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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{9 0. 7 \%}$</tex>, recall of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{9 3. 6 \%}$</tex>, and an F1-score of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{9 2. 1 \%}$</tex>. The model predicts infection peaks with less than 6 % deviation (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$I_{\text {max }}=0.82$</tex> vs. 0.87), a timing error under 2% (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t_{\text {peak }}=11.0 ~\mathrm{s}$</tex> vs. 11.2s), and captures overall infection duration within 0.5 seconds of ground truth. Through iterative defence parameter optimization, the total infection burden <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathscr{J}$</tex> was reduced by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{4 5. 2 \%}$</tex>, thereby validating the model's capacity for real-time threat forecasting and dynamic policy adaptation in cloud security systems.