Differential equations for moments of an interacting particle process on a lattice

Differential equations are derived for a class of interacting lattice particle processes first introduced in [11]. When there is one type of particle, the equations are in terms of the moments of boundary counts and particle counts. When there are two types of particles, the equations are in terms of the moments of certain neighbourhood statistics as well.This work has been supported by grants from the Natural Sciences and Engineering Research Council of Canada. WJB also acknowledges support of an Ontario Graduate Scholarship     

Mathematical model of the development of an epidemic process with aerosol transmission

A model of a pathogenic system with respiratory transmission was proposed. The internal regulation mechanism and the conditions for the onset of seasonal rise in morbidity were studied. The role of asymptomatic carriers in the formation of the virulence potential of the pathogen was shown.     

Contact rate calculation for a basic epidemic model

Mass-action epidemic models are the foundation of the majority of studies of disease dynamics in human and animal populations. Here, a kinetic model of mobile susceptible and infective individuals in a two-dimensional domain is introduced, and an examination of the contact process results in a mass-action-like term for the generation of new infectives. The conditions under which density dependent and frequency dependent transmission terms emerge are clarified. Moreover, this model suggests that epidemics in large mobile spatially distributed populations can be well described by homogeneously mixing mass-action models. The analysis generates an analytic formula for the contact rate (β) and the basic reproductive ratio (R₀) of an infectious pathogen, which contains a mixture of demographic and epidemiological parameters. The analytic results are compared with a simulation and are shown to give good agreement. The simulation permits the exploration of more realistic movement strategies and their consequent effect on epidemic dynamics.     

Asymptotic results for a multi-type contact birth-death process and related SIS epidemic

Exact results concerning the asymptotic speed of propagation of infection have recently been obtained for the multi-type SIS epidemic in continuous space when the contact distributions are assumed to be symmetric with the Laplace transforms finite for all entries. There is a link between the equations for this epidemic and the equations for a multi-type contact birth-death process. This enables methods developed for the epidemic to be used to obtain the asymptotic speed of translation for the contact birth-death process. Symmetry of the contact distributions is required but no existence constraint is placed on their Laplace transforms. The method for removing this constraint may also be used for the SIS epidemic. Results are given for both processes when the basic reproduction ratio is at most one.     

Optimal observation times in experimental epidemic processes

Summary .   This article describes a method for choosing observation times for stochastic processes to maximise the expected information about their parameters. Two commonly used models for epidemiological processes are considered: a simple death process and a susceptible-infected (SI) epidemic process with dual sources for infection spreading within and from outwith the population. The search for the optimal design uses Bayesian computational methods to explore the joint parameter-data-design space, combined with a method known as moment closure to approximate the likelihood to make the acceptance step efficient. For the processes considered, a small number of optimally chosen observations are shown to yield almost as much information as much more intensively observed schemes that are commonly used in epidemiological experiments. Analysis of the simple death process allows a comparison between the full Bayesian approach and locally optimal designs around a point estimate from the prior based on asymptotic results. The robustness of the approach to misspecified priors is demonstrated for the SI epidemic process, for which the computational intractability of the likelihood precludes locally optimal designs. We show that optimal designs derived by the Bayesian approach are similar for observational studies of a single epidemic and for studies involving replicated epidemics in independent subpopulations. Different optima result, however, when the objective is to maximise the gain in information based on informative and non-informative priors: this has implications when an experiment is designed to convince a naïve or sceptical observer rather than consolidate the belief of an informed observer. Some extensions to the methods, including the selection of information criteria and extension to other epidemic processes with transition probabilities, are briefly addressed.     

The effect of contact heterogeneity and multiple routes of transmission on final epidemic size

Heterogeneity in the number of potentially infectious contacts amongst members of a population increases the basic reproduction ratio (R(0)) and markedly alters disease dynamics compared to traditional mean-field models. Most models describing transmission on contact networks only account for one specific route of transmission. However, for many infectious diseases multiple routes of transmission exist. The model presented here captures transmission through a well defined network of contacts, complemented by mean-field type transmission amongst the nodes of the network that accounts for alternative routes of transmission. The impact of these combined transmission mechanisms on the final epidemic size is investigated analytically. The analytic predictions for the purely mean-field case and the transmission through the network-only case are confirmed by individual-based network simulations. There is a critical transmission potential above which an increased contribution of the mean-field type transmission increases the final epidemic size while an increased contribution of the transmission through the network decreases it. Below the critical transmission potential the opposite effect is observed.     

Pandemic bounds for an epidemic on an infinite lattice

Exact results have previously been obtained concerning the spread of infection in continuous space contact models describing a class of multi-type epidemics. Pandemic lower and upper bounds were obtained for the spatial final size. Pandemic results have also been obtained for a discrete space model on the integer lattice using an infinite matrix formulation of the final size equations. However, the proof required restrictive constraints to be placed on the model parameters which do not hold in general and will not be valid when infection modifies behaviour. The purpose of this paper is to remove these constraints and give a general proof of the pandemic results for the multi-type epidemic on the lattice Z(N).     

The identification of the periodic behaviour in an epidemic model

In this paper we study a method for the identification of the unknown parameter of the periodic function and also the first component of the state vector, in a mathematical model which describes the evolution of some diseases with an oro-fecal transmission.To solve the identification problem we use a numerical method to integrate the differential equations system, which reproduces the stability properties of the above mentioned continuous system.The numerical methods which we propose can be applied also to a spatial semi discretization of the reaction-diffusion model which is a diffusive generalization of the system that we consider in this paper.Finally, through an analysis on both the continuous and the discrete system we also obtain a necessary condition on the experimental data in order that a periodic trajectory of the system exists.Work supported by: Progetto Finalizzato Controllo Malattie da Infezione-CNR and by Ministero Pubblica Istruzione     

Stochastic epidemic models: A survey

This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic (relying on a large community) properties are presented. The purpose of modelling is illustrated by studying effects of vaccination and also in terms of inference procedures for important parameters, such as the basic reproduction number and the critical vaccination coverage. Several generalizations towards realism, e.g. multitype and household epidemic models, are also presented, as is a model for endemic diseases.     

Stochastic model of an influenza epidemic with drug resistance

A continuous-time Markov chain (CTMC) model is formulated for an influenza epidemic with drug resistance. This stochastic model is based on an influenza epidemic model, expressed in terms of a system of ordinary differential equations (ODE), developed by Stilianakis, N.I., Perelson, A.S., Hayden, F.G., [1998. Emergence of drug resistance during an influenza epidemic: insights from a mathematical model. J. Inf. Dis. 177, 863-873]. Three different treatments-chemoprophylaxis, treatment after exposure but before symptoms, and treatment after symptoms appear, are considered. The basic reproduction number, R(0), is calculated for the deterministic-model under different treatment strategies. It is shown that chemoprophylaxis always reduces the basic reproduction number. In addition, numerical simulations illustrate that the basic reproduction number is generally reduced with realistic treatment rates. Comparisons are made among the different models and the different treatment strategies with respect to the number of infected individuals during an outbreak. The final size distribution is computed for the CTMC model and, in some cases, it is shown to have a bimodal distribution corresponding to two situations: when there is no outbreak and when an outbreak occurs. Given an outbreak occurs, the total number of cases for the CTMC model is in good agreement with the ODE model. The greatest number of drug resistant cases occurs if treatment is delayed or if only symptomatic individuals are treated.     

A sequential test for assessing the effectiveness of response strategies during an emerging epidemic

In epidemiology, the fatality rate is an important indicator of disease severity and has been used to evaluate the effects of new treatments. During an emerging epidemic with limited resources, monitoring the changes in fatality rate can also provide signals on the evaluation of government policies and healthcare quality, which helps to guide public health decision. A statistical test is developed in this paper to detect changes in fatality rate over time during the course of an emerging infectious disease. A major advantage of the proposed test is that it only requires the regularly reported numbers of deaths and recoveries, which meets the actual need as detailed surveillance data are hard to collect during the course of an emerging epidemic especially the deadly infectious diseases with large magnitude. In addition, with the sequential testing procedure, the effective measures can be detected at the earliest possible time to provide guidance to policymakers for swift action. Simulation studies showed that the proposed test performs well and is sensitive in picking up changes in the fatality rate. The test is applied to the 2014–2016 Ebola outbreak in Sierra Leone for illustration.     

Competitive exclusion and coexistence for pathogens in an epidemic model with variable population size

We study an SIR epidemic model with a variable host population size. We prove that if the model parameters satisfy certain inequalities then competition between n pathogens for a single host leads to exclusion of all pathogens except the one with the largest basic reproduction number. It is shown that a knowledge of the basic reproduction numbers is necessary but not sufficient for determining competitive exclusion. Numerical results illustrate that these inequalities are sufficient but not necessary for competitive exclusion to occur. In addition, an example is given which shows that if such inequalities are not satisfied then coexistence may occur.     

猪流行性腹泻病毒非结构蛋白nsp9互作宿主蛋白对病毒复制的影响

猪流行性腹泻病毒(porcineepidemicdiarrheavirus,PEDV)导致仔猪和育肥猪发生急性肠道传染病,是危害养猪业最重要的病原体之一。目前发现PEDV能够编码至少16个非结构蛋白,其中nsp9能够结合至单链RNA中,但是其功能机制还不清楚。本研究通过免疫沉淀联合蛋白质谱分析,筛选出潜在的与PEDV nsp9宿主互作蛋白。通过进一步免疫共沉淀(co-immunoprecipitation, Co-IP)和激光共聚焦技术确认了nsp9与热休克蛋白HSPA8、Toll相互作用蛋白Tollip、热休克蛋白HSPA9、线粒体外膜蛋白TOMM70互作。其中,过表达HSPA8使nsp9的表达量先上调而后下调,并促进PEDV的增殖;过表达Tollip使nsp9的表达量显著上调,并抑制PEDV的增殖;过表达TOMM70使nsp9的表达量显著下调,但对PEDV的增殖无明显影响;过表达HSPA9对nsp9的表达以及PEDV的增殖均无明显影响。该研究为探索nsp9互作蛋白在PEDV感染过程中的功能提供了重要信息。     

Optimal intervention for an epidemic model under parameter uncertainty

We will be concerned with optimal intervention policies for a continuous-time stochastic SIR (susceptible-->infective-->removed) model for the spread of infection through a closed population. In previous work on such optimal policies, it is common to assume that model parameter values are known; in reality, uncertainty over parameter values exists. We shall consider the effect upon the optimal policy of changes in parameter estimates, and of explicitly taking into account parameter uncertainty via a Bayesian decision-theoretic framework. We consider policies allowing for (i) the isolation of any number of infectives, or (ii) the immunisation of all susceptibles (total immunisation). Numerical examples are given to illustrate our results.     

Global stability of an SIR epidemic model with time delays

An SIR disease transmission model is formulated under the assumption that the force of infection at the present time depends on the number of infectives at the past. It is shown that a disease free equilibrium point is globally stable if no endemic equilibrium point exists. Further the endemic point (if it exists) is globally stable with respect to the whole state space except the neighborhood of the disease free state.Research partly supported by the Ministry of Education, Science and Culture, Japan, Grant 05640256