This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease.This model can be viewed as a nonlocal version of the fre...This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease.This model can be viewed as a nonlocal version of the free boundary problem studied by Kim et al.(An SIR epidemic model with free boundary.Nonlinear Anal RWA,2013,14:1992-2001).We first prove that this problem has a unique solution defined for all time,and then we give sufficient conditions for the disease vanishing and spreading.Our result shows that the disease will not spread if the basic reproduction number R_(0)<1,or the initial infected area h_(0),expanding ability μ and the initial datum S_(0) are all small enough when 1<R_(0)<1+d/μ_(2)+α.Furthermore,we show that if 1<R_(0)<1+d/μ_(2)+α,the disease will spread when h_(0) is large enough or h_(0) is small but μ is large enough.It is expected that the disease will always spread when R_(0)≥1+d/μ_(2)+α which is different from the local model.展开更多
This paper is devoted to the study of a three-dimensional delayed system with nonlocal diffusion and partial quasi-monotonicity. By developing a new definition of upper-lower solutions and a new cross iteration scheme...This paper is devoted to the study of a three-dimensional delayed system with nonlocal diffusion and partial quasi-monotonicity. By developing a new definition of upper-lower solutions and a new cross iteration scheme, we establish some existence results of traveling wave solutions. The results are applied to a nonlocal diffusion model which takes the three-species Lotka-Volterra model as its special case.展开更多
An epidemic model with vaccination and nonlocal diffusion is proposed, and the existence of traveling wave solutions of this model is studied. By the cross-iteration scheme companied with a pair of upper and lower sol...An epidemic model with vaccination and nonlocal diffusion is proposed, and the existence of traveling wave solutions of this model is studied. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder's fixed point theorem,sufficient conditions are obtained for the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.展开更多
In this article, we consider a backward problem in time of the diffusion equation with local and nonlocal operators. This inverse problem is ill-posed because the solution does not depend continuously on the measured ...In this article, we consider a backward problem in time of the diffusion equation with local and nonlocal operators. This inverse problem is ill-posed because the solution does not depend continuously on the measured data. Inspired by the classical Landweber iterative method and Fourier truncation technique, we develops a modified Landweber iterative regularization method to restore the continuous dependence of solution on the measurement data. Under the a-priori and a-posteriori choice rules for the regularized parameter, the convergence estimates for the regularization method are derived. Some results of numerical simulation are provided to verify the stability and feasibility of our method in dealing with the considered problem.展开更多
Foot-and-mouth disease is one of the major contagious zoonotic diseases in the world.It is caused by various species of the genus Aphthovirus of the family Picornavirus,and it always brings a large number of infection...Foot-and-mouth disease is one of the major contagious zoonotic diseases in the world.It is caused by various species of the genus Aphthovirus of the family Picornavirus,and it always brings a large number of infections and heavy financial losses.The disease has become a major public health concern.In this paper,we propose a nonlocal foot-and-mouth disease model in a spatially heterogeneous environment,which couples virus-to-animals and animals-to-animals transmission pathways,and investigate the dynamics of the disperal.The basic reproduction number R_(0)is defined as the spectral radius of the next generation operator R(x)by a renewal equation.The relationship between R_(0)and a principal eigenvalue of an operator L_(0)is built.Moreover,the proposed system exhibits threshold dynamics in terms of R_(0),in the sense that R_(0)determines whether or not foot-and-mouth disease invades the hosts.Through numerical simulations,we have found that increasing animals'movements is an effective control measure for preventing prevalence of the disease.展开更多
We investigate the properties of fundamental,multi-peak,and multi-peaked twisted solitons in three types of finite waveguide lattices imprinted in photorefractive media with asymmetrical diffusion nonlinearity.Two opp...We investigate the properties of fundamental,multi-peak,and multi-peaked twisted solitons in three types of finite waveguide lattices imprinted in photorefractive media with asymmetrical diffusion nonlinearity.Two opposite soliton selfbending signals are considered for different families of solitons.Power thresholdless fundamental and multi-peaked solitons are stable in the low power region.The existence domain of two-peaked twisted solitons can be changed by the soliton self-bending signals.When solitons tend to self-bend toward the waveguide lattice,stable two-peaked twisted solitons can be found in a larger region in the middle of their existence region.Three-peaked twisted solitons are stable in the lower(upper)cutoff region for a shallow(deep)lattice depth.Our results provide an effective guidance for revealing the soliton characteristics supported by a finite waveguide lattice with diffusive nonlocal nonlinearity.展开更多
This research investigates a novel approach to modeling an SIR epidemic in a heterogeneous environment by imposing certain restrictions on population mobility.Our study reveals the influence of partially restricting t...This research investigates a novel approach to modeling an SIR epidemic in a heterogeneous environment by imposing certain restrictions on population mobility.Our study reveals the influence of partially restricting the mobility of the infected population,who are allowed to diffuse locally and can be modeled using random dispersion.In contrast,the non-infective population,which includes susceptible and recovered individuals,has more freedom in their movements.This greater mobility can be modeled using nonlocal dispersion.Our approach is valid for a class of nonlocal dispersion kernels.For the analysis,we first establish the well-posedness of the solution,ensuring the existence,uniqueness,and positivity of this solution.Additionally,we identify the basic reproduction number R0 with its threshold role.Specifically,when R0<1,we prove the global asymptotic stability of the disease-free steady state.Conversely,when R0>1,we demonstrate the corresponding semiflow of the model is uniformly persistent and establish behavior at endemic steady state.Lastly,we examine the asymptotic profiles of the positive steady state as the rate at which susceptible or infected individuals disperse tends to zero or infinity.Our findings reveal that when the movement of infected individuals is restricted,the infection concentrates in specific locations that may be described as the infected preferred spots.展开更多
To study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease, we consider a nonlocal susceptible–infected–susceptibl...To study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease, we consider a nonlocal susceptible–infected–susceptible (SIS) reaction–diffusion model with media coverage, hospital bed numbers and free boundaries. The principal eigenvalue of the integral operator is defined, and the impacts of the diffusion rate of infected individuals and interval length on the principal eigenvalue are analyzed. Furthermore, sufficient conditions for spreading and vanishing of the disease are derived. Our results show that large media coverage and hospital bed numbers are beneficial to the prevention and control of disease. The difference between the model with nonlocal diffusion and that with local diffusion is also discussed and nonlocal diffusion leads to more possibilities.展开更多
We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically ...We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data.We show that for small values of the parameter the corresponding solutions decay to O,while for large values the related solutions converge to 1 uniformly on compacts.Moreover,we prove that the transition from extinction(converging to O)to propagation(converging to 1)is sharp.Numerical results are provided to verify the theoretical results.展开更多
This paper aims to study the existence of traveling wave solutions(TWS)for a three-component noncooperative systems with nonlocal diffusion.Our main results reveal that when a threshold R>1,there exists a critical ...This paper aims to study the existence of traveling wave solutions(TWS)for a three-component noncooperative systems with nonlocal diffusion.Our main results reveal that when a threshold R>1,there exists a critical wave speed c^(*)>0.By using sub-and super-solution methods and Schauder's fixed point theorem,we prove that the system admits a nontrivial TWS for each c≥c^(*).Meanwhile,we show that there exists no nontrivial TWS for c<c^(*)by detailed analysis.Finally,we apply our results to a nonlocal diffusive epidemic model with vaccination,and the boundary asymptotic behavior of TWS for the special case is obtained by constructing a suitable Lyapunov functional.Our research provides some insights on how to deal with the problem of TWS for the nonlocal diffusive epidemic models with bilinear incidence,which extends some results in the previous studies.展开更多
The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Capu...The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)^(-σ)u) = f, 0 < σ <1/2.This paper poses the problem over {t ∈ R^+, x ∈ R^n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.展开更多
基金Zhao was supported by a scholarship from the China Scholarship Council,Li was partially supported by NSF of China(11731005)Cao was partially supported by NSF of China(11901264).
文摘This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease.This model can be viewed as a nonlocal version of the free boundary problem studied by Kim et al.(An SIR epidemic model with free boundary.Nonlinear Anal RWA,2013,14:1992-2001).We first prove that this problem has a unique solution defined for all time,and then we give sufficient conditions for the disease vanishing and spreading.Our result shows that the disease will not spread if the basic reproduction number R_(0)<1,or the initial infected area h_(0),expanding ability μ and the initial datum S_(0) are all small enough when 1<R_(0)<1+d/μ_(2)+α.Furthermore,we show that if 1<R_(0)<1+d/μ_(2)+α,the disease will spread when h_(0) is large enough or h_(0) is small but μ is large enough.It is expected that the disease will always spread when R_(0)≥1+d/μ_(2)+α which is different from the local model.
基金Supported by the Natural Science Foundation of China (11171120)the Doctoral Program of Higher Education of China (20094407110001)Natural Science Foundation of Guangdong Province (10151063101000003)
文摘This paper is devoted to the study of a three-dimensional delayed system with nonlocal diffusion and partial quasi-monotonicity. By developing a new definition of upper-lower solutions and a new cross iteration scheme, we establish some existence results of traveling wave solutions. The results are applied to a nonlocal diffusion model which takes the three-species Lotka-Volterra model as its special case.
基金Supported by the National Natural Science Foundation of China（11371368） Supported by the Natural Science Foundation of Hebei Province（A2013506012） Supported by the Foundation of Shijiazhuang Mechanical Engineering College（JCB1201, YJJXM13008）
文摘An epidemic model with vaccination and nonlocal diffusion is proposed, and the existence of traveling wave solutions of this model is studied. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder's fixed point theorem,sufficient conditions are obtained for the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.
基金supported by the NSF of Ningxia(2022AAC03234)the NSF of China(11761004),the Construction Project of First-Class Disciplines in Ningxia Higher Education(NXYLXK2017B09)the Postgraduate Innovation Project of North Minzu University(YCX23074).
文摘In this article, we consider a backward problem in time of the diffusion equation with local and nonlocal operators. This inverse problem is ill-posed because the solution does not depend continuously on the measured data. Inspired by the classical Landweber iterative method and Fourier truncation technique, we develops a modified Landweber iterative regularization method to restore the continuous dependence of solution on the measurement data. Under the a-priori and a-posteriori choice rules for the regularized parameter, the convergence estimates for the regularization method are derived. Some results of numerical simulation are provided to verify the stability and feasibility of our method in dealing with the considered problem.
基金supported by the National Natural Science Foundation of China(12001339,61573016,11871316)Shanxi Scholarship Council of China(2015-094)+1 种基金the Natural Science Foundation of Shanxi(201801D121006)the Shanxi Province Science Foundation for Youths(201901D211413).
文摘Foot-and-mouth disease is one of the major contagious zoonotic diseases in the world.It is caused by various species of the genus Aphthovirus of the family Picornavirus,and it always brings a large number of infections and heavy financial losses.The disease has become a major public health concern.In this paper,we propose a nonlocal foot-and-mouth disease model in a spatially heterogeneous environment,which couples virus-to-animals and animals-to-animals transmission pathways,and investigate the dynamics of the disperal.The basic reproduction number R_(0)is defined as the spectral radius of the next generation operator R(x)by a renewal equation.The relationship between R_(0)and a principal eigenvalue of an operator L_(0)is built.Moreover,the proposed system exhibits threshold dynamics in terms of R_(0),in the sense that R_(0)determines whether or not foot-and-mouth disease invades the hosts.Through numerical simulations,we have found that increasing animals'movements is an effective control measure for preventing prevalence of the disease.
基金Project supported by the National Natural Science Foundation of China(Grant No.11704339)the Applied Basic Research Program of Shanxi Province,China(Grant No.201901D211466)+1 种基金the Natural Science Basic Research Plan in Shaanxi Province of China(Grant No.2019JM-307)the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(STIP),China(Grant Nos.2019L0896 and 2019L0905)。
文摘We investigate the properties of fundamental,multi-peak,and multi-peaked twisted solitons in three types of finite waveguide lattices imprinted in photorefractive media with asymmetrical diffusion nonlinearity.Two opposite soliton selfbending signals are considered for different families of solitons.Power thresholdless fundamental and multi-peaked solitons are stable in the low power region.The existence domain of two-peaked twisted solitons can be changed by the soliton self-bending signals.When solitons tend to self-bend toward the waveguide lattice,stable two-peaked twisted solitons can be found in a larger region in the middle of their existence region.Three-peaked twisted solitons are stable in the lower(upper)cutoff region for a shallow(deep)lattice depth.Our results provide an effective guidance for revealing the soliton characteristics supported by a finite waveguide lattice with diffusive nonlocal nonlinearity.
文摘This research investigates a novel approach to modeling an SIR epidemic in a heterogeneous environment by imposing certain restrictions on population mobility.Our study reveals the influence of partially restricting the mobility of the infected population,who are allowed to diffuse locally and can be modeled using random dispersion.In contrast,the non-infective population,which includes susceptible and recovered individuals,has more freedom in their movements.This greater mobility can be modeled using nonlocal dispersion.Our approach is valid for a class of nonlocal dispersion kernels.For the analysis,we first establish the well-posedness of the solution,ensuring the existence,uniqueness,and positivity of this solution.Additionally,we identify the basic reproduction number R0 with its threshold role.Specifically,when R0<1,we prove the global asymptotic stability of the disease-free steady state.Conversely,when R0>1,we demonstrate the corresponding semiflow of the model is uniformly persistent and establish behavior at endemic steady state.Lastly,we examine the asymptotic profiles of the positive steady state as the rate at which susceptible or infected individuals disperse tends to zero or infinity.Our findings reveal that when the movement of infected individuals is restricted,the infection concentrates in specific locations that may be described as the infected preferred spots.
基金supported by the Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX21-3188)the I.Ahn is supported by the National Research Foundation of Korea(NRF)grant funded by the Korea government(MSIT)(NRF-2022R1F1A1063068)the Z.Lin is supported by the National Natural Science Foundation of China(Grant No.12271470).
文摘To study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease, we consider a nonlocal susceptible–infected–susceptible (SIS) reaction–diffusion model with media coverage, hospital bed numbers and free boundaries. The principal eigenvalue of the integral operator is defined, and the impacts of the diffusion rate of infected individuals and interval length on the principal eigenvalue are analyzed. Furthermore, sufficient conditions for spreading and vanishing of the disease are derived. Our results show that large media coverage and hospital bed numbers are beneficial to the prevention and control of disease. The difference between the model with nonlocal diffusion and that with local diffusion is also discussed and nonlocal diffusion leads to more possibilities.
基金supported in part by NSFC(Grant Nos.12071175,11171132,11571065)National Research Program of China(Grant No.2013CB834100)+1 种基金by the Natural Science Foundation of jilin Province(Grant Nos.20200201253JC,201902013020JC)by the Project of Science and Technology Development of Jilin Province,China(Grant No.2017C028-1).
文摘We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data.We show that for small values of the parameter the corresponding solutions decay to O,while for large values the related solutions converge to 1 uniformly on compacts.Moreover,we prove that the transition from extinction(converging to O)to propagation(converging to 1)is sharp.Numerical results are provided to verify the theoretical results.
基金supported by National Natural Science Foundation of China(Nos.11971013,12101309)the Fundamental Research Funds for the Colleges and Universities in Heilongjiang Province(No.2022-KYYWF-1113).
文摘This paper aims to study the existence of traveling wave solutions(TWS)for a three-component noncooperative systems with nonlocal diffusion.Our main results reveal that when a threshold R>1,there exists a critical wave speed c^(*)>0.By using sub-and super-solution methods and Schauder's fixed point theorem,we prove that the system admits a nontrivial TWS for each c≥c^(*).Meanwhile,we show that there exists no nontrivial TWS for c<c^(*)by detailed analysis.Finally,we apply our results to a nonlocal diffusive epidemic model with vaccination,and the boundary asymptotic behavior of TWS for the special case is obtained by constructing a suitable Lyapunov functional.Our research provides some insights on how to deal with the problem of TWS for the nonlocal diffusive epidemic models with bilinear incidence,which extends some results in the previous studies.
基金supported by NSG grant DMS-1303632NSF grant DMS-1500871,NSF grant DMS-1209420
文摘The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)^(-σ)u) = f, 0 < σ <1/2.This paper poses the problem over {t ∈ R^+, x ∈ R^n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.
