Considering the significant role of the SIR epidemic model in infectious disease research field, the dynamical analysis of a spatiotemporal SIR epidemiological model under homogeneous Neumann boundary conditions is investigated. First, the Turing instability of the constant equilibrium solution is analyzed via the linearization method, and a priori estimate of the positive nonconstant solution is derived by applying the maximum principle. Second, the nonexistence and existence of positive nonconstant solutions are proved respectively by using energy estimation and the Leray-Schauder degree theory. Furthermore, local and global bifurcations at simple eigenvalues are established by applying bifurcation theory, and the conditions for determining the bifurcation direction are obtained. The local bifurcation at double eigenvalues is discussed by employing spatial decomposition and the implicit function theorem. Finally, the theoretical analysis results are verified through numerical simulations.

To effectively prevent and control HBV infection, a HBV infection model with liver-virus-blood transmission is developed in this paper. By employing the Routh-Hurwitz criterion, Lyapunov-LaSalle invariant set principle and other related theories, it is proven that the threshold dynamic behavior of the model is determined by the basic reproductive number. Furthermore, two control measures of antiviral therapy, interferon therapy and nucleoside (acid) therapy, are introduced to study the corresponding optimal control problem, and the existence and uniqueness of the optimal control strategy are proved by Pontriagin's minimum principle. Finally, the influence of blood transmission and optimal control on HBV infection model is discussed by forward-backward sweeping method to simulate. The results show that ignoring blood transmission will underestimate the level of HBV infection, and the optimal control strategy can significantly improve the concentration of healthy cells and reduce the viral load. Meanwhile, the therapy effect of nucleotide (acid) is better than that using interferons.

In consideration of the time delays required for the transform from potential smokers and passive smokers into active smokers, the paper establishes a smoking and lung cancer growth interaction model with double time delays. By analyzing the distribution of characteristic equation roots and the stability switching curves, as well as the conditions for the equilibrium asymptotic stability and the Hopf bifurcation, the specific effects of smoking on lung growth are studied. On the basis of  the computing formulas for the stability and the direction of Hopf bifurcating periodic solutions, numerical simulations are performed to show the changing laws of lung cancer growth, such as steady states, periodic oscillations and chaos, when the parameters of the environmental carrying capacity, the passive smoking rate and the time delays vary. The theoretical and numerical results are helpful to understand the relationship between smoking and lung cancer growth, which provide theoretical guidelines for preventing and controlling smoking.

Conformal mapping plays an important role in fluid such as fluid mechanics, yet its analytical expression is difficult to derive for complex domains. This paper proposes a numerical method based on the charge simulation method for calculating conformal mappings from the multiply connected regions onto disk with logarithmic spiral slit domains. First, the proposed method approximates the undetermined function in the conformal mapping function by using the linear combination of the fundamental solutions of the Laplace equation. Second, we establish the constraint equation system about the coefficients in the linear combination according to the boundary value conditions. The generalized minimal residual (GMRES(m)) method is used to solve the constraint equation system and obtain the charge, which constructs a high-precision approximate conformal mapping function. Because the complex potential in the problem domain can be derived from that in the regular region via conformal mapping, the flow around of the spiral point vortex in the bounded multiply connected region is further simulated. Numerical experiments show the effectiveness of the proposed algorithm and simulate the spiral point vortex in the bounded multiply connected region.

In view of the dense distribution of multiple roots of nonlinear equation systems, the heuristic optimisation algorithm is used to solve the root loss problem of multiple roots in the process of one operation, and a fuzzy neighborhood-based slime mould algorithm (FASMA) is proposed. The fuzzy neighborhood technique is applied to divide the population into sub-populations, so that the algorithm has the ability to locate multiple roots of nonlinear equations simultaneously. At the same time, the slime mould algorithm (SMA) is used to update the individuals in the fuzzy neighborhood. The SMA simulates the positive and negative feedback of the propagation wave generated by the biological oscillator in the process of slime mould foraging in nature, which makes the proposed algorithm have good local search ability. In addition, the archiving mechanism and individual reinitialization are used to enhance the population diversity and improve the ability of the algorithm to jump out of the local optimum. In order to verify the validity of the proposed algorithm, the special cases of nonlinear equation systems with different density roots and 15 dense-root nonlinear equation systems are tested. The results of FASMA are compared with those of FNODE, DREA and NCDE algorithms. The experimental results show that the proposed algorithm FASMA performs better in term of both the root ratio and the success rate.

In order to investigate the hyperchaos phenomena in nonlinear systems, a new hyperchaotic system based on the hyperchaotic Tang system is constructed. Based on the Lyapunov stability theory, qualitative theory and bifurcation theory, the dynamical characteristics of this new hyperchaotic systems are studied, including dissipation, equilibrium point and its local stability, Lyapunov exponential spectrum and bifurcation diagram of some \mbox{parameters} change. By introducing the generalized Lyapunov function, it is proved that there exists the global exponential attractive set for the new hyperchaotic system. The global exponential synchronization of two hyperchaotic Tang systems is studied by using the global exponential attraction set and a linear controller is given to achieve global exponential synchronization. Then, a nonlinear feedback controller is presented to achieve global exponential synchronization. Finally, through numerical simulation, it is proved that both feedback control methods can achieve global exponential synchronization and the advantages of the two methods are compared.

Hybrid stochastic system serve as an effective tool for describing the dynamic behavior of complex systems.

For the stabilization problem of nonlinear stochastic systems with external disturbances and uncertainties, their stability analysis faces tremendous challenges due to the interference from multiple time delays. Based on the dynamic event-triggered mechanisms, in order to reduce the transmission frequency of feedback control signal and avoid Zeno behavior, the aperiodic intermittent event-triggered control strategy is adopted. The practically input-to-state stability is used to describe the dynamic performance of control target in the event-triggered schemes. Besides, by means of dynamic event-triggered feedback control, the stabilization criteria for nonlinear hybrid stochastic  systems with multiple time-delays are obtained by using Lyapunov function and linear matrix inequality method.  Finally, the effectiveness of the conclusions is verified by several examples and simulations.

Energy dissipation exists in many practical problems of dynamics, yet research on the oscillation problems with energy dissipation remains relatively scarce. Thus, the novel energy dissipative standing wave is simulated with a complex prediction-correction finite difference method (PCFD). The PCFD is used to solve the three dimensional energy dissipation potential flow equation and is applied to numerically simulate the dynamic evolution of dissipative standing wave in free surface. The computational domain is arranged with a staggered mesh-grid system and it is used to capture the flow field variables correctly. An irregular tank is mapped onto a square cubic tank through $\sigma$ coordinate transformation. The PCFD is applied to discrete the potential flow equations, Poisson equation and boundary conditions. From the experimental process, we can conclude that the present results are good agree with previous article. The energy dissipation standing wave is presented with different experimental parameters and the different wave elevation of free surface is detailed displayed in experimental process. The free surface maintains the characteristics of the standing wave well under energy dissipation.

Chaotic system play an important role in fields such as communication, yet the analysis of high-dimensional chaotic system faces severe challenges due to their complex dynamic behavior. In order to address this issue, this paper investigates the Hopf bifurcation of a five-dimensional chaotic system with three quadratic terms. First, by employing the parameter-dependent center manifold theory and Hopf bifurcation theory, the stability of the equilibrium points and local dynamical behaviors such as Hopf bifurcation in the five-dimensional chaotic system are analyzed. The conditions for the existence of equilibrium points and Hopf bifurcation are obtained, indicating the presence of Hopf bifurcation in the system. Then, rigorous mathematical analysis and numerical simulation methods are used to study the conditions for the generation of one-dimensional and two-dimensional bifurcations in the chaotic system. Finally, in response to issues such as weak key sensitivity and strong correlation between adjacent pixels in image encryption schemes, the system is combined with DNA coding and applied to image encryption. The experimental results show that the new algorithm significantly improves the performance and safety of image encryption.

To investigate the nonlinear wave phenomena in magnetized ionic-pair plasma, a $(3+1)$-dimensional Schamel-Zakharov-Kuznetsov-Burgers' equation is established. The physics-informed neural network (PINN) is employed to solve this nonlinear differential equation. In order to address the unbalanced gradients caused by the stiffness of gradient flow dynamics and the inherent spectral bias in PINN, an adaptive weighting coefficient and Fourier feature mapping are introduced to improve the algorithm. By utilizing the modified PINN model, the influence of the viscosity coefficient on shock wave characteristics is systematically studied. Numerical results indicate that an increase in the viscosity coefficient leads to a smoothing effect on the shock wave structure.

Integer-valued time series with finite range are commonly encountered in practice, and many time series exhibit multimodality features on marginal or conditional distributions. However, most integer-valued models assume that the series is driven by a unimodal innovation sequence. Mixture models often perform well on data with multimodality and overdispersion features. A mixture model is considered for finite range integer-valued time series. In particalar, a binomial mixture integer-valued GARCH (B-MINGARCH) model is established, which includes a kind of binomial mixture ARCH (B-MINARCH) model with strict stationary and ergodicity. The necessary and sufficient stationary conditions are derived for B-MINGARCH achieving on- and two-order stability. The performances of the conditional maximum likelihood estimators via the EM algorithm are simulated. A real data example is also given.

In the field of survival analysis, interval censored data is a common data type. Based on the K-type interval censored response variable and the interval censored covarites, the dependent accelerated failure time model is constructed. The dependence relationship between the failure time and the censored time is studied. In this model, the polynomial kernel function is used to express the unknown relationship between the failure time and the censoring time. The Pólya Tree distribution is used to approximate the interval censored covarites. Moreover, the maximum likelihood estimation method is adopted to estimate the parameters. The simulation results show that the method achieves good results in different cases, and the nonlinear model can better fit the relationship between failure time and censoring time when the covarites have strong correlation with each other. Finally, the proposed method and model are applied to clinical trial data to further verify its effectiveness.

Aiming at solving the problem of navigation congestion caused by insufficient navigation service capacity of locks, the linkage mechanism of `congestion charge-overturning subsidy' is studied. The prospect theory is used to analyze the navigation behavior of shipowners, and an optimization model based on congestion charge-overturning subsidy is established to reveal the freight sharing law of the two dam-crossing modes under different congestion charge and overturn subsidy levels. According to the number of ships arriving, the warning level of navigation congestion is then divided. The selection probability of dam-crossing mode, the change trend of total cost of dam-crossing and the benefit of navigation carbon emission reduction are explored. Finally, taking the Three Gorges Project as an example, the validity of the above model is verified. The results show that under different warning levels, with the increase of congestion charges, the probability of dam-crossing selection of the five-level lock decreases continuously. Furthermore, the implementation of a reasonable congestion charging-dumping subsidy policy can reduce the total cost of crossing the dam and carbon emissions by up to 62.7% and 49.5%, respectively. While achieving a balanced freight sharing, it produces both good economic and ecological benefits.