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We develop a new methodology for the efficient computation of epidemic final size distributions for a broad class of Markovian models. We exploit a particular representation of the stochastic epidemic process to derive a method which is both computat ionally efficient and numerically stable. The algorithms we present are also physically transparent and so allow us to extend this method from the basic SIR model to a model with a phase-type infectious period and another with waning immunity. The underlying theory is applicable to many Markovian models where we wish to efficiently calculate hitting probabilities.
28 - Kevin J. Black 2013
Pharmacological challenge imaging has mapped, but rarely quantified, the sensitivity of a biological system to a given drug. We describe a novel method called rapid quantitative pharmacodynamic imaging. This method combines pharmacokinetic-pharmacody namic modeling, repeated small doses of a challenge drug over a short time scale, and functional imaging to rapidly provide quantitative estimates of drug sensitivity including EC50 (the concentration of drug that produces half the maximum possible effect). We first test the method with simulated data, assuming a typical sigmoidal dose-response curve and assuming imperfect imaging that includes artifactual baseline signal drift and random error. With these few assumptions, rapid quantitative pharmacodynamic imaging reliably estimates EC50 from the simulated data, except when noise overwhelms the drug effect or when the effect occurs only at high doses. In preliminary fMRI studies of primate brain using a dopamine agonist, the observed noise level is modest compared with observed drug effects, and a quantitative EC50 can be obtained from some regional time-signal curves. Taken together, these results suggest that research and clinical applications for rapid quantitative pharmacodynamic imaging are realistic.
We calculate both the exponential and pre-factor contributions in a WKB approximation of the master equation for a stochastic SIR model with highly oscillatory dynamics. Fixing the basic parameters of the model we investigate how the outbreak distrib ution changes with the population size. We show that this distribution rapidly becomes highly non-Gaussian, acquiring large tails indicating the presence of rare, but large outbreaks, as the population is made smaller. The analytic results are found to be in excellent agreement with simulations until the systems become so small that the dynamics are dominated by fade-out of the disease.
We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period-doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics.
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