Basic Reproduction Ratios for Periodic Abstract Functional Differential Equations (with Application to a Spatial Model for Lyme Disease)

Liang, Xing; Zhang, Lei; Zhao, Xiao-Qiang

doi:10.1007/s10884-017-9601-7

Cited by 141 publications

(67 citation statements)

References 40 publications

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“…According to Ref. [15], R 0 = s * , where s * is a unique principal eigenvalue of problem (18) , and R 0 increases monotonically with respect to β(x) based on Ref. [50] and the results given in Theorem 2.…”

Section: Remarkmentioning

confidence: 86%

“…It has been universally accepted that spatial diffusion and environmental heterogeneity are crucial factors that should be considered in the infectious disease transmission [17,25,33,20]. Since the pioneering work of Allen et al [2] on the influence of the environmental heterogeneity and the movement of individuals on disease transmission, there are many scholars begin to study the reaction-diffusion equations in a spatially heterogeneous environment [3,4,15,41]. Lou and Zhao [16] proposed a reaction-diffusion malaria model with incubation period in the vector population, in which several system parameters were spatially dependent.…”

Section: Introductionmentioning

confidence: 99%

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A theoretical approach to understanding rumor propagation dynamics in a spatially heterogeneous environment

Zhu¹,

Liu²,

Zhang³

2021

DCDS-B

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Most of the previous work on rumor propagation either focus on ordinary differential equations with temporal dimension or partial differential equations (PDE) with only consideration of spatially independent parameters. Little attention has been given to rumor propagation models in a spatiotemporally heterogeneous environment. This paper is dedicated to investigating a SCIR reaction-diffusion rumor propagation model with a general nonlinear incidence rate in both heterogeneous and homogeneous environments. In spatially heterogeneous case, the well-posedness of global solutions is established first. The basic reproduction number R 0 is introduced, which can be used to reveal the threshold-type dynamics of rumor propagation: if R 0 < 1, the rumor-free steady state is globally asymptotically stable, while R 0 > 1, the rumor is uniformly persistent. In spatially homogeneous case, after introducing the time delay, the stability properties have been extensively studied. Finally, numerical simulations are presented to illustrate the validity of the theoretical analysis and the influence of spatial heterogeneity on rumor propagation is further demonstrated.

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Section: Remarkmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

A theoretical approach to understanding rumor propagation dynamics in a spatially heterogeneous environment

Zhu¹,

Liu²,

Zhang³

2021

DCDS-B

View full text Add to dashboard Cite

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“…These two definitions are consistent for a scalar equation. There are also developments of R 0 in a temporal and spatially heterogeneous environment (see, e.g., [3,10,13,17,20,21,25]).…”

mentioning

confidence: 99%

“…Several approaches have been proposed to compute R 0 numerically in periodic models (see, e.g., [18,21,24] and the references therein). Liang, Zhang, and Zhao [13] proposed a method to compute spectral radius numerically, based on the well-known power method (see, e.g., [12]). Their method can be applied to various types of periodic models, such as ODEs, reaction-diffusion equations, and nonlocal dispersal equations with or without time delay.…”

mentioning

confidence: 99%

“…It is also shown that R 0 on a one-dimensional interval is no more than that in the corresponding two-dimensional case, under the same specific setting. Besides, we propose a numerical algorithm to compute R 0 in a two-dimensional rectangular domain based on the methods in [13,22]. Numerical results reveal the complicated effects of the spatial variables on the basic reproduction ratio, that is, R 0 may be monotone or non-monotone with respect to some variable in different settings.…”

mentioning

confidence: 99%

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The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model

Yang¹,

Qarariyah²,

Yang³

2022

DCDS-B

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In this paper, we study the influence of spatial-dependent variables on the basic reproduction ratio (R 0 ) for a scalar reaction-diffusion equation model. We first investigate the principal eigenvalue of a weighted eigenvalue problem and show the influence of spatial variables. We then apply these results to study the effect of spatial heterogeneity and dimension on the basic reproduction ratio for a spatial model of rabies. Numerical simulations also reveal the complicated effects of the spatial variables on R 0 in two dimensions.

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Dynamics of a dengue fever model with vertical transmission and time periodic in spatially heterogeneous environments

Zhao

2021

Math Methods in App Sciences

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To reflect the impacts of vertical transmission in human and mosquitoes on the transmission of dengue fever, we propose an SIS‐SI dengue fever model with time periodic in spatially heterogeneous environments. Firstly, we obtain the properties of the basic reproduction number R0 for the dengue model by the eigenvalue problem and the spectral radius of next infection operator. Secondly, we investigate the existence and attractivity of the positive ω‐periodic solution for the dengue fever model. We also investigate the stability of ω‐periodic solution. The theoretical results indicate that vertical transmission can enhance the persistence of dengue virus in spatially heterogeneous environments with time periodic when R0 > 1. We perform numerical simulations to illustrate theoretical results of dengue fever model and give some epidemiological explanations. Finally, we combine medical advice to seek some effective prevention and control measures for vertical transmission.

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Basic Reproduction Ratios for Periodic Abstract Functional Differential Equations (with Application to a Spatial Model for Lyme Disease)

Cited by 141 publications

References 40 publications

A theoretical approach to understanding rumor propagation dynamics in a spatially heterogeneous environment

A theoretical approach to understanding rumor propagation dynamics in a spatially heterogeneous environment

The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model

Dynamics of a dengue fever model with vertical transmission and time periodic in spatially heterogeneous environments

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