Optimal control of joint multi-virus infection and information spreading

Nowadays, epidemic models provide an appropriate tool to describe the propagation of biological viruses in human or animal populations, rumors and misinformation in social networks, and malware in both computer and ad hoc networks. It is common that there are multiple types of malware infecting a network of computing devices, and different messages can spread over the social network. Information spreading and virus propagation are interdependent processes. To capture their independencies, we integrate two epidemic models into one holistic framework, known as the modified Susceptible-Warned-Infected-Recovered-Susceptible (SWIRS) model. The first epidemic model describes the information spreading regarding the risk of malware attacks and possible preventive procedures. The second one describes the propagation of multiple viruses over the network of devices. To minimize the impact of the virus spreading and improve the protection of the networks, we consider an optimal control problem with two types of control strategies: information spreading among healthy nodes and the treatment of infected nodes. We obtain the structure of optimal control strategies and study the condition of epidemic outbreaks. The main results are extended to the case of the network of two connected clusters. Numerical examples are used to corroborate the theoretical findings.