scholarly journals Asymptotic Analysis for a nonlinear Reaction-Diffusion System Modeling an Infectious Disease

2021 ◽  
Author(s):  
Hong-Ming Yin ◽  
Jun Zou
Keyword(s):  
Infectious Disease ◽  
System Modeling ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Time Behavior ◽  
Strongly Coupled ◽  
Nonlinear Reaction ◽  
Open Questions ◽  
Long Time

In this paper we study a nonlinear reaction-diffusion system which models an infectious disease caused by bacteria such as cholera. One of the features in this model is that a certain portion of the recovered human hosts lost a lifetime immunity and could be infected again. Another feature in the model is that the mobility for each species is assumed to be dependent upon location and time. We also assume that the whole group is susceptible with the bacteria. This leads to a strongly coupled nonlinear reaction-diffusion system. We prove that the nonlinear system has a unique solution globally in any space dimension under some natural conditions on known parameters and functions. Moreover, the long-time behavior and stability analysis for the solution are carried out rigorously. In particular, we characterize the precise conditions on variable parameters about the stability or instability for all steady-state solutions. These results obtained in this paper answered several open questions raised in the previous literature

scholarly journals Attractors for Multivalued Impulsive Systems: Existence and Applications to Reaction-Diffusion System

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
S. Dashkovskiy ◽  
O. A. Kapustian ◽  
O. V. Kapustyan ◽  
N. V. Gorban
Keyword(s):  
Initial Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Impulsive Systems ◽  
Uniform Attractor ◽  
Infinite Dimensional ◽  
Long Time

In this paper, we develop a general approach to investigate limit dynamics of infinite-dimensional dissipative impulsive systems whose initial conditions do not uniquely determine their long time behavior. Based on the notion of an uniform attractor, we show how to describe limit behavior of such complex systems with the help of properties of their components. More precisely, we prove the existence of the uniform attractor for an impulsive multivalued system in terms of properties of nonimpulsive semiflow and impulsive parameters. We also give an application of these abstract results to the impulsive reaction-diffusion system without uniqueness.

On a reaction–diffusion system modelling infectious diseases without lifetime immunity

2021 ◽  
pp. 1-25
Author(s):  
HONG-MING YIN
Keyword(s):  
Diffusion Coefficients ◽  
A Priori ◽  
Main Tool ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Sobolev Embedding ◽  
System Modelling ◽  
Strongly Coupled ◽  
Elliptic And Parabolic Equations

In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.

A NUMERICAL ANALYSIS OF A REACTION–DIFFUSION SYSTEM MODELING THE DYNAMICS OF GROWTH TUMORS

2010 ◽  
Vol 20 (05) ◽  
pp. 731-756 ◽  
Author(s):  
VERÓNICA ANAYA ◽  
MOSTAFA BENDAHMANE ◽  
MAURICIO SEPÚLVEDA
Keyword(s):  
Weak Solution ◽  
Numerical Analysis ◽  
Numerical Scheme ◽  
System Modeling ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Existence Of Weak Solutions ◽  
Early Tumor ◽  
Volume Method

We consider a reaction–diffusion system of 2 × 2 equations modeling the spread of early tumor cells. The existence of weak solutions is ensured by a classical argument of Faedo–Galerkin method. Then, we present a numerical scheme for this model based on a finite volume method. We establish the existence of discrete solutions to this scheme, and we show that it converges to a weak solution. Finally, some numerical simulations are reported with pattern formation examples.

Unique continuation for a reaction-diffusion system with cross diffusion

2019 ◽  
Vol 27 (4) ◽  
pp. 511-525 ◽  
Author(s):  
Bin Wu ◽  
Ying Gao ◽  
Zewen Wang ◽  
Qun Chen
Keyword(s):  
Unique Continuation ◽  
Reaction Diffusion ◽  
Cauchy Data ◽  
Carleman Estimate ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Lateral Boundary ◽  
Strongly Coupled ◽  
Cross Diffusion ◽  
Coupled Reaction

Abstract This paper concerns unique continuation for a reaction-diffusion system with cross diffusion, which is a drug war reaction-diffusion system describing a simple dynamic model of a drug epidemic in an idealized community. We first establish a Carleman estimate for this strongly coupled reaction-diffusion system. Then we apply the Carleman estimate to prove the unique continuation, which means that the Cauchy data on any lateral boundary determine the solution uniquely in the whole domain.

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