Conservative Numerical Framework for Fractional Stochastic Delay Modeling of CSF-1/EGF Tumor-Macrophage Aggregation.

Tumor-macrophage aggregation mediated by the CSF-1/EGF paracrine loop involves spatial chemotaxis together with potentially history-dependent, delayed, and fluctuating cellular responses. This study asks how those three temporal mechanisms alter the classical aggregation model when they are represented in a common conservative numerical framework. The continuum system is extended by Caputo temporal derivatives, a discrete delay in the CSF-1 and EGF production terms, and component-wise multiplicative perturbations. Diffusive and chemotactic fluxes are discretized in conservative form, the Caputo derivative is approximated by convolution weights, delayed states are obtained from a history buffer with interpolation, and stochastic increments are interpreted in the Itô sense. The deterministic integer-order, zero-delay, and zero-noise limit is used as the reference configuration. The displayed parameter studies indicate that changes in the fractional order and delay can reorganize aggregation and signaling activity, while the stochastic comparison illustrates a sample-path-dependent modification of the spatial state. The deterministic no-flux discretization preserves total cell mass, and the deterministic consistency contributions are identified under smoothness assumptions. Because the memory, delay, and noise parameters are not fitted to experimental data and the stochastic figures do not represent ensemble averages, the results are interpreted as model-based sensitivity evidence rather than quantitative biomedical predictions.