62 research outputs found
The SIR epidemic model from a PDE point of view
We present a derivation of the classical SIR model through a mean-field
approximation from a discrete version of SIR. We then obtain a hyperbolic
forward Kolmogorov equation, and show that its projected characteristics
recover the standard SIR model. Moreover, we show that the long time limit of
the evolution will be a Dirac measure. The exact position will depend on the
well-know parameter, and it will be supported on the corresponding stable
SIR equilibrium
From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics
We study the large population limit of the Moran process, assuming
weak-selection, and for different scalings. Depending on the particular choice
of scalings, we obtain a continuous model that may highlight the genetic-drift
(neutral evolution) or natural selection; for one precise scaling, both effects
are present. For the scalings that take the genetic-drift into account, the
continuous model is given by a singular diffusion equation, together with two
conservation laws that are already present at the discrete level. For scalings
that take into account only natural selection, we obtain a hyperbolic singular
equation that embeds the Replicator Dynamics and satisfies only one
conservation law. The derivation is made in two steps: a formal one, where the
candidate limit model is obtained, and a rigorous one, where convergence of the
probability density is proved. Additional results on the fixation probabilities
are also presented.Comment: 18 pages, 3 figure
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