Abstract
Models of the spread of infectious diseases commonly have to deal with the problem of multiple timescales which naturally occur in the epidemic models. In the most cases, this problem is implicitly avoided with the use of the so-called "constant population size" assumption. However, applicability of this assumption can require a justification (which is typically omitted). In this paper we consider some multiscale phenomena that arise in a reasonably simple SusceptibleInfected-Removed (SIR) model with variable population size. In particular, we discuss examples of the canard cascades and a black swan that arise in this model.
Highlights
Typical mathematical models for the spread of an infectious disease usually implicitly employ an assumption that is so common that in the majority of cases it is used without remarks regarding its justification
From the geometrical theory of singular perturbations viewpoint, a canard may be considered as a result of gluing stable and unstable slow invariant manifolds at one point of the breakdown surface due to the availability of an additional scalar parameter in the differential system
If we take an additional function of a vector variable parameterizing the breakdown surface, we can glue the stable and unstable slow invariant manifolds at all points of the breakdown surface simultaneously
Summary
Typical mathematical models for the spread of an infectious disease usually implicitly employ an assumption that is so common that in the majority of cases it is used without remarks regarding its justification. We deal with such specific objects of the geometric theory of singular perturbations as slow invariant manifold with a change of stability, and in particular canards, canard cascades, and black swans. From the geometrical theory of singular perturbations viewpoint, a canard may be considered as a result of gluing stable and unstable slow invariant manifolds at one point of the breakdown surface due to the availability of an additional scalar parameter in the differential system.
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