The effect of time delay on the dynamics of an SEIR model with nonlinear incidence

In this paper, a SEIR epidemic model with nonlinear incidence rate and time delay is investigated in three cases. The local stability of an endemic equilibrium and a disease-free equilibrium are discussed using stability theory of delay differential equations. The conditions that guarantee the asymptotic stability of corresponding steady-states are investigated. The results show that the introduction of a time delay in the transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation when using the time delay as a bifurcation parameter. Applying the normal form theory and center manifold argument, the explicit formulas determining the properties of the bifurcating periodic solution are derived. In addition, the effect of the inhibitory effect on the properties of the bifurcating periodic solutions is studied. Numerical simulations are provided in order to illustrate the theoretical results and to gain further insight into the behaviors of delayed systems.     

Qualitative study of a quarantine/isolation model with multiple disease stages

Recent studies suggest that, for disease transmission models with latent and infectious periods, the use of gamma distribution assumption seems to provide a better fit for the associated epidemiological data in comparison to the use of exponential distribution assumption. The objective of this study is to carry out a rigorous mathematical analysis of a communicable disease transmission model with quarantine (of latent cases) and isolation (of symptomatic cases), in which the waiting periods in the infected classes are assumed to have gamma distributions. Rigorous analysis of the model reveals that it has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. The model has a unique endemic equilibrium when the threshold quantity exceeds unity. The endemic equilibrium is shown to be locally and globally-asymptotically stable for special cases. Numerical simulations, using data related to the 2003 SARS outbreaks, show that the cumulative number of disease-related mortality increases with increasing number of disease stages. Furthermore, the cumulative number of new cases is higher if the asymptomatic period is distributed such that most of the period is spent in the early stages of the asymptomatic compartments in comparison to the cases where the average time period is equally distributed among the associated stages or if most of the time period is spent in the later (final) stages of the asymptomatic compartments. Finally, it is shown that distributing the average sojourn time in the infectious (asymptomatic) classes equally or unequally does not effect the cumulative number of new cases.     

Global dynamics of a discretized SIRS epidemic model with time delay

We derive a discretized SIRS epidemic model with time delay by applying a nonstandard finite difference scheme. Sufficient conditions for the global dynamics of the solution are obtained by improvements in discretization and applying proofs for continuous epidemic models. These conditions for our discretized model are the same as for the original continuous model.     

The dynamics of pest control pollution model with age structure and time delay

In this paper, by using pollution model and impulsive delay differential equation, we investigate the dynamics of a pest control model with age structure for pest by introducing a constant periodic pesticide input and releasing natural enemies at different fixed moment. We assume only the pests are affected by pesticide. We show that there exists a global attractive pest-extinction periodic solution when the periodic natural enemies release amount μ₁ and pesticide input amount μ₂ are larger than some critical value. Further, the condition for the permanence of the system is also given. By numerical analyses, we also show that constant maturation time delay, pulse pesticide input and pulse releasing of the natural enemies can bring obvious effects on the dynamics of system. We believe that the results will provide reliable tactic basis for the practical pest management.     

The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators

In this paper, we introduce a mutual interference age structured predator-prey (natural enemy-pest) model with constant maturation time delay for the prey, and then propose a pest management strategy by constant periodic releasing for the predator. We show that there exists a global attractive pest-eradication periodic solution when the periodic releasing amount μ₁ and μ₂ are lager than some critical value. Further, to obtain a more effective pest control strategy, we give the conditions (involving the estimate of μ₁ and μ₂) in which the model is uniformly permanent and the pest population is under the economic threshold level. We believe that the results will provide reliable tactic basis for the practical pest management.     

Asymptotic properties of an HIV/AIDS model with a time delay

A mathematical model for HIV/AIDS with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed. The disease-free and endemic equilibria are found and their local stability investigated. We use the Lyapunov functional approach to show the global stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The HIV/AIDS model is numerically analysed to asses the effects of incubation period on the dynamics of HIV/AIDS and the demographic impact of the epidemic using the demographic and epidemiological parameters for Zimbabwe.     

The dynamics of an eco-epidemiological model with distributed delay

In this paper, the dynamical behavior of an eco-epidemiological model with distributed delay is studied. Sufficient conditions for the asymptotical stability of all the equilibria are obtained. We prove that there exists a threshold value of the conversion rate h beyond which the positive equilibrium bifurcates towards a periodic solution. We further analyze the orbital stability of the periodic orbits arising from bifurcation by applying Poore’s condition. Numerical simulation with some hypothetical sets of data has been done to support the analytical findings.     

Traveling waves of a nonlocal diffusion SIRS epidemic model with a class of nonlinear incidence rates and time delay

In this paper, we study the traveling waves of a delayed SIRS epidemic model with nonlocal diffusion and a class of nonlinear incidence rates. When the basic reproduction ratio $\mathscr{R}_0>1$, by using the Schauder''s fixed point theorem associated with upper-lower solutions, we reduce the existence of traveling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of traveling wave solutions connecting the disease-free steady state and the endemic steady state. When $\mathscr{R}_0<1$, the nonexistence of traveling waves is obtained by the comparison principle.     

随机延滞的时序模型的稳定性分析

本文考察了一个随机延滞的时序模型:x(n+ 1) =∑η( n+ 1)l=0al| xn+ 1| s+ εn-1,n 0 ,0 相似文献   

Traveling wavefront in a Hematopoiesis model with time delay

This paper is concerned with a reaction–diffusion equation with time delay, which describes the dynamics of the blood cell production. The existence of the traveling wavefront is given by using the upper–lower solution technique and the monotone iteration.     

The effect of dispersal on the permanence of a predator–prey system with time delay

A predator–prey model with prey dispersal and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of each of the nonnegative equilibria is discussed. By using an iteration technique, a threshold is derived for the permanence and extinction of the proposed model. Numerical simulations are carried out to illustrate the main results.     

Global dynamics of a general brucellosis model with discrete delay

For the prevention and control of brucellosis, it is important to investigate the mechanism of brucellosis transmission. Based on the characteristics of the spread of brucellosis, a susceptible-exposed-infectious-brucella (SEIB) delay dynamic model is proposed with the general incidence, elimination rate and shedding rate of pathogen. Under biologically motivated assumptions, it shows the uniqueness of the endemic equilibrium, and investigates the global asymptotically stability of the disease-free equilibrium and the endemic equilibrium. The results suggest that the global stability of equilibria depends entirely on the basic reproduction number $R_0$ and time delay is harmless for the stability of equilibria. Finally, some specific examples and numerical simulations are used to illustrate the utilization of research results and reveal the biological significance of hypothesis $(H_7)$, which implies that the dynamics of brucellosis transmission depend largely on the development of the prevention and control strategies.     

Effect of time delay on a harvested predator-prey model

The predator-prey systems with harvesting have received a great deal of attentions for last few decades. Incorporating discrete time delays into predator-prey models could induce instability and bifurcation. In this paper we are interested in studying the combined effects of harvesting and discrete time delay on the dynamics of a predator-prey model. A comparative analysis is provided for stability behaviour in absence as well as in presence of time delay. The length of discrete time delay to preserve stability of the model system is obtained. Existence of Hopf-bifurcating small amplitude periodic solutions is derived by taking discrete time delay as a bifurcation parameter.     

Hopf bifurcation of a tumor immune model with time delay

In this paper, a tumor immune model with time delay is studied. First, the stability of nonnegative equilibria is analyzed. Then the time delay τ is selected as a bifurcation parameter and the existence of Hopf bifurcation is proved. Finally, by using the canonical method and the central manifold theory, the criteria for judging the direction and stability of Hopf bifurcation are given.     

一类具时滞和阶段结构的捕食模型的稳定性与Hopf分支

研究一类具有时滞和阶段结构的捕食模型的稳定性和Hopf分支的存在性问题.通过分析特征方程,得到了正平衡点局部稳定的条件.同时,应用中心流形定理和规范型理论,得到了确定Hopf分支方向和分支周期解的稳定性的计算公式.最后对所得理论结果进行了数值模拟.