br The model system br The model
2. The model system
The model is formulated on the basis of the model in . Let X1 and X2 be the concentrations of androgen-dependent cells and androgen-independent cells respectively, and let A be the concentration of androgen in the blood. To separate the influence of stochastic noise from immune response, we don’t include the state variables of immune cells and cytokines related to immunotherapy in . Inspired by Tanaka et al. , we incorporate the stochastic perturbations into the model of . Furthermore, we introduce the different competition intensities between AD cells and AI cells into the model because of the biological data from [16,48]. Let us focus our attention on the treatment of continuous androgen suppression (CAS). Then our model is described by
where a0 is the normal androgen concentration, γ is the androgen clearance and production rate, and u is the e cacy of CAS therapy. Unlike IAS therapy, we assume that u is a positive constant, which becomes reasonable if drugs are more frequently administrated or through an intravenous injection [19,20,37]. In addition, we assume < u < 1 because in practice, the LHRH agonist (drug) could reduce testosterone (resp. DHT) by 60% (resp. 95%) [10,21]. Note that the e cacy u of CAS therapy can be adjusted by the dosage.
All the parameters in system (2.1) are positive. The biological meanings of parameters in the last two equations of (2.1) are interpreted as follows. First, r1 and d1 denote the growth rate and death rate of AD cells respectively, r2 is the
net growth rate of AI cells, α and β are the positive competition coe cients between two types of tumor cells, and K is the Dorsomorphin of these cells. Second, it is assumed that the mutation rate of AD cells (resp. death of AD cells) is androgen-dependent and increases to its maximum m1u (resp. d1u) when the androgen reaches its minimum A∗ = a0 (1 − u), where m1 is the irreversible mutation rate from AD cells to AI cells. Finally, Bi(t), i = 1, 2 are independent Brownian motions; σ 1 and σ 2 denote the intensities of the white noises, respectively.
We point out the major differences between model (2.1) and the model of Rutter and Kuang . Model (2.1) includes the stochastic perturbations in tumor dynamics and allow for distinct competitive coe cients between sensitive tumor cells and resistant tumor cells, whereas immunotherapy is combined with continuous androgen suppression in Rutter and Kuang
 to determine whether tumor elimination is possible under a combination of these two treatments. If the mutation rate from AI cells to AD cells in  is ignored α = β = 1, and σ1 = σ2 = 0, then Rutter and Kuang’s model  without immunotherapy is reduced to system (2.1).
Note that the evolution of androgen dynamics is very fast compared to that of cancer cells and thus reaches the equi-librium point in a much shorter time than that of cancer cells. Thus, we let androgen go to the steady state A∗ = a0 (1 − u) and then system (2.1) is reduced to
In what follows, we focus the mathematical analysis and numerical simulations on the system (2.2). We will begin with the analysis for the system (2.2) without stochastic perturbation, which presents the complete classifications for global convergence of the model. Then we turn to the full model (2.2) to derive the threshold conditions for the persistence and extinction of tumor cells.
3. Analysis of deterministic model
As a preliminary analysis, in this section we consider model (2.2) in the absence of stochastic noise. Let σ1 = σ2 = in (2.2) to reduce the model to the following deterministic system:
Because this is a system of ordinary differential equations, the global convergence of its solutions can be achieved through qualitative analysis. Set
The results concerning the existence of equilibria and their stability are stated in the following theorems. The proofs of these results are given in Appendix A, Appendix B and Appendix C.