Heterogeneous virus propagation in networks: a theoretical study

The node‐based epidemic modeling is an effective approach to the understanding of the impact of the structure of the propagation network on the epidemics of electronic virus. In view of the heterogeneity of the propagation network, a heterogeneous node‐based SIRS model is proposed. Theoretical analysis shows that the maximum eigenvalue of a matrix related to the model determines whether viruses tend to extinction or persist. When viruses persist, the connectedness of the propagation network implies the existence and uniqueness of a viral equilibrium, and a set of sufficient conditions for the global stability of the viral equilibrium are given. Numerical examples verify the correctness of our results. Copyright © 2016 John Wiley & Sons, Ltd.     

On generalized Poisson–Nernst–Planck equations with inhomogeneous boundary conditions: a‐priori estimates and stability

In this paper, we consider the strongly nonlinear Nernst–Planck equations coupled with the quasi‐linear Poisson equation under inhomogeneous, moreover, nonlinear boundary conditions. This system describes joint multi‐component electrokinetics in a pore phase. The system is supplemented by the force balance and by the volume and positivity constraints. We establish well‐posedness of the problem in the variational setting. Namely, we prove the existence theorem supported by the energy and the entropy a‐priori estimates, and we provide the Lyapunov stability of the solution as well as its uniqueness in special cases. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

In this paper, stability and bifurcation of a two‐dimensional ratio‐dependence predator–prey model has been studied in the close first quadrant . It is proved that the model undergoes a period‐doubling bifurcation in a small neighborhood of a boundary equilibrium and moreover, Neimark–Sacker bifurcation occurs at a unique positive equilibrium. We study the Neimark–Sacker bifurcation at unique positive equilibrium by choosing b as a bifurcation parameter. Some numerical simulations are presented to illustrate theocratical results. Copyright © 2017 John Wiley & Sons, Ltd.     

On input-to-state stability for stochastic multi-group models with multi-dispersal

This paper focuses on the input-to-state stability for a general class of stochastic multi-group models with multi-dispersal. By incorporating graph theory with Lyapunov method as well as stochastic analysis techniques, novel sufficient criteria are derived, which are in the form of Lyapunov-type theorem and coefficient-type criterion, respectively. Moreover, to show the applicability of our findings, we employ coefficient-type criterion to analyze the input-to-state stability for stochastic coupled oscillators. Finally, a numerical example and its simulations are offered to demonstrate the validity and feasibility of the theoretic results.     

On the stability of Timoshenko-type systems with internal frictional dampings and discrete time delays

In this paper, we consider a vibrating system of Timoshenko-type in a bounded one-dimensional domain under Dirichlet–Dirichlet or Dirichlet–Neumann boundary conditions with one or two discrete time delays and one or two internal frictional dampings. First, we show that the system is well posed in the sens of semigroup theory. Second, we prove the exponential stability regardless to the speeds of wave propagation of the system if the weights of the time delays are smaller than the ones of the corresponding dampings, respectively. However, when the weight of one time delay is not smaller than the one of the corresponding damping, we prove the exponential stability in case of equal-speed wave propagation, and the polynomial stability in the opposite case.     

The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion

In this paper, we consider a class of impulsive stochastic differential equations driven by G-Brownian motion (IGSDEs in short). By means of the G-Lyapunov function method, some criteria on p-th moment stability and p-th moment asymptotical stability for the trivial solutions of IGSDEs are established. An example is presented to illustrate the efficiency of the obtained results.     

Decoupled schemes for unsteady MHD equations. I. time discretization

In this article, a time discretization decoupled scheme for two‐dimensional magnetohydrodynamics equations is proposed. The almost unconditional stability and convergence of this scheme are provided. The optimal error estimates for velocity and magnet are provided, and the optimal error estimate for pressure are deduced as well. Finite element spatial discretization and numerical implementation are considered in our article (Zhang and He, Comput Math Appl 69 (2015), 1390–1406). © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 956–973, 2017     

Approximation of the unsteady Brinkman‐Forchheimer equations by the pressure stabilization method

In this work, we propose and analyze the pressure stabilization method for the unsteady incompressible Brinkman‐Forchheimer equations. We present a time discretization scheme which can be used with any consistent finite element space approximation. Second‐order error estimate is proven. Some numerical results are also given.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1949–1965, 2017     

Convergence analysis for second‐order accurate schemes for the periodic nonlocal Allen‐Cahn and Cahn‐Hilliard equations

In this paper, we provide a detailed convergence analysis for fully discrete second‐order (in both time and space) numerical schemes for nonlocal Allen‐Cahn and nonlocal Cahn‐Hilliard equations. The unconditional unique solvability and energy stability ensures ? ⁴ stability. The convergence analysis for the nonlocal Allen‐Cahn equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nonlocal Cahn‐Hilliard equation, because of the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken, and an H ^?1 inner‐product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a priori bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (eg, 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O (s ³+h ⁴) convergence in norm, in which s and h denote the time step and spatial mesh sizes, respectively. This in turn leads to the necessary bound under a standard constraint s ≤C h . Here, we also prove convergence of the scheme in the maximum norm under the same constraint.