一类多菌株STD模型的共存性

我们已经研究过一类拥有两种菌株的异性传播的性传染病模型.得到了边界平衡点稳定的充要条件,并确认在边界平衡点的稳定性和正平衡点的存在性之间存在着很强的联系.但是只给出了特殊条件下正平衡点稳定的充要条件,这篇文章将就以前没解决的问题,对这类模型给出完整的分析.     

Evolutionary consequences of harvesting for a two-zooplankton one-phytoplankton system

Considering the impact of harvesting on the coexistence and competitive exclusion of competitive predators, a two-zooplankton one-phytoplankton model with harvesting is proposed and investigated. First, stability criteria of the model is analyzed both from local and global point of view. Second, two types of zooplankton will competitively exclude each other in the absence of harvesting with the zooplankton with the larger threshold persisting. If harvest rates are discriminate, then a dominant zooplankton may occur depending on the harvesting level. Thus, for some harvesting levels, the zooplankton one may persist while for other harvesting levels zooplankton two may persist. Furthermore, the value of the harvesting level and coexistence line are obtained when coexistence occur. Finally, the impact of harvesting is mentioned along with numerical results to provide some support to the analytical findings.     

Dynamics of an epidemic model with host migration

Host migration among discrete geographical regions is demonstrated as an important factor that brings about the diffusion and outbreak of many vector-host diseases. In the paper, we develop a mathematical model to explore the effect of host migration between two patches on the spread of a vector-host disease. Analytical results show that the reproduction number R₀ provides a threshold condition that determines the uniform persistence and extinction of the disease. If both the patches are identical, it is shown that an endemic equilibrium is locally stable. It is also shown that a unique endemic equilibrium, which exists when the disease cannot induce the death of the host, is globally asymptotically stable. Finally, two examples are given to illustrate the effect of host migration on the spread of the vector-host disease.     

Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis

A deterministic model for studying the transmission dynamics of bovine tuberculosis in a single cattle herd is presented and qualitatively analyzed. A notable feature of the model is that it allows for the importation of asymptomatically infected cattle (into the herd) because re‐stocking from outside sources. Rigorous analysis of the model shows that the model has a globally‐asymptotically stable disease‐free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. In the absence of importation of asymptomatically infected cattle, the model has a unique endemic equilibrium whenever the reproduction number exceeds unity (this equilibrium is globally asymptotically stable for a special case). It is further shown that, for the case where asymptomatically infected cattle are imported into the herd, the model has a unique endemic equilibrium. This equilibrium is also shown to be globally asymptotically stable for a special case. Copyright © 2011 John Wiley & Sons, Ltd.     

Global dynamics of a pine wilt disease transmission model with nonlinear incidence rates

In this study, we investigate a pine wilt transmission model with nonlinear incidence rates. The stability of the system is analyzed for disease-free and endemic equilibria. It is proved that the global dynamics are completely by the basic reproduction number _R0$R_0$. If _R0$R_0$ is less than one, the disease-free equilibrium is globally asymptotically stable, and in such a case, the endemic equilibrium does not exist. If _R0$R_0$ is greater than one, the disease persists and the unique endemic equilibrium is globally asymptotically stable.     

A fractional order epidemic model and simulation for avian influenza dynamics

We present a nonlinear fractional order epidemic model to investigate the spreading dynamical behavior of the avian influenza. The population of the model contains susceptible individuals, asymptomatic but infective latent individuals, and infective individuals. We first establish the existence, uniqueness, nonnegativity, and positive invariance of the solution, then we study the reproduction number of the model and the stability of the disease‐free equilibrium. We observe that the reproduction number varies with the order of the fractional derivative ν. In terms of epidemics, this suggests that varying ν induces a change in the avian's epidemic status. Furthermore, we derive the sufficient conditions for the existence and the stability of the endemic equilibrium. Finally, we carry out some numerical simulations to validate the analytical results. We find from simulations that the solution of the fractional order model tends to a stationary state over a longer period of time with decreasing the value of the fractional derivative, and the size of epidemic decreases with decreasing ν.     

Stability analysis for an epidemic model with stage structure

An epidemic model with stage structure is formulated. The period of infection is partitioned into the early and later stages according to the developing process of infection, and the infectious individuals in the different stages have the different ability of transmitting disease. The constant recruitment rate and exponential natural death, as well as the disease-related death, are incorporated into the model. The basic reproduction number of this model is determined by the method of next generation matrix. The global stability of the disease-free equilibrium and the local stability of the endemic equilibrium are obtained; the global stability of the endemic equilibrium is got under the case that the infection is not fatal.     

Global dynamics of a mathematical model for HTLV-I infection of CD4 T-cells

In this paper, a mathematical model for HILV-I infection of CD4⁺ T-cells is investigated. The force of infection is assumed be of a function in general form, and the resulting incidence term contains, as special cases, the bilinear and the saturation incidences. The model can be seen as an extension of the model [Wang et al. Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression, Math. Biosci. 179 (2002) 207-217; Song, Li, Global stability and periodic solution of a model for HTLV-I infection and ATL progression, Appl. Math. Comput. 180(1) (2006) 401-410]. Mathematical analysis establishes that the global dynamics of T-cells infection is completely determined by a basic reproduction number _R0${\mathfrak{R}}_0$. If _R0?1${\mathfrak{R}}_0?1$, the infection-free equilibrium is globally stable; if _R0>1${\mathfrak{R}}_0>1$, the unique infected equilibrium is globally stable in the interior of the feasible region.     

Mathematical model for assessing the impact of vaccination and treatment on measles transmission dynamics

A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease‐free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed if either measles vaccine is assumed to be perfect or disease related mortality rates are negligible. In the latter case, the disease‐free equilibrium is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a nonlinear Lyapunov function of Goh‐Volterra type, to be globally asymptotically stable for a special case.     

Spatial‐temporal basic reproduction number and dynamics for a dengue disease diffusion model

A reaction‐diffusion system with free boundary is proposed to describe the transmission of the dengue disease from mosquitoes to humans. In addition to the classical basic reproduction number R₀, the spatial‐temporal basic reproduction number is introduced to determine the persistence and eradication of the disease. Some sufficient conditions for the disease vanishing or spreading are obtained. The disease will go extinct under one of the conditions: the classical basic reproduction number R₀ < 1 and the spatial‐temporal basic reproduction number with small expanding capability. The spread of the disease in the whole area is possible if for some t≥0. Numerical simulations are also given to illustrate the theoretical results.     

Threshold behaviour of a stochastic SIR model

In this paper, we investigate the threshold behaviour of a susceptible-infected-recovered (SIR) epidemic model with stochastic perturbation. When the noise is small, we show that the threshold determines the extinction and persistence of the epidemic. Compared with the corresponding deterministic system, this value is affected by white noise, which is less than the basic reproduction number of the deterministic system. On the other hand, we obtain that the large noise will also suppress the epidemic to prevail, which never happens in the deterministic system. These results are illustrated by computer simulations.     

Global stability for a model of competition in the chemostat with microbial inputs

We propose a model of competition of n species in a chemostat, with constant input of some species. We mainly emphasize the case that can lead to coexistence in the chemostat in a non-trivial way, i.e., where the n−1 less competitive species are in the input. We prove that if the inputs satisfy a constraint, the coexistence between the species is obtained in the form of a globally asymptotically stable (GAS) positive equilibrium, while a GAS equilibrium without the dominant species is achieved if the constraint is not satisfied. This work is round up with a thorough study of all the situations that can arise when having an arbitrary number of species in the chemostat inputs; this always results in a GAS equilibrium that either does or does not encompass one of the species that is not present in the input.     

Mathematical model for the dynamics of visceral leishmaniasis–malaria co‐infection

A mathematical model to understand the dynamics of malaria–visceral leishmaniasis co‐infection is proposed and analyzed. Results show that both diseases can be eliminated if R₀, the basic reproduction number of the co‐infection, is less than unity, and the system undergoes a backward bifurcation where an endemic equilibrium co‐exists with the disease‐free equilibrium when one of R_m or R_l, the basic reproduction numbers of malaria‐only and visceral leishmaniasis‐only, is precisely less than unity. Results also show that in the case of maximum protection against visceral leishmaniasis (VL), the disease‐free equilibrium is globally asymptotically stable if malaria patients are protected from VL infection; similarly, in the case of maximum protection against malaria, the disease‐free equilibrium is globally asymptotically stable if VL and post‐kala‐azar dermal leishmaniasis patients and the recovered humans after VL are protected from malaria infection. Numerical results show that if R_m and R_l are greater than unity, then we have co‐existence of both disease at an endemic equilibrium, and malaria incidence is higher than visceral leishmaniasis incidence at steady state. Copyright © 2016 John Wiley & Sons, Ltd.     

Global dynamics of a class of SEIRS epidemic models in a periodic environment

In this paper, we study a class of periodic SEIRS epidemic models and it is shown that the global dynamics is determined by the basic reproduction number R₀ which is defined through the spectral radius of a linear integral operator. If R₀<1, then the disease free periodic solution is globally asymptotically stable and if R₀>1, then the disease persists. Our results really improve the results in [T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology Bull. Math. Biol. 69 (8) (2007) 2537-2559] for the periodic case. Moreover, from our results, we see that the eradication policy on the basis of the basic reproduction number of the time-averaged system may overestimate the infectious risk of the periodic disease. Numerical simulations which support our theoretical analysis are also given.