Time-delayed Reaction-diffusion Research Articles

Dynamical analysis of an age-space structured malaria epidemic model

In this paper, we will revisit the model studied in Lou and Zhao (J Math Biol 62:543–568, 2011), where the model takes the form of a nonlocal and time-delayed reaction–diffusion model arising from the fixed incubation period. We consider the infection age to be a continuous variable but without the limitation of the fixed incubation period, leading to an age-space structured malaria model in a bounded domain. By performing the elementary analysis, we investigate the well-posedness of the model by proving the global existence of the solution, define the explicit formula of basic reproduction number when all parameters remain constant. By analyzing the characteristic equations and designing suitable Lyapunov functions, we also establish the threshold dynamics of the constant disease-free and positive equilibria. Our theoretical results are also validated by numerical simulations for 1-dimensional and 2-dimensional domains.

Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

A uniformly convergent numerical method is presented for solving singularly perturbed time delay reaction-diffusion problems. Properties of the continuous solution are discussed. The Crank–Nicolson method is used for discretizing the temporal derivative, and an exponentially fitted tension spline method is applied for the spatial derivative. Using the comparison principle and solution bound, the stability of the method is analyzed. The proposed numerical method is second-order uniformly convergent. The theoretical analysis is supported by numerical test examples for various values of perturbation parameters and mesh size.

Dynamical analysis of a diffusive malaria model with fixed latent period in the human and vector populations

To investigate the impact of the fixed latent periods in the human and vector populations on the disease transmission in heterogenous environment, we formulate a nonlocal and time-delayed reaction-diffusion (NLTD-RD) system. By appealing to the next generation operator (NGO), we define the basic reproduction number (BRN) [Formula: see text], and prove it as a threshold parameter for indicating whether disease persists or not. Specifically, if [Formula: see text], the disease-free equilibrium is globally asymptotically stable, while if [Formula: see text], the disease is shown to be uniformly persistent. In the homogeneous case that all parameters are assumed to be constants, the explicit expression of [Formula: see text] is obtained. We further achieved the global attractivity of the constant equilibria by utilizing Lyapunov functionals. Numerical simulations are performed to verify the theoretical results and the effects of the diffusion rate on disease transmission.

Evolution dynamics of a time-delayed reaction–diffusion HIV latent infection model with two strains and periodic therapies

Multiple HIV stains coinfection within the host has always been a challenging problem in clinical treatment and mathematical modeling. In order to explore joint effects of spatial factors, latent infection, latent period, and periodic therapies on the dynamics of HIV infection, we formulate a time-periodic reaction diffusion HIV latent infection model with multiple strains, four latent periods, and spatial heterogeneity. Applying the next infection operator, we derive the basic reproduction number Ri0 and the invasion number R˜i0(i=1,2), which is shown to be the threshold parameter determining the capability of the ith strain invading the jth strain (i≠j,i,j=1,2), thereby affecting the competitive evolution result of the two HIV stains. More precisely, the infection-free steady state is globally attractive when Ri0 both less than unity; if Ri0 larger than unity and Rj0 less than unity (i≠j), then the ith strain outcompetes the jth strain, i.e., the ith strain persists and the jth stain will extinct eventually; if R˜i0>1(i=1,2), then the infection is uniformly persistent within the host. Numerical simulations are conducted to verify these analytical results.

Dynamic analysis of a delayed COVID-19 epidemic with home quarantine in temporal-spatial heterogeneous via global exponential attractor method.

As the COVID-19 epidemic has entered the normalization stage, the task of prevention and control remains very arduous. This paper constructs a time delay reaction-diffusion model that is closer to the actual spread of the COVID-19 epidemic, including relapse, time delay, home quarantine and temporal-spatial heterogeneous environment that affect the spread of COVID-19. These factors increase the number of equations and the coupling between equations in the system, making it difficult to apply the methods commonly used to discuss global dynamics, such as the Lyapunov function method. Therefore, we use the global exponential attractor theory in the infinite-dimensional dynamic system to study the spreading trend of the COVID-9 epidemic with relapse, time delay, home quarantine in a temporal-spatial heterogeneous environment. Using our latest results of global exponential attractor theory, the global asymptotic stability and the persistence of the COVID-19 epidemic are discussed. We find that due to the influence of relapse in the in temporal-spatial heterogeneity environment, the principal eigenvalue can describe the spread of the epidemic more accurately than the usual basic reproduction number . That is, the non-constant disease-free equilibrium is globally asymptotically stable when and the COVID-19 epidemic is persisting uniformly when . Combine with the latest official data of the COVID-19 and the prevention and control strategies of different countries, some numerical simulations on the stability and global exponential attractiveness of the spread of the COVID-19 epidemic in China and the USA are given. The simulation results fully reflect the impact of the temporal-spatial heterogeneous environment, relapse, time delay and home quarantine strategies on the spread of the epidemic, revealing the significant differences in epidemic prevention strategies and control effects between the East and the West. The results of this study provide a theoretical basis for the current epidemic prevention and control.

Propagation dynamics for monotone evolution systems without spatial translation invariance

In this paper, under an abstract setting we establish the existence of spatially inhomogeneous steady states and the asymptotic propagation properties for a large class of monotone evolution systems without spatial translation invariance. Then we apply the developed theory to study traveling waves and spatio-temporal propagation patterns for time-delayed nonlocal equations, reaction-diffusion equations in a cylinder, and asymptotically homogeneous KPP-type equations. We also obtain the existence of steady state solutions and asymptotic spreading properties of solutions for a time-delayed reaction-diffusion equation subject to the Dirichlet boundary condition.

Analysis of a reaction–diffusion SVIR model with a fixed latent period and non-local infections

This paper concerns with a reaction–diffusion susceptible-vaccinated-infectious-recovered model with a fixed latent period. The model is formulated as a non-local and time-delayed reaction–diffusion model due to the fact that an individual infected by the disease in one place may not stay at the same space in the domain due to the movement of human during the incubation period. We then derive the basic reproduction number as the spectral radius of the next infection operator and show that it serves as a threshold role in predicting whether the disease will spread. Further, the explicit formula of basic reproduction number is obtained when all model parameters to be positive constants and the domain to the one-dimensional case. Moreover, we demonstrate the differences in the form of basic reproduction number between the standard incidence and bilinear incidence rate.

Propagation dynamics of Fisher–KPP equation with time delay and free boundaries

Incorporating free boundary into time-delayed reaction-diffusion equations yields a compatible condition that guarantees the well-posedness of the initial value problem. With the KPP type nonlinearity we then establish a vanishing-spreading dichotomy result. Further, when the spreading happens, we show that the spreading speed and spreading profile are nonlinearly determined by a delay-induced nonlocal semi-wave problem. It turns out that time delay slows down the spreading speed.

Travelling fronts in time-delayed reaction-diffusion systems.

We review a series of key travelling front problems in reaction-diffusion systems with a time-delayed feedback, appearing in ecology, nonlinear optics and neurobiology. For each problem, we determine asymptotic approximations for the wave shape and its speed. Particular attention is devoted to their validity and all analytical solutions are compared to solutions obtained numerically. We also extend the work by Erneux et al. (Erneux et al. 2010 Phil. Trans. R. Soc. A 368, 483-493 (doi:10.1098/rsta.2009.0228)) by considering the case of a slowly propagating front subject to a weak delayed feedback. The delay may either speed up the front in the same direction or reverse its direction. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

A nonlocal and time-delayed reaction–diffusion eco-epidemiological predator–prey model

In this paper, we propose and investigate a nonlocal and time-delayed reaction–diffusion predator–prey model with disease spread in the prey which incorporates spatial heterogeneity and assimilation lag of consumed prey biomass into predator biomass. The SI-type subsystem of prey without predator is first studied, and we establish the threshold dynamics in terms of the basic reproduction number based on epidemiology. Then we show a series of sufficient conditions for the co-existence of prey and predator species, the global extinction of predator species and the global extinction of both the prey and predator species. Finally, a special example is explored via numerical simulations, where the influences of nonlocal effect, spatial heterogeneity and time delay on the invasion of the predator species are probed.

Modeling Cell-to-Cell Spread of HIV-1 with Nonlocal Infections

This paper is devoted to develop a nonlocal and time-delayed reaction-diffusion model for HIV infection within host cell-to-cell viral transmissions. In a bounded spatial domain, we study threshold dynamics in terms of basic reproduction numberR0for the heterogeneous model. Our results show that ifR0<1, the infection-free steady state is globally attractive, implying infection becomes extinct, while ifR0>1, virus will persist in the host environment.

Traveling waves for time-delayed reaction diffusion equations with degenerate diffusion

This paper is concerned with time-delayed reaction–diffusion equations with degenerate diffusion. When the term for birth rate is a nonlocal integral with a heat kernel, the family of minimum wave speeds corresponding to all the degenerate diffusion coefficients is proved to admit a uniform positive infimum. However, when the term for birth rate is local, there is no positive infimum of all the minimum wave speeds. This difference indicates that the nonlocal effect plays a role as Laplacian such that a positive lower bound independent of the degenerate diffusion exists for the minimum wave speeds. The approach adopted for the proof is the monotone technique with the viscosity vanishing method. The degeneracy of diffusion for the equation causes us essential difficulty in the proof. A number of numerical simulations are also carried out at the end of the paper, which further numerically confirm our theoretical results.

Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

<p style='text-indent:20px;'>We study the Turing-Hopf bifurcation and give a simple and explicit calculation formula of the normal forms for a general two-components system of reaction-diffusion equation with time delays. We declare that our formula can be automated by Matlab. At first, we extend the normal forms method given by Faria in 2000 to Hopf-zero singularity. Then, an explicit formula of the normal forms for Turing-Hopf bifurcation are given. Finally, we obtain the possible attractors of the original system near the Turing-Hopf singularity by the further analysis of the normal forms, which mainly include, the spatially non-homogeneous steady-state solutions, periodic solutions and quasi-periodic solutions.

Global dynamics for a class of non-monotone time-delayed reaction-diffusion equations

This paper deals with a large class of non-monotone time-delayed reaction-diffusion equations in which the reaction term can be spatially nonlocal. Nonexistence, existence, uniqueness and global attractivity of positive equilibriums to the equation are addressed. In particular, developed is a technique that combines the method of super-sub solutions, the variation-of-constants formula for the delay differential equation and the estimation of integral kernels, which enables us to obtain some sufficient conditions for the global attractiveness of the unique positive equilibrium. Two examples are given to illustrate the obtained results.

Hopf bifurcation in time-delayed Lotka–Volterra competition systems with advection

ABSTRACTIn this paper, we study time-delayed reaction–diffusion systems with advection subject to Lotka–Volterra competition dynamics over one-dimensional domains. These systems model the population dynamics of two groups of competing species, with one dispersing randomly and the other a combination of random and biased dispersal (to avoid competition). We show that time-delay(s) in the interspecific competition mechanism can induce instability of the homogeneous equilibrium to the reaction–advection–diffusion systems, and further promote the appearance of time-oscillating spatially inhomogeneous distributions of the species. Our results indicate that these time-delayed systems (both single and double time-delays) can be used to model the well-observed time-periodic distributions of interacting species in natural fields, compared to the systems without time-delay(s).

Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations

This paper is concerned with the stability of noncritical/critical traveling waves for nonlocal time-delayed reaction-diffusion equation. When the birth rate function is non-monotone, the solution of the delayed equation is proved to converge time-exponentially to some (monotone or non-monotone) traveling wave profile with wave speed $c&gt;c_*$, where $c_*&gt;0$ is the minimum wave speed, when the initial data is a small perturbation around the wave. However, for the critical traveling waves ($c = c_*$), the time-asymptotical stability is only obtained, and the decay rate is not gotten due to some technical restrictions. The proof approach is based on the combination of the anti-weighted method and the nonlinear Halanay inequality but with some new development.

Patterns in a nonlocal time-delayed reaction–diffusion equation

In this paper, the existence, stability, and multiplicity of nontrivial (spatially homogeneous or nonhomogeneous) steady-state solution and periodic solutions for a reaction–diffusion model with nonlocal delay effect and Dirichlet/Neumann boundary condition are investigated by using Lyapunov–Schmidt reduction. Moreover, we illustrate our general results by applications to population models with one-dimensional spatial domain.

Hepatitis C virus infection is blocked by HMGB1: A new nonlocal and time-delayed reaction–diffusion model

This paper is devoted to developing a nonlocal and time-delayed reaction–diffusion model which makes contributions to understanding the mechanism of block effect on HCV by high-mobility group box 1 (HMGB1). Then basic reproduction number R0 is established. The existence of HMGB1 can make R0<1, that is, can induce the extinction of virus. Further, threshold dynamics are investigated in terms of R0 in a bounded spatial domain. From the biological perspective, threshold-type result can predict whether the disease will die out or persist. To investigate the invasion speed of the infectious disease, the existence of traveling waves is investigated in an unbounded spatial domain. Our results imply that the nonlocal time delay cannot change threshold dynamics, while it can slow down the wave speed of virus.

Threshold dynamics of a reaction-diffusion epidemic model with stage structure

A time-delayed reaction-diffusion epidemic model with stage structure and spatial heterogeneity is investigated, which describes the dynamics of disease spread only proceeding in the adult population. We establish the basic reproduction number $\mathcal{R}_0$ for the model system, which gives the threshold dynamics in the sense that the disease will die out if $\mathcal{R}_0＜1$ and the disease will be uniformly persistent if $\mathcal{R}_0&gt;1.$ Furthermore, it is shown that there is at least one positive steady state when $\mathcal{R}_0&gt;1.$ Finally, in terms of general birth function for adult individuals, through introducing two numbers $\check{\mathcal{R}}_0$ and $\hat{\mathcal{R}}_0$ , we establish sufficient conditions for the persistence and global extinction of the disease, respectively.

Time-delayed reaction–diffusion equations with boundary effect: (I) convergence to non-critical traveling waves

This paper is concerned with time-delayed reaction–diffusion equations on half space with boundary effect. When the birth rate function is non-monotone, the solution of the delayed equation subjected to appropriate boundary condition is proved to converge time-exponentially to a certain (monotone or non-monotone) traveling wave profile with wave speed , where is the minimum wave speed, when the initial data is a small perturbation around the wave, while the wave is shifted far away from the boundary so that the boundary layer is sufficiently small. The adopted method is the technical weighted-energy method with some new flavors to handle the boundary terms. However, when the birth rate function is monotone under consideration, then, for all traveling waves with , no matter what size of the boundary layers is, these monotone traveling waves are always globally stable. The proof approach is the monotone technique and squeeze theorem but with some new development.