Dynamics of neural systems with discrete and distributed time delays

In real-world systems, interactions between elements do not happen instantaneously, due to the time required for a signal to propagate, reaction times of individual elements, and so forth. Moreover, time delays are normally nonconstant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In order to analyze the fundamental properties of neural networks with time-delayed connections, we consider a system of two coupled two-dimensional nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. An exciting feature of the model under consideration is the combination of both discrete and distributed delays, where distributed time delays represent the neural feedback between the two subsystems, and the discrete delays describe the neural interaction within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding results on stability for networks with no distributed delays. It is shown how approximations of the boundary of the stability region of a trivial equilibrium can be obtained analytically for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical findings

Stability and bifurcations in an epidemic model with varying immunity period

An epidemic model with distributed time delay is derived to describe the dynamics of infectious diseases with varying immunity. It is shown that solutions are always positive, and the model has at most two steady states: disease-free and endemic. It is proved that the disease-free equilibrium is locally and globally asymptotically stable. When an endemic equilibrium exists, it is possible to analytically prove its local and global stability using Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and traceDDE to investigate different dynamical regimes in the model using numerical continuation for different values of system parameters and different integral kernels.Comment: 16 pages, 5 figure

Control of unstable steady states in neutral time-delayed systems

We present an analysis of time-delayed feedback control used to stabilize an unstable steady state of a neutral delay differential equation. Stability of the controlled system is addressed by studying the eigenvalue spectrum of a corresponding characteristic equation with two time delays. An analytic expression for the stabilizing control strength is derived in terms of original system parameters and the time delay of the control. Theoretical and numerical results show that the interplay between the control strength and two time delays provides a number of regions in the parameter space where the time-delayed feedback control can successfully stabilize an otherwise unstable steady state.Comment: 11 pages, 8 figure

Enhancing noise-induced switching times in systems with distributed delays

The paper addresses the problem of calculating the noise-induced switching rates in systems with delay-distributed kernels and Gaussian noise. A general variational formulation for the switching rate is derived for any distribution kernel, and the obtained equations of motion and boundary conditions represent the most probable, or optimal, path, which maximizes the probability of escape. Explicit analytical results for the switching rates for small mean time delays are obtained for the uniform and bi-modal (or two-peak) distributions. They suggest that increasing the width of the distribution leads to an increase in the switching times even for longer values of mean time delays for both examples of the distribution kernel, and the increase is higher in the case of the two-peak distribution. Analytical predictions are compared to the direct numerical simulations and show excellent agreement between theory and numerical experiment

Asymptotic properties of the spectrum of neutral delay differential equations

Spectral properties and transition to instability in neutral delay differential equations are investigated in the limit of large delay. An approximation of the upper boundary of stability is found and compared to an analytically derived exact stability boundary. The approximate and exact stability borders agree quite well for the large time delay, and the inclusion of a time-delayed velocity feedback improves this agreement for small delays. Theoretical results are complemented by a numerically computed spectrum of the corresponding characteristic equations.Comment: 14 pages, 6 figure

Time-delayed models of gene regulatory networks

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternativemodelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems

Time-delayed and stochastic effects in a predator-prey model with ratio dependence and Holling type III functional response

In this article, we derive and analyze a novel predator-prey model with account for maturation delay in predators, ratio dependence, and Holling type III functional response. The analysis of the system’s steady states reveals conditions on predation rate, predator growth rate, and maturation time that can result in a prey-only equilibrium or facilitate simultaneous survival of prey and predators in the form of a stable coexistence steady state, or sustain periodic oscillations around this state. Demographic stochasticity in the model is explored by means of deriving a delayed chemical master equation. Using system size expansion, we study the structure of stochastic oscillations around the deterministically stable coexistence state by analyzing the dependence of variance and coherence of stochastic oscillations on system parameters. Numerical simulations of the stochastic model are performed to illustrate stochastic amplification, where individual stochastic realizations can exhibit sustained oscillations in the case, where deterministically the system approaches a stable steady state. These results provide a framework for studying realistic predator-prey systems with Holling type III functional response in the presence of stochasticity, where an important role is played by non-negligible predator maturation delay

Dynamics of vaccination in a time-delayed epidemic model with awareness

This paper investigates the effects of vaccination on the dynamics of infectious disease, which is spreading in a population concurrently with awareness. The model considers contributions to the overall awareness from a global information campaign, direct contacts between unaware and aware individuals, and reported cases of infection. It is assumed that there is some time delay between individuals becoming aware and modifying their behaviour. Vaccination is administered to newborns, as well as to aware individuals, and it is further assumed that vaccine-induced immunity may wane with time. Feasibility and stability of the disease-free and endemic equilibria are studied analytically, and conditions for the Hopf bifurcation of the endemic steady state are found in terms of system parameters and the time delay. Analytical results are supported by numerical continuation of the Hopf bifurcation and numerical simulations of the model to illustrate different types of dynamical behaviour

Mathematical model for the impact of awareness on the dynamics of infectious diseases

This paper analyses an SIRS-type model for infectious diseases with account for behavioural changes associated with the simultaneous spread of awareness in the population. Two types of awareness are included into the model: private awareness associated with direct contacts between unaware and aware populations, and public information campaign. Stability analysis of different steady states in the model provides information about potential spread of disease in a population, and well as about how the disease dynamics is affected by the two types of awareness. Numerical simulations are performed to illustrate the behaviour of the system in different dynamical regimes

Mathematical model of immune response to hepatitis B

A new detailed mathematical model for dynamics of immune response to hepatitis B is proposed, which takes into account contributions from innate and adaptive immune responses, as well as cytokines. Stability analysis of different steady states is performed to identify parameter regions where the model exhibits clearance of infection, maintenance of a chronic infection, or periodic oscillations. Effects of nucleoside analogues and interferon treatments are analysed, and the critical drug efficiency is determined