Modeling the Spread and Mitigation of Cybersecurity Threats using Non-Homogeneous Heat Equations with Boundary Conditions

This study presents a mathematical model using non-homogeneous heat equations with dynamic boundary conditions to simulate the spread and mitigation of cybersecurity threats. Our results demonstrate that incorporating adaptive boundary conditions, which represent real-time adjustments to defense mechanisms, improves the robustness of cybersecurity systems against evolving threats. Stability analysis using the Lyapunov criterion shows that the system remains stable over time, with the vulnerability (represented by the Lyapunov function) decreasing as time progresses. Numerical simulations indicate that varying the intensity of cyber-attacks and adjusting boundary conditions dynamically leads to more efficient and effective defense strategies, optimizing both computational load and system security. The mathematical model, validated through several test cases, reveals that dynamic adaptation of defense mechanisms significantly outperforms traditional static models, offering scalable solutions for real-time cyber defense. Furthermore, our results demonstrate the model's ability to simulate the impact of network topology changes and varying attack intensities, providing insights into how network configurations influence the propagation of vulnerabilities.