14 research outputs found
Of lice and math: using models to understand and control populations of head lice
In this paper we use detailed data about the biology of the head louse
(pediculus humanus capitis) to build a model of the evolution of head lice
colonies. Using theory and computer simulations, we show that the model can be
used to assess the impact of the various strategies usually applied to
eradicate head lice, both conscious (treatments) and unconscious (grooming). In
the case of treatments, we study the difference in performance that arises when
they are applied in systematic and non-systematic ways. Using some reasonable
simplifying assumptions (as random mixing of human groups and the same mobility
for all life stages of head lice other than eggs) we model the contagion of
pediculosis using only one additional parameter. It is shown that this
parameter can be tuned to obtain collective infestations whose variables are
compatible with what is given in the literature on real infestations. We
analyze two scenarios: one where group members begin treatment when a similar
number of lice are present in each head, and another where there is one
individual who starts treatment with a much larger threshold ('superspreader').
For both cases we assess the impact of several collective strategies of
treatment.Comment: manuscript of 23 pages and 13 figures, also a supporting file of 13
pages and 13 figure
La Matemática de los piojos
La epidemiologĂa matemática, que se ocupa de estudiar modelos de propagaciĂłn de enfermedades infecciosas, es ya una disciplina bien establecida que es frecuentemente utilizada para evaluar el posible impacto de polĂticas de salud pĂşblica. Si bien las publicaciones sobre epidemiologĂa matemática de una gran cantidad de enfermedades han crecido muchĂsimo en los Ăşltimos años, una bĂşsqueda en la literatura de modelos de pediculosis arroja sĂłlo dos resultados. Uno de ellos está en una revista rusa (¡y no hay traducciĂłn!) y el otro modela la propagaciĂłn de la pediculosis como si fuera una enfermedad clásica: los individuos se dividen simplemente en dos grupos: el de los “susceptibles” y el de los ”infectados”. Pero la pediculosis no es una enfermedad sino una infestaciĂłn, y por lo tanto debe abordarse de forma diferente a las enfermedades habituales, en las que sĂłlo se considera que la persona está enferma o sana. En el caso de la pediculosis importa saber cuántos piojos tiene cada persona (porque de eso dependerá, por ejemplo, cuánto tiempo le lleve «curarse», o a cuántas personas pueda contagiar). Por eso mismo, es importante conocer cĂłmo evolucionan las poblaciones de piojos. Y para esto es esencial utilizar informaciĂłn lo más detallada posible sobre la biologĂa del piojo. En lo que sigue reseñaremos brevemente sus caracterĂsticas más importantes
Properties of dense partially random graphs
We study the properties of random graphs where for each vertex a {\it
neighbourhood} has been previously defined. The probability of an edge joining
two vertices depends on whether the vertices are neighbours or not, as happens
in Small World Graphs (SWGs). But we consider the case where the average degree
of each node is of order of the size of the graph (unlike SWGs, which are
sparse). This allows us to calculate the mean distance and clustering, that are
qualitatively similar (although not in such a dramatic scale range) to the case
of SWGs. We also obtain analytically the distribution of eigenvalues of the
corresponding adjacency matrices. This distribution is discrete for large
eigenvalues and continuous for small eigenvalues. The continuous part of the
distribution follows a semicircle law, whose width is proportional to the
"disorder" of the graph, whereas the discrete part is simply a rescaling of the
spectrum of the substrate. We apply our results to the calculation of the
mixing rate and the synchronizability threshold.Comment: 14 pages. To be published in Physical Review
Intransitivity and coexistence in four species cyclic games
Intransitivity is a property of connected, oriented graphs representing
species interactions that may drive their coexistence even in the presence of
competition, the standard example being the three species Rock-Paper-Scissors
game. We consider here a generalization with four species, the minimum number
of species allowing other interactions beyond the single loop (one predator,
one prey). We show that, contrary to the mean field prediction, on a square
lattice the model presents a transition, as the parameter setting the rate at
which one species invades another changes, from a coexistence to a state in
which one species gets extinct. Such a dependence on the invasion rates shows
that the interaction graph structure alone is not enough to predict the outcome
of such models. In addition, different invasion rates permit to tune the level
of transitiveness, indicating that for the coexistence of all species to
persist, there must be a minimum amount of intransitivity.Comment: Final, published versio