Wanbiao Ma

Global stability of an SIR epidemicmodel with time delay

An SIR disease transmission model is formulated under the assumption that the force of infection at the present time depends on the number of infectives at the past. It is shown that a disease free equilibrium point is globally stable if no endemic equilibrium point exists. Further the endemic point (if it exists) is globally stable with respect to the whole state space except the neighborhood of the disease free state.

LYAPUNOV FUNCTIONALS FOR DELAY DIFFERENTIAL EQUATIONS MODEL OF VIRAL INFECTIONS

We study global properties of a class of delay differential equations model for virus infections with nonlinear transmissions. Compared with the typical virus infection dynamical model, this model has two important and novel features. To give a more complex and general infection process, a general nonlinear contact rate between target cells and viruses and the removal rate of infected cells are considered, and two constant delays are incorporated into the model, which describe (i) the time needed for a newly infected cell to start producing viruses and (ii) the time needed for a newly produced virus to become infectious (mature), respectively. By the Lyapunov direct method and using the technology of constructing Lyapunov functionals, we establish global asymptotic stability of the infection-free equilibrium and the infected equilibrium. We also discuss the effects of two delays on global dynamical properties by comparing the results with the stability conditions for the model without delays. Further, we ...

Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate.

In this paper, based on SIR and SEIR epidemic models with a general nonlinear incidence rate, we incorporate time delays into the ordinary differential equation models. In particular, we consider two delay differential equation models in which delays are caused (i) by the latency of the infection in a vector, and (ii) by the latent period in an infected host. By constructing suitable Lyapunov functionals and using the Lyapunov–LaSalle invariance principle, we prove the global stability of the endemic equilibrium and the disease-free equilibrium for time delays of any length in each model. Our results show that the global properties of equilibria also only depend on the basic reproductive number and that the latent period in a vector does not affect the stability, but the latent period in an infected host plays a positive role to control disease development.

Global properties for virus dynamics model with Beddington–DeAngelis functional response☆

Abstract This paper investigates the global stability of virus dynamics model with Beddington–DeAngelis infection rate. By constructing Lyapunov functions, the global properties have been analysed. If the basic reproductive ratio of the virus is less than or equal to one, the uninfected steady state is globally asymptotically stable. If the basic reproductive ratio of the virus is more than one, the infected steady state is globally asymptotically stable. The conditions imply that the steady states are always globally asymptotically stable for Holling type II functional response or for a saturation response.

Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response

Abstract A class of virus dynamics model with intracellular delay and nonlinear infection rate of Beddington–DeAngelis functional response is analysed in this paper. By constructing suitable Lyapunov functionals and using LaSalle-type theorem for delay differential equations, we show that the global stability of the infection-free equilibrium and the infected equilibrium depends on the basic reproductive ratio R 0 , that is, the former is globally stable if R 0 ≤ 1 and so is the latter if R 0 > 1 . Our results extend the known results on delay virus dynamics considered in the other papers and suggest useful methods to control virus infection.

Periodic solution of a prey-predator model with nonlinear state feedback control

Assume that when the number of pests reaches the certain threshold, pest management strategy will be taken to control pests. Based on this assumption, in this paper, we propose a pest management model with nonlinear state feedback control. We then analyze the dynamic behavior of the model. More precisely, we first investigate the singularity of the model by using method of qualitative analysis; secondly the existence of periodic solution of the model is studied by using successor functions and Poincare-Bendixson theorem; and then it is followed by the study of the stability of periodic solution; finally, an example with numerical simulations is given to illustrate our conclusions.

Stability analysis on a predator-prey system with distributed delays

Abstract This paper concerns the local and global dynamical properties of the nonnegative and positive equilibria of a Lotka-Volterra predator-prey system with distributed delays. It is shown that, while the positive equilibrium does not exist, the nonnegative equilibrium is globally asymptotically stable or globally attractive as long as the delays are small enough. If the positive equilibrium exists, it is shown that it is locally asymptotically stable when the delays are suitably small. Furthermore, an explicit asymptotic stability region for the positive equilibrium is also obtained based on a Liapunov functional.

Analysis of the dynamics of a delayed HIV pathogenesis model

In this paper, considering full Logistic proliferation of CD4^+ T cells, we study an HIV pathogenesis model with antiretroviral therapy and HIV replication time. We first analyze the existence and stability of the equilibrium, and then investigate the effect of the time delay on the stability of the infected steady state. Sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delay. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold, and investigate the existence of Hopf bifurcation by using a delay @t as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the main results.

M-matrix structure and harmless delays in a Hopfield-type neural network

Abstract Motivated by work in [W. Ma, T. Hara, Y. Takeuchi, Stability of a 2-dimensional neural network with time delays, J. Biol. Syst. 8 (2000) 177–193; S.A. Campbell, Delay independent stability for additive neural networks, in: New Millennium Special Issue on Neural Networks and Neurocomputing—Theory, Models and Applications Part I, Differential Equations Dynam. Syst. 9 (2001) 115–138; S. Zhang, W. Ma, Y. Kuang, Necessary and sufficient conditions for global attractivity of Hopfield-type neural networks with time delays, Rocky Mountain J. Math. 38 (2008) 1829–1840], this work gives some necessary and sufficient conditions for global attractivity of an n -dimensional Hopfield-type neural network model with time delays based on the M -matrix and a class of nonlinear algebra equations with n variables.

STABILITY OF A 2-DIMENSIONAL NEURAL NETWORK WITH TIME DELAYS

A 2-dimensional neural network with time delayed connections between neurons is considered. Based on the construction of Liapunov functionals, we obtain sufficient criteria to ensure local and global asymptotic stability of the equilibrium of the neural network.