Analytic approximation of spatial epidemic models of foot and mouth disease

The effect of spatial heterogeneity in epidemic models has improved with computational advances, yet far less progress has been made in developing analytical tools for understanding such systems. Here, we develop two classes of second-order moment closure methods for approximating the dynamics of a stochastic spatial model of the spread of foot and mouth disease. We consider the performance of such ‘pseudo-spatial’ models as a function of R₀, the locality in disease transmission, farm distribution and geographically-targeted control when an arbitrary number of spatial kernels are incorporated. One advantage of mapping complex spatial models onto simpler deterministic approximations lies in the ability to potentially obtain a better analytical understanding of disease dynamics and the effects of control. We exploit this tractability by deriving analytical results in the invasion stages of an FMD outbreak, highlighting key principles underlying epidemic spread on contact networks and the effect of spatial correlations.     

Cities and villages: infection hierarchies in a measles metapopulation

An important issue in the dynamics of directly transmitted microparasites is the relationship between infection probability and host density. We use models and extensive spatio-temporal data for the incidence of measles to examine evidence for spatial heterogeneity in transmission probability, in terms of urban–rural hierarchies in infection rate. Pre-vaccination measles data for England and Wales show strong evidence for urban–rural heterogeneities in infection rate – the proportion of urban cases rises significantly before major epidemics. The model shows that this effect is consistent with a higher infection rate in large cities, though small towns have epidemic characteristics intermediate between town and country. Surprisingly, urban and rural areas of the same population size have a similar propensity for local extinction of infection. A spatial map of urban–rural correlations reveals complex regional patterns of synchronization of towns and cities. The hierarchical heterogeneities in infection persist into the vaccine era; their implications for disease persistence and control are discussed.     

Insights into the Evolution and Emergence of a Novel Infectious Disease

Many zoonotic, novel infectious diseases in humans appear as sporadic infections with spatially and temporally restricted outbreaks, as seen with influenza A(H5N1). Adaptation is often a key factor for successfully establishing sustained human-to-human transmission. Here we use simple mathematical models to describe different adaptation scenarios with particular reference to spatial heterogeneity within the human population. We present analytical expressions for the probability of emergence per introduction, as well as the waiting time to a successful emergence event. Furthermore, we derive general analytical results for the statistical properties of emergence events, including the probability distribution of outbreak sizes. We compare our analytical results with a stochastic model, which has previously been studied computationally. Our results suggest that, for typical connection strengths between communities, spatial heterogeneity has only a weak effect on outbreak size distributions, and on the risk of emergence per introduction. For example, if or larger, any village connected to a large city by just ten commuters a day is, effectively, just a part of the city when considering the chances of emergence and the outbreak size distribution. We present empirical data on commuting patterns and show that the vast majority of communities for which such data are available are at least this well interconnected. For plausible parameter ranges, the effects of spatial heterogeneity are likely to be dominated by the evolutionary biology of host adaptation. We conclude by discussing implications for surveillance and control of emerging infections.     

Congruent epidemic models for unstructured and structured populations: analytical reconstruction of a 2003 SARS outbreak

Both the threat of bioterrorism and the natural emergence of contagious diseases underscore the importance of quantitatively understanding disease transmission in structured human populations. Over the last few years, researchers have advanced the mathematical theory of scale-free networks and used such theoretical advancements in pilot epidemic models. Scale-free contact networks are particularly interesting in the realm of mathematical epidemiology, primarily because these networks may allow meaningfully structured populations to be incorporated in epidemic models at moderate or intermediate levels of complexity. Moreover, a scale-free contact network with node degree correlation is in accord with the well-known preferred mixing concept. The present author describes a semi-empirical and deterministic epidemic modeling approach that (a) focuses on time-varying rates of disease transmission in both unstructured and structured populations and (b) employs probability density functions to characterize disease progression and outbreak controls. Given an epidemic curve for a historical outbreak, this modeling approach calls for Monte Carlo calculations (that define the average new infection rate) and solutions to integro-differential equations (that describe outbreak dynamics in an aggregate population or across all network connectivity classes). Numerical results are obtained for the 2003 SARS outbreak in Taiwan and the dynamical implications of time-varying transmission rates and scale-free contact networks are discussed in some detail.     

Heterogeneity in multiple transmission pathways: modelling the spread of cholera and other waterborne disease in networks with a common water source

Many factors influencing disease transmission vary throughout and across populations. For diseases spread through multiple transmission pathways, sources of variation may affect each transmission pathway differently. In this paper we consider a disease that can be spread via direct and indirect transmission, such as the waterborne disease cholera. Specifically, we consider a system of multiple patches with direct transmission occurring entirely within patch and indirect transmission via a single shared water source. We investigate the effect of heterogeneity in dual transmission pathways on the spread of the disease. We first present a 2-patch model for which we examine the effect of variation in each pathway separately and propose a measure of heterogeneity that incorporates both transmission mechanisms and is predictive of R₀. We also explore how heterogeneity affects the final outbreak size and the efficacy of intervention measures. We conclude by extending several results to a more general n-patch setting.     

Modeling epidemic spread with awareness and heterogeneous transmission rates in networks

During an epidemic outbreak in a human population, susceptibility to infection can be reduced by raising awareness of the disease. In this paper, we investigate the effects of three forms of awareness (i.e., contact, local, and global) on the spread of a disease in a random network. Connectivity-correlated transmission rates are assumed. By using the mean-field theory and numerical simulation, we show that both local and contact awareness can raise the epidemic thresholds while the global awareness cannot, which mirrors the recent results of Wu et al. The obtained results point out that individual behaviors in the presence of an infectious disease has a great influence on the epidemic dynamics. Our method enriches mean-field analysis in epidemic models.     

A mechanistic model of infection: why duration and intensity of contacts should be included in models of disease spread

Background  

Mathematical models and simulations of disease spread often assume a constant per-contact transmission probability. This assumption ignores the heterogeneity in transmission probabilities, e.g. due to the varying intensity and duration of potentially contagious contacts. Ignoring such heterogeneities might lead to erroneous conclusions from simulation results. In this paper, we show how a mechanistic model of disease transmission differs from this commonly used assumption of a constant per-contact transmission probability.     

A Mathematical Model of Chikungunya Dynamics and Control: The Major Epidemic on Réunion Island

Chikungunya is a re-emerging arboviral disease transmitted by Aedes spp. mosquitoes. Although principally endemic to Africa and Asia, recent outbreaks have occurred in Europe following introductions by returning travellers. A particularly large outbreak occurred on Réunion Island in 2006, the published data from which forms the basis of the current study. A simple, deterministic mathematical model of the transmission of the virus between humans and mosquitoes was constructed and parameterised with the up-to-date literature on infection biology. The model is fitted to the large Réunion epidemic, resulting in an estimate of 4.1 for the type reproduction number of chikungunya. Although simplistic, the model provided a close approximation of both the peak incidence of the outbreak and the final epidemic size. Sensitivity analysis using Monte Carlo simulation demonstrated the strong influence that both the latent period of infection in humans and the pre-patent period have on these two epidemiological outcomes. We show why separating these variables, which are epidemiologically distinct in chikungunya infections, is not only necessary for accurate model fitting but also important in informing control.     

Percolation on heterogeneous networks as a model for epidemics

We consider a spatial model related to bond percolation for the spread of a disease that includes variation in the susceptibility to infection. We work on a lattice with random bond strengths and show that with strong heterogeneity, i.e. a wide range of variation of susceptibility, patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the heterogeneity. These results are qualitatively different from those of standard models in epidemiology, but correspond to real effects. We suggest that heterogeneity in the epidemic will affect the phylogenetic distance distribution of the disease-causing organisms. We also investigate small world lattices, and show that the effects mentioned above are even stronger.     

Epidemics in nonrandomly mixing populations: a simulation

Two stochastic, discrete-time simulation models for the spread of an epidemic through a population are presented. The models explore the effects of nonrandom mixing within the population and are based on an SIR epidemic model without vital statistics. They consider a population of preschool children, some of whom attend child care facilities. Disease transmission occurs both within the home neighborhood and at the child care facility used, if any. The two models differ in population size used, population density, the proportions of children using different kinds of care, and the functions used for calculating the probability of disease transmission. Results are presented for seven different variables--length of the epidemic in weeks, number of cases, number of cases in each kind of care (two day care centers, private homes, and children staying at home), and the number of private home providers affected by the epidemic. In addition, the distribution of total epidemic size and the progress of an epidemic are estimated from 25 epidemic trials. The effects of the location of homes of initial cases, the type of care used by initial cases, and the density of the population are discussed. Results from the simulation confirmed the importance of type of care on the risk for disease transmission. Results from all runs of the simulation showed that children who attended a day care center were most likely to become infected, children who went to a private home were intermediate, and children who did not use any day care facility were at the lowest risk. The size and length of the epidemics were related to the presence of the disease in day care centers, regardless of the location of the initial case, and the time at which the disease entered the center(s). The simulations also showed that the geographical distribution of the homes of children attending a particular center was a critical feature involved in the production of epidemics. The center with more widely distributed homes of students was less likely to experience a major epidemic than the center with clustering of student's homes within a neighborhood. This indicates that it is not simply attendance at a day care center that is critical for disease spread, but that the nature of the population of children attending a center is also of critical importance in the actual risk for disease spread within the center. These results are discussed with reference to the spread of hepatitis A among day care centers in Albuquerque, New Mexico.     

Mathematical modeling of severe acute respiratory syndrome nosocomial transmission in Japan: the dynamics of incident cases and prevalent cases

An outbreak of Severe Acute Respiratory Syndrome (SARS) occurred in Hong Kong in late February 2003, resulting in 8,096 cumulative cases with 774 deaths. The outbreak was amplified by nosocomial transmission in many hospitals. Using mathematical modeling, we simulated the number of new incident and prevalent cases of SARS after one infected person was admitted to a hospital (index case). The simulation was tested stochastically using the SEIR model based on previously reported Gamma distributions. We estimated the duration time until 10 beds in negative pressure rooms in Chiyoda-ku, one of the 23 wards in Tokyo, were fully occupied with SARS-infected patients. We determined the impact of an increasing number of days on the number of prevalent cases until the index case was isolated. The prevalent cases increase exponentially along with the increase of the non-isolation period of the index case, and all the beds were fully occupied if the index case was not isolated until more than 6 days. However even 2 days non-isolation period of the index case could fill up all the beds when 16% of secondary infections are transmitted outside the hospital. There is a possibility that an epidemic will occur with the isolation of the index case even at early days if the infection is transmitted outside the hospital. The simulation results revealed that it was important to recognize and isolate SARS patients as early as possible and also to prevent the transmission spreading outside the hospital to control an epidemic.     

Epidemiology, transmission dynamics and control of SARS: the 2002-2003 epidemic

This paper reviews current understanding of the epidemiology, transmission dynamics and control of the aetiological agent of severe acute respiratory syndrome (SARS). We present analyses of data on key parameters and distributions and discuss the processes of data capture, analysis and public health policy formulation during the SARS epidemic are discussed. The low transmissibility of the virus, combined with the onset of peak infectiousness following the onset of clinical symptoms of disease, transpired to make simple public health measures, such as isolating patients and quarantining their contacts, very effective in the control of the SARS epidemic. We conclude that we were lucky this time round, but may not be so with the next epidemic outbreak of a novel aetiological agent. We present analyses that help to further understanding of what intervention measures are likely to work best with infectious agents of defined biological and epidemiological properties. These lessons learnt from the SARS experience are presented in an epidemiological and public health context.     

Evaluations of Interventions Using Mathematical Models with Exponential and Non-exponential Distributions for Disease Stages: The Case of Ebola

Many mathematical models for the disease transmission dynamics of Ebola have been developed and studied, particularly during and after the 2014 outbreak in West Africa. Most of these models are systems of ordinary differential equations (ODEs). One of the common assumptions made in these ODE models is that the duration of disease stages, such as latent and infectious periods, follows an exponential distribution. Gamma distributions have also been used in some of these models. It has been demonstrated that, when the models are used to evaluate disease control strategies such as quarantine or isolation, the models with exponential and Gamma distribution assumptions may generate contradictory results (Feng et al. in Bull Math Biol 69(5):1511–1536, 2007). Several Ebola models are considered in this paper with various stage distributions, including exponential, Gamma and arbitrary distributions. These models are used to evaluate control strategies such as isolation (or hospitalization) and timely burial and to identify potential discrepancies between the results from models with exponential and Gamma distributions.     

Fundamental limits on inferring epidemic resurgence in real time using effective reproduction numbers

We find that epidemic resurgence, defined as an upswing in the effective reproduction number (R) of the contagion from subcritical to supercritical values, is fundamentally difficult to detect in real time. Inherent latencies in pathogen transmission, coupled with smaller and intrinsically noisier case incidence across periods of subcritical spread, mean that resurgence cannot be reliably detected without significant delays of the order of the generation time of the disease, even when case reporting is perfect. In contrast, epidemic suppression (where R falls from supercritical to subcritical values) may be ascertained 5–10 times faster due to the naturally larger incidence at which control actions are generally applied. We prove that these innate limits on detecting resurgence only worsen when spatial or demographic heterogeneities are incorporated. Consequently, we argue that resurgence is more effectively handled proactively, potentially at the expense of false alarms. Timely responses to recrudescent infections or emerging variants of concern are more likely to be possible when policy is informed by a greater quality and diversity of surveillance data than by further optimisation of the statistical models used to process routine outbreak data.     

Stage-structured transmission of phocine distemper virus in the Dutch 2002 outbreak

Heterogeneities in transmission among hosts can be very important in shaping infectious disease dynamics. In mammals with strong social organization, such heterogeneities are often structured by functional stage: juveniles, subadults and adults. We investigate the importance of such stage-related heterogeneities in shaping the 2002 phocine distemper virus (PDV) outbreak in the Dutch Wadden Sea, when more than 40 per cent of the harbour seals were killed. We do this by comparing the statistical fit of a hierarchy of models with varying transmission complexity: homogeneous versus heterogeneous mixing and density- versus frequency-dependent transmission. We use the stranding data as a proxy for incidence and use Poisson likelihoods to estimate the ‘who acquires infection from whom’ (WAIFW) matrix. Statistically, the model with strong heterogeneous mixing and density-dependent transmission was found to best describe the transmission dynamics. However, patterns of incidence support a model of frequency-dependent transmission among adults and juveniles. Based on the maximum-likelihood WAIFW matrix estimates, we use the next-generation formalism to calculate an R₀ between 2 and 2.5 for the Dutch 2002 PDV epidemic.     

基于多源地理数据精细尺度的武汉市人居环境新型冠状病毒肺炎疫情传播风险评估

新型冠状病毒肺炎的迅速传播和扩散警示着疾病风险评估的重要性。但现有的风险评估方法受数据限制,缺少实时性和准确性。此外,多数研究以行政统计单元作为分析尺度,存在可变面元问题。为解决这些问题,耦合精细尺度下武汉市疫情数据及多源地理数据,基于随机森林算法构建社区尺度的市域疫情传播风险评估模型并进行了疫情风险制图。模型测试精度达到0.85,Kappa系数达到0.70。此外,本研究还建立基于随机森林算法的社区及场所尺度的"空间变量-感染风险"模型,评估了不同场所设施疫情传播的风险程度。研究表明,（1）武汉中心区域感染风险最高并呈现出向外围递减的趋势;（2）感染风险排名前五的一级场所类型分别为购物服务、医疗服务、金融服务、交通设施以及公共设施;（3）小学、中学的疫情传播风险较低,而高等院校传播风险较高;（4）社区尺度下的疫情风险程度,预测购物场所与交通场所是疫情传播风险最高的驱动因子。本研究基于精细尺度提出风险评估新方法,可为未来疾病风险评估提供新思路,为疫情防控提供决策支持,人民群众提供安全保障。     

Effects of behavioral changes in a smallpox attack model

The impact of individual and community behavioral changes in response to an outbreak of a disease with high mortality is often not appreciated. Response strategies to a smallpox bioterrorist attack have focused on interventions such as isolation of infectives, contact tracing, quarantine of contacts, ring vaccination, and mass vaccination. We formulate and analyze a mathematical model in which some individuals lower their daily contact activity rates once an epidemic has been identified in a community. Transmission parameters are estimated from data and an expression is derived for the effective reproduction number. We use computer simulations to analyze the effects of behavior change alone and in combination with other control measures. We demonstrate that the spread of the disease is highly sensitive to how rapidly people reduce their contact activity rates and to the precautions that the population takes to reduce the transmission of the disease. Even gradual and mild behavioral changes can have a dramatic impact in slowing an epidemic. When behavioral changes are combined with other interventions, the epidemic is shortened and the number of smallpox cases is reduced. We conclude that for simulations of a smallpox outbreak to be useful, they must consider the impact of behavioral changes. This is especially true if the model predictions are being used to guide public health policy.     

Different forms of superspreading lead to different outcomes: Heterogeneity in infectiousness and contact behavior relevant for the case of SARS-CoV-2

Superspreading events play an important role in the spread of several pathogens, such as SARS-CoV-2. While the basic reproduction number of the original Wuhan SARS-CoV-2 is estimated to be about 3 for Belgium, there is substantial inter-individual variation in the number of secondary cases each infected individual causes—with most infectious individuals generating no or only a few secondary cases, while about 20% of infectious individuals is responsible for 80% of new infections. Multiple factors contribute to the occurrence of superspreading events: heterogeneity in infectiousness, individual variations in susceptibility, differences in contact behavior, and the environment in which transmission takes place. While superspreading has been included in several infectious disease transmission models, research into the effects of different forms of superspreading on the spread of pathogens remains limited. To disentangle the effects of infectiousness-related heterogeneity on the one hand and contact-related heterogeneity on the other, we implemented both forms of superspreading in an individual-based model describing the transmission and spread of SARS-CoV-2 in a synthetic Belgian population. We considered its impact on viral spread as well as on epidemic resurgence after a period of social distancing. We found that the effects of superspreading driven by heterogeneity in infectiousness are different from the effects of superspreading driven by heterogeneity in contact behavior. On the one hand, a higher level of infectiousness-related heterogeneity results in a lower risk of an outbreak persisting following the introduction of one infected individual into the population. Outbreaks that did persist led to fewer total cases and were slower, with a lower peak which occurred at a later point in time, and a lower herd immunity threshold. Finally, the risk of resurgence of an outbreak following a period of lockdown decreased. On the other hand, when contact-related heterogeneity was high, this also led to fewer cases in total during persistent outbreaks, but caused outbreaks to be more explosive in regard to other aspects (such as higher peaks which occurred earlier, and a higher herd immunity threshold). Finally, the risk of resurgence of an outbreak following a period of lockdown increased. We found that these effects were conserved when testing combinations of infectiousness-related and contact-related heterogeneity.     

Estimating epidemiological parameters for bovine tuberculosis in British cattle using a Bayesian partial-likelihood approach

Fitting models with Bayesian likelihood-based parameter inference is becoming increasingly important in infectious disease epidemiology. Detailed datasets present the opportunity to identify subsets of these data that capture important characteristics of the underlying epidemiology. One such dataset describes the epidemic of bovine tuberculosis (bTB) in British cattle, which is also an important exemplar of a disease with a wildlife reservoir (the Eurasian badger). Here, we evaluate a set of nested dynamic models of bTB transmission, including individual- and herd-level transmission heterogeneity and assuming minimal prior knowledge of the transmission and diagnostic test parameters. We performed a likelihood-based bootstrapping operation on the model to infer parameters based only on the recorded numbers of cattle testing positive for bTB at the start of each herd outbreak considering high- and low-risk areas separately. Models without herd heterogeneity are preferred in both areas though there is some evidence for super-spreading cattle. Similar to previous studies, we found low test sensitivities and high within-herd basic reproduction numbers (R₀), suggesting that there may be many unobserved infections in cattle, even though the current testing regime is sufficient to control within-herd epidemics in most cases. Compared with other, more data-heavy approaches, the summary data used in our approach are easily collected, making our approach attractive for other systems.