Anticipating critical transitions

I am broadly interested in critical phenomena in epidemiology and ecology. I focus on characterizing infectious disease emergence and elimination, specifically how demographic and environmental drivers operating over long time scales impact transmission processes and outbreak dynamics.  

View a recent seminar I gave at NIMBioS on the topic of forecasting critical transitions in infectious disease dynamics.

My postdoctoral work has developed theory of early warning signals of infectious disease emergence and leading indicators of elimination.

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ROC analysis was used to assess the performance of the statistics over a moving window for the SIR system rapidly approaching elimination.

Anticipating infectious disease emergence and documenting progress in disease elimination are important applications for the theory of critical transitions. The dynamical process of emergence of an infectious disease involves the transition from small sporadic outbreaks and stuttering chains of transmission to sustained outbreaks, whereas infectious disease elimination is the reverse transition. The basic reproduction number, the threshold for persistence of an infection, determines a tipping point.  For an infection to transition from emerging to endemic, or vice versa, this tipping point must be crossed. Mathematically, infectious disease systems undergo a transcritical bifurcation when the basic reproduction number equals one. Prior to the tipping point being crossed, the system may undergo a dynamical process called critical slowing down – perturbations become increasingly slow in returning to equilibrium. Critical slowing down may be detected in time series data using indicator statistics.

In this work, I considered compartmental susceptible–infectious–susceptible (SIS) and susceptible–infectious–recovered (SIR) models that are slowly forced through a critical transition. I derived analytical expressions for several candidate indicators of elimination and emergence, including the autocorrelation coefficient, variance, coefficient of variation, and power spectrum. I 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, the approach to emergence was much more difficult to detect. This project was carried out in collaboration with John Drake.

Leading indicators of elimination for mosquito-borne diseases.

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Anopheles mosquito. Image from Wikimedia.

Mosquito-borne diseases contribute significantly to the global disease burden. High-profile elimination campaigns are currently underway for mosquito-borne infections such as 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. In this project, I studied compartmental Ross-McDonald models that are slowly forced through a critical transition through gradually deployed control measures. I derived 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. I simulated the models 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. My results suggest that tipping points in mosquito-borne infectious disease systems may be foreshadowed by characteristic temporal patterns of disease prevalence. This project was carried out in collaboration with Jonathan Lillie of Northhall High School, Georgia, and John Drake.

Characterizing transient dynamics of immunizing childhood infections approaching elimination.

Signatures of critical transitions in time series depend on how the long-term dynamics of the system change as a bifurcation is approached. The majority of early warning indicators available at  present rely on  critical slowing down, that is,  the return rate of perturbations tends to zero as a critical threshold is approached. The return rate of perturbations is characterized by the dominant eigenvalue of the Jacobian matrix of the system evaluated at the equilibrium and is thus an asymptotic property of the system. In this work, I showed that the amplification envelope and reactivity, measures that characterize short-term behavior of the system, also change systematically as a bifurcation is approached in an important class of models for epidemics of infectious diseases.  Using Floquet theory, I extended the transient analysis to seasonally forced systems, and thus showed mathematically that pertussis systems are more prone to long transients than measles. I am currently testing the performance of the reactivity statistic as an early warning signal for elimination and reemergence of measles and pertussis. I am collaborating with Pejman Rohani and John Drake on this project.

How does altering the way noise is added (e.g., assuming demographic stochasticity vs. environmental stochasticity) influence early warning signals?

To answer this question, I considered the simplest possible models that exhibit transcritical, saddle-node and pitchfork bifurcations. I added stochasticity to each model by assuming noise was either additive, multiplicative or demographic in nature. I show that linearization of the resulting stochastic differential equation for each bifurcation leads to an Ornstein-Uhlenbeck process for the fluctuations around equilibrium. I derived expressions for the variance, autocorrelation and power spectrum for all cases. Trends in the leading indicators as the bifurcation is approached depend on noise type. For example, models with demographic stochasticity predict a decline in variance as the bifurcation parameter changes whereas models with additive noise predict an increase in variance. Models with multiplicative noise in a parameter may predict a rise in variance, decline in variance or that the variance depends on the strength of noise only. The next steps for this project are to test the theoretical predictions by numerical simulations of relevant biological models. The ability to classify trends in summary statistics for a broad class of models will enhance our understanding of how critical slowing down manifests in systems approaching a transition. I am supervising Danielle Burton (Graduate student in Mathematics, University of Tennessee) on this project.

The onset of spatial synchrony as early warning of metapopulation collapse.

Spatial synchrony is a subject of intense interest to ecologists due to its importance for the persistence of metapopulations. However, it is not well recognized that the onset of spatial synchrony in oscillating ecological systems (a nonlinear phenomenon known as phase locking) is a critical transition. The goal of this project is to investigate if the onset of phase locking is detectable in advance for a broad class of oscillating ecological systems. Development of early warning signals of metapopulation extinction that are based on synchrony onset may have important implications for conservation of endangered species and for elimination of infectious diseases. In collaboration with Vasilis Dakos, I am building a suite of mechanistic models that incorporate various hypotheses for synchrony and investigating the performance of leading indicators of synchrony using these models. This work was assisted by attendance as a Short-term Visitor at the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundation, the U.S. Department of Homeland Security, and the U.S. Department of Agriculture through NSF Award #DBI-1300426, with additional support from The University of Tennessee, Knoxville.

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