(untitled) (European Journal of Pure and Applied Mathematics, 2026)
This paper investigates the dynamical behavior of a discrete-time bladder cancer–immune–virotherapy model. The model describes the interactions among uninfected tumor cells, infected tumor cells, free virus particles, tumor-specific immune cells, and virus-specific immune cells. The discrete formulation is obtained via Euler discretization of a biologically motivated continuous system, allowing the exploration of dynamical effects inherent to temporal discretization. We establish the existence and local stability conditions of biologically relevant equilibrium points and determine the basic reproduction number governing viral invasion. Furthermore, we analyze the occurrence of flip (period-doubling), fold, and Neimark–Sacker (N–S) bifurcations to characterize qualitative transitions in system dynamics. The results reveal distinct dynamical regimes induced by variations in the viral infection rate and time step size, including stable equilibria, sustained oscillations, and more complex discrete behaviors. Numerical simulations support the analytical findings and illustrate how parameter changes influence tumor–virus–immune interactions. The study highlights the sensitivity of discrete tumor virotherapy models to parameter variation and emphasizes the role of discrete-time dynamics in shaping treatment-related outcomes.