Dynamical analysis of two fractional-order SIQRA malware propagation models and their discretizations

The aim of this work is to propose and study dynamics of two fractional-order SIQRA malware propagation models and their discretizations. Positivity, boundedness and asymptotic stability properties of the proposed fractional-order models are analyzed rigorously. It is worthy noting that the global and uniform asymptotic stability properties of the fractional-order models are investigated based on appropriate Lyapunov functions. As an important consequence, the global asymptotic stability properties of the original integer-order models are also established completely. In addition, the fractional forward Euler method is utilized to discretize the fractional-order models. By rigorously mathematical analyses, we obtain step size thresholds which guarantee that the positivity, boundedness and asymptotic stability properties of the fractional-order models are preserved correctly by the discrete models. Consequently, simple conditions for reliable approximations for the fractional-order models are determined. Finally, a set of numerical examples is performed to illustrate and support the theoretical findings. The results show that the numerical examples are consistent with the constructed theoretical assertions.