## 1 Introduction  
Chronic myeloid leukemia (CML) is a hematologic cancer that accounts for about 15% of all leukemias in adults and is characterized by uncontrolled expansion of myeloid cells in the bone marrow and their accumulation in the blood [1]. The progression of the disease can be divided into three phases denoted _chronic, accelerated_ and _blast_[1]. The chronic phase can last several years with levels of immature white blood cells (blasts) growing steadily but at a low rate. Once the disease enters the accelerated or blast phase, cells proliferate rapidly and the disease can be lethal within a few months if not treated. Current standard of care includes targeted tyrosine kinase inhibitors (TKIs), which have significantly improved long-term survival rates [2].  
Responses to certain treatments have offered evidence of an immune component in the disease [3]. Early indications were provided by a correlation between incidence of graft-vs-host disease and improved leukemia-free survival in CML patients who had received allogeneic stem cell transplants [4]. Additionally, treatment with interferons (which are known to be immunomodulatory) has led to complete or partial responses in some fraction of CML patients [5]. More recently, studies that include immunomodulatory therapies such as nivolumab have been initiated [6].  
Mathematical modeling of CML dynamics has a history dating back to the late 1960s with early work of [PERSON] and [PERSON] [7, 8]. Models by [PERSON] et al. [9] in the 1990s focused on maturation and proliferation of T-cell precursors. In 2004, [PERSON] and [PERSON] [10] published a model of CML dynamics, which accounts for the actions of naive and effector T-cells separately. In [11], this model was analyzed as an optimal control problem. The model presented here first appeared in [12] and models the immune system effects with one compartment, and separates the CML cells into quiescent and proliferating classes. The rationale behind this new model is the ability to represent certain types of therapies for use in combination treatment. These therapies are: a BCR-ABL1 tyrosine kinase inhibitor (e.g., a therapy such as imatinib), an immunomodulatory therapy (e.g., a therapy such as nivolumab), and a therapy that combines both actions (e.g., a therapy such as dasatinib).  
The model introduced in [12] is reviewed in Section 2 and then analyzed as a dynamical system with constant drug concentrations in Section 3. The analysis is carried out theoretically for values of parameters covering a range of dynamic possibilities. As will be seen, there are parameter values for which the model can have an asymptotically stable equilibrium point in which all the state variables are positive. This could be interpreted as disease control through continuous therapy. As parameters change, the system can become unstable and undergo exponential growth, representing the accelerated or blast phases of the disease. Our analysis incorporates constant drug concentrations, and thus provide insights into the dynamics both without and with treatment. In particular, we analyze how an increase in the levels of each of the three treatments affects the values of all three populations, the two types of leukemia cells and the strength of the immune effect. The combination of theoretical analysis and simulations is intended to shed some light on understanding the long-term dynamics of this disease under treatment.