equations to describe the combinations of a dendritic cell vaccine and intermittent androgen therapy with white noise. The tumour-free solution is obtained and the unique global positive solution of the system is explored. Then it is proved that the solutions of the system are stochastic ultimately bounded and stochastically permanent. In addition, threshold conditions for the extinction and persistence of prostate cancer cells are derived, and the stationary distribution and ergodicity of the system are investigated. Finally, numerical studies and the biological significance of the results are discussed.