Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells
Hongying Shu Lin Wang
Discrete & Continuous Dynamical Systems - B 2014, 19(6): 1749-1768 doi: 10.3934/dcdsb.2014.19.1749
In this paper, a general viral model with virus-driven proliferation of target cells is studied. Global stability results are established by employing the Lyapunov method and a geometric approach developed by Li and Muldowney. It is shown that under certain conditions, the model exhibits a global threshold dynamics, while if these conditions are not met, then backward bifurcation and bistability are possible. An example is presented to provide some insights on how the virus-driven proliferation of target cells influences the virus dynamics and the drug therapy strategies.
keywords: global stability Virus dynamics backward bifurcation. in-host model
Analyzing the causes of alpine meadow degradation and the efficiency of restoration strategies through a mathematical modelling exercise
Hanwu Liu Lin Wang Fengqin Zhang Qiuying Li Huakun Zhou
Mathematical Biosciences & Engineering 2018, 15(3): 765-773 doi: 10.3934/mbe.2018034

As an important ecosystem, alpine meadow in China has been degraded severely over the past few decades. In order to restore degraded alpine meadows efficiently, the underlying causes of alpine meadow degradation should be identified and the efficiency of restoration strategies should be evaluated. For this purpose, a mathematical modeling exercise is carried out in this paper. Our mathematical analysis shows that the increasing of raptor mortality and the decreasing of livestock mortality (or the increasing of the rate at which livestock increases by consuming forage grass) are the major causes of alpine meadow degradation. We find that controlling the amount of livestock according to the grass yield or ecological migration, together with protecting raptor, is an effective strategy to restore degraded alpine meadows; while meliorating vegetation and controlling rodent population with rodenticide are conducive to restoring degraded alpine meadows. Our analysis also suggests that providing supplementary food to livestock and building greenhouse shelters to protect livestock in winter may contribute to alpine meadow degradation.

keywords: Alpine meadow degradation restoration strategy mathematical modelling stability
Modeling the spread of bed bug infestation and optimal resource allocation for disinfestation
Ali Gharouni Lin Wang
Mathematical Biosciences & Engineering 2016, 13(5): 969-980 doi: 10.3934/mbe.2016025
A patch-structured multigroup-like $SIS$ epidemiological model is proposed to study the spread of the common bed bug infestation. It is shown that the model exhibits global threshold dynamics with the basic reproduction number as the threshold parameter. Costs associated with the disinfestation process are incorporated into setting up the optimization problems. Procedures are proposed and simulated for finding optimal resource allocation strategies to achieve the infestation free state. Our analysis and simulations provide useful insights on how to efficiently distribute the available exterminators among the infested patches for optimal disinfestation management.
keywords: Bed bug infestation threshold dynamics optimal resource allocation disinfestation. $SIS$ model
Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy
Shengqiang Liu Lin Wang
Mathematical Biosciences & Engineering 2010, 7(3): 675-685 doi: 10.3934/mbe.2010.7.675
Global stability is analyzed for a general mathematical model of HIV-1 pathogenesis proposed by Nelson and Perelson [11]. The general model include two distributed intracellular delays and a combination therapy with a reverse transcriptase inhibitor and a protease inhibitor. It is shown that the model exhibits a threshold dynamics: if the basic reproduction number is less than or equal to one, then the HIV-1 infection is cleared from the T-cell population; whereas if the basic reproduction number is larger than one, then the HIV-1 infection persists and the viral concentration maintains at a constant level.
keywords: HIV-1; Global stability; delay; steady state; Lyapunov functional.
Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions
Lin Wang James Watmough Fang Yu
Mathematical Biosciences & Engineering 2015, 12(4): 699-715 doi: 10.3934/mbe.2015.12.699
In this paper, we study a diffusive plant-herbivore system with homogeneous and nonhomogeneous Dirichlet boundary conditions. Stability of spatially homogeneous steady states is established. We also derive conditions ensuring the occurrence of Hopf bifurcation and steady state bifurcation. Interesting transient spatio-temporal behaviors including oscillations in one or both of space and time are observed through numerical simulations.
keywords: Plant-herbivore interaction transient dynamics. Hopf bifurcation diffusion steady state bifurcation
Modeling diseases with latency and relapse
P. van den Driessche Lin Wang Xingfu Zou
Mathematical Biosciences & Engineering 2007, 4(2): 205-219 doi: 10.3934/mbe.2007.4.205
A general mathematical model for a disease with an exposed (latent) period and relapse is proposed. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. For this model with a general probability of remaining in the exposed class, the basic reproduction number $\R_0$ is identified and its threshold property is discussed. In particular, the disease-free equilibrium is proved to be globally asymptotically stable if $\R_0<1$. If the probability of remaining in the exposed class is assumed to be negatively exponentially distributed, then $\R_0=1$ is a sharp threshold between disease extinction and endemic disease. A delay differential equation system is obtained if the probability function is assumed to be a step-function. For this system, the endemic equilibrium is locally asymptotically stable if $\R_0>1$, and the disease is shown to be uniformly persistent with the infective population size either approaching or oscillating about the endemic level. Numerical simulations (for parameters appropriate for bovine tuberculosis in cattle) with $\mathcal{R}_0>1$ indicate that solutions tend to this endemic state.
keywords: bovine tuberculosis delay differential equation uniform persistence. endemic equilibrium global asymptotic stability disease-free equilibrium tuberculosis
Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response
Renji Han Binxiang Dai Lin Wang
Mathematical Biosciences & Engineering 2018, 15(3): 595-627 doi: 10.3934/mbe.2018027

A diffusive intraguild predation model with delay and Beddington-DeAngelis functional response is considered. Dynamics including stability and Hopf bifurcation near the spatially homogeneous steady states are investigated in detail. Further, it is numerically demonstrated that delay can trigger the emergence of irregular spatial patterns including chaos. The impacts of diffusion and functional response on the model's dynamics are also numerically explored.

keywords: Intraguild predation delay diffusion Beddington-DeAngelis functional response spatiotemporal dynamics Hopf bifurcation chaos
Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting
Xinsheng Wang Lin Wang Yujun Zhu
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 2125-2140 doi: 10.3934/dcds.2018087

Metric entropies along a hierarchy of unstable foliations are investigated for $C^1 $ diffeomorphisms with dominated splitting. The analogues of Ruelle's inequality and Pesin's formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given.

keywords: Metric entropy along unstable foliations dominated splitting Lyapunov exponents Ruelle inequality Pesin's entropy formula

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