A threshold theorem for the general epidemic in discrete time

Summary A theorem, analogous to the continuous time Threshold Theorem of Kermack and McKendrick, is proved for a certain discrete time epidemic model. This model, in contrast to its continuous time analogue, leads to some solutions in which the total population of susceptibles may become infected in a finite time.     

A standard form for the Kermack-McKendrick epidemic equations

The classical epidemic equations of Kermack and McKendrick are cast into dimensionless form. This allows discussion of the assumptions underlying the standard approximate solutions.     

Generality of the Final Size Formula for an Epidemic of a Newly Invading Infectious Disease

The well-known formula for the final size of an epidemic was published by Kermack and McKendrick in 1927. Their analysis was based on a simple susceptible-infected-recovered (SIR) model that assumes exponentially distributed infectious periods. More recent analyses have established that the standard final size formula is valid regardless of the distribution of infectious periods, but that it fails to be correct in the presence of certain kinds of heterogeneous mixing (e.g., if there is a core group, as for sexually transmitted diseases). We review previous work and establish more general conditions under which Kermack and McKendrick's formula is valid. We show that the final size formula is unchanged if there is a latent stage, any number of distinct infectious stages and/or a stage during which infectives are isolated (the durations of each stage can be drawn from any integrable distribution). We also consider the possibility that the transmission rates of infectious individuals are arbitrarily distributed—allowing, in particular, for the existence of super-spreaders—and prove that this potential complexity has no impact on the final size formula. Finally, we show that the final size formula is unchanged even for a general class of spatial contact structures. We conclude that whenever a new respiratory pathogen emerges, an estimate of the expected magnitude of the epidemic can be made as soon the basic reproduction number ℝ₀ can be approximated, and this estimate is likely to be improved only by more accurate estimates of ℝ₀, not by knowledge of any other epidemiological details.     

Discussion: The Kermack-McKendrick epidemic threshold theorem

The paper reviews the work of Kermack and McKendrick on the development of simple mathematical models of the transmission dynamics of viral and bacterial infectious agents within population of hosts. The focus of attention is centred on the notion of a threshold density of susceptible hosts to trigger an epidemic and recent extensions of this idea as expressed in the definition of a basic or case reproductive rate of infection. The main body of the paper examines recent developments of the basic Kermack-McKendrick model with an emphasis on deterministic models that describe various types of heterogeneity in the processes that determine transmission between infected and susceptible persons. Particular attention is given to the role of behavioural heterogeneity within the framework of a contact or mixing matrix which defines “who acquires infection from whom”.     

Resurgence of malaria in Bombay (Mumbai) in the 1990s: a historical perspective

Bombay has achieved extraordinary success in controlling its malaria problem for nearly six decades by relying primarily on legislative measures and non-insecticidal methods of mosquito abatement. In 1992, however, malaria reemerged in Bombay with a vengeance. During 1992-1997, the city witnessed a manifold increase in the number of malaria cases diagnosed and treated by the public health system. The large number of malaria patients treated by private practitioners was not recorded by the municipal malaria surveillance system during this period. In 1995, at the peak of the resurgence, public health officials of the Municipal Corporation of Greater Bombay (MCGB) confirmed that 170 persons in the city had died due to malaria. The crisis was unprecedented in Bombay's modern public health history. In response to intense criticism from the media, the city's public health officials attributed the resurgence to the global phenomenon of mosquito-vector resistance to insecticides, and Plasmodium resistance to antimalarial chemoprophylaxis and treatment. Local scientists who investigated the problem offered no support to this explanation. So what might explain the resurgence? What factors led the problem to reach an epidemic level in a matter of two or three years? In addressing the above principal questions, this paper adopts a historical perspective and argues that in the resurgence of malaria in Bombay in the 1990s, there is an element of the 'presence of the past'. In many ways the present public health crisis in Bombay resembles the health scenario that characterized the city at the turn of the 19th century. It is possible to draw parallels between the early public health history of malaria control in Bombay, which was punctuated by events that followed the bubonic plague epidemic of 1896, and the present-day malaria epidemic punctuated by the threat of a plague epidemic in 1994. As such, the paper covers a long period, of almost 100 years. This time-depth is used to illustrate how malaria control programs in Bombay and in other parts of India have evolved through a combination of local historical forces and political expediencies in the context of technological developments. The boom in construction activities in Bombay following the liberalization of the Indian economy in 1991, and the local politics affecting administrative practices of the MCGB, are discussed as crucial factors in the crystallization of the present-day malaria resurgence in Bombay. The paper concludes by arguing that malaria in urban India is a serious problem that cannot be neglected. In the case of Bombay, the solution to the crisis can be found, in part, by reexamining the historical and political issues that have determined the nature and magnitude of the problem over the last century.     

Demography and Diffusion in Epidemics: Malaria and Black Death Spread

The classical models of epidemics dynamics by Ross and McKendrick have to be revisited in order to incorporate elements coming from the demography (fecundity, mortality and migration) both of host and vector populations and from the diffusion and mutation of infectious agents. The classical approach is indeed dealing with populations supposed to be constant during the epidemic wave, but the presently observed pandemics show duration of their spread during years imposing to take into account the host and vector population changes as well as the transient or permanent migration and diffusion of hosts (susceptible or infected), as well as vectors and infectious agents. Two examples are presented, one concerning the malaria in Mali and the other the plague at the middle-age.     

Stochastic modeling of nonlinear epidemiology

The objectives of this paper to analyse, model and simulate the spread of an infectious disease by resorting to modern stochastic algorithms. The approach renders it possible to circumvent the simplifying assumption of linearity imposed in the majority of the past works on stochastic analysis of epidemic processes. Infectious diseases are often transmitted through contacts of those infected with those susceptible; hence the processes are inherently nonlinear. According to the classical model of Kermack and McKendrick, or the SIR model, three classes of populations are involved in two types of processes: conversion of susceptibles (S) to infectives (I) and conversion of infectives to removed (R). The master equations of the SIR process have been formulated through the probabilistic population balance around a particular state by considering the mutually exclusive events. The efficacy of the present methodology is mainly attributable to its ability to derive the governing equations for the means, variances and covariance of the random variables by the method of system-size expansion of the nonlinear master equations. Solving these equations simultaneously along with rates associated influenza epidemic data yields information concerning not only the means of the three populations but also the minimal uncertainties of these populations inherent in the epidemic. The stochastic pathways of the three different classes of populations during an epidemic, i.e. their means and the fluctuations around these means, have also been numerically simulated independently by the algorithm derived from the master equations, as well as by an event-driven Monte Carlo algorithm. The master equation and Monte Carlo algorithms have given rise to the identical results.     

Epidemic models with uncertainty in the reproduction number

One of the first quantities to be estimated at the start of an epidemic is the basic reproduction number, ${\mathcal{R}_0}$ . The progress of an epidemic is sensitive to the value of ${\mathcal{R}_0}$ , hence we need methods for exploring the consequences of uncertainty in the estimate. We begin with an analysis of the SIR model, with ${\mathcal{R}_0}$ specified by a probability distribution instead of a single value. We derive probability distributions for the prevalence and incidence of infection during the initial exponential phase, the peaks in prevalence and incidence and their timing, and the final size of the epidemic. Then, by expanding the state variables in orthogonal polynomials in uncertainty space, we construct a set of deterministic equations for the distribution of the solution throughout the time-course of the epidemic. The resulting dynamical system need only be solved once to produce a deterministic stochastic solution. The method is illustrated with ${\mathcal{R}_0}$ specified by uniform, beta and normal distributions. We then apply the method to data from the New Zealand epidemic of H1N1 influenza in 2009. We apply the polynomial expansion method to a Kermack–McKendrick model, to simulate a forecasting system that could be used in real time. The results demonstrate the level of uncertainty when making parameter estimates and projections based on a limited amount of data, as would be the case during the initial stages of an epidemic. In solving both problems we demonstrate how the dynamical system is derived automatically via recurrence relationships, then solved numerically.     

鼠疫传播动力学模型的构建研究

目的:采用疾病传播动力学模型描述鼠疫的流行特征,为科学地制定鼠疫的防控措施提供理论依据。方法:运用微分方程建立鼠间鼠疫和人间鼠疫的传播动力学模型,分析模型中各参数与疫情发展变化的关系。结果:使用基本繁殖率,平均鼠密度,感染者平均病死率,感染者平均治愈率等参数描述感染率随时间的变化趋势。结论:根据鼠间传播期、疫区传播期和人群扩散期的传播特点,分别开展各项防控工作,能够更好地控制鼠疫疫情。     

Suncus murinus. Observations on ecology, distribution, status to plague in Bombay

The common house shrew Suncus murinus has been shown to play an important role in maintenance and perpetuation of plague infection by earlier plague workers. With the control of human plague there is no knowledge about foci of plague in small mammals associated with man. Present study was carried out to fill in this Lacuna. Studies carried out in the present paper reveal that S. murinus does not harbour any plague infection in Bombay. This species is widely distributed in Bombay and is found to be associated with man throughout the year. The principal species of fleas harboured by this mammal is Xenopsylla cheopis. The insectivore mainly feeds on tine animals and insects and breeds throughout the year.     

Pulse vaccination strategy in the SIR epidemic model

Theoretical results show that the measles ‘pulse’ vaccination strategy can be distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination. Using the SIR epidemic model we showed that under a planned pulse vaccination regime the system converges to a stable solution with the number of infectious individuals equal to zero. We showed that pulse vaccination leads to epidemics eradication if certain conditions regarding the magnitude of vaccination proportion and on the period of the pulses are adhered to. Our theoretical results are confirmed by numerical simulations. The introduction of seasonal variation into the basic SIR model leads to periodic and chaotic dynamics of epidemics. We showed that under seasonal variation, in spite of the complex dynamics of the system, pulse vaccination still leads to epidemic eradication. We derived the conditions for epidemic eradication under various constraints and showed their dependence on the parameters of the epidemic. We compared effectiveness and cost of constant, pulse and mixed vaccination policies.     

Validation of Inverse Seasonal Peak Mortality in Medieval Plagues,Including the Black Death,in Comparison to Modern Yersinia pestis-Variant Diseases

Background

Recent studies have noted myriad qualitative and quantitative inconsistencies between the medieval Black Death (and subsequent “plagues”) and modern empirical Y. pestis plague data, most of which is derived from the Indian and Chinese plague outbreaks of A.D. 1900±15 years. Previous works have noted apparent differences in seasonal mortality peaks during Black Death outbreaks versus peaks of bubonic and pneumonic plagues attributed to Y. pestis infection, but have not provided spatiotemporal statistical support. Our objective here was to validate individual observations of this seasonal discrepancy in peak mortality between historical epidemics and modern empirical data.

Methodology/Principal Findings

We compiled and aggregated multiple daily, weekly and monthly datasets of both Y. pestis plague epidemics and suspected Black Death epidemics to compare seasonal differences in mortality peaks at a monthly resolution. Statistical and time series analyses of the epidemic data indicate that a seasonal inversion in peak mortality does exist between known Y. pestis plague and suspected Black Death epidemics. We provide possible explanations for this seasonal inversion.

Conclusions/Significance

These results add further evidence of inconsistency between historical plagues, including the Black Death, and our current understanding of Y. pestis-variant disease. We expect that the line of inquiry into the disputed cause of the greatest recorded epidemic will continue to intensify. Given the rapid pace of environmental change in the modern world, it is crucial that we understand past lethal outbreaks as fully as possible in order to prepare for future deadly pandemics.     

Strain Interactions as a Mechanism for Dominant Strain Alternation and Incidence Oscillation in Infectious Diseases: Seasonal Influenza as a Case Study

Background

Many human infectious diseases are caused by pathogens that have multiple strains and show oscillation in infection incidence and alternation of dominant strains which together are referred to as epidemic cycling. Understanding the underlying mechanisms of epidemic cycling is essential for forecasting outbreaks of epidemics and therefore important for public health planning. Current theoretical effort is mainly focused on the factors that are extrinsic to the pathogens themselves (“extrinsic factors”) such as environmental variation and seasonal change in human behaviours and susceptibility. Nevertheless, co-circulation of different strains of a pathogen was usually observed and thus strains interact with one another within concurrent infection and during sequential infection. The existence of these intrinsic factors is common and may be involved in the generation of epidemic cycling of multi-strain pathogens.

Methods and Findings

To explore the mechanisms that are intrinsic to the pathogens themselves (“intrinsic factors”) for epidemic cycling, we consider a multi-strain SIRS model including cross-immunity and infectivity enhancement and use seasonal influenza as an example to parameterize the model. The Kullback-Leibler information distance was calculated to measure the match between the model outputs and the typical features of seasonal flu (an outbreak duration of 11 weeks and an annual attack rate of 15%). Results show that interactions among strains can generate seasonal influenza with these characteristic features, provided that: the infectivity of a single strain within concurrent infection is enhanced 2−7 times that within a single infection; cross-immunity as a result of past infection is 0.5–0.8 and lasts 2–9 years; while other parameters are within their widely accepted ranges (such as a 2–3 day infectious period and the basic reproductive number of 1.8–3.0). Moreover, the observed alternation of the dominant strain among epidemics emerges naturally from the best fit model. Alternative modelling that also includes seasonal forcing in transmissibility shows that both external mechanisms (i.e. seasonal forcing) and the intrinsic mechanisms (i.e., strain interactions) are equally able to generate the observed time-series in seasonal flu.

Conclusions

The intrinsic mechanism of strain interactions alone can generate the observed patterns of seasonal flu epidemics, but according to Kullback-Leibler information distance the importance of extrinsic mechanisms cannot be excluded. The intrinsic mechanism illustrated here to explain seasonal flu may also apply to other infectious diseases caused by polymorphic pathogens.     

Leslie matrix models as “stroboscopic snapshots” of McKendrick PDE models

 High dimensional Leslie matrix models have long been viewed as discretizations of McKendrick PDE models. However, these two fundamental classes of models can be linked in a completely different way. For populations with periodic birth pulses, Leslie models of any dimension can be viewed as “stroboscopic snapshots” (in time) of an associated impulsive McKendrick model; that is, the solution of the discrete model matches the solution of the corresponding continuous model at every discrete time step. In application, McKendrick models of populations with birth pulses can be used to identify the state of the population between the discrete census times of the associated Leslie model. Furthermore, McKendrick models describing populations with near-synchronous birth pulses can be viewed as realistic perturbations of the associated Leslie model. Received: 7 August 1997 / Revised version: 15 January 1998     

鼠疫的流行病学概述

鼠疫是由鼠疫耶尔森菌(Yersinia pestis)引起严重危害人类健康的烈性传染病。本文介绍了鼠疫病原体——鼠疫耶尔森菌的一般特性及生物学特性, 并对国内、外鼠疫疫情现状进行总结。目前鼠疫在全球范围内的流行已进入新的活跃期,世界卫生组织将鼠疫列为近20年来重新流行的急性传染病之一。当前,全球疫区主要分布在非洲、亚洲和南美洲。我国人间鼠疫自20世纪80年代开始处于明显回升势态,近10年流行逐渐下降,但防控形势依然艰巨。     

Allometric scaling and seasonality in the epidemics of wildlife diseases

We present a susceptibles-exposed-infectives (SEI) model to analyze the effects of seasonality on epidemics, mainly of rabies, in a wide range of wildlife species. Model parameters are cast as simple allometric functions of host body size. Via nonlinear analysis, we investigate the dynamical behavior of the disease for different levels of seasonality in the transmission rate and for different values of the pathogen basic reproduction number (R(0)) over a broad range of body sizes. While the unforced SEI model exhibits long-term epizootic cycles only for large values of R(0), the seasonal model exhibits multiyear periodicity for small values of R(0). The oscillation period predicted by the seasonal model is consistent with those observed in the field for different host species. These conclusions are not affected by alternative assumptions for the shape of seasonality or for the parameters that exhibit seasonal variations. However, the introduction of host immunity (which occurs for rabies in some species and is typical of many other wildlife diseases) significantly modifies the epidemic dynamics; in this case, multiyear cycling requires a large level of seasonal forcing. Our analysis suggests that the explicit inclusion of periodic forcing in models of wildlife disease may be crucial to correctly describe the epidemics of wildlife that live in strongly seasonal environments.     

Structural validation of an individual-based model for plague epidemics simulation

Plague remains endemic in many countries in the world and Madagascar is currently the country where the highest number of human plague cases is reported every year. The investigation of causal factors, which command the disease dynamics in rodent populations, is a crucial step to forecast, control and anticipate the infection extension to humans. This paper presents simulation results obtained from an epidemic model, SIMPEST, designed to simulate bubonic plague in a rodent population at a high level of spatial and temporal resolution. We developed a structurally realistic individual-based model, mobilizing knowledge about fleas and rats behaviour, inter-individual plague transmission, and disease evolution in individual organisms, so that the model reflects the way the real system operates and to generate spatial and temporal patterns of disease spread. To assess the structural validity of our simulations, we perform sensitivity analyses on the initial population size and spatial distribution, and compare our results with theoretical statements, garnered from both previous modelling experiences and repeated field observations. We show our results are consistent with referents about population size conditions for a disease to invade and persist and the effect of the contact network on disease dynamics.