Nonlinear dynamics and stability of a delayed leukemia model with real-world applications.

Leukemia is a complex hematological disorder whose progression involves interactions between healthy, infected, and immune cell populations. Time delays naturally arise in leukemic dynamics due to diagnostic lag, immune activation, and treatment response. Motivated by these biological considerations, a delayed model of leukemia is formulated and analyzed using a system of delayed differential equations. In the model, the population is divided into three compartments, namely, susceptible, infected, and recovered cells, with a nonlinear incidence term that includes natural and artificial delay effects. The positivity, boundedness, and uniqueness of the solutions are established and explicit expressions are derived for the leukemia-free and leukemia-existing equilibria. The basic reproduction number [Formula: see text] is computed using the next-generation matrix method, and local and global stability are investigated using the Routh–Hurwitz criterion and the Lyapunov function techniques. Sensitivity analysis identifies the key parameters that influence [Formula: see text], focusing on delayed and treatment-related terms as effective control mechanisms. The results suggest that increasing the intrinsic biological delay (e.g. cell-cycle progression or treatment-response latency) reduces the effective proliferation of leukemic cells and may shift the system toward a leukemia-free equilibrium. This reflects mechanistic regulation within the model rather than a direct clinical strategy. The study applies its findings to real-world data by analyzing leukemia incidence records from Portugal which show accurate results that match the projected results of the model. The study results demonstrate how essential it is to include delay patterns when creating leukemia models because this approach helps develop better treatment methods.