Applying fractional calculus to malware spread: A fractal-based approach to threat analysis

Malware is a common word in modern era. Everyone using computer is aware of it. Some users have to face the problem known as Cyber crimes. Nobody can survive without use of modern technologies based on computer networking. To avoid threat of malware, different companies provide antivirus strategies on a high cost. To prevent the data and keep privacy, companies using computers have to buy these antivirus programs (software). Software varies due to types of malware and is developed on structure of malware with a deep insight on behavior of nodes. We selected a mathematical malware propagation model having variable infection rate. We were interested in examining the impact of memory effects in this dynamical system in the sense of fractal fractional (FF) derivatives. In this paper, theoretical analysis is performed by concepts of fixed point theory. Existence, uniqueness and stability conditions are investigated for FF model. Numerical algorithm based on Lagrange two points interpolation polynomial is formed and simulation is done using Matlab R2016a on the deterministic model. We see the impact of different FF orders using power law kernel. Sensitivity analysis of different parameters such as initial infection rate, variable adjustment to sensitivity of infected nodes, immune rate of antivirus strategies and loss rate of immunity of removed nodes is investigated under FF model and is compared with classical. On investigation, we find that FF model describes the effects of memory on nodes in detail. Antivirus software can be developed considering the effect of FF orders and parameters to reduce persistence and eradication of infection. Small changes cause significant perturbation in infected nodes and malware can be driven into passive mode by understanding its propagation by FF derivatives and may take necessary actions to prevent the disaster caused by cyber crimes.