How can neural circuits maintain stable activity states when they are constantly being modified by Hebbian processes that are notorious for being unstable? A new synaptic plasticity mechanism is presented here that enables a neuron to obtain homeostasis of its firing rate over longer timescales while leaving the neuron free to exhibit fluctuating dynamics in response to external inputs. Mathema...

The spread of tuberculosis is studied through two models which include fast and slow progression to the infected class. For each model, Lyapunov functions are used to show that when the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is globally asymptoti...

We study a susceptible-infected-susceptible model with distributed delays. By constructing suitable Lyapunov functionals, we demonstrate that the global dynamics of this model is fully determined by the basic reproductive ratio R0. To be specific, we prove that if R0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable. On the other hand, if R0>1, then the endemic equilibriu...

In this paper, the sliding mode observers design techniques for MIMO and as a simple example for SISO systems are systematically advanced as a first purpose. Design parameters are selected such that on the defined switching surface always is generated asymptotically stable sliding mode. Moreover, observer state error dynamics is globally robustly asymptotically stable. Then, advanced design tec...

This paper presents a nonlinear controller based on Sontag’s feedback to render the electromechanical valve actuator (EVA) globally asymptotically stable (GAS). Electromechanical valve actuators have received much attention recently due to their potential for improving the performance of the internal combustion engine. Various control schemes for the EVA have been proposed, however stability is...

We investigate a class of multigroup dengue epidemic model. We show that the global dynamics are determined by the basic reproductive number R0. We present that when R0 ≤ 1, there is a unique disease-free equilibrium which is globally asymptotically stable; when R0 > 1, there exists a unique endemic equilibrium and it is globally asymptotically stable proved by a graph-theoretic approach to the...

In this paper, we consider global stability for a heroin model with two distributed delays. The basic reproduction number of the heroin spread is obtained, which completely determine the stability of equilibria. Using the direct Lyapunov method with Volterra type Lyapunov function, we show that the drug use-free equilibrium is globally asymptotically stable if the basic reproduction number is l...

A general discrete juvenile-adult population model with time-dependent birth rate and nonlinear survivorship rates is studied. When breeding is continuous, it is shown that the model has a unique globally asymptotically stable positive equilibrium provided the net reproductive number is larger than one. If it is smaller than one, then the extinction equilibrium is globally asymptotically stable...

*Correspondence: [email protected] 1Department of Mathematics, Xinjiang Institute of Engineering, Urumqi, Xinjiang 830091, P.R. China Full list of author information is available at the end of the article Abstract In this paper, a diffusive and delayed virus dynamics model with Crowley-Martin incidence function and CTL immune response is investigated. By constructing the Lyapunov functionals...