A Fractional Caputo Kolmogorov Differential Equation with Hawkes Birth Intensity: Application to Cancer Progression (Journal of Pure and Applied Sciences (Science Forum), 2026)
Cancer progression is inherently stochastic, driven by cumulative genetic and epigenetic alterations, temporally clustered mutation events, and long-range biological memory effects that violate the Markov property assumed in conventional models. Classical birth–death and diffusion frameworks therefore fail to capture the observed heterogeneity, burst dynamics, and history dependence of tumor evolution. In this study, a novel fractional stochastic formulation is developed based on the Caputo fractional Kolmogorov forward differential equation coupled with a Hawkes self-exciting birth intensity to describe cancer cell population dynamics. The Caputo fractional derivative of order ( ) 0 1   introduces nonlocal temporal memory and hereditary effects, enabling the system to retain information about past states, while the Hawkes process models mutation-induced proliferative bursts via endogenous selfexcitation and event clustering. Together, these components yield a non-Markovian birthdeath process with transition rates that depend explicitly on the entire evolutionary history. Analytical characteristics of the model are derived using Laplace transform methods and properties of fractional operators, establishing existence, stability conditions, and asymptotic behavior. Numerical simulations are conducted to explore sensitivity to key parameters, including the fractional order, excitation kernel, and baseline proliferation rate. The results reproduce biologically realistic tumor growth patterns, such as latency phases, sub-exponential expansion, heavy-tailed inter-event times, and intermittent episodes of rapid proliferation consistent with clonal sweeps. Comparative evaluation against classical Markovian and purely fractional models demonstrates superior flexibility, predictive capability, and mechanistic interpretability of the hybrid framework. This approach provides a rigorous mathematical platform for modeling tumor evolution and offers potential applications in quantitative oncology, including prognosis, treatment scheduling, and optimization of therapeutic interventions under uncertainty.