Dynamics of a diffusive HBV model with delayed Beddington–DeAngelis response

Zhang, Yiyi; Xu, Zhiting

doi:10.1016/j.nonrwa.2013.06.005

Cited by 93 publications

(52 citation statements)

References 31 publications

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“…The spatiotemporal dynamics have been investigated without or with time delay in [16,17], respectively. Then, some modified diffusive models with time delay and nonlinear functional response have been proposed; see [18][19][20][21][22]. These results mainly focus on the local and global stabilities of uniform steady states and the existence of traveling wave solutions.…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal Dynamics of a Delayed and Diffusive Viral Infection Model with Logistic Growth

Zhuang

2017

MCA

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Abstract:Viruses have important influences on human health: they not only cause some common diseases, but also cause serious illnesses. Moreover, the conventional medicines usually fail to prevent or treat them, and viral infections are hard to treat because viruses live inside the body's cells. However, some mathematical models can help to understand the viral transmission mechanism and control viral diseases. In this paper, a delayed viral infection model with spatial diffusion and logistic growth is presented. The asymptotic stability of nonnegative uniform steady states is investigated by utilizing the linearized method and constructing the proper Lyapunov functional, respectively. The existence of Hopf bifurcation from the positive equilibrium point is established by analyzing the corresponding characteristic equation and the direction of bifurcation, and the properties of bifurcating periodic solutions are derived by the aid of the normal form theory for partial functional differential equations. Then, the cross-diffusion system is introduced. Furthermore, some numerical simulations are carried out and discussions are given.

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Section: Introductionmentioning

confidence: 99%

Spatiotemporal Dynamics of a Delayed and Diffusive Viral Infection Model with Logistic Growth

Zhuang

2017

MCA

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“…The rate of infection in the above diffusive HBV models is assumed to be bilinear in the virus and uninfected target cells, which is not reasonable to describe the HBV infection. For this reason, this bilinear incidence rate is replaced by saturation response in [8], by standard incidence function in [9], by Beddington-DeAngelis response in [10] and by a specific generalized functional response in [11].…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, the model presented by Hattaf and Yousfi in [11] is a particular case of our model (1) if we choose f (u, w, v) = βu 1+δ 1 u+δ 2 v+δ 3 uv , where δ 1 , δ 2 , δ 3 ≥ 0 are constants. Note that the models considered in [8,10] is a particular case of [11]. In this paper, we consider system (1) with Neumann boundary conditions as follows ∂v ∂ν = 0, on ∂Ω × (0, +∞), (2) and initial conditions…”

Section: Introductionmentioning

confidence: 99%

A generalized HBV model with diffusion and two delays

Hattaf

Yousfi

2015

Computers & Mathematics with Applications

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a b s t r a c tAn hepatitis B virus (HBV) model with spatial diffusion, general incidence rate and time delays subject to homogeneous Neumann boundary conditions is formulated. The stability of the disease-free equilibrium and the chronic infection equilibrium is analyzed by using the linearization method and by constructing appropriate Lyapunov functionals. Furthermore, several diffusive HBV models existing in the literature are generalized and extended.

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“…Traveling wave solution of spatial epidemic models represents the transition process of outbreak from the initial disease-free equilibrium to another disease-free equilibrium. Recently, it has been drawing much attention for infectious models, and there are many works using spatially dependent models to study the disease transmission (see other works, 2,7,10,11,15,16,22,25,[27][28][29][33][34][35][36][37][38][39][40][41][42][43][44][45]48,49,[51][52][53][54] for example). However, to the best of our knowledge, there is quite a few issues on the existence and nonexistence of traveling waves for epidemic models consisting of four equations.…”

Section: Introductionmentioning

confidence: 99%

Traveling waves in a diffusive epidemic model with criss‐cross mechanism

Guo

2019

Math Methods in App Sciences

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In this paper, we propose a reaction‐diffusion system to describe the spread of infectious diseases within two population groups by self and criss‐cross infection mechanism. Firstly, based on the eigenvalues, we give two methods for the calculation of the critical wave speed c∗. Secondly, by constructing a pair of upper‐lower solutions and using the Schauder fixed‐point theorem, we prove that the system admits positive traveling wave solutions, which connect the initial disease‐free equilibrium false(u10,0,u30,0false) at t = −∞, but the traveling waves need not connect the final disease‐free equilibrium false(u1∗,0,u3∗,0false) at t = +∞. Hence, we study the asymptotic behaviors of the traveling wave solutions to show that the traveling wave solutions converge to false(u1∗,0,u3∗,0false) at t = +∞. Finally, by the two‐sided Laplace transform, we establish the nonexistence of traveling waves for the model. The approach in this paper provides an effective method to deal with the existence of traveling wave solutions for the nonmonotone reaction‐diffusion systems consisting of four equations.

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Dynamics of a diffusive HBV model with delayed Beddington–DeAngelis response

Cited by 93 publications

References 31 publications

Spatiotemporal Dynamics of a Delayed and Diffusive Viral Infection Model with Logistic Growth

Spatiotemporal Dynamics of a Delayed and Diffusive Viral Infection Model with Logistic Growth

A generalized HBV model with diffusion and two delays

Traveling waves in a diffusive epidemic model with criss‐cross mechanism

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