Optimal Impulsive Control of Epidemic Spreading of Heterogeneous Malware

As the number of devices connected to the Internet is growing, the epidemics of malware spreading imposes a serious cyber security problem. It is common that there exist multiple types of malware infecting a network of devices. Periodically scheduled patching is a common way to protect the devices and thwart the malware spreading over a large population of devices. In this paper, we study the heterogeneous SIR model where two types of malware spread over the network and formulate an impulse optimal control problem to describe the optimal strategy of periodic patching that happens at discrete points of time. We obtain the structure of optimal impulse controls and consider the hybrid case in which we combine the discrete impulses and the continuous components of the control. Numerical simulations are used to corroborate the theoretical results.