Progression age enhanced backward bifurcation in an epidemic model with super-infection

 We consider a model for a disease with a progressing and a quiescent exposed class and variable susceptibility to super-infection. The model exhibits backward bifurcations under certain conditions, which allow for both stable and unstable endemic states when the basic reproduction number is smaller than one. Received: 11 October 2001 / Revised version: 17 September 2002 / Published online: 17 January 2003 Present address: Department of Biological Statistics and Computational Biology, 434 Warren Hall, Cornell University, Ithaca, NY 14853-7801 This author was visiting Arizona State University when most of the research was done. Research partially supported by NSF grant DMS-0137687. This author's research was partially supported by NSF grant DMS-9706787. Key words or phrases: Backward bifurcation – Multiple endemic equilibria – Alternating stability – Break-point density – Super-infection – Dose-dependent latent period – Progressive and quiescent latent stages – Progression age structure – Threshold type disease activation – Operator semigroups – Hille-Yosida operators – Dynamical systems – Persistence – Global compact attractor     

A Toy Model for the Generation of Homochirality during Polymerization

This article examines a toy model of polymerization which though artificial and unphysical has some interesting chiral features. Two key elements, enantiomeric cross inhibition and chiral feedback, are shown to lead to bifurcation, so that the end product can become homo-chiral. We find that the bifurcation is driven by the cross-inhibition but is not strongly dependant on its strength, which for perfect feedback fidelity mainly determines the time scale. We also find that bifurcation with a high degree of chiral polarization remains even when the fidelity of the chiral feedback is substantially less than unity. For small values of the feedback fidelity the polarization drops below unity and at a critical value falls sharply to zero in a `phase transition'. The value at which this happens depends on the cross-inhibition in a complex way. By comparing the behaviour of polymers differing only in their final length, N, we find that the bifurcation process is enhanced as N increases. The symmetry breaking which we find is clearly a particular manifestation of general bifurcation theory. In addition it has the specific interest that, at least in our model, long homochiral polymers are possible even in the presence of substantial enantiomeric cross-inhibition.     

Backward bifurcation,equilibrium and stability phenomena in a three-stage extended BRSV epidemic model

In this paper we consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for Bovine Respiratory Syncytial Virus (BRSV) amongst cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for R ₀ < 1, where R ₀ is the basic reproductive number, and a unique supercritical endemic equilibrium for R ₀ > 1. In our three-stage extended model we find that more complex bifurcation diagrams are possible. The paper starts with a review of some of the previous work on backward bifurcation then describes our three-stage model. We give equilibrium and stability results, and also provide some biological motivation for the model being studied. It is shown that backward bifurcation can occur in the three-stage model for small b, where b is the common per capita birth and death rate. We are able to classify the possible bifurcation diagrams. Some realistic numerical examples are discussed at the end of the paper, both for b small and for larger values of b.      

Modeling the impact of early therapy for latent tuberculosis patients and its optimal control analysis

Effective tuberculosis (TB) control depends on case findings to discover infectious cases, investigation of contacts of those with TB, as well as appropriate treatment. Adherence and successful completion of the treatment are equally important. Unfortunately, due to a number of personal, psychosocial, economic, medical, and health service factors, a significant number of TB patients become irregular and default from treatment. In this paper, a mathematical model is developed to assess the impact of early therapy for latent TB and non-adherence on controlling TB transmission dynamics. Equilibrium states of the model are determined and their local stability is examined. With the aid of the center manifold theory, it is established that the model undergoes a backward bifurcation. Qualitative mathematical analysis of the model suggests that a high level of latent tuberculosis case findings, coupled with a decrease of defaulting rate, may be effective in controlling TB transmission dynamics in the community. Population-level effects of organized campaigns to improve early therapy and to guarantee successful completion of each treatment are evaluated through numerical simulations and presented in support of the analytical results.     

Modeling Spatial Spread of Infectious Diseases with a Fixed Latent Period in a Spatially Continuous Domain

In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may diffuse, we formulate an SIR model with a simple demographic structure for the population living in a spatially continuous environment. The model is given by a system of reaction-diffusion equations with a discrete delay accounting for the latency and a spatially non-local term caused by the mobility of the individuals during the latent period. We address the existence, uniqueness, and positivity of solution to the initial-value problem for this type of system. Moreover, we investigate the traveling wave fronts of the system and obtain a critical value c ^* which is a lower bound for the wave speed of the traveling wave fronts. Although we can not prove that this value is exactly the minimal wave speed, numeric simulations seem to suggest that it is. Furthermore, the simulations on the PDE model also suggest that the spread speed of the disease indeed coincides with c ^*. We also discuss how the model parameters affect c ^*.     

Modeling and dynamics of Wolbachia-infected male releases and mating competition on mosquito control

Despite centuries of continuous efforts, mosquito-borne diseases (MBDs) remain enormous health threat of human life worldwide. Lately, the USA government has approved an innovative technology of releasing Wolbachia-infected male mosquitoes to suppress the wild mosquito population. In this paper we first introduce a stage-structured model for natural mosquitos, then we establish a new model considering the releasing of Wolbachia-infected male mosquitoes and the mating competition between the natural male mosquitoes and infected males on the suppression of natural mosquitoes. Dynamical analysis of the two models, including the existence and local stability of the equilibria and bifurcation analysis, reveals the existence of a forward bifurcation or a backward bifurcation with multiple attractors. Moreover, globally dynamical properties are further explored by using Lyapunov function and theory of monotone operators, respectively. Our findings suggest that infected male augmentation itself cannot always guarantee the success of population eradication, but leads to three possible levels of population suppression, so we define the corresponding suppression rate and estimate the minimum release ratio for population eradication. Furthermore, we study how the release ratio of infected males and natural ones, mating competition, the rate of cytoplasmic incompatibility and the basic offspring number affect the suppression rate of natural mosquitoes. Our results show that the successful eradication relies on assessing the reproductive capacity of natural mosquitoes, a selection of suitable Wolbachia strains and an appropriate release amount of infected males. This study will be helpful for public health authorities in designing proper strategies to control vector mosquitoes and prevent the epidemics of MBDs.

     

Backward bifurcation and oscillations in a nested immuno-eco-epidemiological model

This paper introduces a novel partial differential equation immuno-eco-epidemiological model of competition in which one species is affected by a disease while another can compete with it directly and by lowering the first species' immune response to the infection, a mode of competition termed stress-induced competition. When the disease is chronic, and the within-host dynamics are rapid, we reduce the partial differential equation model (PDE) to a three-dimensional ordinary differential equation (ODE) model. The ODE model exhibits backward bifurcation and sustained oscillations caused by the stress-induced competition. Furthermore, the ODE model, although not a special case of the PDE model, is useful for detecting backward bifurcation and oscillations in the PDE model. Backward bifurcation related to stress-induced competition allows the second species to persist for values of its invasion number below one. Furthermore, stress-induced competition leads to destabilization of the coexistence equilibrium and sustained oscillations in the PDE model. We suggest that complex systems such as this one may be studied by appropriately designed simple ODE models.     

Analysis of a tuberculosis model with a case study in Uganda

We consider a four-compartment tuberculosis model including exogenous reinfection. We derive sufficient conditions, in terms of the parameters of the system, which guarantee the occurrence of backward bifurcation. We also discuss the global stability of the endemic state by using a generalization of the Poincaré–Bendixson criterion. An application is given for the case of Internally Displaced People's Camps in North Uganda. The study suggests how important it is to provide qualitative indications on the threshold value of the population density in the area occupied by the camps, in order to possibly eradicate the disease.     

Hopf bifurcation in a structured population model for the sexual phase of monogonont rotifers

 We are studying a population of monogonont rotifers in the context of non-linear age-dependent models. In the sexual phase of their reproductive cycle we consider the population structured by age, and composed of three subclasses: virgin mictic females, mated mictic females, and haploid males. The model system has a unique stationary population density which is stable as long as a parameter, related to male-female encounter rate, remains below a critical value. When the parameter increases beyond this critical value, the stationary solution becomes unstable and a stable limit cycle (isolated periodic orbit) appears. The occurrence of this supercritical Hopf bifurcation is shown analytically. Received: 2 August 2001 / Revised version: 3 January 2002 / Published online: 26 June 2002     

A simple vaccination model with multiple endemic states

A simple two-dimensional SIS model with vaccination exhibits a backward bifurcation for some parameter values. A two-population version of the model leads to the consideration of vaccination policies in paired border towns. The results of our mathematical analysis indicate that a vaccination campaign φ meant to reduce a disease's reproduction number R(φ) below one may fail to control the disease. If the aim is to prevent an epidemic outbreak, a large initial number of infective persons can cause a high endemicity level to arise rather suddenly even if the vaccine-reduced reproduction number is below threshold. If the aim is to eradicate an already established disease, bringing the vaccine-reduced reproduction number below one may not be sufficient to do so. The complete bifurcation analysis of the model in terms of the vaccine-reduced reproduction number is given, and some extensions are considered.     

Bubble splitting in bifurcating tubes: a model study of cardiovascular gas emboli transport.

The transport of long gas bubbles, suspended in liquid, through symmetric bifurcations, is investigated experimentally and theoretically as a model of cardiovascular gas bubble transport in air embolism and gas embolotherapy. The relevant dimensionless parameters in the models match the corresponding values for arteries and arterioles. The effects of roll angle (the angle the plane of the bifurcation makes with the horizontal), capillary number (a dimensionless indicator of flow), and bubble volume (or length) on the splitting of bubbles as they pass through the bifurcation are examined. Splitting is observed to be more homogenous at higher capillary numbers and lower roll angles. It is shown that, at nonzero roll angles, there is a critical value of the capillary number below which the bubbles do not split and are transported entirely into the upper branch. The value of the critical capillary number increases with roll angle and parent tube diameter. A unique bubble motion is observed at the critical capillary number and for slightly slower flows: the bubble begins to split, the meniscus in the lower branch then moves backward, and finally the entire bubble enters the upper branch. These findings suggest that, in large vessels, emboli tend to be transported upward unless flow is unusually strong but that a more homogeneous distribution of emboli occurs in smaller vessels. This corresponds to previous observations that air emboli tend to lodge in the upper regions of the lungs and suggests that relatively uniform infarction of tumors by gas embolotherapy may be possible.     

Development of Mathematical Models for the Analysis of Hepatitis Delta Virus Viral Dynamics

Background

Mathematical models have shown to be extremely helpful in understanding the dynamics of different virus diseases, including hepatitis B. Hepatitis D virus (HDV) is a satellite virus of the hepatitis B virus (HBV). In the liver, production of new HDV virions depends on the presence of HBV. There are two ways in which HDV can occur in an individual: co-infection and super-infection. Co-infection occurs when an individual is simultaneously infected by HBV and HDV, while super-infection occurs in persons with an existing chronic HBV infection.

Methodology/Principal Findings

In this work a mathematical model based on differential equations is proposed for the viral dynamics of the hepatitis D virus (HDV) across different scenarios. This model takes into consideration the knowledge of the biology of the virus and its interaction with the host. In this work we will present the results of a simulation study where two scenarios were considered, co-infection and super-infection, together with different antiviral therapies. Although, in general the predicted course of HDV infection is similar to that observed for HBV, we observe a faster increase in the number of HBV infected cells and viral load. In most tested scenarios, the number of HDV infected cells and viral load values remain below corresponding predicted values for HBV.

Conclusions/Significance

The simulation study shows that, under the most commonly used and generally accepted therapy approaches for HDV infection, such as lamivudine (LMV) or ribavirine, peggylated alpha-interferon (IFN) or a combination of both, LMV monotherapy and combination therapy of LMV and IFN were predicted to more effectively reduce the HBV and HDV viral loads in the case of super-infection scenarios when compared with the co-infection. In contrast, IFN monotherapy was found to reduce the HDV viral load more efficiently in the case of super-infection while the effect on the HBV viral load was more pronounced during co-infection. The results suggest that there is a need for development of high efficacy therapeutic approaches towards the specific inhibition of HDV replication. These approaches may additionally be directed to the reduction of the half-life of infected cells and life-span of newly produced circulating virions.     

Trade-off between BCG vaccination and the ability to detect and treat latent tuberculosis

Policies regarding the use of the Bacille Calmette-Guérin (BCG) vaccine for tuberculosis vary greatly throughout the international community. In several countries, consideration of discontinuing universal vaccination programs is currently under way. The arguments against mass vaccination are that the effectiveness of BCG in preventing tuberculosis is uncertain and that BCG vaccination can interfere with the detection and treatment of latent tuberculosis.In this work, we pose a dynamical systems model for the population-level dynamics of tuberculosis in order to study the trade-off which occurs between vaccination and detection/treatment of latent tuberculosis. We assume that latent infection in vaccinated individuals is completely undetectable. For the case of a country with very low levels of tuberculosis, we establish analytic thresholds, via stability analysis and the basic reproductive number, which determine the optimal vaccination policy, given the effectiveness of the vaccine and the detection/treatment rate of latent tuberculosis.The results of this work suggest that it is unlikely that a country detects and treats latent tuberculosis at a high enough rate to justify the discontinuation of mass vaccination from this perspective.     

A model for tuberculosis with exogenous reinfection

Following primary tuberculosis (TB) infection, only approximately 10% of individuals develop active T.B. Most people are assumed to mount an effective immune response to the initial infection that limits proliferation of the bacilli and leads to long-lasting partial immunity both to further infection and to reactivation of latent bacilli remaining from the original infection. Infected individuals may develop active TB as a consequence of exogenous reinfection, i.e., acquiring a new infection from another infectious individual. Our results in this paper suggest that exogenous reinfection has a drastic effect on the qualitative dynamics of TB. The incorporation of exogenous reinfection into our TB model allows the possibility of a subcritical bifurcation at the critical value of the basic reproductive number R(0)=1, and hence the existence of multiple endemic equilibria for R(0)<1 and the exogenous reinfection rate larger than a threshold. Our results suggest that reducing R(0) to be smaller than one may not be sufficient to eradicate the disease. An additional reduction in reinfection rate may be required. These results may also partially explain the recently observed resurgence of TB.     

The backward bifurcation in compartmental models for West Nile virus

In all of the West Nile virus (WNV) compartmental models in the literature, the basic reproduction number serves as a crucial control threshold for the eradication of the virus. However, our study suggests that backward bifurcation is a common property shared by the available compartmental models with a logistic type of growth for the population of host birds. There exists a subthreshold condition for the outbreak of the virus due to the existence of backward bifurcation. In this paper, we first review and give a comparison study of the four available compartmental models for the virus, and focus on the analysis of the model proposed by Cruz-Pacheco et al. to explore the backward bifurcation in the model. Our comparison study suggests that the mosquito population dynamics itself cannot explain the occurrence of the backward bifurcation, it is the higher mortality rate of the avian host due to the infection that determines the existence of backward bifurcation.     

Analysis of a tuberculosis model with a case study in Uganda

We consider a four-compartment tuberculosis model including exogenous reinfection. We derive sufficient conditions, in terms of the parameters of the system, which guarantee the occurrence of backward bifurcation. We also discuss the global stability of the endemic state by using a generalization of the Poincaré-Bendixson criterion. An application is given for the case of Internally Displaced People's Camps in North Uganda. The study suggests how important it is to provide qualitative indications on the threshold value of the population density in the area occupied by the camps, in order to possibly eradicate the disease.     

耐药结核分枝杆菌潜伏-复发感染动物模型的研究进展

潜伏结核感染(latent tuberculosis infection,LTBI)复发是新发结核病的主要来源,其中耐药结核病所占比例较大,使耐药LTBI复发的防控成为结核病研究的重点。耐药结核分枝杆菌潜伏-复发感染动物模型是开展耐药结核病防控相关机制研究、抗耐药结核分枝杆菌药物和疫苗研究的基础。目前耐药结核分枝杆菌感染动物模型缺乏,而已有的结核分枝杆菌标准株H37Rv潜伏-复发感染模型存在缺陷,如小鼠模型的潜伏期荷菌量偏高、复发期变异大,而猴模型的潜伏期和复发期不可预测。模型的可控性差使其应用困难,且缺乏可用的免疫学评价指标,导致远期复发无法预测。因此,基于现有H37Rv潜伏-复发感染动物模型的制备方法,展望耐药结核分枝杆菌潜伏-复发感染动物模型可能存在的缺陷,通过选用新的抑菌剂和诱导剂,制备有稳定潜伏期、潜伏时长适中、复发起点和复发水平变异小的动物模型,是未来耐药结核分枝杆菌潜伏-复发感染动物模型研究的方向。     

The minimum effort required to eradicate infections in models with backward bifurcation

We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model shows backward bifurcations, where μ is the death rate and α is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant β. This eradication effort is explicitly expressed in terms of α,r, and μ As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R ₀ is smaller than the eradication effort for r > 1 + (μ/α) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.Dedicated to Karl Peter Hadeler on the occasion of his 70th birthday.     

Turing instability in pioneer/climax species interactions

Systems of pioneer and climax species are used to model interactions of species whose reproductive capacity is sensitive to population density in their shared ecosystem. Intraspecies interaction coefficients can be adjusted so that spatially homogeneous solutions are stable to small perturbations. In a reaction-diffusion pioneer/climax model we will determine the critical value of the diffusion rate of the climax species, below which the equilibrium solution is unstable to non-homogeneous perturbations. For diffusion rates smaller than this critical value, an equilibrium solution remains stable to spatially homogeneous perturbations but is unstable to non-homogeneous perturbations. A Turing (diffusional) bifurcation leads to the formation of spatial patterns in species' densities. Forcing, interpreted as stocking or harvesting of the species, can reverse the bifurcation and establish equilibrium solutions which are stable to small perturbations. The implicit function theorem is used to determine whether stocking or harvesting of one of the species in the model is the appropriate remedy for diffusional instability. The use of stocking or harvesting by a natural resource manager thus influences the long-term dynamics and spatial distribution of species in a pioneer/climax ecosystem.     

ABCA2基因mRNA在结核分枝杆菌不同感染阶段的表达及其作为诊断标志物的研究

摘要 目的：比较ATP结合转运蛋白A2（ATP-binding cassette transporterA2,ABCA2）基因mRNA在活动性结核病和结核分枝杆菌潜伏感染状态下的表达差异及其作为诊断标志物的能力。方法：收集活动性结核病患者、结核分枝杆菌潜伏感染者和健康人外周血,荧光定量PCR检测受试者外周血单个核细胞中ABCA2的mRNA水平,统计学分析各组ABCA2基因表达差异,及其鉴别活动性结核和潜伏感染的能力。结果：活动性结核病患者外周血单个核细胞中ABCA2的mRNA表达显著低于潜伏感染者和健康人（（H=83.38,P＜0.0001）。ABCA2鉴别诊断活动性结核和潜伏感染的曲线下面积为0.9235,灵敏度为73.53%（95%的置信区间为63.87%~81.78%）,特异性为93.55%（95%的置信区间为78.58%~99.21%）。结论：外周血单个核细胞中ABCA2基因mRNA是鉴别活动性结核病和潜伏感染者的潜在标志物,有助于活动性结核病的辅助诊断。