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.     

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.     

Qualitative and bifurcation analysis using a computer virus model with a saturated recovery function

In this paper, we introduce a saturated treatment function into the computer virus propagation model, where the treatment function is limited for increasing number of infected computers. By carrying out global qualitative and bifurcation analysis, it is shown that the system exhibits some new and complicated behaviors: if the basic reproduction number is larger than unity, the number of infected computers will show persistent behavior, either converging to some positive constant or oscillating; and if the basic reproduction number is below unity, the model may exhibit complicated behaviors including: (i) backward bifurcation; (ii) almost sure virus eradication where the number of infective computers tends to zero for all initial positions except the interior equilibria; (iii) ‘‘oscillating’’ backward bifurcation where either the number of infective computers oscillates persistently, if the initial position lies in a region covering the stable virus equilibrium, or virus eradication, if the initial position lies outside this region; (iv) virus eradication for all initial positions if the basic reproduction number is less than a turning point value.     

Global analysis of a vector-host epidemic model with nonlinear incidences

In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number R₀. It is shown that if R₀ 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 is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.     

Dynamical analysis of age-structured pertussis model with covert infection

An age-structured pertussis model with covert infection is proposed to understand the effect of covert infection on the recurrence of pertussis. It is found that vaccination only for young children does not have a decisive effect on whooping cough control. It is shown that although the vaccine coverage rate is relatively high, the model has a backward bifurcation for a larger covert infection rate. In addition, sufficient conditions for the disease-free steady state to be globally asymptotically stable are obtained.     

Dynamics of a population in two patches with dispersal

A two-dimensional discrete system of a species in two patches proposed by Newman et al. is studied. It is shown that the unique interior steady state is globally asymptotically stable if the active population has a Beverton–Holt type growth rate. If the population is also subject to Allee effects, then the system has two interior steady states whenever the density-independent growth rate is large. In addition, the model has period-two solutions if the symmetric dispersal exceeds a critical threshold. For small dispersal, populations may either go extinct or eventually stabilize. However, populations are oscillating over time if dispersal is beyond the critical value and the initial populations are large.     

Dynamics of a structured population on two patches

We study a model for a creature which has a life cycle with two stages and which inhabits two patches. Our examination involves three different choices of birth function, namely linear, Ricker, and Allee. We discover conditions on the model parameters such that extinction will occur on both patches. We also find conditions on the parameters, and additionally in some cases the initial conditions, such that the creature will remain endemic on both patches. When the birth function is of Allee type on both patches, we show that there is always a population level beneath which extinction becomes inevitable on either patch. Simulations corroborate our theoretical results.     

Development and analysis of a malaria transmission mathematical model with seasonal mosquito life-history traits

In this paper, we develop and analyze a malaria model with seasonality of mosquito life-history traits: periodic-mosquitoes per capita birth rate, -mosquitoes death rate, -probability of mosquito to human disease transmission, -probability of human to mosquito disease transmission, and -mosquitoes biting rate. All these parameters are assumed to be time dependent leading to a nonautonomous differential equation system. We provide a global analysis of the model depending on two threshold parameters and (with ). When , then the disease-free stationary state is locally asymptotically stable. In the presence of the human disease-induced mortality, the global stability of the disease-free stationary state is guarantied when . On the contrary, if , the disease persists in the host population in the long term and the model admits at least one positive periodic solution. Moreover, by a numerical simulation, we show that a sub-critical (backward) bifurcation is possible at . Finally, the simulation results are in accordance with the seasonal variation of the reported cases of a malaria-epidemic region in Mpumalanga province in South Africa.     

Dynamic analysis of pest control model with population dispersal in two patches and impulsive effect

In this paper, we investigate the pest control model with population dispersal in two patches and impulsive effect. By exploiting the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we can obtain that the susceptible pest eradication periodic solution is globally asymptotically stable if the impulsive periodic τ is less than the critical value τ₀ . Further, we also prove that the system is permanent when the impulsive periodic τ is larger than the critical value τ₀. Hence, in order to drive the susceptible pest to extinction, we can take impulsive control strategy such that τ < τ₀ according to the effect of the viruses on the environment and the cost of the releasing pest infected in a laboratory. Finally, numerical simulations validate the obtained theoretical results for the pest control model with population dispersal in two patches and impulsive effect.     

辐射板强化喷管换热与红外抑制效果试验研究

试验研究了在发动机喷管中加装金属辐射板前后,喷管壁面温度、热喷流温度与喷管红外辐射特征的变化。结果表明,加装金属辐射板后,热喷流与喷管壁面之间的热量传递显著增强,热喷流中心温度降低,壁面温度明显升高,在90°方向上,热喷流3～5μm波段的红外辐射强度降低了38.5%。文中从热喷流、喷管壁面以及金属辐射板等相关部件的温度变化情况对红外辐射强度的变化原因进行了解释。      

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.     

Birds movement impact on the transmission of West Nile virus between patches

Spatial heterogeneity plays an important role in the distribution and persistence of many infectious disease. In the paper, a multi-patch model for the spread of West Nile virus among $n$ discrete geographic regions is presented that incorporates a mobility process. In the mobility process, we assume that the birds can move among regions, but not the mosquitoes based on scale-space. We show that the movement of birds between patches is sufficient to maintain disease persistence in patches. We compute the basic reproduction number $R_{0}$. We prove that if $R_{0}<1$, then the disease-free equilibrium of the model 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. Finally, numerical simulations demonstrate that the disease becomes endemic in both patches when birds move back and forth between two regions.     

Mathematical Analysis of West Nile Virus Model with Discrete Delays

The paper presents the basic model for the transmission dynamics of West Nile virus (WNV). The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally-asymptotically stable disease-free equilibrium whenever the associated reproduction number (?₀) is less than unity. As reveal in [3, 20], the analyses of the model show the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity). It is shown, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. Analysis of the reproduction number of the model shows that, the disease will persist, whenever ?₀ > 1, and increase in the length of incubation period can help reduce WNV burden in the community if a certain threshold quantities, denoted by Δ_b and Δ_v are negative. On the other hand, increasing the length of the incubation period increases disease burden if Δ_b > 0 and Δ_v > 0. Furthermore, it is shown that adding time delay to the corresponding autonomous model with standard incidence (considered in [2]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease).     

Dynamics of a Diffusive SIR Epidemic Model with Time Delay

This paper is devoted to a reaction-diffusion system for a SIR epidemic model with time delay and incidence rate. Firstly, the nonnegativity and boundedness of solutions determined by nonnegative initial values are obtained. Secondly, the existence and local stability of the disease-free equilibrium as well as the endemic equilibrium are investigated by analyzing the characteristic equations. Finally, the global asymptotical stability are obtained via Lyapunov functionals.     

Dynamics and steady-state analysis of a consumer-resource model

This paper is concerned with the dynamics of a consumer-resource reaction diffusion model in the heterogeneous environment, proposed by Zhang et al. (2017). We use the comparison principle to improve the ultimate bounds step by step, and show that the unique steady state is globally asymptotically stable if the resources are fully limited uniformly in space and consumer population abundance is homogeneous in space.     

一类具有饱和发生率和CTL免疫反应的HIV-1感染时滞模型的全局稳定性分析

通过构造合适的Lyapunov函数证明了一类具有饱和发生率和CTL免疫反应的HIV-1感染时滞模型各可能平衡点的全局稳定性.     

Stability analysis of an HIV/AIDS epidemic model with treatment

An HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are completely determined by the basic reproduction number _ℜ0${\Re }_0$. If _ℜ0≤1${\Re }_0\le 1$, the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if _ℜ0>1${\Re }_0>1$. Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results.     

The effect of vaccines on backward bifurcation in a fractional order HIV model

In this paper, a homogeneous-mixing population fractional model for human immunodeficiency virus (HIV) transmission, which incorporates anti-HIV preventive vaccines, is proposed. The dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when there is no vaccine. However, it has been shown that when the efficacy or dosage of vaccines is low, the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium point (DFE) coexists with a stable endemic equilibrium point (EE) when the associated reproduction number is less than unity. Therefore, driving the basic reproduction number below the unity is not enough to eradicate the disease. A new critical value at the turning point should be deduced as a new threshold of disease eradication. We have generalized the integer LaSalle invariant set theorem into fractional system and given some sufficient conditions for the disease-free equilibrium point being globally asymptotical stability. Mathematical results in this paper suggest that improving the efficiency and dosage of vaccines are all valid methods for the control of disease.     

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.