Dynamics of an HIV-1 therapy model of fighting a virus with another virus

In this paper, we rigorously analyse an ordinary differential equation system that models fighting the HIV-1 virus with a genetically modified virus. We show that when the basic reproduction ratio ?₀<1, then the infection-free equilibrium E ₀ is globally asymptotically stable; when ?₀>1, E ₀ loses its stability and there is the single-infection equilibrium E _s. If ?₀∈(1, 1+δ) where δ is a positive constant explicitly depending on system parameters, then the single-infection equilibrium E _s that is globally asymptotically stable, while when ?₀>1+δ, E _s becomes unstable and the double-infection equilibrium E _d comes into existence. When ?₀ is slightly larger than 1+δ, E _d is stable and it loses its stability via Hopf bifurcation when ?₀ is further increased in some ways. Through a numerical example and by applying a normal form theory, we demonstrate how to determine the bifurcation direction and stability, as well as the estimates of the amplitudes and the periods of the bifurcated periodic solutions. We also perform numerical simulations which agree with the theoretical results. The approaches we use here are a combination of analysis of characteristic equations, fluctuation lemma, Lyapunov function and normal form theory.     

Oscillations in a patchy environment disease model

For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R(0) is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)<1. For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.     

Dynamical behavior of a hepatitis B virus transmission model with vaccination

Hepatitis B virus (HBV) infection is a globally health problem. In 2005, the WHO Western Pacific Regional Office set a goal of reducing chronic HBV infection rate to less than 2% among children five years of age by 2012, as an interim milestone towards the final goal of less than 1%. Many countries made some plans (such as free HBV vaccination program for all neonates in China now) to control the transmission HBV. We develop a model to explore the impact of vaccination and other controlling measures of HBV infection. The model has simple dynamical behavior which has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R₀≤1, and a globally asymptotically stable endemic equilibrium when R₀>1. Numerical simulation results show that the vaccination is a very effective measure to control the infection and they also give some useful comments on controlling the transmission of HBV.     

Mathematical model for Zika virus dynamics with sexual transmission route

Zika virus is a flavivirus transmitted to humans primarily through the bite of infected Aedes mosquitoes. In addition to vector-borne spread, however, the virus can also be transmitted through sexual contact. In this paper, we formulate and analyze a new system of ordinary differential equations which incorporates both vector and sexual transmission routes. Theoretical analysis of this model when there is no disease induced mortality shows that the disease-free equilibrium is locally and globally asymptotically stable whenever the associated reproduction number is less than unity and unstable otherwise. However, when we extend this same model to include Zika induced mortality, which have been documented in Latin America, we find that the model exhibits a backward bifurcation. Specifically, a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. To further explore model predictions, we use numerical simulations to assess the importance of sexual transmission to disease dynamics. This analysis shows that risky behavior involving multiple sexual partners, particularly among male populations, substantially increases the number of infected individuals in the population, contributing significantly to the disease burden in the community.     

具有一般形式饱和接触率SEIS模型渐近分析

研究具有一般形式饱和接触率SEIS模型渐近性态,得到决定疾病绝灭和持续的阈值-基本再生数R0。当R0 ≤ 1时,仅存在无病平衡点P^0;当R0＞1时,除存在无病平衡点P^0外,还存在惟一的地方病平衡点P^＊。当R0＜1时,无病平衡点P^0全局渐近稳定;当R0＞1时,地方病平衡点P^＊局部渐近稳定。特别地,无因病死亡时,极限方程地方病平衡点P^-＊全局渐近稳定。     

Pest management through continuous and impulsive control strategies

In this paper, we propose two mathematical models concerning continuous and, respectively, impulsive pest control strategies. In the case in which a continuous control is used, it is shown that the model admits a globally asymptotically stable positive equilibrium under appropriate conditions which involve parameter estimations. As a result, the global asymptotic stability of the unique positive equilibrium is used to establish a procedure to maintain the pests at an acceptably low level in the long term. In the case in which an impulsive control is used, it is observed that there exists a globally asymptotically stable susceptible pest-eradication periodic solution on condition that the amount of infective pests released periodically is larger than some critical value. When the amount of infective pests released is less than this critical value, the system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its stability. Further, the existence of a nontrivial periodic solution is also studied by means of numerical simulation. Finally, the efficiency of continuous and impulsive control policies is compared.     

Modeling and analysis of a predator-prey model with disease in the prey

A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator k=k(0) is constant (independent of delay tau;, gestation period), we show that positive equilibrium is locally asymptotically stable when time delay tau; is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If k=k(0)e(-dtau;) (d is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delay. A concluding discussion is then presented.     

Effect of a sharp change of the incidence function on the dynamics of a simple disease

We investigate two cases of a sharp change of incidencec functions on the dynamics of a susceptible-infective-susceptible epidemic model. In the first case, low population levels have mass action incidence, while high population levels have proportional incidence, the switch occurring when the total population reaches a certain threshold. Using a modified Dulac theorem, we prove that this system has a single equilibrium which attracts all solutions for which the disease is present and the population remains bounded. In the second case, an increase of the number of infectives leads to a mass action term being added to a standard incidence term. We show that this allows a Hopf bifurcation to occur, with periodic orbits being generated when a locally asymptotically stable equilibrium loses stability.     

改进的病毒感染动力学模型的稳定性分析

提出一个改进的乙肝病毒感染动力学模型.本模型有三个平衡点.对于HBV感染人群,三个平衡点分别对应于三类人群：感染病毒后自愈人群、健康带毒人群、慢性乙肝患者人群.证明了当模型导出的基本复制数R_0〈1时病毒清除平衡点具有局部稳定性和全局渐近稳定性,当1〈R_0〈k_3d/（k_2λ-k_3a）＋1时持续带毒平衡点具有局部稳定性.     

The Ross-Macdonald model in a patchy environment

We generalize to n patches the Ross-Macdonald model which describes the dynamics of malaria. We incorporate in our model the fact that some patches can be vector free. We assume that the hosts can migrate between patches, but not the vectors. The susceptible and infectious individuals have the same dispersal rate. We compute the basic reproduction ratio R(0). We prove that if R(0)1, then the disease-free equilibrium is globally asymptotically stable. When R(0)>1, we prove that there exists a unique endemic equilibrium, which is globally asymptotically stable on the biological domain minus the disease-free equilibrium.     

Global Dynamics of an In-host Viral Model with Intracellular Delay

The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R ₀ for the viral infection, and establish that the global dynamics are completely determined by the values of R ₀. If R ₀≤1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R ₀>1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functional, we prove that the chronic-infection equilibrium is globally asymptotically stable when R ₀>1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments.     

一类带有非线性感染率传染病模型的动力学性质（英文）

主要介绍了一类带有非线性感染率的传染病模型.并且证明了当基本再生数Ro≤1时,无病平衡点是全局稳定的,当基本再生数R_0〉1时,疾病持续.     

具有周期传染率的SIR传染病模型的周期解

考虑了具有周期传染率的SIR流行病模型,定义了基本再生数^-R0=β／（μ＋γ）,分析了该模型的动力学性态,证明了当^-R0〈1时无病平衡点是全局稳定的;^-R0〉1时,无病平衡点是不稳定的,模型至少存在一个周期解。对小振幅的周期传染率模型,给出了模型周期解的近似表达式,证明了该周期解的稳定性,最后做了数值模拟,结果显示周期解可能是全局稳定的。     

Dynamics induced by delay in a nutrient–phytoplankton model with diffusion

In this paper, a nutrient–phytoplankton model described by a couple of reaction-diffusion equations with delay is studied analytically and numerically. The aim of this research is to provide an understanding of the impact of delay on the nutrient–phytoplankton dynamics. Significantly, the delay can not only induce instability of a positive equilibrium, but also promote the formation of patchiness (an irregular pattern) via Hopf bifurcation. However, if the delay does not exist, the positive equilibrium is always globally asymptotically stable when it exists. In addition, the numerical analysis indicates that the input rate and the loss rate of nutrient also play an important role in the growth of phytoplankton, which supports that eutrophic conditions may be a significant reason inducing phytoplankton blooms. Numerical results are consistent with the analytical results.     

The Onset and Extinction of Neural Spiking: A Numerical Bifurcation Approach

We study the onset of neural spiking when the equilibrium rest state loses stability by the change of a critical parameter, the applied current. In the case of the well-known Morris-Lecar model, we start from a complete numerical study of the bifurcation diagram in the most relevant two-parameter range. This diagram includes all equilibrium and limit cycle bifurcations, thus correcting and completing earlier studies.We discuss and classify the behavior of the spiking orbits, when increasing or decreasing the applied current. A complete classification can be extracted from the complete bifurcation diagram. It is based on three components: bifurcation type of the equilibrium at the loss of stability, subcritical behavior in the limit of decreasing the applied current and supercritical behavior in the limit of increasing the applied current.     

Global dynamics of a staged-progression model with amelioration for infectious diseases

We analyze the global dynamics of a mathematical model for infectious diseases that progress through distinct stages within infected hosts with possibility of amelioration. An example of such diseases is HIV/AIDS that progresses through several stages with varying degrees of infectivity; amelioration can result from a host's immune action or more commonly from antiretroviral therapies, such as highly active antiretroviral therapy. For a general n-stage model with constant recruitment and bilinear incidence that incorporates amelioration, we prove that the global dynamics are completely determined by the basic reproduction number R(0). If R(0)≤1, then the disease-free equilibrium P(0) is globally asymptotically stable, and the disease always dies out. If R(0)>1, P(0) is unstable, a unique endemic equilibrium P* is globally asymptotically stable, and the disease persists at the endemic equilibrium. Impacts of amelioration on the basic reproduction number are also investigated.     

Effect of hunting cooperation and fear in a predator-prey model

Dynamics of predator-prey systems under the influence of cooperative hunting among predators and the fear thus imposed on the prey population is of great importance from ecological point of view. The role of hunting cooperation and the fear effect in the predator-prey system is gaining considerable attention by the researchers recently. But the study on combined effect of hunting cooperation and fear in the predator-prey system is not yet studied. In the present paper, we investigate the impact of hunting cooperation among predators and predator induced fear in prey population by using the classical predator-prey model. We consider that predator populations cooperate during hunting. We also consider that hunting cooperation induces fear among prey, which has far richer and complex dynamics. We observe that without hunting cooperation, the unique coexistence equilibrium point is globally asymptotically stable. However, an increase in the hunting cooperation induced fear may destabilize the system and produce periodic solution via Hopf-bifurcation. The stability of the Hopf-bifurcating periodic solution is obtained by computing the Lyapunov coefficient. The limit cycles thus obtained may be supercritical or subcritical. We also observe that the system undergoes the Bogdanov-Takens bifurcation in two-parameter space. Further, we observe that the system exhibits backward bifurcation between predator-free equilibrium and coexisting equilibrium. The system also exhibits two different types of bi-stabilities due to subcritical Hopf-bifurcation (between interior equilibrium and stable limit cycle) and backward bifurcation (between predator-free and interior equilibrium points). Further, we observe strong demographic Allee phenomenon in the system. To visualize the dynamical behavior of the system, extensive numerical experiments are performed by using MATLAB and MATCONT softwares.     

A Within-Host Virus Model with Periodic Multidrug Therapy

This paper is devoted to the investigation of the effects of periodic drug treatment on a standard within-host virus model. We first introduce the basic reproduction ratio for the model, and then show that the infection free equilibrium is globally asymptotically stable, and the disease eventually disappears if $\mathcal{R}_{0} < 1$ , while there exists at least one positive periodic state and the disease persists when $\mathcal{R}_{0}>1$ . We also consider an optimization problem by shifting the phase of these drug efficacy functions. It turns out that shifting the phase can certainly affect the stability of the infection free steady state. A numerical study is performed to illustrate our analytic results.     

相互干扰的捕食与被捕食者种群的Hassall模型定性分析

本文证明了Hassall模型的渐近稳定性、有界性、全局稳定性。解决了文献[1]对该模型的三个猜想：（1）模型在某种条件下,正平衡点（x^*,y^*）是渐近稳定的;（2）模型的一切正初始条件的解有界;（3）模型在一定条件下,正平衡点（x^*,y^*）是全局稳定的。     

Qualitative analysis of models with different treatment protocols to prevent antibiotic resistance

This paper is concerned with the qualitative analysis of two models [S. Bonhoeffer, M. Lipsitch, B.R. Levin, Evaluating treatment protocols to prevent antibiotic resistance, Proc. Natl. Acad. Sci. USA 94 (1997) 12106] for different treatment protocols to prevent antibiotic resistance. Detailed qualitative analysis about the local or global stability of the equilibria of both models is carried out in term of the basic reproduction number R₀. For the model with a single antibiotic therapy, we show that if R₀ < 1, then the disease-free equilibrium is globally asymptotically stable; if R₀ > 1, then the disease-endemic equilibrium is globally asymptotically stable. For the model with multiple antibiotic therapies, stabilities of various equilibria are analyzed and combining treatment is shown better than cycling treatment. Numerical simulations are performed to show that the dynamical properties depend intimately upon the parameters.