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In the new and sophisticated cyber attacks (mainly, advanced persistent threats) the advanced specimens of malware such that zero-day malware play a crucial role. Due to its stealthy behavior it is very important to study and analyze its propagation process by designing mathematical models that could predict in an efficient way its spread on a network. With no doubt the computational implementation of these theoretical models leads to the develop of solutions to be used in the Security Operation Centers (SOC) with forensic purposes. The main goal of this work is to introduce a novel mathematical model to simulate advanced malware. Specifically, it is a compartmental and global SCIRAS (Susceptible-Carrier-Infectious-Recovered-Attacked-Susceptible) model where susceptible, carrier, infectious, recovered and attacked devices are considered. The local and global stability of its equilibrium points are studied and the basic reproductive number is computed. From the analysis of this epidemiological threshold, the most efficient security countermeasures are derived.INDEX TERMS Basic reproductive number, malware spread, mathematical model, advanced persistent threats, zero-day malware.

An important way considered to control malware epidemic processes is to take into account security measures that are associated to the systems of ordinary differential equations that governs the dynamics of such systems. We can observe two types of control measures: the analysis of the basic reproductive number and the study of control measure functions. The first one is taken at the beginning of the epidemic process and, therefore, we can consider this to be a prevention measure. The second one is taken during the epidemic process. In this work, we use the theory of optimal control that is associated to systems of ordinary equations in order to find a new function to control malware epidemic through time. Specifically, this approach is evaluate on a particular compartmental malware model that considers carrier devices.

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