Dynamical analysis and optimal control for a malware propagation model in an information network

• A delayed malware propagation model with optimal control strategy is regarded.
• Stability and Hopf bifurcation are proved in a strict mathematical way.
• Hopf bifurcation causes an oscillation phenomenon in this model.
• An optimal control strategy is obtained by means of the Pontryagin׳s Maximum Principle.
• Numerical simulations provide a new insight into malware propagation in WSNs.

With the rapid development of network information technology, information networks security has become a very critical issue in our work and daily life. This paper investigates a nonlinear malware propagation model in wireless sensor networks (WSNs) based on SIR epidemic model. Sufficient conditions for the local stability of the positive equilibrium point and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation. Moreover, formulas for determining the properties of the bifurcating periodic oscillations are derived by applying the normal form method and center manifold theorem. Furthermore, with the help of the Maximum Principle of Pontryagin, we design an optimal control strategy for the previous model to extend the region of stability and reduce the density of infected nodes in WSNs. Finally, we conduct extensive simulations to evaluate the proposed model. Numerical evidence shows that the dynamic characteristics of malware propagation in WSNs are closely related to the immune period of a recovered node and the rate constant for nodes becoming susceptible again after recovered. Besides, we obtain that the optimal control strategy effectively improves the performance of the networks.