Global Analysis of a Time Fractional Order Spatio-Temporal SIR Model

2021 ◽  
Author(s):  
Moulay Rchid Sidi Ammi ◽  
Mostafa Tahiri ◽  
Mouhcine Tilioua ◽  
Anwar Zeb ◽  
Ilyas Khan
Keyword(s):  
Direct Method ◽  
Global Analysis ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Positive Equilibrium ◽  
Lyapunov Direct Method ◽  
Diffusion Constants ◽  
Uniqueness Of The Solution ◽  
Spatio Temporal ◽  
Globally Stable

Abstract We deal in this paper with a diffusive SIR epidemic model described by reaction-diffusion equations involving a fractional derivative. The existence and uniqueness of the solution are shown, next to the boundedness of the solution. Further, it has been shown that the global behavior of the solution is governed by the value of R0 , which is known in epidemiology by the basic reproduction number. Indeed, using the Lyapunov direct method it has been proved that the disease will extinct for R0 < 1 for any value of the diffusion constants. For R0 > 1, the disease will persist and the unique positive equilibrium is globally stable. Some numerical illustrations have been used to confirm our theoretical results.Subject classification: 26A33; 34A08; 92D30; 35K57.

OSCILLATIONS ON THE EDGE OF CHAOS VIA DISSIPATION AND DIFFUSION

2007 ◽  
Vol 17 (05) ◽  
pp. 1531-1573 ◽  
Author(s):  
MAKOTO ITOH ◽  
LEON O. CHUA
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Reaction Diffusion Equations ◽  
Nonlinear Dynamical ◽  
Edge Of Chaos ◽  
Secondary Purpose ◽  
Spatio Temporal ◽  
And Diffusion

The primary purpose of this paper is to show that simple dissipation can bring about oscillations in certain kinds of asymptotically stable nonlinear dynamical systems; namely when the system is locally active where the dissipation is introduced. Furthermore, if these nonlinear dynamical systems are coupled with appropriate choice of diffusion coefficients, then the coupled system can exhibit spatio-temporal oscillations. The secondary purpose of this paper is to show that spatio-temporal oscillations can occur in spatially discrete reaction diffusion equations operating on the edge of chaos, provided the array size is sufficiently large.

Noise-induced suppression of periodic travelling waves in oscillatory reaction–diffusion systems

2010 ◽  
Vol 466 (2119) ◽  
pp. 1903-1917 ◽  
Author(s):  
Michael Sieber ◽  
Horst Malchow ◽  
Sergei V. Petrovskii
Keyword(s):  
Travelling Waves ◽  
Natural Populations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Oscillatory Reaction ◽  
Stochastic Forcing ◽  
Spatial Direction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

Ecological field data suggest that some species show periodic changes in abundance over time and in a specific spatial direction. Periodic travelling waves as solutions to reaction–diffusion equations have helped to identify possible scenarios, by which such spatio-temporal patterns may arise. In this paper, such solutions are tested for their robustness against an irregular temporal forcing, since most natural populations can be expected to be subject to erratic fluctuations imposed by the environment. It is found that small environmental noise is able to suppress periodic travelling waves in stochastic variants of oscillatory reaction–diffusion systems. Irregular spatio-temporal oscillations, however, appear to be more robust and persist under the same stochastic forcing.

scholarly journals Global dynamics of alcoholism epidemic model with distributed delays

2021 ◽  
Vol 18 (6) ◽  
pp. 8245-8256
Author(s):  
Salih Djillali ◽  
◽  
Soufiane Bentout ◽  
Tarik Mohammed Touaoula ◽  
Abdessamad Tridane ◽  
...  
Keyword(s):  
Epidemic Model ◽  
Direct Method ◽  
Distributed Delays ◽  
Positive Equilibrium ◽  
Model Parameters ◽  
Threshold Condition ◽  
Global Dynamics ◽  
Lyapunov Direct Method ◽  
Globally Asymptotically Stable ◽  
The Social

<abstract><p>This paper aims to investigate the global dynamics of an alcoholism epidemic model with distributed delays. The main feature of this model is that it includes the effect of the social pressure as a factor of drinking. As a result, our global stability is obtained without a "basic reproduction number" nor threshold condition. Hence, we prove that the alcohol addiction will be always uniformly persistent in the population. This means that the investigated model has only one positive equilibrium, and it is globally asymptotically stable independent on the model parameters. This result is shown by proving that the unique equilibrium is locally stable, and the global attraction is shown using Lyapunov direct method.</p></abstract>

Fragmentation–diffusion model. Existence of solutions and their asymptotic behaviour

1998 ◽  
Vol 128 (4) ◽  
pp. 759-774 ◽  
Author(s):  
Philippe Laurençot ◽  
Dariusz Wrzosek
Keyword(s):  
Infinite System ◽  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Fragmentation Model ◽  
Convergence Of Solutions ◽  
Diffusion Constants ◽  
Spatially Homogeneous ◽  
Model Existence

An infinite system of reaction–diffusion equations that represents a particular case of the discrete coagulation–fragmentation model with diffusion is studied. The reaction part of the model describes the rate of clusters break-up into smaller particles. Diffusion constants are assumed to be different in each equation and concentration-dependent fragmentation coefficients are considered. Existence of solutions is studied under fairly general assumptions on fragmentation coefficients and initial data. Uniqueness in the class of mass-preserving solutions is proved. Convergence of solutions to spatially homogeneous equilibrium state is obtained.

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 Accelerating the Finite-Element Method for Reaction-Diffusion Simulations on GPUs with CUDA

Micromachines ◽  
2020 ◽  
Vol 11 (9) ◽  
pp. 881 ◽  
Author(s):  
Hedi Sellami ◽  
Leo Cazenille ◽  
Teruo Fujii ◽  
Masami Hagiya ◽  
Nathanael Aubert-Kato ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Temporal Dynamics ◽  
Dna Nanotechnology ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
The Finite Element Method ◽  
Highly Nonlinear ◽  
Spatio Temporal ◽  
Element Method

DNA nanotechnology offers a fine control over biochemistry by programming chemical reactions in DNA templates. Coupled to microfluidics, it has enabled DNA-based reaction-diffusion microsystems with advanced spatio-temporal dynamics such as traveling waves. The Finite Element Method (FEM) is a standard tool to simulate the physics of such systems where boundary conditions play a crucial role. However, a fine discretization in time and space is required for complex geometries (like sharp corners) and highly nonlinear chemistry. Graphical Processing Units (GPUs) are increasingly used to speed up scientific computing, but their application to accelerate simulations of reaction-diffusion in DNA nanotechnology has been little investigated. Here we study reaction-diffusion equations (a DNA-based predator-prey system) in a tortuous geometry (a maze), which was shown experimentally to generate subtle geometric effects. We solve the partial differential equations on a GPU, demonstrating a speedup of ∼100 over the same resolution on a 20 cores CPU.

Map dynamics versus dynamics of associated delay reaction–diffusion equations with a Neumann condition

2010 ◽  
Vol 466 (2122) ◽  
pp. 2955-2973 ◽  
Author(s):  
Taishan Yi ◽  
Xingfu Zou
Keyword(s):  
Steady State ◽  
Global Stability ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Neumann Condition ◽  
Main Concern ◽  
Homogeneous Neumann Boundary Condition ◽  
Globally Stable

In this paper, we consider a class of delay reaction–diffusion equations (DRDEs) with a parameter ε >0. A homogeneous Neumann boundary condition and non-negative initial functions are posed to the equation. By letting , such an equation is formally reduced to a scalar difference equation (or map dynamical system). The main concern is the relation of the absolute (or delay-independent) global stability of a steady state of the equation and the dynamics of the nonlinear map in the equation. By employing the idea of attracting intervals for solution semiflows of the DRDEs, we prove that the globally stable dynamics of the map indeed ensures the delay-independent global stability of a constant steady state of the DRDEs. We also give a counterexample to show that the delay-independent global stability of DRDEs cannot guarantee the globally stable dynamics of the map. Finally, we apply the abstract results to the diffusive delay Nicholson blowfly equation and the diffusive Mackey–Glass haematopoiesis equation. The resulting criteria for both model equations are amazingly simple and are optimal in some sense (although there is no existing result to compare with for the latter).

scholarly journals Existence and uniqueness of the solution to granuloma model in visceral leishmaniasi

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3561-3575
Author(s):  
Chunyan Liu ◽  
Xuemei Wei
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Reaction Diffusion ◽  
Local Solution ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Banach Fixed Point Theorem ◽  
Extension Method ◽  
Uniqueness Of The Solution ◽  
Coupled Reaction

In this paper we study a granuloma model in visceral leishmaniasis, which contains eleven coupled reaction diffusion equations. The existence and uniqueness of the model in the local solution is proved by using the Banach Fixed Point Theorem and theory of parabolic equation. Then, the existence and uniqueness of the global solution are obtained by using the extension method.

scholarly journals Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models

2008 ◽  
Vol 5 (22) ◽  
pp. 483-505 ◽  
Author(s):  
Jonathan A Sherratt ◽  
Matthew J Smith
Keyword(s):  
Travelling Waves ◽  
Empirical Studies ◽  
Reaction Diffusion ◽  
Field Studies ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Oscillatory Reaction ◽  
Relevant Field ◽  
Spatio Temporal ◽  
Cyclic Populations

Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reaction–diffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reaction–diffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical.

scholarly journals Multiplier method and exact solutions for a density dependent reaction-diffusion equation

2016 ◽  
Vol 1 (2) ◽  
pp. 311-320 ◽  
Author(s):  
M. Rosa ◽  
M. L Gandarias
Keyword(s):  
Conservation Laws ◽  
Direct Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
Density Dependent ◽  
Population Dynamics Models ◽  
Dynamics Models

AbstractReaction-diffusion equations have enjoyed a considerable amount of scientific interest. The reason for the large amount of work put into studying these equations is not only their practical relevance, but also interesting phenomena that can arise from such equations. Fisher equation is commonly used in biology for population dynamics models and in bacterial growth problems as well as development and growth of solid tumours. The physical aspects of this equation are not fully understood without getting deeper into the concept of conservation laws. In [4], Anco and Bluman gave a general treatment of a direct conservation law method for partial differential equations expressed in a standard Cauchy-Kovaleskaya form. In this work we study the well known density dependent diffusion-reaction equation. We derive conservation laws by using the direct method of the multipliers.

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