Network Virus Propagation under Planar Cross-Diffusion: Spatiotemporal Pattern Analysis

An important security problem faced by information networks is virus transmission. The traditional ordinary differential equation models of virus propagation take less consideration of the impact of diffusion factors in computer networks. A few reaction-diffusion models proposed to simulate virus spreading only take into account self-diffusion in one-dimensional space. As far as we know, there is not any model considering the cross-diffusion in two-dimensional space. In this paper, a novel virus transmission model with cross-diffusion in two-dimensional space is put forward. The introduction of cross-diffusion terms leads to diverse pattern dynamics and Turing instability in virus transmission, aspects that have not been previously studied. The sufficient conditions of Hopf bifurcation and Turing instability are given. Then, the amplitude equation near the critical value of Turing instability is constructed to identify various space-time patterns, such as spotted pattern, coexistence pattern and striped pattern. Finally, the numerical simulations show the correctness of theoretical analysis.