8 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
A Markov chain model to investigate the spread of antibiotic-resistant bacteria in hospitals
This paper proposes a Markov chain model to describe the spread of a single bacterial species in a hospital ward where patients may be free of bacteria or may carry bacterial strains that are either sensitive or resistant to antimicrobial agents. The aim is to determine the probability law of the exact reproduction number Rexact,0 which is here defined as the random number of secondary infections generated by those patients who are accommodated in a predetermined bed before a patient who is free of bacteria is accommodated in this bed for the first time. Specifically, we decompose the exact reproduction number Rexact,0 into two contributions allowing us to distinguish between infections due to the sensitive and the resistant bacterial strains. Our methodology is mainly based on structured Markov chains and the use of related matrix-analytic methods.Depto. de EstadÃstica e Investigación OperativaFac. de Ciencias MatemáticasFALSEMinisterio de Ciencia e InnovaciónFundação para a Ciência e a Tecnologia (Portugal)unpu
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
On Huygens' principle for Dirac operators associated to electromagnetic fields
We study the behavior of massless Dirac particles, i.e., solutions of the Dirac equation with m = 0 in the presence of an electromagnetic field. Our main result (Theorem 1) is that for purely real or imaginary fields any Huygens type (in Hadamard's sense) Dirac operators is equivalent to the free Dirac operator, equivalence given by changes of variables and multiplication (right and left) by nonzero functions