DCDS-B
Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay
Jinling Zhou Yu Yang
Discrete & Continuous Dynamical Systems - B 2017, 22(4): 1719-1741 doi: 10.3934/dcdsb.2017082

In this paper, we study a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. By Schauder's fixed point theorem and Laplace transform, we show that the existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. Some examples are listed to illustrate the theoretical results. Our results generalize some known results.

keywords: Traveling wave solution SIR model nonlocal dispersal spatio-temporal delay nonlinear incidence rate
DCDS-B
Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models
Yu Yang Dongmei Xiao
Discrete & Continuous Dynamical Systems - B 2010, 13(1): 195-211 doi: 10.3934/dcdsb.2010.13.195
A disease transmission model of SIRS type with latent period and nonlinear incidence rate is considered. Latent period is assumed to be a constant $\tau$, and the incidence rate is assumed to be of a specific nonlinear form, namely, $\frac{kI(t-\tau)S(t)}{1+\alpha I^{h}(t-\tau)}$, where $h\ge 1$. Stability of the disease-free equilibrium, and existence, uniqueness and stability of an endemic equilibrium for the model, are investigated. It is shown that, there exists the basic reproduction number $R_0$ which is independent of the form of the nonlinear incidence rate, if $R_0\le 1$, then the disease-free equilibrium is globally asymptotically stable, whereas if $R_0>1$, then the unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region for the model in which there is no latency, and periodic solutions can arise by Hopf bifurcation from the endemic equilibrium for the model at a critical latency. Some numerical simulations are provided to support our analytical conclusions.
keywords: SIRS model; latent period; Nonlinear incidence rate; global stable; Hopf bifurcation.
DCDS-B
Global dynamics of a latent HIV infection model with general incidence function and multiple delays
Yu Yang Yueping Dong Yasuhiro Takeuchi
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-18 doi: 10.3934/dcdsb.2018207

In this paper, we propose a latent HIV infection model with general incidence function and multiple delays. We derive the positivity and boundedness of solutions, as well as the existence and local stability of the infection-free and infected equilibria. By constructing Lyapunov functionals, we establish the global stability of the equilibria based on the basic reproduction number. We further study the global dynamics of this model with Holling type-Ⅱ incidence function through numerical simulations. Our results improve and generalize some existing ones. The results show that the prolonged time delay period of the maturation of the newly produced viruses may lead to the elimination of the viruses.

keywords: Virus dynamics general incidence multiple delays stability analysis Lyapunov functional
MBE
Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function
Yu Yang Shigui Ruan Dongmei Xiao
Mathematical Biosciences & Engineering 2015, 12(4): 859-877 doi: 10.3934/mbe.2015.12.859
In this paper, we study an age-structured virus dynamics model with Beddington-DeAngelis infection function. An explicit formula for the basic reproductive number $\mathcal{R}_{0}$ of the model is obtained. We investigate the global behavior of the model in terms of $\mathcal{R}_{0}$: if $\mathcal{R}_{0}\leq1$, then the infection-free equilibrium is globally asymptotically stable, whereas if $\mathcal{R}_{0}>1$, then the infection equilibrium is globally asymptotically stable. Finally, some special cases, which reduce to some known HIV infection models studied by other researchers, are considered.
keywords: Age structure virus dynamics model Liapunov function infection equilibrium global stability.

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