Dynamics of an infection age-space structured cholera model with Neumann boundary condition

2021 ◽  
pp. 1-30
Author(s):  
WEIWEI LIU ◽  
JINLIANG WANG ◽  
RAN ZHANG
Keyword(s):  
Vibrio Cholerae ◽  
Disease Transmission ◽  
Global Attractivity ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Human Populations ◽  
Global Dynamics ◽  
Infection Age ◽  
Reaction Diffusion Model ◽  
Disease Persistence

This paper investigates global dynamics of an infection age-space structured cholera model. The model describes the vibrio cholerae transmission in human population, where infection-age structure of vibrio cholerae and infectious individuals are incorporated to measure the infectivity during the different stage of disease transmission. The model is described by reaction–diffusion models involving the spatial dispersal of vibrios and the mobility of human populations in the same domain Ω ⊂ ℝ n . We first give the well-posedness of the model by converting the model to a reaction–diffusion model and two Volterra integral equations and obtain two constant equilibria. Our result suggest that the basic reproduction number determines the dichotomy of disease persistence and extinction, which is achieved by studying the local stability of equilibria, disease persistence and global attractivity of equilibria.

Dynamics of a susceptible–infected–susceptible epidemic reaction–diffusion model

2016 ◽  
Vol 146 (5) ◽  
pp. 929-946 ◽  
Author(s):  
Keng Deng ◽  
Yixiang Wu
Keyword(s):  
Diffusion Model ◽  
Disease Transmission ◽  
Global Attractivity ◽  
Diffusion Rate ◽  
Reaction Diffusion ◽  
Endemic Equilibrium ◽  
Recovery Rates ◽  
Reaction Diffusion Model ◽  
Globally Attractive ◽  
Disease Free

We study a susceptible–infected–susceptible reaction–diffusion model with spatially heterogeneous disease transmission and recovery rates. A basic reproduction number is defined for the model. We first prove that there exists a unique endemic equilibrium if . We then consider the global attractivity of the disease-free equilibrium and the endemic equilibrium for two cases. If the disease transmission and recovery rates are constants or the diffusion rate of the susceptible individuals is equal to the diffusion rate of the infected individuals, we show that the disease-free equilibrium is globally attractive if , while the endemic equilibrium is globally attractive if .

scholarly journals Combating Cholera

F1000Research ◽  
2019 ◽  
Vol 8 ◽  
pp. 589 ◽  
Author(s):  
Brian Y. Hsueh ◽  
Christopher M. Waters
Keyword(s):  
Vibrio Cholerae ◽  
Disease Transmission ◽  
Phage Therapy ◽  
Emerging Technologies ◽  
Oral Rehydration Therapy ◽  
Health Surveillance ◽  
Human Populations ◽  
Flexible Approach ◽  
Oral Rehydration ◽  
Current Therapies

Cholera infections caused by the gamma-proteobacterium Vibrio cholerae have ravaged human populations for centuries, and cholera pandemics have afflicted every corner of the globe. Fortunately, interventions such as oral rehydration therapy, antibiotics/antimicrobials, and vaccines have saved countless people afflicted with cholera, and new interventions such as probiotics and phage therapy are being developed as promising approaches to treat even more cholera infections. Although current therapies are mostly effective and can reduce disease transmission, cholera outbreaks remain deadly, as was seen during recent outbreaks in Haiti, Ethiopia, and Yemen. This is due to significant underlying political and socioeconomic complications, including shortages of vaccines and clean food and water and a lack of health surveillance. In this review, we highlight the strengths and weaknesses of current cholera therapies, discuss emerging technologies, and argue that a multi-pronged, flexible approach is needed to continue to reduce the worldwide burden of cholera.

scholarly journals Spatio-temporal Brusselator Model and Biological Pattern Formation

2021 ◽  
pp. 88-99
Author(s):  
Zakir Hossine ◽  
Oishi Khanam ◽  
Md. Mashih Ibn Yasin Adan ◽  
Md. Kamrujjaman
Keyword(s):  
Pattern Formation ◽  
Diffusion Model ◽  
Initial Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Brusselator Model ◽  
Reaction Diffusion Model ◽  
Biological Pattern Formation ◽  
Spatio Temporal

This paper explores a two-species non-homogeneous reaction-diffusion model for the study of pattern formation with the Brusselator model. We scrutinize the pattern formation with initial conditions and Neumann boundary conditions in a spatially heterogeneous environment. In the whole investigation, we assume the case for random diffusion strategy. The dynamics of model behaviors show that the nature of pattern formation with varying parameters and initial conditions thoroughly. The model also studies in the absence of diffusion terms. The theoretical and numerical observations explain pattern formation using the reaction-diffusion model in both one and two dimensions.

scholarly journals Domain-Dependent Stability Analysis of a Reaction–Diffusion Model on Compact Circular Geometries

2018 ◽  
Vol 28 (08) ◽  
pp. 1830024
Author(s):  
Wakil Sarfaraz ◽  
Anotida Madzvamuse
Keyword(s):  
Numerical Solutions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Domain Size ◽  
Reaction Rates ◽  
Polar Coordinates ◽  
Angular Variable ◽  
Diffusion Operator ◽  
Nodal Lines ◽  
Reaction Diffusion Model

In this work an activator-depleted reaction–diffusion system is investigated on polar coordinates with the aim of exploring the relationship and the corresponding influence of domain size on the types of possible diffusion-driven instabilities. Quantitative relationships are found in the form of necessary conditions on the area of a disk-shape domain with respect to the diffusion and reaction rates for certain types of diffusion-driven instabilities to occur. Robust analytical methods are applied to find explicit expressions for the eigenvalues and eigenfunctions of the diffusion operator on a disk-shape domain with homogenous Neumann boundary conditions in polar coordinates. Spectral methods are applied using Chebyshev nonperiodic grid for the radial variable and Fourier periodic grid for the angular variable to verify the nodal lines and eigen-surfaces subject to the proposed analytical findings. The full classification of the parameter space in light of the bifurcation analysis is obtained and numerically verified by finding the solutions of the partitioning curves inducing such a classification. Spatiotemporal periodic behavior is demonstrated in the numerical solutions of the system for a proposed choice of parameters and a rigorous proof of the existence of infinitely many such points in the parameter plane is presented under a restriction on the area of the domain, with a lower bound in terms of reaction–diffusion rates.

Dynamics Analysis in a Gierer–Meinhardt Reaction–Diffusion Model with Homogeneous Neumann Boundary Condition

2019 ◽  
Vol 29 (09) ◽  
pp. 1930025 ◽  
Author(s):  
Xiang-Ping Yan ◽  
Ya-Jun Ding ◽  
Cun-Hua Zhang
Keyword(s):  
Boundary Condition ◽  
Hopf Bifurcation ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Equations ◽  
Positive Equilibrium ◽  
Reaction Diffusion Model ◽  
Homogeneous Neumann Boundary Condition

A reaction–diffusion Gierer–Meinhardt system with homogeneous Neumann boundary condition on one-dimensional bounded spatial domain is considered in the present article. Local asymptotic stability, Turing instability and existence of Hopf bifurcation of the constant positive equilibrium are explored by analyzing in detail the associated eigenvalue problem. Moreover, properties of spatially homogeneous Hopf bifurcation are carried out by employing the normal form method and the center manifold technique for reaction–diffusion equations. Finally, numerical simulations are also provided in order to check the obtained theoretical conclusions.

scholarly journals Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 91
Author(s):  
Costică Moroşanu ◽  
Silviu Pavăl
Keyword(s):  
Mathematical Model ◽  
Image Segmentation ◽  
A Priori Estimates ◽  
A Priori ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Second Order ◽  
Reaction Diffusion Model ◽  
Anisotropic Reaction Diffusion ◽  
Physical Phenomena

In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f(t,x), w(t,x) and v0(x), we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space Wp1,2(Q), facilitating for the present model to be a more complete description of certain classes of physical phenomena. The second topic refers to the construction of two numerical schemes in order to approximate the solution of a particular mathematical model (local and nonlocal case). To illustrate the effectiveness of the new mathematical model, we present some numerical experiments by applying the model to image segmentation tasks.

scholarly journals Analysis on a diffusive SEI epidemic model with/without immigration of infected hosts

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chengxia Lei ◽  
Yi Shen ◽  
Guanghui Zhang ◽  
Yuxiang Zhang
Keyword(s):  
Epidemic Model ◽  
Disease Transmission ◽  
Lyapunov Functions ◽  
Migration Rate ◽  
Global Attractivity ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Transmission Risk ◽  
Threshold Behaviour ◽  
Homogeneous Environment

<p style='text-indent:20px;'>In this paper, we study a reaction-diffusion SEI epidemic model with/without immigration of infected hosts. Our results show that if there is no immigration for the infected (exposed) individuals, the model admits a threshold behaviour in terms of the basic reproduction number, and if the system includes the immigration, the disease always persists. In each case, we explore the global attractivity of the equilibrium via Lyapunov functions in the case of spatially homogeneous environment, and investigate the asymptotic behavior of the endemic equilibrium (when it exists) with respect to the small migration rate of the susceptible, exposed or infected population in the case of spatially heterogeneous environment. Our results suggest that the strategy of controlling the migration rate of population can not eradicate the disease, and the disease transmission risk will be underestimated if the immigration of infected hosts is ignored.</p>

scholarly journals Numerical continuation of spiral waves in heteroclinic networks of cyclic dominance

2021 ◽  
Author(s):  
Cris R Hasan ◽  
Hinke M Osinga ◽  
Claire M Postlethwaite ◽  
Alastair M Rucklidge
Keyword(s):  
Boundary Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Spiral Waves ◽  
Numerical Continuation ◽  
Heteroclinic Cycles ◽  
Reaction Diffusion Model ◽  
Inner Radius ◽  
Set Up ◽  
Low Dimensional

Abstract Heteroclinic-induced spiral waves may arise in systems of partial differential equations that exhibit robust heteroclinic cycles between spatially uniform equilibria. Robust heteroclinic cycles arise naturally in systems with invariant subspaces, and their robustness is considered with respect to perturbations that preserve these invariances. We make use of particular symmetries in the system to formulate a relatively low-dimensional spatial two-point boundary-value problem in Fourier space that can be solved efficiently in conjunction with numerical continuation. The standard numerical set-up is formulated on an annulus with small inner radius, and Neumann boundary conditions are used on both inner and outer radial boundaries. We derive and implement alternative boundary conditions that allow for continuing the inner radius to zero and so compute spiral waves on a full disk. As our primary example, we investigate the formation of heteroclinic-induced spiral waves in a reaction–diffusion model that describes the spatiotemporal evolution of three competing populations in a 2D spatial domain—much like the Rock–Paper–Scissors game. We further illustrate the efficiency of our method with the computation of spiral waves in a larger network of cyclic dominance between five competing species, which describes the so-called Rock–Paper–Scissors–Lizard–Spock game.

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