A Delayed Malware Propagation Model Under a Distributed Patching Mechanism: Stability Analysis

Antivirus (patch) is one of the most powerful tools for defending against malware spread. Distributed patching is superior to its centralized counterpart in terms of significantly lower bandwidth requirement. Under the distributed patching mechanism, a novel malware propagation model with double delays and double saturation effects is proposed. The basic properties of the model are discussed. A pair of thresholds, i.e., the first threshold R0 and the second threshold R1, are determined. It is shown that (a) the model admits no malware-endemic equilibrium if R0≤1, (b) the model admits a unique patch-free malware-endemic equilibrium and admits no patch-endemic malware-endemic equilibrium if 1<R0≤R1, and (c) the model admits a unique patch-free malware-endemic equilibrium and a unique patch-endemic malware-endemic equilibrium if R0>R1. A criterion for the global asymptotic stability of the malware-free equilibrium is given. A pair of criteria for the local asymptotic stability of the patch-free malware-endemic equilibrium are presented. A pair of criteria for the local asymptotic stability of the patch-endemic malware-endemic equilibrium are derived. Using cybersecurity terms, these theoretical outcomes have the following explanations: (a) In the case where the first threshold can be kept below unity, the malware can be eradicated through distributed patching. (b) In the case where the first threshold can only be kept between unity and the second threshold, the patches may fail completely, and the malware cannot be eradicated through distributed patching. (c) In the case where the first threshold cannot be kept below the second threshold, the patches may work permanently, but the malware cannot be eradicated through distributed patching. The influence of the delays and the saturation effects on malware propagation is examined experimentally. The relevant conclusions reveal the way the delays and saturation effects modulate these outcomes.