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

Expansion of net correlations in terms of partial correlations

Graphical models are usually employed to represent statistical relationships between pairs of variables when all the remaining variables are fixed. In this picture, conditionally independent pairs are disconnected. In the real world, however, strict conditional independence is almost impossible to prove. Here we use a weaker version of the concept of graphical models, in which only the linear component of the conditional dependencies is represented. This notion enables us to relate the marginal Pearson correlation coefficient (a measure of linear marginal dependence) with the partial correlations (a measure of linear conditional dependence). Specifically, we use the graphical model to express the marginal Pearson correlation $\rho_{ij}$ between variables $X_i$ and $X_j$ as a sum of the efficacies with which messages propagate along all the paths connecting the variables in the graph. The expansion is convergent, and provides a mechanistic interpretation of how global correlations arise from local interactions. Moreover, by weighing the relevance of each path and of each intermediate node, an intuitive way to imagine interventions is enabled, revealing for example what happens when a given edge is pruned, or the weight of an edge is modified. The expansion is also useful to construct minimal equivalent models, in which latent variables are introduced to replace a larger number of marginalised variables. In addition, the expansion yields an alternative algorithm to calculate marginal Pearson correlations, particularly beneficial when partial correlation matrix inversion is difficult. Finally, for Gaussian variables, the mutual information is also related to message-passing efficacies along paths in the graph.Comment: 21 pages, 7 figure

La Matemática de los piojos

La epidemiología matemática, que se ocupa de estudiar modelos de propagación de enfermedades infecciosas, es ya una disciplina bien establecida que es frecuentemente utilizada para evaluar el posible impacto de políticas de salud pública. Si bien las publicaciones sobre epidemiología matemática de una gran cantidad de enfermedades han crecido muchísimo en los últimos años, una búsqueda en la literatura de modelos de pediculosis arroja sólo dos resultados. Uno de ellos está en una revista rusa (¡y no hay traducción!) y el otro modela la propagación 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 infestación, y por lo tanto debe abordarse de forma diferente a las enfermedades habituales, en las que sólo 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 cómo evolucionan las poblaciones de piojos. Y para esto es esencial utilizar información lo más detallada posible sobre la biología del piojo. En lo que sigue reseñaremos brevemente sus características 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