PS 80-73 - Transient analysis of an SIR epidemic model

Friday, August 12, 2011
Exhibit Hall 3, Austin Convention Center
Suzanne M. O'Regan, Odum School of Ecology, University of Georgia, Athens, GA and John M. Drake, Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA

The standard technique for analyzing a Susceptible-Infected-Recovered (SIR) epidemic model is linearization about a steady state and then determining the eigenvalues of the Jacobian matrix. Linearization about an equilibrium provides an accurate description of the local dynamics about a steady state. However, eigenvalues do not measure transient dynamics. A precise description of transient dynamics in epidemic models has not been achieved. Furthermore, it is well known that SIR models undergo a transcritical bifurcation, i.e., the disease-free and endemic steady states meet and exchange stability at a critical threshold determined by the basic reproduction number R0. In this poster, we present methods based on reactivity and the amplification envelope applied to an SIR vaccination model for an endemic disease. These methods allow for the prediction of the maximum initial growth rate of the infected population, or the maximum growth rate of an epidemic in the short-term.


We show that the recovery rate to equilibrium decreases, and in turn, the characteristic return time to equilibrium increases, as the bifurcation point is approached. This means that the system exhibits critical slowing down, which is a clear signal of an impending critical transition in its behavior. In addition, the SIR system becomes more reactive as the fraction of vaccinated cohorts, p, approaches the threshold. The amplification envelope, the maximum amplification that any perturbation can achieve, increases as p approaches the critical value and the time to maximum amplification becomes more delayed. These results describe the type of transient behavior that one may expect from disease systems and may offer an alternative to “resilience” (recovery rate to equilibrium) as a means of detecting critical transitions in data.

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