scholarly journals Dynamics of Spatial Epidemic Model with Infected Population Repulsion

Author(s):  
Meng Yan ◽  
Qingshan Zhang

Abstract In this paper, we are concerned with the spatial epidemic model with infected-taxis in which the susceptible individuals could avoid the infected ones. The spatial pattern for the resulted model is investigated under homogeneous Neumann boundary condition. We gain the condition for spatial pattern induced by diffusion term and infected-taxis term. Moreover, we obtain the condition for the occurrence of pattern formations induced by infected-taxis, in which the diffusion-driven Turing instability case is excluded. We give numerical examples to support the theoretical scheme.

2019 ◽  
Vol 29 (09) ◽  
pp. 1930025 ◽  
Author(s):  
Xiang-Ping Yan ◽  
Ya-Jun Ding ◽  
Cun-Hua Zhang

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.


2021 ◽  
Vol 31 (09) ◽  
pp. 2150129
Author(s):  
Shihong Zhong ◽  
Jinliang Wang ◽  
Juandi Xia ◽  
You Li

By using center manifold theory, Poincaré–Bendixson theorem, spatiotemporal spectrum and dispersion relation of linear operators, the spatiotemporal dynamics of an activator-substrate model with double saturation terms under the homogeneous Neumann boundary condition are considered in the present paper. It is surprising to find that the system can induce new dynamics, such as subcritical Hopf bifurcation and the coexistence of two limit cycles. Moreover, Turing instability in equilibrium mainly generates stripe patterns, while homogeneous periodic solutions mainly generate spot patterns or spot-stripe patterns, where the pattern formations are enormously consistent with the theoretical results. Interestingly, Turing instability can create equilibrium and periodic solution simultaneously in the subcritical Hopf bifurcation, which is the new finding of the diffusion-driven instability. In fact, those theoretical methods are also valid for finding the patterns of other models in one-dimensional space.


2008 ◽  
Vol 25 (6) ◽  
pp. 2296-2299 ◽  
Author(s):  
Sun Gui-Quan ◽  
Jin Zhen ◽  
Liu Quan-Xing ◽  
Li Li

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Wenzhen Gan ◽  
Canrong Tian ◽  
Qunying Zhang ◽  
Zhigui Lin

This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species' moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results.


PLoS ONE ◽  
2016 ◽  
Vol 11 (12) ◽  
pp. e0168127 ◽  
Author(s):  
Chiyori T. Urabe ◽  
Gouhei Tanaka ◽  
Kazuyuki Aihara ◽  
Masayasu Mimura

2018 ◽  
Vol 2018 ◽  
pp. 1-19 ◽  
Author(s):  
Jiao Wang ◽  
Lijun Su ◽  
Xinqiang Qin

Due to the nonlinear diffusion term, it is hard to use the collocation method to solve the unsaturated soil water movement equation directly. In this paper, a nonmesh Hermite collocation method with radial basis functions was proposed to solve the nonlinear unsaturated soil water movement equation with the Neumann boundary condition. By preprocessing the nonlinear diffusion term and using the Hermite radial basis function to deal with the Neumann boundary, the phenomenon that the collocation method cannot be used directly is avoided. The numerical results of unsaturated soil moisture movement with Neumann boundary conditions on the regular and nonregular regions show that the new method improved the accuracy significantly, which can be used to solve the low precision problem for the traditional collocation method when simulating the Neumann boundary condition problem. Moreover, the effectiveness and reliability of the algorithm are proved by the one-dimensional and two-dimensional engineering problem of soil water infiltration in arid area. It can be applied to engineering problems.


1997 ◽  
Vol 34 (3) ◽  
pp. 698-710 ◽  
Author(s):  
Håkan Andersson ◽  
Boualem Djehiche

We study the long-term behaviour of a sequence of multitype general stochastic epidemics, converging in probability to a deterministic spatial epidemic model, proposed by D. G. Kendall. More precisely, we use branching and deterministic approximations in order to study the asymptotic behaviour of the total size of the epidemics as the number of types and the number of individuals of each type both grow to infinity.


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