Abstract
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.
Highlights
Infectious diseases are among the most visible and costly threats to individual and public health
The epidemic models we investigate are generalizations of the closed population SIS and susceptible–infectious– recovered (SIR) models that allow for immigration from external sources, of which the more familiar models without immigration that are characterized by a transcritical bifurcation are special cases
Demographic stochasticity is expected to excite the transient oscillations of the SIR model (Bauch and Earn 2003), and we will see this in detail later when we examine the power spectrum of the fluctuations
Summary
Infectious diseases are among the most visible and costly threats to individual and public health. Millions of persons die every year from treatable ancient diseases, such as malaria (Gething et al 2011; WHO 2012), tuberculosis (Dye et al 2008), and measles (Orenstein and Hinman 2012; Simons et al 2012). Sometimes, elimination of these diseases through vaccination, prophylaxis, and/or vector control is possible, but sustaining elimination campaigns is difficult as pathogens approach the point of elimination (Cohen et al 2012). The benefits of accurately forecasting disease emergence would be tremendous: in the case of a low-incidence SARSlike pathogen, the savings could be tens of billions of $US (Rossi and Walker 2005; Smith et al 2009); an illness resembling the 1918 influenza virus might take millions of lives and impose costs of the same order as a year’s gross domestic product (Osterholm 2005)
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