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Abstract We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative and satisfies the equation classically. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closedC1manifold with cylindrical singularities. © 2017 Wiley Periodicals, Inc.
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Necessary and sufficient conditions for exact closures of epidemic equations on configuration model networks
Abstract We prove that it is possible to obtain the exact closure of SIR pairwise epidemic equations on a configuration model network if and only if the degree distribution follows a Poisson, binomial, or negative binomial distribution. The proof relies on establishing the equivalence, for these specific degree distributions, between the closed pairwise model and a dynamical survival analysis (DSA) model that was previously shown to be exact. Specifically, we demonstrate that the DSA model is equivalent to the well-known edge-based Volz model. Using this result, we also provide reductions of the closed pairwise and Volz models to a single equation that involves only susceptibles. This equation has a useful statistical interpretation in terms of times to infection. We provide some numerical examples to illustrate our results.
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- Award ID(s):
- 1853587
- PAR ID:
- 10438024
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Journal of Mathematical Biology
- Volume:
- 87
- Issue:
- 2
- ISSN:
- 0303-6812
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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