The purpose of this paper is to establish some new criteria for the oscillation of the second-order nonlinear noncanonical difference equations of the form $$∆ (a (n) ∆x (n)) + q(n)x^β (g(n)) = 0, n ≥ n_0$$ under the assumption $$\sum\limits^∞_{s=n} \frac{1}{a(s)}< ∞.$$ Corresponding difference equations of both retarded and advanced type are studied. A particular example of Euler type equation is provided in order to illustrate the significance of our main results.

The paper established a so-called analogue-difference method (ADM) to compute the numerical solutions for boundary value problems of higher-order differential equations, which can be a fundamental method and performs much better than the finite difference method (FDM), even for second-order boundary value problems. Numerical examples and results illustrate the simplicity, efficiency and applicability of the method, which also show that the proposed method has obvious advantages over the methods presented by recent state-of-the-art work for induction motor models.

In this paper, we establish stochastic differential equations on the basis of a nonlinear deterministic model and study the global dynamics. For the deterministic model, we show that the basic reproduction number $\mathfrak{R}_0$ determines whether there is an endemic outbreak or not: if $\mathfrak{R}_0<1,$ the disease dies out; while if $\mathfrak{R}_0>1,$ the disease persists. For the stochastic model, we provide analytic results regarding the stochastic boundedness, perturbation, permanence and extinction. Finally, some numerical examples are carried out to confirm the analytical results. One of the most interesting findings is that stochastic fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics.

The novel coronavirus pneumonia 2019 (COVID-19) has swept the globe in just a few months with negative social and psychological consequences for public health. So far, the United States has been one of the countries most affected by the epidemic. In this study, 51 states in the United States are divided into 10 state clusters according to relevant factors, and a difference equation model with spatio-temporal dynamic characteristics is established to predict the transmission dynamics of COVID-19 in the 10 state clusters and obtain data on regional aggregation levels (the United States). The study showed that the Pearson Correlation Coefficient between the actual data and the predicted data in the 10 state clusters is between 0.6 and 0.96 (mean $R^2$=0.8448), and the mean absolute error (MAE) of the newly confirmed cases in each cluster is between 300 and 1650 (mean MAE=878) and the average forecasting error rate (AFER) of the total confirmed cases in each cluster is between 0.9% and 3% (mean AFER=1.57%). These results show that the difference equation model can well predict the changes in the recent confirmed cases of infectious diseases such as COVID-19.

In this paper, we establish a novel delayed SIQS epidemic model on scale-free networks, where time delay represents the average quarantine period. Through mathematical analysis, we present the basic reproduction number $R_0.$ Then, we provide the global asymptotical stability of the disease-free equilibrium and the local asymptotical stability of the endemic equilibrium. Finally, we perform numerical simulations to verify the correctness of the main results and analyze the sensitivity of parameters. Our research shows that when $R_0 > 1,$ lengthening the quarantine period can slow the spread of the disease and reduce the number of infected individuals.

In this paper, we consider Cauchy problem of the axially symmetric Magnetohydrodynamic (MHD) system. By using energy method, we establish some regularity criteria of the solutions for the axisymmetric solutions of the three dimensional incompressible MHD system.

In this paper, a singular nonlinear high order fractional differential problem involving multi-point boundary conditions is solved by means of the fixed point index theory. Some properties of the first eigenvalue corresponding to relevant operator and some new height functions are also used to prove the existence and multiplicity of positive solutions. The nonlinearity depends on arbitrary fractional derivative.

Mussel beds are important habitats and food sources for biodiversity in coastal ecosystems. The predation of mussel on algae depends not only on the current time and location, but also on the quantity of algae at other spatial location and time. To know the impacts of such predation behavior on the dynamics of mussel beds well, we pose a reaction-diffusion mussel-algae model coupling nonlocal interaction with kernel function. By calculating the critical conditions of Hopf bifurcation and Turing bifurcation, the conditions for the generation of Turing pattern are obtained. We find that the diffusion rate and predation rate of mussels have effect on the structure and density of spatial pattern of mussels under the nonlocal interaction, and the predation rate of mussels can produce different pattern types, while the diffusion rate plays a more important role on the pattern density. Moreover, the nonlocal interaction promotes the stability of the mussel beds. These results suggest that the nonlocal interaction between mussels and algaes is one of the important mechanisms for the formation of the spatial structure of mussel beds.

In this paper, we initiate the solvability and stability for a class of singular fractional $(p, q)$-difference equations. First, we obtain an existence theorem of solution for the fractional $(p, q)$-difference equation. Then, by using a fractional $(p, q)$-Gronwall inequality, some stability criteria of solution are established, which also implies the uniqueness of solution.

In this paper, we focus on stability analysis of steady-state solutions of a predator-prey system with hunting cooperation functional response. The results show that the Turing instability can be affected not only the existence of hunting cooperation, but also the diffusion coefficients: (1) in the absence of predator diffusion, diffusion-driven instability can be induced by hunting cooperation, but no stable patterns appear; (2) the system can occur diffusion-driven instability and Turing patterns, when both predator and prey have diffusion, and the diffusion coefficient of prey is greater than that of the predator. The numerical simulations of two cases are presented to verify the validity of our theoretical results.