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.     

Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity

We derive and analyse a deterministic model for the transmission of malaria disease with mass action form of infection. Firstly, we calculate the basic reproduction number, R(0), and investigate the existence and stability of equilibria. The system is found to exhibit backward bifurcation. The implication of this occurrence is that the classical epidemiological requirement for effective eradication of malaria, R(0)<1, is no longer sufficient, even though necessary. Secondly, by using optimal control theory we derive the conditions under which it is optimal to eradicate the disease and examine the impact of a possible combined vaccination and treatment strategy on the disease transmission. When eradication is impossible, we derive the necessary conditions for optimal control of the disease using Pontryagin's Maximum Principle. The results obtained from the numerical simulations of the model show that a possible vaccination combined with effective treatment regime would reduce the spread of the disease appreciably.     

Analysis of a dengue disease transmission model

A model for the transmission of dengue fever in a constant human population and variable vector population is discussed. A complete global analysis is given, which uses the results of the theory of competitive systems and stability of periodic orbits, to establish the global stability of the endemic equilibrium. The control measures of the vector population are discussed in terms of the threshold condition, which governs the existence and stability of the endemic equilibrium.     

A disease transmission model in a nonconstant population

A general SIRS disease transmission model is formulated under assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. For a class of incidence functions it is shown that the model has no periodic solutions. By contrast, for a particular incidence function, a combination of analytical and numerical techniques are used to show that (for some parameters) periodic solutions can arise through homoclinic loops or saddle connections and disappear through Hopf bifurcations.Supported in part by NSERC grant A-8965, the University of Victoria Committee on Faculty Research & Travel, and the Institute for Mathematics and its Applications, Minneapolis, MN, with funds provided by NSF     

The impact of economic crises on communicable disease transmission and control: a systematic review of the evidence

There is concern among public health professionals that the current economic downturn, initiated by the financial crisis that started in 2007, could precipitate the transmission of infectious diseases while also limiting capacity for control. Although studies have reviewed the potential effects of economic downturns on overall health, to our knowledge such an analysis has yet to be done focusing on infectious diseases. We performed a systematic literature review of studies examining changes in infectious disease burden subsequent to periods of crisis. The review identified 230 studies of which 37 met our inclusion criteria. Of these, 30 found evidence of worse infectious disease outcomes during recession, often resulting from higher rates of infectious contact under poorer living circumstances, worsened access to therapy, or poorer retention in treatment. The remaining studies found either reductions in infectious disease or no significant effect. Using the paradigm of the "SIR" (susceptible-infected-recovered) model of infectious disease transmission, we examined the implications of these findings for infectious disease transmission and control. Key susceptible groups include infants and the elderly. We identified certain high-risk groups, including migrants, homeless persons, and prison populations, as particularly vulnerable conduits of epidemics during situations of economic duress. We also observed that the long-term impacts of crises on infectious disease are not inevitable: considerable evidence suggests that the magnitude of effect depends critically on budgetary responses by governments. Like other emergencies and natural disasters, preparedness for financial crises should include consideration of consequences for communicable disease control.     

Vaccine efficacy, and immunity affecting transmission

New candidate malaria vaccines being developed experimentally can only be assessed properly after they have been through the full range of clinical trials and extended follow-up. Defining vaccine efficacy, especially that of multi-component vaccines, is difficult, in particular because the outcome is determined by finely defined epitope specificity. Immune responses can have contrasting effects, some protective, some promoting infection, but concerns that interventions that reduce transmission from a high to a lower intensity will lead to greater risks of severe malaria may not have taken into account all of the many factors that determine the outcome of infection.     

A canonical model of multistability and scale-invariance in biological systems

Multistability and scale-invariant fluctuations occur in a wide variety of biological organisms from bacteria to humans as well as financial, chemical and complex physical systems. Multistability refers to noise driven switches between multiple weakly stable states. Scale-invariant fluctuations arise when there is an approximately constant ratio between the mean and standard deviation of a system's fluctuations. Both are an important property of human perception, movement, decision making and computation and they occur together in the human alpha rhythm, imparting it with complex dynamical behavior. Here, we elucidate their fundamental dynamical mechanisms in a canonical model of nonlinear bifurcations under stochastic fluctuations. We find that the co-occurrence of multistability and scale-invariant fluctuations mandates two important dynamical properties: Multistability arises in the presence of a subcritical Hopf bifurcation, which generates co-existing attractors, whilst the introduction of multiplicative (state-dependent) noise ensures that as the system jumps between these attractors, fluctuations remain in constant proportion to their mean and their temporal statistics become long-tailed. The simple algebraic construction of this model affords a systematic analysis of the contribution of stochastic and nonlinear processes to cortical rhythms, complementing a recently proposed biophysical model. Similar dynamics also occur in a kinetic model of gene regulation, suggesting universality across a broad class of biological phenomena.     

Determination of the processes driving the acquisition of immunity to malaria using a mathematical transmission model

Acquisition of partially protective immunity is a dominant feature of the epidemiology of malaria among exposed individuals. The processes that determine the acquisition of immunity to clinical disease and to asymptomatic carriage of malaria parasites are poorly understood, in part because of a lack of validated immunological markers of protection. Using mathematical models, we seek to better understand the processes that determine observed epidemiological patterns. We have developed an age-structured mathematical model of malaria transmission in which acquired immunity can act in three ways (“immunity functions”): reducing the probability of clinical disease, speeding the clearance of parasites, and increasing tolerance to subpatent infections. Each immunity function was allowed to vary in efficacy depending on both age and malaria transmission intensity. The results were compared to age patterns of parasite prevalence and clinical disease in endemic settings in northeastern Tanzania and The Gambia. Two types of immune function were required to reproduce the epidemiological age-prevalence curves seen in the empirical data; a form of clinical immunity that reduces susceptibility to clinical disease and develops with age and exposure (with half-life of the order of five years or more) and a form of anti-parasite immunity which results in more rapid clearance of parasitaemia, is acquired later in life and is longer lasting (half-life of >20 y). The development of anti-parasite immunity better reproduced observed epidemiological patterns if it was dominated by age-dependent physiological processes rather than by the magnitude of exposure (provided some exposure occurs). Tolerance to subpatent infections was not required to explain the empirical data. The model comprising immunity to clinical disease which develops early in life and is exposure-dependent, and anti-parasite immunity which develops later in life and is not dependent on the magnitude of exposure, appears to best reproduce the pattern of parasite prevalence and clinical disease by age in different malaria transmission settings. Understanding the effector mechanisms underlying these two immune functions will assist in the design of transmission-reducing interventions against malaria.     

Analysis of a disease transmission model in a population with varying size

An S I R S epidemiological model with vital dynamics in a population of varying size is discussed. A complete global analysis is given which uses a new result to establish the nonexistence of periodic solutions. Results are discussed in terms of three explicit threshold parameters which respectively govern the increase of the total population, the existence and stability of an endemic proportion equilibrium and the growth of the infective population. These lead to two distinct concepts of disease eradication which involve the total number of infectives and their proportion in the population.Partially supported by NSF Grant No. DMS-8703631. This work was done while this author was visiting the University of VictoriaResearch supported in part by NSERC A-8965     

Defining dysbiosis and its influence on host immunity and disease

Mammalian immune system development depends on instruction from resident commensal microorganisms. Diseases associated with abnormal immune responses towards environmental and self antigens have been rapidly increasing over the last 50 years. These diseases include inflammatory bowel disease (IBD), multiple sclerosis (MS), type I diabetes (T1D), allergies and asthma. The observation that people with immune mediated diseases house a different microbial community when compared to healthy individuals suggests that pathogenesis arises from improper training of the immune system by the microbiota. However, with hundreds of different microorganisms on our bodies it is hard to know which of these contribute to health and more importantly how? Microbiologists studying pathogenic organisms have long adhered to Koch's postulates to directly relate a certain disease to a specific microbe, raising the question of whether this might be true of commensal–host relationships as well. Emerging evidence supports that rather than one or two dominant organisms inducing host health, the composition of the entire community of microbial residents influences a balanced immune response. Thus, perturbations to the structure of complex commensal communities (referred to as dysbiosis) can lead to deficient education of the host immune system and subsequent development of immune mediated diseases. Here we will overview the literature that describes the causes of dysbiosis and the mechanisms evolved by the host to prevent these changes to community structure. Building off these studies, we will categorize the different types of dysbiosis and define how collections of microorganisms can influence the host response. This research has broad implications for future therapies that go beyond the introduction of a single organism to induce health. We propose that identifying mechanisms to re‐establish a healthy complex microbiota after dysbiosis has occurred, a process we will refer to as rebiosis, will be fundamental to treating complex immune diseases.     

Six types of multistability in a neuronal model based on slow calcium current

Background

Multistability of oscillatory and silent regimes is a ubiquitous phenomenon exhibited by excitable systems such as neurons and cardiac cells. Multistability can play functional roles in short-term memory and maintaining posture. It seems to pose an evolutionary advantage for neurons which are part of multifunctional Central Pattern Generators to possess multistability. The mechanisms supporting multistability of bursting regimes are not well understood or classified.

Methodology/Principal Findings

Our study is focused on determining the bio-physical mechanisms underlying different types of co-existence of the oscillatory and silent regimes observed in a neuronal model. We develop a low-dimensional model typifying the dynamics of a single leech heart interneuron. We carry out a bifurcation analysis of the model and show that it possesses six different types of multistability of dynamical regimes. These types are the co-existence of 1) bursting and silence, 2) tonic spiking and silence, 3) tonic spiking and subthreshold oscillations, 4) bursting and subthreshold oscillations, 5) bursting, subthreshold oscillations and silence, and 6) bursting and tonic spiking. These first five types of multistability occur due to the presence of a separating regime that is either a saddle periodic orbit or a saddle equilibrium. We found that the parameter range wherein multistability is observed is limited by the parameter values at which the separating regimes emerge and terminate.

Conclusions

We developed a neuronal model which exhibits a rich variety of different types of multistability. We described a novel mechanism supporting the bistability of bursting and silence. This neuronal model provides a unique opportunity to study the dynamics of networks with neurons possessing different types of multistability.     

Tsetse immunity and the transmission of trypanosomiasis

Cyclical transmission of African trypanosomes - Trypanosoma congolense and subspecies of T. brucei - depends on their uptake by and development within their tsetse fly vectors. Tsetse susceptibility to such trypanosome infection seems to be controlled by maternally inherited rickettsia-like organisms (RLOs) (Fig. 1) and it now seems that the RLOs may exert this effect by controlling midgut lectins in the fly. Ian Maudlin and Susan Welburn explain the latest findings.     

A fully coupled, mechanistic model for infectious disease dynamics in a metapopulation: movement and epidemic duration

The drive to understand the invasion, spread and fade out of infectious disease in structured populations has produced a variety of mathematical models for pathogen dynamics in metapopulations. Very rarely are these models fully coupled, by which we mean that the spread of an infection within a subpopulation affects the transmission between subpopulations and vice versa. It is also rare that these models are accessible to biologists, in the sense that all parameters have a clear biological meaning and the biological assumptions are explained. Here we present an accessible model that is fully coupled without being an individual-based model. We use the model to show that the duration of an epidemic has a highly non-linear relationship with the movement rate between subpopulations, with a peak in epidemic duration appearing at small movement rates and a global maximum at large movement rates. Intuitively, the first peak is due to asynchrony in the dynamics of infection between subpopulations; we confirm this intuition and also show the peak coincides with successful invasion of the infection into most subpopulations. The global maximum at relatively large movement rates occurs because then the infectious agent perceives the metapopulation as if it is a single well-mixed population wherein the effective population size is greater than the critical community size.     

Innate immunity in Alzheimer's disease: a model for local inflammatory reactions

Over the past fifteen years, evidence has been accumulating that there is a chronic inflammatory reaction in areas of the brain affected by Alzheimer's disease. Chronic inflammation, which arises in reaction to an underlying pathology, represents a threat in its own right, wherever it may occur, and can in fact surpass primary affronts upon tissues. The brain, however, is particularly vulnerable because neurons are generally irreplaceable. In the case of Alzheimer's disease, inflammatory processes thus have the potential for turning a relatively slowly progressing condition into one characterized by rapid neurodegeneration.     

A hierarchical Bayesian model to predict the duration of immunity to Haemophilus influenzae type b

A hierarchical Bayesian regression model is fitted to longitudinal data on Haemophilus influenzae type b (Hib) serum antibodies. To estimate the decline rate of the antibody concentration, the model accommodates the possibility of unobserved subclinical infections with Hib bacteria that cause increasing concentrations during the study period. The computations rely on Markov chain Monte Carlo simulation of the joint posterior distribution of the model parameters. The model is used to predict the duration of immunity to subclinical Hib infection and to a serious invasive Hib disease.     

Stage-structured infection transmission and a spatial epidemic: a model for Lyme disease

A greater understanding of the rate at which emerging disease advances spatially has both ecological and applied significance. Analyzing the spread of vector-borne disease can be relatively complex when the vector's acquisition of a pathogen and subsequent transmission to a host occur in different life stages. A contemporary example is Lyme disease. A long-lived tick vector acquires infection during the larval blood meal and transmits it as a nymph. We present a reaction-diffusion model for the ecological dynamics governing the velocity of the current epidemic's spread. We find that the equilibrium density of infectious tick nymphs (hence the risk of human disease) can depend on density-independent survival interacting with biotic effects on the tick's stage structure. The local risk of infection reaches a maximum at an intermediate level of adult tick mortality and at an intermediate rate of juvenile tick attacks on mammalian hosts. If the juvenile tick attack rate is low, an increase generates both a greater density of infectious nymphs and an increased spatial velocity. However, if the juvenile attack rate is relatively high, nymph density may decline while the epidemic's velocity still increases. Velocities of simulated two-dimensional epidemics correlate with the model pathogen's basic reproductive number (R0), but calculating R0 involves parameters of both host infection dynamics and the vector's stage-structured dynamics.