Analyzing Cyber Threat Propagation in Cloud Environments Using Green's Function and Laplace Transforms

This paper presents a mathematically grounded model for malware propagation in cloud-based virtualized infrastructures using an extended susceptible-infected-recovered (SIR) framework with spatial diffusion and Green's function convolution. The resulting reaction-diffusion PDE system was solved using a semi-implicit Crank-Nicolson scheme optimized via control parameter tuning. Evaluation against the CICIDS2017 and CTU-13 datasets demonstrates strong alignment with empirical attack dynamics, achieving a classification accuracy of 92.3 %, precision of $\mathbf{9 0. 7 \%}$, recall of $\mathbf{9 3. 6 \%}$, and an F1-score of $\mathbf{9 2. 1 \%}$. The model predicts infection peaks with less than 6 % deviation ($I_{\text {max }}=0.82$ vs. 0.87), a timing error under 2% ($t_{\text {peak }}=11.0 ~\mathrm{s}$ vs. 11.2s), and captures overall infection duration within 0.5 seconds of ground truth. Through iterative defence parameter optimization, the total infection burden $\mathscr{J}$ was reduced by $\mathbf{4 5. 2 \%}$, thereby validating the model's capacity for real-time threat forecasting and dynamic policy adaptation in cloud security systems.