Periodic Reaction Diffusion Research Articles

A periodic reaction-diffusion model of hospital infection with crowding effects

In this paper, we propose a periodic reaction-diffusion model of hospital infection with crowding effects. We introduce the basic reproduction number R0 for this model and show that the infection-free periodic solution is globally asymptotically stable if R0≤1, while the system admits a globally asymptotically stable positive periodic solution if R0>1. We also obtain the asymptotic behavior of the basic reproduction number when the diffusion rates go to infinity and zero, respectively. Further, we numerically study the effects of diffusion rates, the outflow rate by patient crowding and other parameters on R0 with and without seasonality, and found that the neglect of seasonality does overestimate the infection risk and the disease may be well controlled by avoiding overcrowding of patients rather than increasing the mobility of individuals or bacteria.

A reaction–diffusion vector-borne disease model with incubation period in almost periodic environments

To study the multiple effects of seasonal fluctuations, spatial heterogeneity and extrinsic incubation period on vector-borne disease transmission, a nonlocal almost periodic reaction–diffusion model of vector-borne diseases is proposed and studied. We first give a characterization of the upper Lyapunov exponent λ∗ for a class of linear almost periodic reaction–diffusion systems with time delay, and provide a numerical scheme to compute it. Then we show that λ∗ is a threshold value determining the uniform persistence and extinction of our model. Specifically, the disease will die out when λ∗<0, while the disease is uniformly persistent when λ∗>0. Some numerical simulations finally are presented to verify our theoretical results, and to investigate the effects of diffusion rates, extrinsic incubation period and spatial heterogeneity on disease transmission.

Pulsating traveling fronts of a discrete periodic system with a quiescent stage

In this paper, we study the pulsating traveling fronts for a spatially discrete periodic reaction–diffusion system with a quiescent stage. It is known that there exists a critical number [Formula: see text] (called minimal wave speed) such that a pulsating traveling front exists if and only if its speed is above [Formula: see text]. In this paper, we derive a uniqueness theorem for supercritical pulsating traveling fronts. Further, we show that all supercritical pulsating traveling fronts are exponentially stable.

The principal eigenvalue for partially degenerate and periodic reaction-diffusion systems with time delay

In this paper, we first establish the theory of principal eigenvalues for a large class of partially degenerate, linear and periodic parabolic cooperative systems with time delay. Then we apply these theoretical results to study the global dynamics of a blacklegged tick Ixodes scapularis population model. To present a threshold-type result in terms of basic reproduction ratio R0 for such a model, we also extend the earlier theory of R0 to abstract functional differential equations with time-delayed internal transition.

Dynamics of a periodic benthic-drift model for two species competition

This paper is devoted to the study of a time-periodic model for two species competition in river environments, where the populations reproduce and compete in the benthic zone and diffuse in the drifting water zone. We obtain a complete threshold result on the persistence and extinction of two species in terms of principal eigenvalues for degenerate periodic reaction-diffusion systems, and establish the coexistence of two competing populations under persistence conditions.

Pyramidal traveling fronts of a time periodic diffusion equation with degenerate monostable nonlinearity

This article focuses on the nonplanar traveling fronts of degenerate monostable time periodic reaction-diffusion equations in Rn with n≥3. By constructing a couple of proper supersolution and subsolution, we prove the existence of periodic pyramidal traveling front in R3 and then in Rn with n&gt;3.

On positive periodic solutions of the time–space periodic Lotka–Volterra cooperating system in multi-dimensional media

This work is devoted to the study of the time–space periodic reaction–diffusion–advection Lotka–Volterra cooperating system in multi-dimensional media. By using the method of sub-super solutions and its associated iterations, we prove the existence and uniqueness of the positive periodic solution under appropriate conditions. Finally, we are able to derive the asymptotic behavior of the solutions to the associated Cauchy problem.

Basic reproduction ratios for almost periodic reaction-diffusion epidemic models

The theory of the basic reproduction ratio R0 and its computation formula for the almost periodic reaction-diffusion epidemic models are established. First, we present some dynamical properties for linear almost periodic reaction-diffusion systems. By using evolution semigroup approach, we show that R0−1 has the same sign as the exponential growth bound of an associated linear system. Then, we also apply the developed theory to an almost periodic reaction-diffusion model of vector-borne disease to obtain a threshold type for the uniform persistence and global extinction of the disease. Finally, we illustrate the above results by numerical simulations and show the relationship between the diffusion rate and R0. We also present that the periodic epidemic models may overestimate or underestimate the basic reproduction ratio.

A numerical efficient splitting method for the solution of HIV time periodic reaction–diffusion model having spatial heterogeneity

This study examines a novel reaction–diffusion model for the existence and treatment of acquired immunodeficiency syndrome. This model is a spatial extension of the recent HIV model and human immunodeficiency viruses cause this disorder. The most significant barrier to eradicating this virus is latency and the virus’ subsequent viral reservoir in CD4+ T cells. A nonstandard operator splitting strategy is proposed to approximate the solution of the time-periodic reaction–diffusion model. The main advantages of employing this approach over other techniques are its low computational costs, high accuracy and ease of implementation. The results are truly solid and match those available in the literature. The nature of the solution for the threshold parameter is demonstrated graphically using numerical results. Finally, the M-matrix theory and the positivity of the proposed scheme are discussed.

Bistable traveling waves for time periodic reaction-diffusion equations in strip with Dirichlet boundary condition

We study the existence of bistable traveling wave solutions for time periodic reaction-diffusion equations in straight strips subject to the Dirichlet boundary condition. We first employ a monotone dynamical system framework to establish the existence of such a wave by assuming a bistability structure in terms of multiplicity and stability of periodic solutions in the section of the strip, which is then realized under a set of sufficient explicit conditions; in particular, the bistability structure appears if the reaction term is a time periodic smooth perturbation of the nonlinearity λu(1−u)(u−a) when a∈(0,1/2) and λ>λ⁎ for some λ⁎>0.

Threshold dynamics of a nonlocal and time-delayed West Nile virus model with seasonality

In this paper, we study a nonlocal periodic reaction–diffusion system modeling the West Nile virus transmission between mosquitoes and birds. For the subsystem of mosquito growth, we establish a global stability result on the extinction and persistence in terms of mosquito reproduction number RM, which is defined as the cone spectral radius of a monotone homogeneous map. For the full model, we introduce the basic reproduction number R0 and show that the disease transmission dynamics is determined by the sign of R0−1 under the assumption that RM>1. Moreover, we obtain the global attractivity of the positive constant steady state in the case where all the coefficients are constants. We also conduct numerical simulations to reveal the effects of diffusion, heterogeneity and periodic delay on R0.

Propagation dynamics of a Lotka–Volterra competition model with stage structure in time–space periodic environment

We consider a diffusive Lotka–Volterra competition model with stage structure and time–space periodic habitats. Firstly, we investigate that the system has two periodic semi-trivial solutions (u∗(t,x),0) and (0,v∗(t,x)). Next, the existence of the principal eigenvalue for the eigenvalue problem associated with a linear time–space periodic reaction–diffusion cooperative system with time delay is established. By using eigenvalue functions and semi-trivial solutions, we construct upper and lower solutions. Then by Schauder’s fixed point, there is a pulsating wave solution (U(t,x,x+ct),V(t,x,x+ct)) connecting (0,v∗(t,x)) and (u∗(t,x),0).

Propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay

&lt;p style='text-indent:20px;'&gt;This paper is concerned with a nonlocal time-space periodic reaction diffusion model with age structure. We first prove the existence and global attractivity of time-space periodic solution for the model. Next, by a family of principal eigenvalues associated with linear operators, we characterize the asymptotic speed of spread of the model in the monotone and non-monotone cases. Furthermore, we introduce a notion of transition semi-waves for the model, and then by constructing appropriate upper and lower solutions, and using the results of the asymptotic speed of spread, we show that transition semi-waves of the model in the non-monotone case exist when their wave speed is above a critical speed, and transition semi-waves do not exist anymore when their wave speed is less than the critical speed. It turns out that the asymptotic speed of spread coincides with the critical wave speed of transition semi-waves in the non-monotone case. In addition, we show that the obtained transition semi-waves are actually transition waves in the monotone case. Finally, numerical simulations for various cases are carried out to support our theoretical results.&lt;/p&gt;

Admissible speeds in spatially periodic bistable reaction-diffusion equations

Spatially periodic reaction-diffusion equations typically admit pulsating waves which describe the transition from one steady state to another. Due to the heterogeneity, in general such an equation is not invariant by rotation and therefore the speed of the pulsating wave may a priori depend on its direction. However, little is actually known in the literature about whether it truly does: surprisingly, it is even known in the one-dimensional monostable Fisher-KPP case that the speed is the same in the opposite directions despite the lack of symmetry. Here we investigate this issue in the bistable case and show that the set of admissible speeds is actually rather large, which means that the shape of propagation may indeed be asymmetrical. More precisely, we show in any spatial dimension that one can choose an arbitrary large number of directions, and find a spatially periodic bistable type equation to achieve any combination of speeds in those directions, provided those speeds have the same sign. In particular, in spatial dimension 1 and unlike the Fisher-KPP case, any pair of (either nonnegative or nonpositive) rightward and leftward wave speeds is admissible. We also show that these variations in the speeds of bistable pulsating waves lead to strongly asymmetrical situations in the multistable equations.

Global Dynamics of a Reaction-Diffusion Model of Zika Virus Transmission with Seasonality.

In this paper, we propose a periodic reaction-diffusion model of Zika virus with seasonal and spatial heterogeneous structure in host and vector population. We introduce the basic reproduction ratio [Formula: see text] for this model and show that the disease-free periodic solution is globally asymptotically stable if [Formula: see text], while the system admits a globally asymptotically stable positive periodic solution if [Formula: see text]. Numerically, we study the Zika transmission in Rio de Janeiro Municipality, Brazil, and investigate the effects of some model parameters on [Formula: see text]. We find that the neglect of seasonality underestimates the value of [Formula: see text] and the maximum carrying capacity affects the spread of Zika virus.

Global dynamics of a nonlocal periodic reaction-diffusion model of bluetongue disease

In this paper, we propose a nonlocal reaction-diffusion model of bluetongue disease with seasonality, spatial heterogeneous structure, and periodic extrinsic incubation period. We introduce the basic reproduction ratio R0 for this model and show that the disease-free periodic solution is globally attractive if R0<1, while the disease is uniformly persistent if R0>1. Further, we obtain the global attractivity of the positive steady state in the case where all the coefficients are constants. Numerically, we study the bluetongue transmission in Corsica Island, France, and investigate the impact of some model parameters on R0. We find that the disease risk may be underestimated if the spatial heterogeneity is ignored.

A reaction–diffusion epidemic model with incubation period in almost periodic environments

In this paper, we propose and study an almost periodic reaction–diffusion epidemic model in which disease latency, spatial heterogeneity and general seasonal fluctuations are incorporated. The model is given by a spatially nonlocal reaction–diffusion system with a fixed time delay. We first characterise the upper Lyapunov exponent λ* for a class of almost periodic reaction–diffusion equations with a fixed time delay and provide a numerical method to compute it. On this basis, the global threshold dynamics of this model is established in terms of λ* It is shown that the disease-free almost periodic solution is globally attractive if λ* &lt; 0, while the disease is persistent if λ* &gt; 0. By virtue of numerical simulations, we investigate the effects of diffusion rate, incubation period and spatial heterogeneity on disease transmission.

Time periodic reaction–diffusion equations for modeling 2-LTR dynamics in HIV-infected patients

Considering multiple effects of spatial factors (spatial heterogeneity and mobility) and temporal heterogeneity (the periodic administration of the drug) in HIV infection and treatment, we formulate a time periodic reaction–diffusion equation model to study the HIV viral dynamics. The basic reproduction number R0 is derived and the global threshold dynamics is explored in terms of R0. When R0<1, the infection-free periodic solution is globally attractive. When R0>1, the model system is uniformly persistent. The global stability analysis is also performed in the critical case of R0=1 with spatial heterogeneity but without time periodicity. The global asymptotic stability of the infection steady state for the spatial homogeneous model is discussed. Furthermore, numerical simulations reveal that (i) the value of R0 depends on spatial heterogeneity, which indicates that different human individuals or different organs in one individual may have different infection states even if the drug efficacy is the same; and (ii) the 2-LTR level at the steady state may increase and the steady-state viral load may decrease dramatically if raltegravir is administered from the beginning of thetreatment.

The dynamics of a zooplankton–fish system in aquatic habitats

Diel vertical migration is a common movement pattern of zooplankton in marine and freshwater habitats. In this paper, we use a temporally periodic reaction–diffusion–advection system to describe the dynamics of zooplankton and fish in aquatic habitats. Zooplankton live in both the surface water and the deep water, while fish only live in the surface water. Zooplankton undertake diel vertical migration to avoid predation by fish during the day and to consume sufficient food in the surface water during the night. We establish the persistence theory for both species as well as the existence of a time-periodic positive solution to investigate how zooplankton manage to maintain a balance with their predators via vertical migration. Numerical simulations discover the effects of migration strategy, advection rates, domain boundary conditions, as well as spatially varying growth rates, on persistence of the system.

Complex dynamics of a time periodic nonlocal and time-delayed model of reaction–diffusion equations for modeling CD4[formula omitted] T cells decline

Pyroptosis, an intensively inflammatory form of programmed cell death triggered during abortive HIV infection, is associated with the release of inflammatory cytokines. The inflammatory cytokines can attract more CD4 + T cells to be infected. Based on the new biological perspective, a time periodic reaction–diffusion equation model with spatial heterogeneity and latent period is developed to investigate whether or not pyroptosis can explain CD4+ T cells decline during HIV infection. Threshold dynamics is explored in terms of the basic reproduction number ℛ0. It is shown that the infection-free periodic solution is globally attractive if ℛ0<1, while there exists positive infection periodic solution and the virus is uniformly persistent if ℛ0>1, which might be a new finding for viral infection dynamical models. Theoretical analyses and simulations for the spatially homogeneous model demonstrate rich dynamics, including the occurrence of Turing instability, which may result in Hopf bifurcation and spatially inhomogeneous pattern formations. It turns out that the new model issues some new challenges due to the enhancement infection term in the global stability problem and Turing instability analysis for the high dimensional system. Our results reveal that the inflammatory cytokines can make the CTLs level increase, which is a new phenomenon not presented in the existing literature.