A Fractional Caputo Kolmogorov Differential Equation with Hawkes Birth Intensity: Application to Cancer Progression

Abstract

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 self
excitation and event clustering. Together, these components yield a non-Markovian birth
death 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.