O’Regan, S.M. & J.M. Drake. 2013. “Theory of early warning signals of disease emergence and leading indicators of elimination.” Theoretical Ecology. [online] [PDF]

  • This paper presents the basic theory of early warning signals in slow-fast disease transmission systems.

Lofgren, E.T. , M.E. Halloran, C.M. Rivers, J.M. Drake, T.C. Porco, B. Lewis, W. Yang, A. Vespignani, J. Shaman, J.N.S. Eisenberg, M.C. Eisenberg, M.Marathe, S.V. Scarpino, K.A. Alexander, R. Meza, Matthew J. Ferrari, J.M. Hyman, L.A. Meyers & S. Eubank. 2014. “Opinion: Mathematical models: A key tool for outbreak response.” Proceedings of the National Academy of Sciences. [online]


Halloran, M.E., A. Vespignani, N. Bharti, L.R. Feldstein, K.A. Alexander, M. Ferrari, J. Shaman, J. M. Drake, T. Porco, J.N.S. Eisenberg, S.Y. Del Valle, E. Lofgren, S.V. Scarpino, M.C. Eisenberg, D. Gao, J.M. Hyman, S. Eubank & I.M. Longini Jr.. 2014. “Ebola: Mobility data.” Science. [online]


Drake, J.M., R.B. Kaul, L.W. Alexander, S.M. O’Regan, A.M. Kramer, J.T. Pulliam, M.J. Ferrari & A.W. Park. 2015. “Ebola Cases and Health System Demand in Liberia.” PLOS Biology. [online]

  • AUTHOR SUMMARY: There is considerable uncertainty regarding the steps needed to contain the ongoing Ebola crisis in West Africa, the timeline required to achieve control, and the projected burden of mortality. To address these issues, we develop a branching process model for Ebola transmission that focuses on offspring distributions (i.e., the numbers of new infections caused by each case). We use the model to assess the likely progression of Ebola in Liberia. The model assesses the feedback between new cases and hospital demand under a range of plausible intervention scenarios, particularly ramping-up of treatment facilities over time and increasing the number of individuals seeking hospital treatment through outreach and education. Transmission scenarios—to health care workers in hospitals, to caregivers in the community, to hospital visitors, and to individuals preparing bodies for funerals—are described by distinct offspring distributions based on available data. Results suggest that the outcome of the epidemic depends on both hospital capacity and individual behavior. Additionally, the model highlights the conditions under which transmission might have outpaced hospital capacity, and projects possible epidemic trajectories into 2015.

Drake, J.M., I.Bakach, M.R. Just, S.M. O’Regan, M.Gambhir & I.C. Fung. 2015. “Transmission Models of Historical Ebola Outbreaks.” Emerging Infectious Diseases. [online]


Alexander, K.A., C.E. Sanderson, M. Marathe, B.L. Lewis, C.M. Rivers, J. Shaman, J.M. Drake, E. Lofgren, V.M. Dato, M.C. Eisenberg & S. Eubank. 2015. “What Factors Might Have Led to the Emergence of Ebola in West Africa?.” PLOS Neglected Tropical Diseases. [online]


O’Regan, S. M., J.W. Lillie & J.M. Drake. 2016. “Leading indicators of mosquito-borne disease elimination.” Theoretical Ecology. [online]

  • This paper extends the theory of critical slowing down to vector-borne disease systems.

Dibble, C.J., E.B. O’Dea, A.W. Park & J.M. Drake. 2016. “Waiting time to infectious disease emergence.” Journal of the Royal Society Interface. [online]

  • This paper established the theory of bifurcation delay, derives the probability distribution of the lag that occurs between the time at which an infectious disease becomes critical and the time that outbreak occurs, and shows how these are important to managing emerging and resurgent infectious diseases.

Chen, S. & B. Epureanu. 2016. “Regular biennial cycles in epidemics caused by parametric resonance.” Journal of Theoretical Biology. [online]

  • A novel explanation – parametric resonance – is proposed for the existence of biennial cycles in measles prior to the introduction of vaccines. Stochastic analysis shows that stochasticity can trigger parametric resonance even when the system is in the non-resonance regime of the parameter space.

Han, B.A., A.M. Kramer & J.M. Drake. 2016. “Global Patterns of Zoonotic Disease in Mammals.” Trends in Parasitology. [online]


Han, B.A., J.P. Schmidt, L.A. Alexander, S.E. Bowden, D.T.S. Hayman & J.M. Drake. 2016. “Undiscovered Bat Hosts of Filoviruses.” PLOS Neglected Tropical Diseases. [online]


Kramer, A.M., J.T. Pulliam, L.W. Alexander, A.W. Park, P. Rohani, J.M. Drake. 2016. “Spatial spread of the West Africa Ebola epidemic.” Royal Society Open Science. [online]


Vinson, J.E., J.M. Drake, P. Rohani & A.W. Park. 2016. “The potential for sexual transmission to compromise control of Ebola virus outbreaks.” Royal Society Open Science. [online]


Drake, J.M.& A.W. Park. 2016. “Chapter: A Model for Coupled Outbreaks Contained by Behavior Change.” Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases.
G. Chowel & J.M.Hyman, editors. Springer International Publishing. [online]

  • Large epidemics such as the recent Ebola crisis in West Africa occur when local efforts to contain outbreaks fail to overcome the probabilistic onward transmission to new locations. As a result, there may be large differences in total epidemic size from similar initial conditions. This work seeks to determine the extent to which the effects of behavior changes and metapopulation coupling on epidemic size can be characterized. While mathematical models have been developed to study local containment by social distancing, intervention and other behavior changes, their connection to larger-scale transmission is relatively underdeveloped. We make use of the assumption that behavior changesBehavior changes limit local transmission before susceptible depletion to develop a time-varying birth-death processBirth-death process capturing the dynamic decrease of the transmission rateTransmission rate associated with behavior changes. We derive an expression for the mean outbreak size of this model and show that the distribution of outbreak sizes is approximately geometric. This allows a probabilistic extension whereby infected individuals may initiate new outbreaks. From this model we characterize the overall epidemic size as a function of the behavior change rate and the probability that an infected individual starts a new outbreak. We find good agreement between the analytical results and stochastic simulations leading to novel findings including critical learning rates that demarcate large and small epidemic sizes.

Evans M.V., T.A. Dallas , B.A. Han, C.C. Murdock & J.M. Drake. 2017. “Data-driven identification of potential Zika virus vectors.” eLIFE. [online]

  • ABSTRACT: Zika is an emerging virus whose rapid spread is of great public health concern. Knowledge about transmission remains incomplete, especially concerning potential transmission in geographic areas in which it has not yet been introduced. To identify unknown vectors of Zika, we developed a data-driven model linking vector species and the Zika virus via vector-virus trait combinations that confer a propensity toward associations in an ecological network connecting flaviviruses and their mosquito vectors. Our model predicts that thirty-five species may be able to transmit the virus, seven of which are found in the continental United States, including Culex quinquefasciatus and Cx. pipiens. We suggest that empirical studies prioritize these species to confirm predictions of vector competence, enabling the correct identification of populations at risk for transmission within the United States.

Bhattacharyya, S. & M.J. Ferrari. 2017. “Age-specific mixing generates transient outbreak risk following critical-level vaccination.” Epidemiology & Infection. [online]

  • This paper establishes that achieving the theoretical critical vaccination point is insufficient to achieve disease elimination.

Li, S., C. Ma, L. Hao, Q. Su, Z. An, F. Ma, S. Xie, A. Xu, Y. Zhang, Z. Ding, H. Li, L. Cairns, H. Wang, H. Luo, N. Wang, L. Li & M.J. Ferrari. 2017. “Demographic transition and the dynamics of measles in six provinces in China: A modeling study.” PLOS Medicine. [online]

  • This paper shows that herd immunity and sustained transmission are key determinants of measles dynamics in  China over 30 years.

Drake, J.M. & S.I. Hay. 2017. “Monitoring the path to the elimination of infectious diseases.” Tropical Medicine & Infectious Disease. [online]

  • This paper suggests that critical slowing down can be used to document the approach to disease elimination and shows through simulation that critical slowing down can be robustly detected even in the  presence of significant under-reporting.

Brett, T.S., J.M. Drake & P. Rohani. 2017. “Anticipating the emergence of infectious diseases.” Journal of the Royal Society Interface. [online]

  • This paper presents system- and model-independent candidate approaches for anticipating disease emergence prior to large-scale outbreaks using ideas from the theories of dynamical systems and stochastic processes. The paper examines a set of early-warning signals based around the theory of critical slowing down and a likelihood-based approach, and tests the reliability of these two approaches by contrasting theoretical predictions with simulated data.

Miller, P.B., E.B. O’Dea, P. Rohani & J.M. Drake. 2017. “Forecasting infectious disease emergence subject to seasonal forcing.” Theoretical Biology and Medical Modelling. [online]

  • ABSTRACT: Background: Despite high vaccination coverage, many childhood infections pose a growing threat to human populations. Accurate disease forecasting would be of tremendous value to public health. Forecasting disease emergence using early warning signals (EWS) is possible in non-seasonal models of infectious diseases. Here, we assessed whether EWS also anticipate disease emergence in seasonal models. Methods: We simulated the dynamics of an immunizing infectious pathogen approaching the tipping point to disease endemicity. To explore the effect of seasonality on the reliability of early warning statistics, we varied the amplitude of fluctuations around the average transmission. We proposed and analyzed two new early warning signals based on the wavelet spectrum. We measured the reliability of the early warning signals depending on the strength of their trend preceding the tipping point and then calculated the Area Under the Curve (AUC) statistic. Results: Early warning signals were reliable when disease transmission was subject to seasonal forcing. Wavelet-based early warning signals were as reliable as other conventional early warning signals. We found that removing seasonal trends, prior to analysis, did not improve early warning statistics uniformly. Conclusions: Early warning signals anticipate the onset of critical transitions for infectious diseases which are subject to seasonal forcing. Wavelet-based early warning statistics can also be used to forecast infectious disease.

Schmidt, J.P., A.W. Park, A.M. Kramer, B.A. Han, L.W. Alexander & John M. Drake. 2017. “Spatiotemporal Fluctuations and Triggers of Ebola Virus Spillover.” Emerging Infectious Diseases. [online]


Chen, S. & B.I. Epureanu. 2017. “Forecasting bifurcations of multi-degree-of-freedom nonlinear systems with parametric resonance.” Nonlinear Dynamics. [online]


Evans M.V., C.C. Murdock & J.M. Drake. 2018. “Anticipating Emerging Mosquito-borne Flaviviruses in the USA: What Comes after Zika?.” Trends in Parasitology. [online]

  • New mosquito-borne diseases have emerged on multiple occasions over the last several decades, raising fears that there are yet more poorly understood viruses that may emerge in the USA. Here, we provide a data-driven ‘watch list’ of viruses in the Flaviviridae family with high potential to emerge in the USA, identified using statistical techniques, to enable the public health community to better target surveillance. We suggest that public health authorities further incorporate predictive modeling techniques into disease-prevention strategies.

Brett, T.S., E.B. O’Dea, É. Marty, P.B. Miller, A.W. Park, J.M. Drake & P. Rohani. 2018. “Anticipating epidemic transitions with imperfect data.” PLoS Computational Biology.

  • This paper examines the impacts of imperfect data (reporting error and case aggregation) on the performance of early-warning signals in anticipating disease emergence through a simulation study combining a stochastic SIR model and a model of reporting error. The study finds seven of ten commonly used Early Warning Signals perform well for realistic reporting scenarios, and are strong candidates for incorporation in disease emergence monitoring systems.

O’Dea, E.B., A.W. Park, J.M. Drake. 2018. “Estimating the distance to an epidemic threshold.” Journal of the Royal Society Interface.

  • This paper proposes an approach to identifying suitable variables in a multivariate system for estimation of how close the system is to a bifurcation. The approach is applied to the susceptible-infected-recovered model of an infectious disease to provide an example. The time series of case reports and the number infected are show to be suitable variables over a broad range of parameters.

O’Dea, E.B., J.M. Drake. 2018. “Disentangling Reporting and Disease Transmission.” Theoretical Ecology.

  • This paper attempts to distinguish changes in disease transmission from changes in reporting of cases in a time series of case reports. This paper shows that a moving window “normalized second factorial moment” of the number of case reports is insensitive to the reporting probability but sensitive to disease transmission rate, and therefore has potential as real-world early warning signal of an approaching epidemic threshold and critical transition.

Chen, S., E.B. O’Dea, J.M. Drake, & B.I. Epureanu.. 2019. “Eigenvalues of the covariance matrix as early warning signals for critical transitions in ecological systems.” Scientific Reports.

  • In this study, we propose the use of eigenvalues of the covariance matrix of multiple time series as early warning signals of ciritcal transition. These indicators take into account both temporal and spatial aspects of an epidemic. The paper provide the theoretical basis for their use and demonstrates their applicability wih simulated data from several spatial ecology models.

J.M. Drake , T.S. Brett, S. Chen, B.I. Epureanu, M.J. Ferrari, É.Marty, P.B. Miller, E.B. O’Dea, S.M. O’Regan, A.W. Park, P. Rohani. 2019. “The statistics of epidemic transitions.” PLOS Computational Biology.

  • In the paper, we show that when perturbations in a stochastic SIR model are relatively small, the dynamics are approximated by those of a damped harmonic oscillator. The paper lays out the statistical properties of such a system as it undergoes critical slowing down on approach to a critical transition. We demonstrate how two moving window statistics–variance and the autocorrelation function–have potential as early warning signals for anticipating disease emergence or elimination.