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.     

Modeling and dynamics of HIV transmission among high-risk groups in Guangzhou city, China

In this paper, a multicompartmental model is formulated to study how HIV is transmitted among different HIV high-risk groups, including MSM (men who have sex with men), FRs (foreigner residents), FSWs (female sex workers), and IDUs (injection drug users). The explicit expression for the basic reproduction number is obtained via the next generation matrix approach. We show that the disease free equilibrium is locally as well as globally asymptotically stable (the disease goes to extinction) when the basic reproduction number is less than unity, and the disease is always present when the basic reproduction number is larger than unity. As an illustration of our theoretical results, we conduct numerical simulations. We also conduct a case study where model parameters are estimated from the demographic and epidemiological data from Guangzhou. Using the parameter estimates, we predict the HIV/AIDS trend for each high-risk group. Furthermore, our study suggests that reducing the transmission routes of the disease and increasing condom use will be useful for control of HIV transmission.     

Modeling the role of government efforts in controlling extremism in a society

People having extreme idealogies affect the process in a region using fear of terror acts, money power, and the word of mouth communication network to change individuals to their way of thinking. This forces government to divert its limited financial resources for controlling extremism and thus affecting development. In this paper, therefore, a nonlinear mathematical model is proposed to study the dynamics of extremism governed by four dependent variables, namely, number of people in the general population having no extreme ideology, number of extreme ideologists, number of isolated ideologists (prisoners), and the cumulative density of government efforts and their interactions. The model is analyzed using the stability theory of differential equations and computer simulation. The analysis shows that if appropriate level of government efforts is applied on extremists, the spread of their ideology can be controlled in the general population. A numerical study of the model is also carried out to investigate the effects of certain parameters on the spread of extremism confirming the analytical results.Copyright © 2014 John Wiley & Sons, Ltd.     

Global dynamics of a two-patch SIS model with infection during transport

A two-patch SIS model is formulated and studied. The standard incidence rate and mass-action incidence rate are used within each patch and during transport, respectively. The basic reproduction number is calculated and the global dynamics is investigated. The simulation results show the influence of travel rates, the different dynamics by using standard incidence rate and mass-action incidence rate. The importance of border screening is also explored by numerical simulation.     

一类两菌株传染病模型的稳定性分析

建立和讨论一类具有比例接种疫苗丧失率的两菌株SIJVS传染病模型,给出了该模型基本再生数和侵入再生数的表达式,分析了无病平衡点、菌株占优平衡点、共存平衡点的存在性和稳定性.     

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.     

Modeling the Transmission of West Nile Virus with {\it Wolbachia} in a Heterogeneous Environment

{\it Wolbachia} are maternally transmitted endosymbiotic bacteria. To investigate the effect of {\it Wolbachia} on the spreading and vanishing of West Nile virus, we construct a reaction-diffusion model associated with the {\it Wolbachia} parameter in a heterogeneous environment, which has nonlinear infectious disease parameters. Based on the spectral radius of next infection operator and the related eigenvalue problem, we present a corresponding explicit expression describing the basic reproduction number. Furthermore, utilizing this number, we not only give out the stability of disease-free equilibrium, but also analyze the uniqueness and globally asymptotic behavior of endemic equilibrium. Our theoretical results and numerical simulations indicate that only if {\it Wolbachia} reach a certain magnitude in mosquitoes, it can be effective in the control of West Nile virus.     

一类潜伏期和染病期均具有传染性的手足口病模型

根据手足口病的病理特性及传播特点,建立一类描述其传播的数学模型并对模型的动力学性态进行分析.首先利用再生矩阵的方法定义了模型的基本再生数R_0,同时通过构造Lyapunov函数和Routh-Hurwitz判据证明了当R_0≤1时无病平衡点E_0的金局渐近稳定性,R_0>1时地方病平衡点E_*的局部渐近稳定性,并进一步证明了在一定条件下地方病平衡点的全局渐近稳定性.     

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.     

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.     

Dynamics analysis of a quarantine model in two patches

A new deterministic model for assessing the impact of quarantine on the transmission dynamics of a communicable disease in a two‐patch community is designed. Rigorous analysis of the model shows that the imperfect nature of quarantine (in the two patches) could induce the phenomenon of backward bifurcation when the associated reproduction number of the model is less than unity. For the case when quarantined susceptible individuals do not acquire infection during quarantine, the disease‐free equilibrium of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model has a unique Patch i‐only boundary equilibrium (i = 1,2) whenever the associated reproduction number for Patch i is greater than unity. The unique Patch i‐only boundary equilibrium is locally asymptotically stable whenever the invasion reproduction number of Patch 3 ? i is less than unity (and the associated reproduction number for Patch i exceeds unity). The model has at least one endemic equilibrium when its reproduction number exceeds unity (and the disease persists in both patches in this case). It is shown that adding multi‐patch dynamics to a single‐patch quarantine model (which allow the quarantine of susceptible individuals) in a single patch does not alter its quantitative dynamics (with respect to the existence and asymptotic stability of its associated equilibria as well as its backward bifurcation property). Copyright © 2014 John Wiley & Sons, Ltd.     

Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter

Our aim in this work was to examine the model underpinning the spread of the Rubella virus using the novel derivative called beta‐derivative. The study of the equilibrium points together with the analysis of the disease free equilibrium points was presented. Due to the complexity of the modified equation, we introduced a new operator based on the Sumudu transform. The properties of this operator were proposed and proved in detail. We made used of this operator together with the idea of perturbation method to derive a special solution of the extended model. The stability of the method for solving this model was presented. The uniqueness of the special solution was presented, and numerical simulations were done. The graphical representations show that the model depends on both parameters and the fractional order. © 2015 Wiley Periodicals, Inc. Complexity 21: 442–451, 2016     

Transmission dynamics and the control of hepatitis B in China: a population dynamics view

Though the prevalence of hepatitis B began to decline for the first time in 2010, it remains unclear whether this downward trend is permanent and the disease will be eradicated in mainland China under the current measures. Because a large number of hepatitis B virus (HBV) carriers and unknown HBV infections is characteristic of HBV infections in China, a mathematical model was designed and fitted to the reported hepatitis B data. The estimated basic reproduction number is 1.2861 (95\% confidence interval (CI) 1.2386-1.3302), which remains greater than one. Thus, the decline in 2010 may be part of the temporary benefits of public policy measures and should not be interpreted as indicative of successful intervention, although interventions do provide some benefits. To assess the effects of various interventions, the global uncertainty and sensitivity analyses revealed that the contribution of carriers is always greater than that of acute infections, and the prevalence of hepatitis B in China may be primarily a result of transmission by unknown patients. Therefore, strategies for controlling the HBV endemic, which target known patients, are unlikely to be highly effective. Additionally, three feasible strategies are proposed, although the benefits of these strategies may change radically over time.     

一类带有接种的SIR传染病模型的全局分析

基于经典的SIR传染病模型,建立了一类具有接种的SIR-V传染病模型,考虑了被接种者具有确定免疫期和免疫力按指数消失两种情形,得到了相应的基本再生数,并证明了其全局渐近稳定性.