Global Stability for Novel Complicated SIR Epidemic Models with the Nonlinear Recovery Rate and Transfer from Being Infectious to Being Susceptible to Analyze the Transmission of COVID-19

Epidemiological models play pivotal roles in predicting, anticipating, understanding, and controlling present and future epidemics. The dynamics of infectious diseases is complex, and therefore, researchers need to consider more complicated mathematical models. In this paper, we first describe the dynamics of a complex SIR epidemic model with nonstandard nonlinear incidence and recovery rates. In this model, we consider the rate at which individuals lose immunity. Rigorous mathematical results have been established from the point of view of stability and bifurcation. The basic reproduction number ( R 0 ) is determined. We then apply LaSalle’s invariance principle and Lyapunov’s direct method to prove that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 . The model has a unique endemic equilibrium when R 0 > 1 . A nonlinear Lyapunov function is used together with LaSalle’s invariance principle to show that the endemic equilibrium is globally asymptotically stable under some conditions. Further, for the case when R 0 = 1 , we analyze the model and show a backward bifurcation under certain conditions. In the second part of this paper, we analyze a modified SIR model with a vaccination term, which must be a function of time. We show that the modified model agrees well with COVID-19 data in Saudi Arabia. We then investigate different future scenarios. Simulation results suggest that a two-pronged strategy is crucial to control the COVID-19 pandemic in Saudi Arabia.

Cluster Anticonsensus of Multiagent Systems Based on theQ-Theory

We investigate the problem of cluster anticonsensus of multiagent systems. For multiagent continuous systems, a new control protocol is designed based on theQ-theory. Then by LaSalle's invariance principle we prove that if the graph is connected and bipartite, then the cluster anticonsensus is achieved by the proposed control protocol. On the other hand, a similar control protocol is designed for multiagent discrete-time systems. Then, sufficient conditions are given to guarantee the cluster anticonsensus of multiagent discrete-time systems by using theQ-theory and LaSalle's invariance principle. Numerical simulations show the effectiveness of our theoretical results.

Asymptotic Behavior of a Class of Evolution Variational Inequalities

We establish a new LaSalle's invariance principle and discuss the asymptotic behavior of a class of first-order evolution variational inequalities.