Spatial Heterogeneous Environment Research Articles

Global Dynamics Analysis of Non-Local Delayed Reaction-Diffusion Avian Influenza Model with Vaccination and Multiple Transmission Routes in the Spatial Heterogeneous Environment

Global Dynamics Analysis of Non-Local Delayed Reaction-Diffusion Avian Influenza Model with Vaccination and Multiple Transmission Routes in the Spatial Heterogeneous Environment

Qualitative analysis on a reaction–diffusion SIS epidemic model with nonlinear incidence and Dirichlet boundary

In this paper, the dynamical behavior in a spatially heterogeneous reaction–diffusion SIS epidemic model with general nonlinear incidence and Dirichlet boundary condition is investigated. The well-posedness of solutions, including the global existence, nonnegativity, ultimate boundedness, as well as the existence of compact global attractor, are first established, then the basic reproduction number R0 is calculated by defining the next generation operator. Secondly, the threshold dynamics of the model with respect to R0 are studied. That is, when R0<1 the disease-free steady state is globally asymptotically stable, and when R0>1 the model is uniformly persistent and admits one positive steady state, and under some additional conditions the uniqueness of positive steady state is obtained. Furthermore, some interesting properties of R0 are established, including the calculating formula of R0, the asymptotic profiles of R0 with respect to diffusion rate dI, and the monotonicity of R0 with diffusion rate dI and domain Ω. In addition, the bang–bang-type configuration optimization of R0 also is obtained. This rare result in diffusive equation reveals that we can control disease diffusion at least at one peak. Finally, the numerical examples and simulations are carried out to illustrate the rationality of open problems proposed in this paper, and explore the influence of spatial heterogeneous environment on the disease spread and make a comparison on dynamics between Dirichlet boundary condition and Neumann boundary condition.

The bifurcation analysis for a degenerate reaction–diffusion host–pathogen model

In this paper, we mainly concern with the bifurcation problem of positive steady states for a degenerate reaction–diffusion host–pathogen model with spatial heterogeneous environment. A criterion on the existence of bifurcation of positive steady states from the disease-free steady state is established. Our results reveal that the mortality rate of the incubation host affects the existence of the positive steady state solutions. This provides the extensive insight on controlling disease by adjusting the mortality of incubation host.

Role of media coverage in a SVEIR-I epidemic model with nonlinear incidence and spatial heterogeneous environment.

In this paper, we propose a SVEIR-I epidemic model with media coverage in a spatially heterogeneous environment, and study the role of media coverage in the spread of diseases in a spatially heterogeneous environment. In a spatially heterogeneous environment, we first set up the well-posedness of the model. Then, we define the basic reproduction number $ R_0 $ of the model and establish the global dynamic threshold criteria: when $ R_0 < 1 $, disease-free steady state is globally asymptotically stable, while when $ R_0 > 1 $, the model is uniformly persistent. In addition, the existence and uniqueness of the equilibrium state of endemic diseases were obtained when $ R_0 > 1 $ in homogeneous space and heterogeneous diffusion environment. Further, by constructing appropriate Lyapunov functions, the global asymptotic stability of disease-free and positive steady states was established. Finally, through numerical simulations, it is shown that spatial heterogeneity can increase the risk of disease transmission, and can even change the threshold for disease transmission; media coverage can make people more widely understand disease information, and then reduce the effective contact rate to control the spread of disease.

Hyperbolic limit for a biological invasion

In a spatially heterogeneous environment the propagation speed of a biological invasion varies in space. The traveling wave theory in a homogeneous case is not extended to a heterogeneous case. Taking a singular limit in a hyperbolic scale is a good way to study such a wave propagation with constant speed. The goal of this project is to understand the effect of biological diffusion on the wave speed in a spatial heterogeneous environment. For this purpose, we consider \begin{document}$ U_t = {\varepsilon}(\gamma(s)U)_{xx}+{1\over{\varepsilon}}U(1-{U/m(x)}), $\end{document} where $ m $ is a nonconstant carrying capacity, $ s = {U\over m} $ is a starvation measure and $ \gamma(s) = s^{\tilde{k}}, \tilde{k} \ge 1 $. The diffusion is a starvation driven diffusion. We show that the diffusion speed is constant even if $ m $ is nonconstant.

Spatial dynamics and optimization method for a rumor propagation model in both homogeneous and heterogeneous environment.

Considering the influence of environmental capacity and forgetting on rumor spreading, we improve the traditional SIR (susceptible–infected–removed) rumor propagation model and give two dynamic models of rumor propagation in heterogeneous environment and homogeneous environment, respectively. The main purpose of this paper is to make a dynamic analysis of rumor propagation models. In the spatial heterogeneous environment, we have analyzed the uniform persistence of the rumor propagation model and the asymptotic behavior of the positive equilibrium point when the diffusion rate of rumor–susceptible tends to zero. In the spatial homogeneous environment, we discuss the stability of rumor propagation model. Further, optimal control and the necessary optimality conditions are obtained by using the maximum principle. Finally, we study the Hopf bifurcation phenomenon through inducing time delay in the reaction–diffusion model. In addition, the existence of Hopf bifurcation is verified and the influence of diffusion coefficients is studied by numerical simulations.

Global dynamics analysis of a Zika transmission model with environment transmission route and spatial heterogeneity

&lt;abstract&gt;&lt;p&gt;Zika virus, a recurring mosquito-borne flavivirus, became a global public health agency in 2016. It is mainly transmitted through mosquito bites. Recently, experimental result demonstrated that $ Aedes $ mosquitoes can acquire and transmit Zika virus by breeding in contaminated aquatic environments. The environmental transmission route is unprecedented discovery for the Zika virus. Therefore, it is necessary to introduce environment transmission route into Zika model. Furthermore, we consider diffusive terms in order to capture the movement of humans and mosquitoes. In this paper, we propose a novel reaction-diffusion Zika model with environment transmission route in a spatial heterogeneous environment, which is different from all Zika models mentioned earlier. We introduce the basic offspring number $ R_{0}^{m} $ and basic reproduction number $ R_{0} $ for this spatial model. By using comparison arguments and the theory of uniform persistence, we prove that disease free equilibrium with the absence of mosquitoes is globally attractive when $ R_{0}^{m} &amp;lt; 1 $, disease free equilibrium with the presence of mosquitoes is globally attractive when $ R_{0}^{m} &amp;gt; 1 $ and $ R_{0} &amp;lt; 1 $, the model is uniformly persistent when $ R_{0}^{m} &amp;gt; 1 $ and $ R_{0} &amp;gt; 1 $. Finally, numerical simulations conform these analytical results.&lt;/p&gt;&lt;/abstract&gt;

Viral infection dynamics in a spatial heterogeneous environment with cell-free and cell-to-cell transmissions.

In this paper, we investigate a diffusive viral infection model in a spatial heterogeneous environment with two types of infection mechanisms and distinct dispersal rates for the susceptible and infected target cells. After establishing well-posedness of the model system, we identify the basic reproduction number R0 and explore the properties of R0 when the dispersal rate for infected target cells varies from zero to infinity. Moreover, we demonstrate that the basic reproduction number is a threshold parameter: the infection and virus will be cleared out if R0 ≤ 1, while if R0 > 1, the infection will persist and the model system admits at least one positive (chronic infection) steady state. For the special case when all model parameters are spatial homogeneous, this chronic infection steady state is unique and globally asymptotically stable.

Spread trend of COVID-19 epidemic outbreak in China: using exponential attractor method in a spatial heterogeneous SEIQR model.

In this paper we introduce a method of global exponential attractor in the reaction-diffusion epidemic model in spatial heterogeneous environment to study the spread trend and long-term dynamic behavior of the COVID-19 epidemic. First, we prove the existence of the global exponential attractor of general dissipative evolution systems. Then, by using the existence theorem, the global asymptotic stability and the persistence of epidemic are discussed. Finally, combine with the official data of the COVID-19 and the national control strategy, some numerical simulations on the stability and global exponential attractiveness of the COVID-19 epidemic are given. Simulations show that the spread trend of the epidemic is in line with our theoretical results, and the preventive measures taken by the Chinese government are effective.

Dynamics of a reaction–diffusion SVIR model in a spatial heterogeneous environment

In this paper, a reaction–diffusion SVIR epidemic model in a spatial heterogeneous environment is proposed. We defined the basic reproduction number ℜ0 and showed that it is a threshold parameter, which determines the disease extinction or persistence in the case of a bounded domain. The global attractiveness of the constant positive steady state and the explicit formula of ℜ0 are obtained when the space is homogeneous. Simulation results reveal that the spatial heterogeneity can enhance the spread risk of the disease. It is found that the distribution of infected individuals is affected by different diffusion rate and its prevalence becomes higher when a larger diffusion rate is used. The relationship among the basic reproduction number, the vaccination rate and recovery rate of vaccinated individuals and infected individuals are also addressed. If the recovery rates of vaccinated individuals and infected individuals are sufficiently large, disease can be eradicated by conducting suitable vaccination strategy, which reveals that increasing the recovery rates of vaccinated individuals and infected individuals seems more important than increasing vaccination rate.

Influence of spatial heterogeneous environment on long-term dynamics of a reaction-diffusion SVIR epidemic model with relaps.

In this paper by adding the factors of disease relapse and vaccination in the space hetero-geneous environment, we establish and discuss a class of reaction-diffusion SVIR model with relapse and a varying external source in spatial heterogeneous environment. By applying a different method than the Lyapunov function, we study the long-term dynamic behavior of this model by means of global exponential attractor theory and gradient flow method. The global asymptotic stability and the persistence of epidemic are proved. To test the validity of our theoretical results, we choose some specific epidemic disease with some more practical and more definitive official data to simulate the global stability and exponential attraction of the model. The simulation results showed that the factors of disease relapse, vaccination and spatial heterogeneity had a great influence on the persists uniformly of the disease.

Complex Dynamics of a host–parasite model with both horizontal and vertical transmissions in a spatial heterogeneous environment

In this paper, we investigate the dynamical outcomes of a host–parasite model incorporating both horizontal and vertical transmissions in a spatial heterogeneous environment analytically and numerically. Our study provides valuable insights in two aspects: Mathematically, we propose three threshold parameters, the demographic reproduction number Rd, the horizontal transmission reproduction number R0h and the vertical transmission reproduction number R0v, to identify the conditions that lead to disease-free dynamics, or susceptible-free dynamics, or endemic dynamics. Epidemiologically, we find that both host population movements and spatial heterogeneity strongly affect the disease dynamics of our proposed epidemic model: (1) the larger random mobility can result in 100% infection prevalence; and (2) the heterogeneity tends to enhance the persistence of the infected hosts with uninfected ones. As a consequence, our work suggests that, in order to control the invasion of the parasite, different preventive measures can be implemented in different regions.

A rare mutation model in a spatial heterogeneous environment

We propose a stochastic model in evolutionary game theory where individuals (or subpopulations) can mutate changing their strategies randomly (but rarely) and explore the external environment. This environment affects the selective pressure by modifying the payoff arising from the interactions between strategies. We derive a Fokker–Planck integro-differential equation and provide Monte Carlo simulations for the Hawks vs Doves game. In particular we show that, in some cases, taking into account the external environment favors the persistence of the low-fitness strategy.

A nonlocal diffusion model with free boundaries in spatial heterogeneous environment

This paper is concerned with the nonlocal diffusive model with double free boundaries in spatial heterogeneous environment, where the spatial heterogeneity is described by the sign indefinite coefficients. Such a model can be used to illustrate the spreading or vanishing of a new or invasive species. Due to the lack of comparison principle in the nonlocal reaction–diffusion equation, many classical methods cannot be used directly to this nonlocal problem. This motivates us to find new techniques. We first establish the spreading–vanishing dichotomy as well as some criteria that ensure the species spreading or vanishing by principal eigenvalues of associated scalar elliptic eigenvalue problems. And then we determine the spreading speed when spreading occurs.

SPATIAL HETEROGENEITY, FAST MIGRATION AND COEXISTENCE OF INTRAGUILD PREDATION DYNAMICS

In this paper, we investigate effects of spatial heterogeneous environment and fast migration of individuals on the coexistence of the intraguild predation (IGP) dynamics. We present a two-patch model. We assume that on one patch two species compete for a common resource, and on the other patch one species can capture the other one for the maintenance. We also assume IGP individuals are able to migrate between the two patches and the migration process acts on a fast time scale in comparison with demography, predation and competition processes. We show that under certain conditions the heterogeneous environment and fast migration can lead to coexistence of the two species.

Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment

In this paper, we explore a parasite-host epidemiological model incorporating demographic and epidemiologicalprocesses in a spatially heterogeneous environment in which the individuals are subject to a random movement. We show the global stability of the extinction equilibrium in three different cases, and prove the existence, uniqueness and the global stability of the disease--free equilibrium. When the death rate in the model becomes a constant, we give the existence of the endemic equilibrium and the global stability of the endemic equilibrium in a special case. Furthermore, we perform a series of numerical simulations to display the effects of the movement of hosts and the heterogeneous environment on the disease dynamics. Our analytical and numerical results reveal that the disease extinction/outbreak can be ignited by both individual mobility and the environmental heterogeneity.

Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment

In this paper, we propose a mathematical model to describe the avianinfluenza dynamics in wild birds with bird mobility andheterogeneous environment incorporated. In addition to establishingthe basic properties of solutions to the model, we also prove thethreshold dynamics which can be expressed either by the basicreproductive number or by the principal eigenvalue of thelinearization at the disease free equilibrium. When the environmentfactor in the model becomes a constant (homogeneous environment), weare able to find explicit formulas for the basic reproductivenumber and the principal eigenvalue. We also perform numericalsimulation to explore the impact of the heterogeneous environment onthe disease dynamics. Our analytical and numerical results revealthat the avian influenza dynamics in wild birds is highly affectedby both bird mobility and environmental heterogeneity.

DYNAMICAL ANALYSIS AND CONTROL IN A DELAYED DIFFERENTIAL-ALGEBRAIC BIO-ECONOMIC MODEL WITH STAGE STRUCTURE AND DIFFUSION

Nowadays, biological resource in prey-predator ecosystem is commercially harvested and sold with aim of achieving economic interest. Furthermore, harvest effort is usually influenced by variation of economic interest of harvesting and spatial heterogeneous environment. In this paper, a delayed differential-algebraic bio-economic model is proposed, which is utilized to investigate interaction and coexistence mechanism of biological population in the harvested ecosystem due to the variation of economic interest of harvesting as well as the change of population spatial diffusion and gestation delay. Local stability analysis of the proposed model without gestation delay and diffusion reveals that there is a phenomenon of singularity induced bifurcation due to the variation of economic interest of harvesting, and state feedback controllers are designed to stabilize the proposed model at the interior equilibrium. Furthermore, local stability of the proposed model with gestation delay and diffusion is studied, which reveals that the interior equilibrium loses its stability at some critical values of gestation delay and corresponding cycle occurs. It is also shown that population spatial diffusion and harvesting have a stabilizing effect on population dynamics. Finally, numerical simulations are carried out to show consistency with theoretical analysis obtained in this paper.