Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough

Although many stochastic models can accurately capture the qualitative epidemic patterns of many childhood diseases, there is still considerable discussion concerning the basic mechanisms generating these patterns; much of this stems from the use of deterministic models to try to understand stochastic simulations. We argue that a systematic method of analysing models of the spread of childhood diseases is required in order to consistently separate out the effects of demographic stochasticity, external forcing and modelling choices. Such a technique is provided by formulating the models as master equations and using the van Kampen system-size expansion to provide analytical expressions for quantities of interest. We apply this method to the susceptible–exposed–infected–recovered (SEIR) model with distributed exposed and infectious periods and calculate the form that stochastic oscillations take on in terms of the model parameters. With the use of a suitable approximation, we apply the formalism to analyse a model of whooping cough which includes seasonal forcing. This allows us to more accurately interpret the results of simulations and to make a more quantitative assessment of the predictions of the model. We show that the observed dynamics are a result of a macroscopic limit cycle induced by the external forcing and resonant stochastic oscillations about this cycle.     

Multifractal signatures of infectious diseases

Incidence of infection time-series data for the childhood diseases measles, chicken pox, rubella and whooping cough are described in the language of multifractals. We explore the potential of using the wavelet transform maximum modulus (WTMM) method to characterize the multiscale structure of the observed time series and of simulated data generated by the stochastic susceptible-exposed-infectious-recovered (SEIR) epidemic model. The singularity spectra of the observed time series suggest that each disease is characterized by a unique multifractal signature, which distinguishes that particular disease from the others. The wavelet scaling functions confirm that the time series of measles, rubella and whooping cough are clearly multifractal, while chicken pox has a more monofractal structure in time. The stochastic SEIR epidemic model is unable to reproduce the qualitative singularity structure of the reported incidence data: it is too smooth and does not appear to have a multifractal singularity structure. The precise reasons for the failure of the SEIR epidemic model to reproduce the correct multiscale structure of the reported incidence data remain unclear.     

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

The population dynamics of infectious diseases occasionally undergo rapid qualitative changes, such as transitions from annual to biennial cycles or to irregular dynamics. Previous work, based on the standard seasonally forced ‘susceptible–exposed–infectious–removed’ (SEIR) model has found that transitions in the dynamics of many childhood diseases result from bifurcations induced by slow changes in birth and vaccination rates. However, the standard SEIR formulation assumes that the stage durations (latent and infectious periods) are exponentially distributed, whereas real distributions are narrower and centred around the mean. Much recent work has indicated that realistically distributed stage durations strongly affect the dynamical structure of seasonally forced epidemic models. We investigate whether inferences drawn from previous analyses of transitions in patterns of measles dynamics are robust to the shapes of the stage duration distributions. As an illustrative example, we analyse measles dynamics in New York City from 1928 to 1972. We find that with a fixed mean infectious period in the susceptible–infectious–removed (SIR) model, the dynamical structure and predicted transitions vary substantially as a function of the shape of the infectious period distribution. By contrast, with fixed mean latent and infectious periods in the SEIR model, the shapes of the stage duration distributions have a less dramatic effect on model dynamical structure and predicted transitions. All these results can be understood more easily by considering the distribution of the disease generation time as opposed to the distributions of individual disease stages. Numerical bifurcation analysis reveals that for a given mean generation time the dynamics of the SIR and SEIR models for measles are nearly equivalent and are insensitive to the shapes of the disease stage distributions.     

Modelling the influence of human behaviour on the spread of infectious diseases: a review

Human behaviour plays an important role in the spread of infectious diseases, and understanding the influence of behaviour on the spread of diseases can be key to improving control efforts. While behavioural responses to the spread of a disease have often been reported anecdotally, there has been relatively little systematic investigation into how behavioural changes can affect disease dynamics. Mathematical models for the spread of infectious diseases are an important tool for investigating and quantifying such effects, not least because the spread of a disease among humans is not amenable to direct experimental study. Here, we review recent efforts to incorporate human behaviour into disease models, and propose that such models can be broadly classified according to the type and source of information which individuals are assumed to base their behaviour on, and according to the assumed effects of such behaviour. We highlight recent advances as well as gaps in our understanding of the interplay between infectious disease dynamics and human behaviour, and suggest what kind of data taking efforts would be helpful in filling these gaps.     

The Human Microbiota in Health and Disease

Trillions of microbes have evolved with and continue to live on and within human beings. A variety of environmental factors can affect intestinal microbial imbalance, which has a close relationship with human health and disease. Here, we focus on the interactions between the human microbiota and the host in order to provide an overview of the microbial role in basic biological processes and in the development and progression of major human diseases such as infectious diseases, liver diseases, gastrointestinal cancers, metabolic diseases, respiratory diseases, mental or psychological diseases, and autoimmune diseases. We also review important advances in techniques associated with microbial research, such as DNA sequencing, metabonomics, and proteomics combined with computation-based bioinformatics. Current research on the human microbiota has become much more sophisticated and more comprehensive. Therefore, we propose that research should focus on the host-microbe interaction and on cause-effect mechanisms, which could pave the way to an understanding of the role of gut microbiota in health and disease, and provide new therapeutic targets and treatment approaches in clinical practice.     

Dissecting the determinants of malaria chronicity: why within-host models struggle to reproduce infection dynamics

The duration of infection is fundamental to the epidemiological behaviour of any infectious disease, but remains one of the most poorly understood aspects of malaria. In endemic areas, the malaria parasite Plasmodium falciparum can cause both acute, severe infections and asymptomatic, chronic infections through its interaction with the host immune system. Frequent superinfection and massive parasite genetic diversity make it extremely difficult to accurately measure the distribution of infection lengths, complicating the estimation of basic epidemiological parameters and the prediction of the impact of interventions. Mathematical models have qualitatively reproduced parasite dynamics early during infection, but reproducing long-lived chronic infections remains much more challenging. Here, we construct a model of infection dynamics to examine the consequences of common biological assumptions for the generation of chronicity and the impact of co-infection. We find that although a combination of host and parasite heterogeneities are capable of generating chronic infections, they do so only under restricted parameter choices. Furthermore, under biologically plausible assumptions, co-infection of parasite genotypes can alter the course of infection of both the resident and co-infecting strain in complex non-intuitive ways. We outline the most important puzzles for within-host models of malaria arising from our analysis, and their implications for malaria epidemiology and control.     

Spatio-temporal waves and targeted vaccination in recurrent epidemic network models

The success of an infectious disease to invade a population is strongly controlled by the population''s specific connectivity structure. Here, a network model is presented as an aid in understanding the role of social behaviour and heterogeneous connectivity in determining the spatio-temporal patterns of disease dynamics. We explore the controversial origins of long-term recurrent oscillations believed to be characteristic of diseases that have a period of temporary immunity after infection. In particular, we focus on sexually transmitted diseases such as syphilis, where this controversy is currently under review. Although temporary immunity plays a key role, it is found that, in realistic small-world networks, the social and sexual behaviour of individuals also has a great influence in generating long-term cycles. The model generates circular waves of infection with unusual spatial dynamics that depend on focal areas that act as pacemakers in the population. Eradication of the disease can be efficiently achieved by eliminating the pacemakers with a targeted vaccination scheme. A simple difference equation model is derived, which captures the infection dynamics of the network model and gives insights into their origins and their eradication through vaccination. Illustrative videos may be found in the electronic supplementary material.     

Stochastic amplification in epidemics

The role of stochasticity and its interplay with nonlinearity are central current issues in studies of the complex population patterns observed in nature, including the pronounced oscillations of wildlife and infectious diseases. The dynamics of childhood diseases have provided influential case studies to develop and test mathematical models with practical application to epidemiology, but are also of general relevance to the central question of whether simple nonlinear systems can explain and predict the complex temporal and spatial patterns observed in nature outside laboratory conditions. Here, we present a stochastic theory for the major dynamical transitions in epidemics from regular to irregular cycles, which relies on the discrete nature of disease transmission and low spatial coupling. The full spectrum of stochastic fluctuations is derived analytically to show how the amplification of noise varies across these transitions. The changes in noise amplification and coherence appear robust to seasonal forcing, questioning the role of seasonality and its interplay with deterministic components of epidemiological models. Childhood diseases are shown to fall into regions of parameter space of high noise amplification. This type of "endogenous" stochastic resonance may be relevant to population oscillations in nonlinear ecological systems in general.     

Biomedical and social contributions to sustainability

Over the past two or three centuries, biomedical advances have provided methods to prevent and treat infectious diseases. These changes have greatly reduced human suffering and enhanced sustainability by allowing people to live longer and healthier lives. The challenge for the coming centuries will be to ensure that these longer, healthier lives are also more productive lives. We must build on the gains of the past by translating new discoveries in regenerative medicine into therapies for degenerative and genetic diseases. Stem cells may be used to identify drugs that prevent the development of symptoms or to replace cells that have either died or lost their physiological function. In the case of genetic diseases, it may be possible to correct the genetic error. While most conditions that might be treated in these ways are common to all communities, some are more prevalent in specific races. Provision of these and other benefits depends not only on attainment of the research objectives, but also upon our ability to make treatment opportunities available throughout both developed and developing communities. The long history of researching and treating infectious diseases shows that it may take many decades to reap the full benefit of the new biological understanding.     

Infection dynamics: from organ to host population.

A symposium discussing collaborative research work on infectious diseases dynamics was held at Queens' College, University of Cambridge on 25 October 2006.     

Insights from unifying modern approximations to infections on networks

Networks are increasingly central to modern science owing to their ability to conceptualize multiple interacting components of a complex system. As a specific example of this, understanding the implications of contact network structure for the transmission of infectious diseases remains a key issue in epidemiology. Three broad approaches to this problem exist: explicit simulation; derivation of exact results for special networks; and dynamical approximations. This paper focuses on the last of these approaches, and makes two main contributions. Firstly, formal mathematical links are demonstrated between several prima facie unrelated dynamical approximations. And secondly, these links are used to derive two novel dynamical models for network epidemiology, which are compared against explicit stochastic simulation. The success of these new models provides improved understanding about the interaction of network structure and transmission dynamics.     

Disease dynamics over very different time-scales: foot-and-mouth disease and scrapie on the network of livestock movements in the UK.

We analyse the relationship between the network of livestock movements in the UK and the dynamics of two diseases: foot-and-mouth disease (FMD), which has an incubation period of days, and scrapie, which incubates over years. For FMD, the time-scale of expected epidemics is similar to the time-scale of the evolution of the network. We argue that, under appropriate conditions, a static network analysis can be an appropriate tool for gaining insights into disease dynamics even when the relevant time-scales are similar, as with FMD. We show that a subclass of 'linkage moves' maintains the network structure, and so removing these links has a dramatic effect on the number of potentially infected farms, an effect corroborated by simulations. In contrast, because scrapie has a low probability of transmission per contact and a long incubation period, a static network representation is probably appropriate; however, the signature of the network in the pattern of transmission is likely to be faint. Scrapie-notifying farms were more likely to be associated with each other via trading at markets than were control farms; however, network community structure proves to be less representative of prevalence patterns than geographical region. These contradictory indicators emphasize that appropriate observation time frames and good discrimination among types of potentially infectious contacts are vital in order for network analysis to be a valuable epidemiological tool.     

An avian influenza model with latency and vaccination

Avian influenza, caused by influenza A viruses, has received worldwide attention in recent years. The viruses can spread from birds to humans as well as transmit among human hosts. In this study, we formulate a mathematical model for avian influenza that includes bird-human interaction and that incorporates the effects of infection latency and human vaccination. We investigate the essential dynamics of the model through an equilibrium analysis. Meanwhile, we explore effective vaccination strategy to control avian influenza outbreaks using optimal control theory. Our results indicate that strategically deployed human vaccination can significantly reduce the numbers of exposed and infectious people.     

Time-dependent spectral analysis of epidemiological time-series with wavelets.

In the current context of global infectious disease risks, a better understanding of the dynamics of major epidemics is urgently needed. Time-series analysis has appeared as an interesting approach to explore the dynamics of numerous diseases. Classical time-series methods can only be used for stationary time-series (in which the statistical properties do not vary with time). However, epidemiological time-series are typically noisy, complex and strongly non-stationary. Given this specific nature, wavelet analysis appears particularly attractive because it is well suited to the analysis of non-stationary signals. Here, we review the basic properties of the wavelet approach as an appropriate and elegant method for time-series analysis in epidemiological studies. The wavelet decomposition offers several advantages that are discussed in this paper based on epidemiological examples. In particular, the wavelet approach permits analysis of transient relationships between two signals and is especially suitable for gradual change in force by exogenous variables.     

Emerging diseases,re-emerging histories

The notion of “emerging infectious diseases” (EID) as a category of global health concerns was created in the 1990s to acknowledge that, although public health interventions, vaccines, and antibiotics since the late 19th century had given wealthier parts of the world control over most infectious diseases, the experience of Ebola and HIV/AIDS showed that new human diseases could still arise. “Emerging diseases” have clear histories, and the task the field of EID set itself has been to make those histories as short as possible—to catch emerging diseases as close to their origin as possible and snuff them out before they can become pandemics. In contrast, many of the known infectious diseases, those that were allegedly “conquered” by the rise of biomedicine in the late 19th and early 20th centuries and which (aside from smallpox) nonetheless still persist today, were assumed to have existed “since time immemorial.” Their histories had no specific time-depth, no documentable origins, and few discernible narrative arcs. The advent of a new field of research, palaeogenetics, has made possible an evolutionary perspective on pathogenic microorganisms, bringing hitherto unimaginable specificity to their documentable histories. Many origin stories, and nearly all globalization stories, fall within periods that historians and archaeologists study intensively. I argue that the diseases that have proved best suited to global expansion since the advent of the Holocene—those that became pandemics, like COVID-19—are those that have best exploited humans' global networks and behaviors in a given age. This recognition, in turn, gives the fields of both global health and history of medicine a new agenda.     

Dynamics of infectious disease transmission by inhalable respiratory droplets

Transmission of respiratory infectious diseases in humans, for instance influenza, occurs by several modes. Respiratory droplets provide a vector of transmission of an infectious pathogen that may contribute to different transmission modes. An epidemiological model incorporating the dynamics of inhalable respiratory droplets is developed to assess their relevance in the infectious process. Inhalable respiratory droplets are divided into respirable droplets, with droplet diameter less than 10 µm, and inspirable droplets, with diameter in the range 10–100 µm: both droplet classes may be inhaled or settle. Droplet dynamics is determined by their physical properties (size), whereas population dynamics is determined by, among other parameters, the pathogen infectivity and the host contact rates. Three model influenza epidemic scenarios, mediated by different airborne or settled droplet classes, are analysed. The scenarios are distinguished by the characteristic times associated with breathing at contact and with hand-to-face contact. The scenarios suggest that airborne transmission, mediated by respirable droplets, provides the dominant transmission mode in middle and long-term epidemics, whereas inspirable droplets, be they airborne or settled, characterize short-term epidemics with high attack rates. The model neglects close-contact transmission by droplet sprays (direct projection onto facial mucous membranes), retaining close-contact transmission by inspirable droplets.     

Anticancer efficacy and subcellular site of action investigated by real-time monitoring of cellular responses to localized drug delivery in single cells

Subcellular-targeted drug delivery has much potential to defeat infectious diseases and cancers. Medical and/or biochemical effects of drugs/bioactive molecules delivered to subcellular compartments and their subcellular sites of action need to be investigated but have not been explored. Here the subcellular location-dependent biochemical responses of a potent anticancer drug, β-lapachone (β-lap), is investigated by a reduced graphene oxide (rGO)-functionalized optical nanoprobe, which can deliver and simultaneously monitor the drug effects at nanoscales. For the first time, distinct oxidative responses and calcium alterations in three selected subcellular domains are observed and clearly pinpoint that the perinuclear region is the optimal subcellular site for β-lap to have the best anticancer efficacy. The results presented here provide not only scientific insights of subcellular drug-cell interaction that is not obtainable from conventional methods, but they also provide valuable knowledge for rational design of subcellular-targeted delivery or spatially resolved signal intervention.     

Strong seasonality produces spatial asynchrony in the outbreak of infectious diseases

Models for infectious diseases usually assume a fixed demographic structure. Yet, a disease can spread over a region encountering different local demographic variations that may significantly alter local dynamics. Spatial heterogeneity in the resulting dynamics can lead to important differences in the design of surveillance and control strategies. We illustrate this by exploring the north–south gradient in the seasonal demography of raccoon rabies over the eastern USA. We find that the greater variance in the timing of spring births characteristic of southern populations can lead to the spatial synchronization of southern epidemics, while the narrow birth-pulse associated with northern populations can lead to an irregular patchwork of epidemics. These results indicate that surveillance in the southern states can be reduced relative to northern locations without loss of detection ability. This approach could yield significant savings in vaccination programmes. The importance of seasonality in many widely distributed diseases indicates that our findings will find applications beyond raccoon rabies.     

Applications of Gold Nanoparticles in the Detection and Identification of Infectious Diseases and Biothreats

The situation of infectious diseases and biothreats all over the world remains serious. The effective identification of such diseases plays a very important role. In recent years, gold nanoparticles have been widely used in biosensor design to improve the performance for the detection of infectious diseases and biothreats. Here, recent advances of gold‐nanoparticle‐based biosensors in this field are summarized.