Two SIS epidemiologic models with delays

 The SIS epidemiologic models have a delay corresponding to the infectious period, and disease-related deaths, so that the population size is variable. The population dynamics structures are either logistic or recruitment with natural deaths. Here the thresholds and equilibria are determined, and stabilities are examined. In a similar SIS model with exponential population dynamics, the delay destabilized the endemic equilibrium and led to periodic solutions. In the model with logistic dynamics, periodic solutions in the infectious fraction can occur as the population approaches extinction for a small set of parameter values. Received: 10 January 1997 / 18 November 1997     

Evolution of airborne infectious diseases according to changes in characteristics of the host population

The authors predicted evolutionary changes in airborne infectious diseases according to changes in the characteristics of the host population. The predictions were based upon a mathematical model of infectious diseases and the validity of the predictions was verified against the history of man and pathogens. The feature of this model is that it involves a density of pathogens in the environment as an additional variable which can be regarded as more suitable to airborne infectious diseases. In spite of this modification, this study reached a similar conclusion to the threshold density theory: that is, susceptible host density in the absence of the pathogen must be larger than that in the presence of the pathogen, for the pathogen to be persistent. Moreover the authors concluded that one type of pathogen cannot be replaced by another type of pathogen as long as the susceptible host density of the former type is the mininum one. The predictions were considered to be valid for a wide range of infectous diseases. Making use of these principles, the authors predicted that the variety of infectious diseases should increase as host density increases and that pathogens should evolve to be less virulent as the host life-span increases. The finalidea discussed is whether or nor the history of man and pathogen can be verified by the predictions.     

Modelling the effect of urbanization on the transmission of an infectious disease

This paper models the impact of urbanization on infectious disease transmission by integrating a CA land use development model, population projection matrix model and CA epidemic model in S-Plus. The innovative feature of this model lies in both its explicit treatment of spatial land use development, demographic changes, infectious disease transmission and their combination in a dynamic, stochastic model. Heuristically-defined transition rules in cellular automata (CA) were used to capture the processes of both land use development with urban sprawl and infectious disease transmission. A population surface model and dwelling distribution surface were used to bridge the gap between urbanization and infectious disease transmission. A case study is presented involving modelling influenza transmission in Southampton, a dynamically evolving city in the UK. The simulation results for Southampton over a 30-year period show that the pattern of the average number of infection cases per day can depend on land use and demographic changes. The modelling framework presents a useful tool that may be of use in planning applications.     

Diseases with chronic stage in a population with varying size

An epidemiological model of hepatitis C with a chronic infectious stage and variable population size is introduced. A non-structured baseline ODE model which supports exponential solutions is discussed. The normalized version where the unknown functions are the proportions of the susceptible, infected, and chronic individuals in the total population is analyzed. It is shown that sustained oscillations are not possible and the endemic proportions either approach the disease-free or an endemic equilibrium. The expanded model incorporates the chronic age of the individuals. Partial analysis of this age-structured model is carried out. The global asymptotic stability of the infection-free state is established as well as local asymptotic stability of the endemic non-uniform steady state distribution under some additional conditions. A numerical method for the chronic-age-structured model is introduced. It is shown that this numerical scheme is consistent and convergent of first order. Simulations based on the numerical method suggest that in the structured case the endemic equilibrium may be unstable and sustained oscillations are possible. Closer look at the reproduction number reveals that treatment strategies directed towards speeding up the transition from acute to chronic stage in effect contribute to the eradication of the disease.     

Resonance effects and outbreaks in ecological time series

Organismal response to environmental variability is an important aspect of ecological processes. We propose new mechanisms whereby environmental variability can cause cyclic population outbreaks due to the nonlinearity of the organismal response. We consider stage-structured populations that respond to variable environments with variable diapause or dormancy, and in which cyclic changes of the environment induce a resonance-like boost in the population size. If there is also a stochastic component of variation in the environment, the population outbreaks are magnified by the phenomenon of "stochastic resonance". The results show that large population fluctuations may not be due to extrinsic or intrinsic factors alone, but to a nonlinear interaction between the external environment and internal population processes. Indeed, in the presence of such nonlinearities even very small environmental fluctuations can cause massive fluctuations in population size. Our theoretical results may help to explain periodic population cycles and outbreak dynamics found in many infectious diseases and pest species. We also discuss the evolution of the response parameters that regulate diapause or dormancy and promote the outbreak dynamics in variable environments.     

Stochastic dynamics of immunity in small populations: a general framework

Assessment of immunological status is a powerful tool in the surveillance and control of infectious pathogens in livestock and human populations. The distribution of immunity levels in the population provides information on time and age dependent transmission. A stochastic model is developed for a livestock population which relates the dynamics of the distribution of immunity levels at the population level to those of pathogen transmission. A general model with K immunity level categories is first proposed, taking into account the increase of the immunity level due to an infection or a re-exposure, the decrease of the immunity level with time since infection or exposure, and the effect of immunity level on the susceptibility and the infectivity of individuals. Numerical results are presented in the particular cases with K=2 and K=3 immunity level categories. We demonstrate that for a given distribution of the immunity levels at the population level, the model can be used to identify quantities such as most likely periods of time since introduction of infection. We discuss this approach in relation to analysis of serological data.     

On the Identifiability of Transmission Dynamic Models for Infectious Diseases

Understanding the transmission dynamics of infectious diseases is important for both biological research and public health applications. It has been widely demonstrated that statistical modeling provides a firm basis for inferring relevant epidemiological quantities from incidence and molecular data. However, the complexity of transmission dynamic models presents two challenges: (1) the likelihood function of the models is generally not computable, and computationally intensive simulation-based inference methods need to be employed, and (2) the model may not be fully identifiable from the available data. While the first difficulty can be tackled by computational and algorithmic advances, the second obstacle is more fundamental. Identifiability issues may lead to inferences that are driven more by prior assumptions than by the data themselves. We consider a popular and relatively simple yet analytically intractable model for the spread of tuberculosis based on classical IS6110 fingerprinting data. We report on the identifiability of the model, also presenting some methodological advances regarding the inference. Using likelihood approximations, we show that the reproductive value cannot be identified from the data available and that the posterior distributions obtained in previous work have likely been substantially dominated by the assumed prior distribution. Further, we show that the inferences are influenced by the assumed infectious population size, which generally has been kept fixed in previous work. We demonstrate that the infectious population size can be inferred if the remaining epidemiological parameters are already known with sufficient precision.     

Statistical Properties of a DNA Sample under the Finite-Sites Model

Statistical properties of a DNA sample from a random-mating population of constant size are studied under the finite-sites model. It is assumed that there is no migration and no recombination occurs within the locus. A Markov process model is used for nucleotide substitution, allowing for multiple substitutions at a single site. The evolutionary rates among sites are treated as either constant or variable. The general likelihood calculation using numerical integration involves intensive computation and is feasible for three or four sequences only; it may be used for validating approximate algorithms. Methods are developed to approximate the probability distribution of the number of segregating sites in a random sample of n sequences, with either constant or variable substitution rates across sites. Calculations using parameter estimates obtained for human D-loop mitochondrial DNAs show that among-site rate variation has a major effect on the distribution of the number of segregating sites; the distribution under the finite-sites model with variable rates among sites is quite different from that under the infinite-sites model.     

Times from Infection to Disease-Induced Death and their Influence on Final Population Sizes After Epidemic Outbreaks

For epidemic models, it is shown that fatal infectious diseases cannot drive the host population into extinction if the incidence function is upper density-dependent. This finding holds even if a latency period is included and the time from infection to disease-induced death has an arbitrary length distribution. However, if the incidence function is also lower density-dependent, very infectious diseases can lead to a drastic decline of the host population. Further, the final population size after an epidemic outbreak can possibly be substantially affected by the infection-age distribution of the initial infectives if the life expectations of infected individuals are an unbounded function of infection age (time since infection). This is the case for lognormal distributions, which fit data from infection experiments involving tiger salamander larvae and ranavirus better than gamma distributions and Weibull distributions.     

Improved procedures for estimation of disease prevalence using ranked set sampling

Ranked set sampling (RSS) is a sampling procedure that can be considerably more efficient than simple random sampling (SRS). When the variable of interest is binary, ranking of the sample observations can be implemented using the estimated probabilities of success obtained from a logistic regression model developed for the binary variable. The main objective of this study is to use substantial data sets to investigate the application of RSS to estimation of a proportion for a population that is different from the one that provides the logistic regression. Our results indicate that precision in estimation of a population proportion is improved through the use of logistic regression to carry out the RSS ranking and, hence, the sample size required to achieve a desired precision is reduced. Further, the choice and the distribution of covariates in the logistic regression model are not overly crucial for the performance of a balanced RSS procedure.     

Pathogen adaptation under imperfect vaccination: implications for pertussis

Mass vaccination campaigns have drastically reduced the burden of infectious diseases. Unfortunately, in recent years several infectious diseases have re-emerged. Pertussis poses a well-known example. Inspired by pertussis, we study, by means of an epidemic model, the population and evolutionary dynamics of a pathogen population under the pressure of vaccination. A distinction is made between infection in immunologically naive individuals (primary infection) and infection in individuals whose immune system has been primed by vaccination or infection (secondary infection). The results show that (i) vaccination with an imperfect vaccine may not succeed in reducing the infection pressure if the transmissibility of secondary infections is higher than that of primary infections; (ii) pathogen strains that are able to evade the immunity induced by vaccination can only spread if escape mutants incur no or only a modest fitness cost and (iii) the direction of evolution depends crucially on the distribution of the different types of susceptibles in the population. We discuss the implications of these results for the design and use of vaccines that provide temporary immunity.     

Effects of heterogeneous and clustered contact patterns on infectious disease dynamics

The spread of infectious diseases fundamentally depends on the pattern of contacts between individuals. Although studies of contact networks have shown that heterogeneity in the number of contacts and the duration of contacts can have far-reaching epidemiological consequences, models often assume that contacts are chosen at random and thereby ignore the sociological, temporal and/or spatial clustering of contacts. Here we investigate the simultaneous effects of heterogeneous and clustered contact patterns on epidemic dynamics. To model population structure, we generalize the configuration model which has a tunable degree distribution (number of contacts per node) and level of clustering (number of three cliques). To model epidemic dynamics for this class of random graph, we derive a tractable, low-dimensional system of ordinary differential equations that accounts for the effects of network structure on the course of the epidemic. We find that the interaction between clustering and the degree distribution is complex. Clustering always slows an epidemic, but simultaneously increasing clustering and the variance of the degree distribution can increase final epidemic size. We also show that bond percolation-based approximations can be highly biased if one incorrectly assumes that infectious periods are homogeneous, and the magnitude of this bias increases with the amount of clustering in the network. We apply this approach to model the high clustering of contacts within households, using contact parameters estimated from survey data of social interactions, and we identify conditions under which network models that do not account for household structure will be biased.     

Scaling properties of childhood infectious diseases epidemics before and after mass vaccination in Canada

The goal of this paper is to analyse the scaling properties of childhood infectious disease time-series data. We present a scaling analysis of the distribution of epidemic sizes of measles, rubella, pertussis, and mumps outbreaks in Canada. This application provides a new approach in assessing infectious disease dynamics in a large vaccinated population. An inverse power-law (IPL) distribution function has been fit to the time series of epidemic sizes, and the results assessed against an exponential benchmark model. We have found that the rubella epidemic size distribution and that of measles in highly vaccinated periods follow an IPL. The IPL suggests the presence of a scale-invariant network for these diseases as a result of the heterogeneity of the individual contact rates. By contrast, it was found that pertussis and mumps were characterized by a uniform network of transmission of the exponential type, which suggests homogeneity in the contact rate or, more likely, boiled down heterogeneity by large intermixing in the population. We conclude that the topology of the network of infectious contacts depends on the disease type and its infection rate. It also appears that the socio-demographic structure of the population may play a part (e.g. pattern of contacts according to age) in the structuring of the topology of the network. The findings suggest that there is relevant information hidden in the variation of the common contagious disease time-series data, and that this information can have a bearing on the strategy of vaccination programs.     

A general model for stochastic SIR epidemics with two levels of mixing

This paper is concerned with a general stochastic model for susceptible→infective→removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution ‘centred’ on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R_* governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective’s household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.     

气候变化对传染病爆发流行的影响研究进展

全球气候变化已影响到传染病发生、传播与变化的各个环节,从病原体及其携带者、传播途径和人体自身抵抗力等方面直接或间接影响传染病的发病趋势,从而对人类健康造成了巨大的威胁。所以加强对气候变化与传染病间关系、预测预报研究,对进一步认识、预防和控制传染病的爆发流行具有重要意义。本文首先阐述了全球气候变化对生物物种的地理分布和人类健康的影响,气候变化改变了生物物种的地理分布范围,增加了某些物种的潜在分布区域,并造成生物物侯期的改变;同时,极端气候事件成为导致种群数量波动的一个重要驱动力。气候变化对人类健康有直接和间接影响,它使得传染病发病率增加、传染病分布范围扩大、人群对疾病易感性增强。文章重点评述了气候变化对疟疾、登革热、霍乱、流行性乙型脑炎、流感、SARS、肠道传染病、鼠疫、血吸虫病等常见传染病流行机制和传播过程的影响研究进展。评述了传染病和气象因子关系分析中常用的定性和定量分析方法,传统的研究多以定性分析为主,方法较单一;目前,利用流行病学资料与同期的气象因子进行单因素相关分析、多元回归分析是常用的研究方法;主成分回归分析、逐步判别分析、灰色关联分析法、RS和GIS等方法近年来逐渐得到应用;数学建模、实验室生物学仿真实验方法是今后需强化的方向。提出了该研究领域国内外研究普遍存在和亟待解决的问题,针对目前的研究现状和存在的问题,提出了未来的研究重点和发展方向。     

A multi-stage population model

This paper extends the two-stage population model treated by Tognetti &; Mazanov (1970) to any number of stages of development before the final, mature, reproductive stage. The multi-stage model is shown to behave like the two-stage model. Biological realism is introduced through variable functions for population processes such as rates of development, fecundities and mortalities. Stability is achieved by introducing driving functions such as food limitations, variable food accessibility and variable “weather” components (such as temperature).This model is applied to a bat-guano mite population for which most of these functions are known or can be estimated. The population is well simulated by the multi-stage model's system of equations.     

How generation intervals shape the relationship between growth rates and reproductive numbers

Mathematical models of transmission have become invaluable management tools in planning for the control of emerging infectious diseases. A key variable in such models is the reproductive number R. For new emerging infectious diseases, the value of the reproductive number can only be inferred indirectly from the observed exponential epidemic growth rate r. Such inference is ambiguous as several different equations exist that relate the reproductive number to the growth rate, and it is unclear which of these equations might apply to a new infection. Here, we show that these different equations differ only with respect to their assumed shape of the generation interval distribution. Therefore, the shape of the generation interval distribution determines which equation is appropriate for inferring the reproductive number from the observed growth rate. We show that by assuming all generation intervals to be equal to the mean, we obtain an upper bound to the range of possible values that the reproductive number may attain for a given growth rate. Furthermore, we show that by taking the generation interval distribution equal to the observed distribution, it is possible to obtain an empirical estimate of the reproductive number.     

Network epidemic models with two levels of mixing

The study of epidemics on social networks has attracted considerable attention recently. In this paper, we consider a stochastic SIR (susceptible-->infective-->removed) model for the spread of an epidemic on a finite network, having an arbitrary but specified degree distribution, in which individuals also make casual contacts, i.e. with people chosen uniformly from the population. The behaviour of the model as the network size tends to infinity is investigated. In particular, the basic reproduction number R(0), that governs whether or not an epidemic with few initial infectives can become established is determined, as are the probability that an epidemic becomes established and the proportion of the population who are ultimately infected by such an epidemic. For the case when the infectious period is constant and all individuals in the network have the same degree, the asymptotic variance and a central limit theorem for the size of an epidemic that becomes established are obtained. Letting the rate at which individuals make casual contacts decrease to zero yields, heuristically, corresponding results for the model without casual contacts, i.e. for the standard SIR network epidemic model. A deterministic model that approximates the spread of an epidemic that becomes established in a large population is also derived. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations work well, even for only moderately sized networks, and that the degree distribution and the inclusion of casual contacts can each have a major impact on the outcome of an epidemic.