Chemical reaction networks with generalized mass-action kinetics lead to power-law dynamical systems. As a simple example, we consider the Lotka reactions and the resulting planar ODE. We characterize the parameters (positive coefficients and real exponents) for which the unique positive equilibrium is a center.

We study a virus dynamics model with reaction-diffusion, logistic growth terms and a general non-linear infection rate functional response. The model has a distributed delay, including the case of state-selective delay. We construct a dynamical system in a Hilbert space and prove the existence of a finite-dimensional global attractor.

In the current era, information dissemination is more convenient, the harm of rumors is more serious than ever. At the beginning of 2020, COVID-19 is a biochemical weapon made by a laboratory, which has caused a very bad impact on the world. It is very important to control the spread of these untrue statements to reduce their impact on people's lives. In this paper, a new rumor spreading model with comprehensive interventions (background detection, public education, official debunking, legal punishment) is proposed for qualitative and quantitative analysis. The basic reproduction number with important biological significance is calculated, and the stability of equilibria is proved. Through the optimal control theory, the expression of optimal control pairs is obtained. In the following numerical simulation, the optimal control under 11 control strategies are simulated. Through the data analysis of incremental cost-effectiveness ratio and infection averted ratio of all control strategies, if we consider the control problem from different perspectives, we will get different optimal control strategies. Our results provide a flexible control strategy for the security management department.

This paper introduces a theory of Thermodynamic Formalism for Iterated Function Systems with Measures (IFSm). We study the spectral properties of the Transfer and Markov operators associated to a IFSm. We introduce variational formulations for the topological entropy of holonomic measures and the topological pressure of IFSm given by a potential. A definition of equilibrium state is then natural and we prove its existence for any continuous potential. We show, in this setting, a uniqueness result for the equilibrium state requiring only the Gâteaux differentiability of the pressure functional.

In this paper, we analyze a stochastic SIRC model with Ornstein-Uhlenbeck process. Firstly, we give the existence and uniqueness of global solution of stochastic SIRC model and prove it. In addition, the existence of ergodic stationary distributions for stochastic SIRC system is proved by constructing a suitable series of Lyapunov functions. A quasi-endemic equilibrium related to endemic equilibrium of deterministic systems is defined by considering randomness. And we obtain the probability density function of the linearized system near the equilibrium point. After the proof of probability density function, the sufficient condition of disease extinction is given and proved. We prove the theoretical results in the paper by numerical simulation at the end of the paper.

A tuberculosis (TB) epidemic model with Beddington-DeAngelis incidence and distributed delay is proposed to characterize the interaction between latent period, endogenous reactivation, treatment of latent TB infection, as well as relapse. The basic reproduction number is defined, and the globally asymptotic stability of disease-free equilibrium is shown when , while if     the globally asymptotic stability of endemic equilibrium is also acquired. Theoretical results are validated through performing numerical simulations, wherein we detect that TB dynamic behavior between models with discrete and distributed delays could be same and opposite, and TB is more persistent in the model with distributed delay. Besides, increasing the protection level of susceptible and infectious individuals is crucial for the control of TB.

To investigate the influence of human behavior on the spread of COVID-19, we propose a reaction-diffusion model that incorporates contact rate functions related to human behavior. The basic reproduction number is derived and a threshold-type result on its global dynamics in terms of is established. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if ; while there exists a positive stationary solution and the disease is uniformly persistent if . By the numerical simulations of the analytic results, we find that human behavior changes may lower infection levels and reduce the number of exposed and infected humans.

Motivated by the study of recurrent orbits and dynamics within a Morse set of a Morse decomposition we introduce the concept of Morse predecomposition of an isolated invariant set within the setting of both combinatorial and classical dynamical systems. While Morse decomposition summarizes solely the gradient part of a dynamical system, the developed generalization extends to the recurrent component as well. In particular, a chain recurrent set, which is indecomposable in terms of Morse decomposition, can be represented more finely in the Morse predecomposition framework. This generalization is achieved by forgoing the poset structure inherent to Morse decomposition and relaxing the notion of connection between Morse sets (elements of Morse decomposition) in favor of what we term 'links'. We prove that a Morse decomposition is a special case of Morse predecomposition indexed by a poset. Additionally, we show how a Morse predecomposition may be condensed back to retrieve a Morse decomposition.