Global dynamics for a Filippov epidemic system with imperfect vaccination

Dynamics of an SVEIR transmission model with protection awareness and two strains

As of May 2024, the main strains of COVID-19 caused hundreds of millions of infection cases and millions of deaths worldwide. In this study, we consider the COVID-19 epidemics with the main strains in the Chinese mainland. We study complex interactions among hosts, non-pharmaceutical interventions, and vaccinations for the main strains by a differential equation model called SVEIR. The disease transmission model incorporates two strains and protection awareness of the susceptible population. Results of this study show that the protection awareness plays a crucial role against infection of the population, and that the vaccines are effective against the circulation of the earlier strains, but ineffective for emerging strains. By using the next generation matrix method, the basic reproduction number of the SVEIR model is firstly obtained. Our analysis by Hurwitz criterion and LaSalle's invariance principle shows that the disease free-equilibrium point is locally and globally asymptotically stable when the threshold value is below one. The existences of endemic equilibrium points are also established, and the global asymptotic stabilities are analyzed using the Lyapunov function method. Further, the SVEIR model is confirmed to satisfy the principle of competitive exclusion, of which the strain with the larger value of the basic reproduction number is dominant. Numerically, the surveillance data with the Omicron strain and the XBB strain are split by the cubic spline interpolation method. The fitting curves against the surveillance data are plotted using the least-squares method from MATLAB. The results indicate that the XBB strain dominates in this study. Moreover, a global sensitivity analysis of the key parameters is performed by using of PRCC. The numerical simulations imply that combination control strategy positively impacts on the infection scale than what separate control strategy does, and that the earlier time producing protection awareness for the public creates less infection scale, further that the increment of protection awareness also reduces the infection scale. Therefore, the policymakers of the local government are suggested to concern the changes of protection awareness of the public.

Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications

Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic "Bristol book" [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.

Global dynamics for a Filippov system with media effects

In the process of spreading infectious diseases, the media accelerates the dissemination of information, and people have a deeper understanding of the disease, which will significantly change their behavior and reduce the disease transmission; it is very beneficial for people to prevent and control diseases effectively. We propose a Filippov epidemic model with nonlinear incidence to describe media's influence in the epidemic transmission process. Our proposed model extends existing models by introducing a threshold strategy to describe the effects of media coverage once the number of infected individuals exceeds a threshold. Meanwhile, we perform the stability of the equilibriua, boundary equilibrium bifurcation, and global dynamics. The system shows complex dynamical behaviors and eventually stabilizes at the equilibrium points of the subsystem or pseudo equilibrium. In addition, numerical simulation results show that choosing appropriate thresholds and control intensity can stop infectious disease outbreaks, and media coverage can reduce the burden of disease outbreaks and shorten the duration of disease eruptions.