Transient indicators of tipping points in infectious diseases.

The majority of known early warning indicators of critical transitions rely on asymptotic resilience and critical slowing down. In continuous systems, critical slowing down is mathematically described by a decrease in magnitude of the dominant eigenvalue of the Jacobian matrix on the approach to a critical transition. Here, we show that measures of transient dynamics, specifically, reactivity and the maximum of the amplification envelope, also change systematically as a bifurcation is approached in an important class of models for epidemics of infectious diseases. Furthermore, we introduce indicators designed to detect trends in these measures and find that they reliably classify time series of case notifications simulated from stochastic models according to levels of vaccine uptake. Greater attention should be focused on the potential for systems to exhibit transient amplification of perturbations as a critical threshold is approached, and should be considered when searching for generic leading indicators of tipping points. Awareness of this phenomenon will enrich understanding of the dynamics of complex systems on the verge of a critical transition.

Integrating data mining and transmission theory in the ecology of infectious diseases.

Our understanding of ecological processes is built on patterns inferred from data. Applying modern analytical tools such as machine learning to increasingly high dimensional data offers the potential to expand our perspectives on these processes, shedding new light on complex ecological phenomena such as pathogen transmission in wild populations. Here, we propose a novel approach that combines data mining with theoretical models of disease dynamics. Using rodents as an example, we incorporate statistical differences in the life history features of zoonotic reservoir hosts into pathogen transmission models, enabling us to bound the range of dynamical phenomena associated with hosts, based on their traits. We then test for associations between equilibrium prevalence, a key epidemiological metric and data on human outbreaks of rodent-borne zoonoses, identifying matches between empirical evidence and theoretical predictions of transmission dynamics. We show how this framework can be generalized to other systems through a rubric of disease models and parameters that can be derived from empirical data. By linking life history components directly to their effects on disease dynamics, our mining-modelling approach integrates machine learning and theoretical models to explore mechanisms in the macroecology of pathogen transmission and their consequences for spillover infection to humans.

The statistics of epidemic transitions.

Emerging and re-emerging pathogens exhibit very complex dynamics, are hard to model and difficult to predict. Their dynamics might appear intractable. However, new statistical approaches-rooted in dynamical systems and the theory of stochastic processes-have yielded insight into the dynamics of emerging and re-emerging pathogens. We argue that these approaches may lead to new methods for predicting epidemics. This perspective views pathogen emergence and re-emergence as a "critical transition," and uses the concept of noisy dynamic bifurcation to understand the relationship between the system observables and the distance to this transition. Because the system dynamics exhibit characteristic fluctuations in response to perturbations for a system in the vicinity of a critical point, we propose this information may be harnessed to develop early warning signals. Specifically, the motion of perturbations slows as the system approaches the transition.

How Stochasticity Influences Leading Indicators of Critical Transitions.

Many complex systems exhibit critical transitions. Of considerable interest are bifurcations, small smooth changes in underlying drivers that produce abrupt shifts in system state. Before reaching the bifurcation point, the system gradually loses stability ('critical slowing down'). Signals of critical slowing down may be detected through measurement of summary statistics, but how extrinsic and intrinsic noises influence statistical patterns prior to a transition is unclear. Here, we consider a range of stochastic models that exhibit transcritical, saddle-node and pitchfork bifurcations. Noise was assumed to be either intrinsic or extrinsic. We derived expressions for the stationary variance, autocorrelation and power spectrum for all cases. Trends in summary statistics signaling the approach of each bifurcation depend on the form of noise. For example, models with intrinsic stochasticity may predict an increase in or a decline in variance as the bifurcation parameter changes, whereas models with extrinsic noise applied additively predict an increase in variance. The ability to classify trends of summary statistics for a broad class of models enhances our understanding of how critical slowing down manifests in complex systems approaching a transition.

How noise and coupling influence leading indicators of population extinction in a spatially extended ecological system.

Anticipating critical transitions in spatially extended systems is a key topic of interest to ecologists. Gradually declining metapopulations are an important example of a spatially extended biological system that may exhibit a critical transition. Theory for spatially extended systems approaching extinction that accounts for environmental stochasticity and coupling is currently lacking. Here, we develop spatially implicit two-patch models with additive and multiplicative forms of environmental stochasticity that are slowly forced through population collapse, through changing environmental conditions. We derive patch-specific expressions for candidate indicators of extinction and test their performance via a simulation study. Coupling and spatial heterogeneities decrease the magnitude of the proposed indicators in coupled populations relative to isolated populations, and the noise regime and the degree of coupling together determine trends in summary statistics. This theory may be readily applied to other spatially extended ecological systems, such as coupled infectious disease systems on the verge of elimination.

Leading indicators of mosquito-borne disease elimination.

Mosquito-borne diseases contribute significantly to the global disease burden. High-profile elimination campaigns are currently underway for many parasites, e.g., Plasmodium spp., the causal agent of malaria. Sustaining momentum near the end of elimination programs is often difficult to achieve and consequently quantitative tools that enable monitoring the effectiveness of elimination activities after the initial reduction of cases has occurred are needed. Documenting progress in vector-borne disease elimination is a potentially important application for the theory of critical transitions. Non-parametric approaches that are independent of model-fitting would advance infectious disease forecasting significantly. In this paper, we consider compartmental Ross-McDonald models that are slowly forced through a critical transition through gradually deployed control measures. We derive expressions for the behavior of candidate indicators, including the autocorrelation coefficient, variance, and coefficient of variation in the number of human cases during the approach to elimination. We conducted a simulation study to test the performance of each summary statistic as an early warning system of mosquito-borne disease elimination. Variance and coefficient of variation were highly predictive of elimination but autocorrelation performed poorly as an indicator in some control contexts. Our results suggest that tipping points (bifurcations) in mosquito-borne infectious disease systems may be foreshadowed by characteristic temporal patterns of disease prevalence.

Interspecific Contact and Competition May Affect the Strength and Direction of Disease-Diversity Relationships for Directly Transmitted Microparasites.

The frequency of opportunities for transmission is key to the severity of directly transmitted disease outbreaks in multihost communities. Transmission opportunities for generalist microparasites often arise from competitive and trophic interactions. Additionally, contact heterogeneities within and between species either hinder or promote transmission. General theory incorporating competition and contact heterogeneities for disease-diversity relationships is underdeveloped. Here, we present a formal framework to explore disease-diversity relationships for directly transmitted parasites that infect multiple host species, including influenza viruses, rabies virus, distemper viruses, and hantaviruses. We explicitly include host regulation via intra- and interspecific competition, where the latter can be dependent on or independent of interspecific contact rates (covering resource utilization overlap, habitat selection preferences, and temporal niche partitioning). We examine how these factors interact with frequency- and density-dependent transmission along with traits of the hosts in the assemblage, culminating in the derivation of a relationship describing the propensity for parasite fitness to decrease in species assemblages relative to that in single-host species. This relationship reveals that increases in biodiversity do not necessarily suppress frequency-dependent parasite transmission and that regulation of hosts via interspecific competition does not always lead to a reduction in parasite fitness. Our approach explicitly shows that species identity and ecological interactions between hosts together determine microparasite transmission outcomes in multispecies communities.

Transmission Models of Historical Ebola Outbreaks.

To guide the collection of data under emergent epidemic conditions, we reviewed compartmental models of historical Ebola outbreaks to determine their implications and limitations. We identified future modeling directions and propose that the minimal epidemiologic dataset for Ebola model construction comprises duration of incubation period and symptomatic period, distribution of secondary cases by infection setting, and compliance with intervention recommendations.

Ebola cases and health system demand in Liberia.

In 2014, a major epidemic of human Ebola virus disease emerged in West Africa, where human-to-human transmission has now been sustained for greater than 12 months. In the summer of 2014, there was great uncertainty about the answers to several key policy questions concerning the path to containment. What is the relative importance of nosocomial transmission compared with community-acquired infection? How much must hospital capacity increase to provide care for the anticipated patient burden? To which interventions will Ebola transmission be most responsive? What must be done to achieve containment? In recent years, epidemic models have been used to guide public health interventions. But, model-based policy relies on high quality causal understanding of transmission, including the availability of appropriate dynamic transmission models and reliable reporting about the sequence of case incidence for model fitting, which were lacking for this epidemic. To investigate the range of potential transmission scenarios, we developed a multi-type branching process model that incorporates key heterogeneities and time-varying parameters to reflect changing human behavior and deliberate interventions in Liberia. Ensembles of this model were evaluated at a set of parameters that were both epidemiologically plausible and capable of reproducing the observed trajectory. Results of this model suggested that epidemic outcome would depend on both hospital capacity and individual behavior. Simulations suggested that if hospital capacity was not increased, then transmission might outpace the rate of isolation and the ability to provide care for the ill, infectious, and dying. Similarly, the model suggested that containment would require individuals to adopt behaviors that increase the rates of case identification and isolation and secure burial of the deceased. As of mid-October, it was unclear that this epidemic would be contained even by 99% hospitalization at the planned hospital capacity. A new version of the model, updated to reflect information collected during October and November 2014, predicts a significantly more constrained set of possible futures. This model suggests that epidemic outcome still depends very heavily on individual behavior. Particularly, if future patient hospitalization rates return to background levels (estimated to be around 70%), then transmission is predicted to remain just below the critical point around Reff = 1. At the higher hospitalization rate of 85%, this model predicts near complete elimination in March to June, 2015.

Chaos in a seasonally perturbed SIR model: avian influenza in a seabird colony as a paradigm.

Seasonality is a complex force in nature that affects multiple processes in wild animal populations. In particular, seasonal variations in demographic processes may considerably affect the persistence of a pathogen in these populations. Furthermore, it has been long observed in computer simulations that under seasonal perturbations, a host-pathogen system can exhibit complex dynamics, including the transition to chaos, as the magnitude of the seasonal perturbation increases. In this paper, we develop a seasonally perturbed Susceptible-Infected-Recovered model of avian influenza in a seabird colony. Numerical simulations of the model give rise to chaotic recurrent epidemics for parameters that reflect the ecology of avian influenza in a seabird population, thereby providing a case study for chaos in a host- pathogen system. We give a computer-assisted exposition of the existence of chaos in the model using methods that are based on the concept of topological hyperbolicity. Our approach elucidates the geometry of the chaos in the phase space of the model, thereby offering a mechanism for the persistence of the infection. Finally, the methods described in this paper may be immediately extended to other infections and hosts, including humans.

Theory of early warning signals of disease emergenceand leading indicators of elimination.

Anticipating infectious disease emergence and documenting progress in disease elimination are important applications for the theory of critical transitions. A key problem is the development of theory relating the dynamical processes of transmission to observable phenomena. In this paper, we consider compartmental susceptible-infectious-susceptible (SIS) and susceptible-infectious-recovered (SIR) models that are slowly forced through a critical transition. We derive expressions for the behavior of several candidate indicators, including the autocorrelation coefficient, variance, coefficient of variation, and power spectra of SIS and SIR epidemics during the approach to emergence or elimination. We validated these expressions using individual-based simulations. We further showed that moving-window estimates of these quantities may be used for anticipating critical transitions in infectious disease systems. Although leading indicators of elimination were highly predictive, we found the approach to emergence to be much more difficult to detect. It is hoped that these results, which show the anticipation of critical transitions in infectious disease systems to be theoretically possible, may be used to guide the construction of online algorithms for processing surveillance data.

Increased hexokinase II expression in the renal glomerulus of mice in response to arsenic.

Epidemiological studies link arsenic exposure to increased risks of cancers of the skin, kidney, lung, bladder and liver. Additionally, a variety of non-cancerous conditions such as diabetes mellitus, hypertension, and cardiovascular disease have been associated with chronic ingestion of low levels of arsenic. However, the biological and molecular mechanisms by which arsenic exerts its effects remain elusive. Here we report increased renal hexokinase II (HKII) expression in response to arsenic exposure both in vivo and in vitro. In our model, HKII was up-regulated in the renal glomeruli of mice exposed to low levels of arsenic (10 ppb or 50 ppb) via their drinking water for up to 21 days. Additionally, a similar effect was observed in cultured renal mesangial cells exposed to arsenic. This correlation between our in vivo and in vitro data provides further evidence for a direct link between altered renal HKII expression and arsenic exposure. Thus, our data suggest that alterations in renal HKII expression may be involved in arsenic-induced pathological conditions involving the kidney. More importantly, these results were obtained using environmentally relevant arsenic concentrations.