Public health services are constantly searching for ways to reduce the spread of infectious diseases, such as public vaccination of asymptomatic individuals, quarantine and treatment of symptomatic individuals. In this paper, we introduce epidemic models including variable population size, degree-related imperfect vaccination and quarantine on scale-free networks. More specifically, the models are formulated both on the population with and without permanent natural immunity to infection, which corresponds respectively to the susceptible-vaccinated-infected-quarantined-recovered (SVIQR) model and the susceptible-vaccinated-infected-quarantined (SVIQS) model. We develop different mathematical methods to study the dynamics of two models, including the basic reproduction number, the global stability of disease-free and endemic equilibria. For the SVIQR model, we show that the system exhibits a forward bifurcation. Meanwhile, the disease-free and unique endemic equilibria are shown to be globally asymptotically stable by constructing suitable Lyapunov functions. For the SVIQS model, conditions ensuring the occurrence of multiple endemic equilibria are derived. Under certain conditions, this system cannot undergo a backward bifurcation. The global asymptotical stability of disease-free equilibrium, and the persistence of the disease are proved. The endemic equilibrium is shown to be globally attractive by using monotone iterative technique. Finally, stochastic network simulations yield quantitative agreement with the deterministic mean-field approach.

In this paper, we first introduce the model of single-driving double-response system (SDDRS), which consists of a driving system (subsystem) and two response systems (subsystems). By applying the theory of Lyapunov stability, we study the projective synchronization of SDDRS between the driving and response systems. The sufficient conditions for achieving projective synchronization are obtained when the driving system has either a globally stable equilibrium point or a chaotic attractor. Furthermore, we use the SDDRS for cryptography in secure communication and present a novel scheme for encryption and decryption based on its projection synchronization. The results of numerical simulations verify the efficiency of the presented control schemes and the excellence of cryptography.

In this paper, to be more precise in modeling the real epidemic spread, an SEIR model with delay describing the fixed latent period is studied on scale-free networks. The basic reproduction number, which serves as a critical threshold of epidemic spread, is calculated. The formula for the basic reproduction number shows that adding delay decreases the basic reproduction number. By constructing appropriate Lyapunov functions, the global stability of disease-free and endemic equilibria is investigated. Finally, numerical simulations are performed to illustrate the effect of network structures and model parameters on the basic reproduction numbers and disease spreading processes.