scholarly journals Dynamics of an SIRS epidemic model with cross-diffusion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yaru Hu ◽  
Jinfeng Wang
Keyword(s):  
Dynamical Behavior ◽  
Disease Model ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Threshold Dynamics ◽  
Cross Diffusion ◽  
Asymptotic Profile ◽  
Reaction Diffusion Model ◽  
Special Cases ◽  
Globally Stable

<p style='text-indent:20px;'>The dynamical behavior of an SIRS epidemic reaction-diffusion model with frequency-dependent mechanism in a spatially heterogeneous environment is studied, with a chemotaxis effect that susceptible individuals tend to move away from higher concentration of infected individuals. Regardless of the strength of the chemotactic coefficient and the spatial dimension <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>, it is established the unique global classical solution which is uniformly-in-time bounded. The model still recognizes the threshold dynamics in terms of the basic reproduction number <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}_{0} $\end{document}</tex-math></inline-formula> even in the case of chemotaxis effects: if <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{R}_{0}&lt;1 $\end{document}</tex-math></inline-formula>, the unique disease free equilibrium is globally stable; if <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_{0}&gt;1 $\end{document}</tex-math></inline-formula>, the disease is uniformly persistent and there is at least one endemic equilibrium, which is globally stable in some special cases with weak chemotactic sensitivity. We also show the asymptotic profile of endemic equilibria (when exists) if the diffusion (migration) rate of the susceptible is small, which indicates that the disease always exists in the entire habitat in this case. Our results suggest that one cannot eradicate the SIRS disease model by only controlling the diffusion rate of susceptible individuals.</p>

scholarly journals Modeling Cell-to-Cell Spread of HIV-1 with Nonlocal Infections

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Xiaoting Fan ◽  
Yi Song ◽  
Wencai Zhao
Keyword(s):  
Reproduction Number ◽  
Reaction Diffusion ◽  
Threshold Dynamics ◽  
Delayed Reaction ◽  
Heterogeneous Model ◽  
Host Environment ◽  
Reaction Diffusion Model ◽  
Globally Attractive ◽  
Cell Spread ◽  
Hiv 1

This paper is devoted to develop a nonlocal and time-delayed reaction-diffusion model for HIV infection within host cell-to-cell viral transmissions. In a bounded spatial domain, we study threshold dynamics in terms of basic reproduction number R0 for the heterogeneous model. Our results show that if R0<1, the infection-free steady state is globally attractive, implying infection becomes extinct, while if R0>1, virus will persist in the host environment.

scholarly journals Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion

2018 ◽  
Vol 31 (1) ◽  
pp. 26-56 ◽  
Author(s):  
HUICONG LI ◽  
RUI PENG ◽  
TIAN XIANG
Keyword(s):  
Infectious Disease ◽  
Total Population ◽  
Initial Boundary ◽  
Endemic Equilibrium ◽  
Diffusion Term ◽  
Frequency Dependent ◽  
Susceptible Population ◽  
Cross Diffusion ◽  
Special Cases ◽  
Globally Stable

This paper is concerned with two frequency-dependent susceptible–infected–susceptible epidemic reaction–diffusion models in heterogeneous environment, with a cross-diffusion term modelling the effect that susceptible individuals tend to move away from higher concentration of infected individuals. It is first shown that the corresponding Neumann initial-boundary value problem in an n-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly in-time bounded regardless of the strength of the cross-diffusion and the spatial dimension n. It is further shown that, even in the presence of cross-diffusion, the models still admit threshold-type dynamics in terms of the basic reproduction number $\mathcal {R}_0$ – i.e. the unique disease-free equilibrium is globally stable if $\mathcal {R}_0\lt1$, while if $\mathcal {R}_0\gt1$, the disease is uniformly persistent and there is an endemic equilibrium (EE), which is globally stable in some special cases with weak chemotactic sensitivity. Our results on the asymptotic profiles of EE illustrate that restricting the motility of susceptible population may eliminate the infectious disease entirely for the first model with constant total population but fails for the second model with varying total population. In particular, this implies that such cross-diffusion does not contribute to the elimination of the infectious disease modelled by the second one.

Existence and Stability of Stationary States of a Reaction–Diffusion-Advection Model for Two Competing Species

2020 ◽  
Vol 30 (05) ◽  
pp. 2050065
Author(s):  
Li Ma ◽  
De Tang
Keyword(s):  
Dynamical Behavior ◽  
Reaction Diffusion ◽  
Similar Species ◽  
Resident Species ◽  
Competition System ◽  
Special Cases ◽  
Existence And Stability ◽  
Maximal Growth Rate ◽  
Steady State Solutions ◽  
Monotone Dynamical Systems

It is well known that the research of two species in the Lotka–Volterra competition system could create very interesting dynamics. In our paper, we investigate the global dynamical behavior of a classic Lotka–Volterra competition system by studying the steady states and corresponding stability by mainly employing the methods of monotone dynamical systems theory, Lyapunov–Schmidt reduction and spectral theory and so on. It illustrates that the dynamical behavior substantially relies on certain variable of the maximal growth rate. Furthermore, we obtain that one of the semi-trivial steady state solutions is a global attractor in some special cases. In biology, these results show that both of the species do not coexist and the mutant forces the extinction of resident species under some condition for two similar species system.

scholarly journals Theory of mechano-chemical patterning in biphasic biological tissues

2018 ◽  
Author(s):  
Pierre Recho ◽  
Adrien Hallou ◽  
Edouard Hannezo
Keyword(s):  
Pattern Formation ◽  
Extracellular Fluid ◽  
Reaction Diffusion ◽  
Single Mode ◽  
Biological Tissues ◽  
Tissue Mechanics ◽  
Chemical Patterning ◽  
Biphasic Model ◽  
Cross Diffusion ◽  
Reaction Diffusion Model

The formation of self-organized patterns is key to the morphogenesis of multicellular organisms, although a comprehensive theory of biological pattern formation is still lacking. Here, we propose a biologically realistic and unifying approach to emergent pattern formation. Our biphasic model of multicellular tissues incorporates turnover and transport of morphogens controlling cell differentiation and tissue mechanics in a single framework, where one tissue phase consists of a poroelastic network made of cells and the other is the extracellular fluid permeating between cells. While this model encompasses previous theories approximating tissues to inert monophasic media, such as Turing’s reaction-diffusion model, it overcomes some of their key limitations permitting pattern formation via any two-species biochemical kinetics thanks to mechanically induced cross-diffusion flows. Moreover, we unravel a qualitatively different advection-driven instability which allows for the formation of patterns with a single morphogen and which single mode pattern scales with tissue size. We discuss the potential relevance of these findings for tissue morphogenesis.

scholarly journals Theory of mechanochemical patterning in biphasic biological tissues

2019 ◽  
Vol 116 (12) ◽  
pp. 5344-5349 ◽  
Author(s):  
Pierre Recho ◽  
Adrien Hallou ◽  
Edouard Hannezo
Keyword(s):  
Pattern Formation ◽  
Extracellular Fluid ◽  
Reaction Diffusion ◽  
Biological Tissues ◽  
Tissue Mechanics ◽  
Two Phase ◽  
Cell Network ◽  
Cross Diffusion ◽  
Model Combining ◽  
Reaction Diffusion Model

The formation of self-organized patterns is key to the morphogenesis of multicellular organisms, although a comprehensive theory of biological pattern formation is still lacking. Here, we propose a minimal model combining tissue mechanics with morphogen turnover and transport to explore routes to patterning. Our active description couples morphogen reaction and diffusion, which impact cell differentiation and tissue mechanics, to a two-phase poroelastic rheology, where one tissue phase consists of a poroelastic cell network and the other one of a permeating extracellular fluid, which provides a feedback by actively transporting morphogens. While this model encompasses previous theories approximating tissues to inert monophasic media, such as Turing’s reaction–diffusion model, it overcomes some of their key limitations permitting pattern formation via any two-species biochemical kinetics due to mechanically induced cross-diffusion flows. Moreover, we describe a qualitatively different advection-driven Keller–Segel instability which allows for the formation of patterns with a single morphogen and whose fundamental mode pattern robustly scales with tissue size. We discuss the potential relevance of these findings for tissue morphogenesis.

Global asymptotic stability for a SEI reaction–diffusion model of infectious diseases with immigration

2018 ◽  
Vol 11 (03) ◽  
pp. 1850044
Author(s):  
Salem Abdelmalek ◽  
Samir Bendoukha
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Reaction Diffusion ◽  
Stability Of Solutions ◽  
Reaction Diffusion Model ◽  
Globally Stable

This paper studies the local and global stability of solutions for a spatially spread SEI epidemic model with immigration of individuals using a Lyapunov functional. It is shown that in the presence of diffusion, the unique steady state remains globally stable. Numerical results obtained through Matlab simulations are presented to confirm the findings of this study.

scholarly journals Spatiotemporal Pattern in a Self- and Cross-Diffusive Predation Model with the Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Feng Rao
Keyword(s):  
Allee Effect ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Reaction Diffusion Model ◽  
Flux Boundary Conditions ◽  
Prey Interaction

This paper proposes and analyzes a mathematical model for a predator-prey interaction with the Allee effect on prey species and with self- and cross-diffusion. The effect of diffusion which can drive the model with zero-flux boundary conditions to Turing instability is investigated. We present numerical evidence of time evolution of patterns controlled by self- and cross-diffusion in the model and find that the model dynamics exhibits a cross-diffusion controlled formation growth to spotted and striped-like coexisting and spotted pattern replication. Moreover, we discuss the effect of cross-diffusivity on the stability of the nontrivial equilibrium of the model, which depends upon the magnitudes of the self- and cross-diffusion coefficients. The obtained results show that cross-diffusion plays an important role in the pattern formation of the predator-prey model. It is also useful to apply the reaction-diffusion model to reveal the spatial predation in the real world.

scholarly journals Dynamic analysis of a malaria reaction-diffusion model with periodic delays and vector bias

2022 ◽  
Vol 19 (3) ◽  
pp. 2538-2574
Author(s):  
Hongyong Zhao ◽  
◽  
Yangyang Shi ◽  
Xuebing Zhang ◽  
◽  
...  
Keyword(s):  
Numerical Simulations ◽  
Diffusion Model ◽  
Disease Transmission ◽  
Reaction Diffusion ◽  
Major Effect ◽  
Threshold Dynamics ◽  
Seasonal Temperature ◽  
Reaction Diffusion Model ◽  
Maturity Period ◽  
Globally Attractive

<abstract><p>One of the most important vector-borne disease in humans is malaria, caused by <italic>Plasmodium</italic> parasite. Seasonal temperature elements have a major effect on the life development of mosquitoes and the development of parasites. In this paper, we establish and analyze a reaction-diffusion model, which includes seasonality, vector-bias, temperature-dependent extrinsic incubation period (EIP) and maturation delay in mosquitoes. In order to get the model threshold dynamics, a threshold parameter, the basic reproduction number $ R_{0} $ is introduced, which is the spectral radius of the next generation operator. Quantitative analysis indicates that when $ R_{0} &lt; 1 $, there is a globally attractive disease-free $ \omega $-periodic solution; disease is uniformly persistent in humans and mosquitoes if $ R_{0} &gt; 1 $. Numerical simulations verify the results of the theoretical analysis and discuss the effects of diffusion and seasonality. We study the relationship between the parameters in the model and $ R_{0} $. More importantly, how to allocate medical resources to reduce the spread of disease is explored through numerical simulations. Last but not least, we discover that when studying malaria transmission, ignoring vector-bias or assuming that the maturity period is not affected by temperature, the risk of disease transmission will be underestimate.</p></abstract>

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