62 research outputs found
Dels jeroglífics egipcis a la compartició de secrets
A la secció 1 parlarem d'un jeroglífic inscrit als laterals de la llinda de Sant Andreu de la Barroca, una petita església perduda en un indret recòndit de la Vall del Llémena, a la Garrotxa. A propòsit d'aquest petit enigma ludolingüístic, plantejarem (i finalment resoldrem) un problema de naturalesa combinatòria que el lector interessat pot provar de resoldre com a exercici. A la secció 2 farem un breu repàs de la història del desxiframent d'escriptures de civilitzacions antigues. Aprofitarem l'avinentesa per introduir els conceptes d'entropia i redundància del llenguatge, que ens ajudaran a entendre per què, tard o d'hora, qualsevol escriptura antiga finalment acaba essent desxifrada quan es té una conjectura raonable sobre quina és la llengua que transcriu. Finalment, a la secció 3 definirem el concepte d'esquema de compartició de secrets, un problema de criptografia moderna en el qual un conjunt de persones gestiona petits fragments d'informació compartida, i que es resol aplicant de manera brillant una idea que prové del món de l'àlgebra lineal més elemental
A network epidemic model with preventive rewiring: comparative analysis of the initial phase
This paper is concerned with stochastic SIR and SEIR epidemic models on
random networks in which individuals may rewire away from infected neighbors at
some rate (and reconnect to non-infectious individuals with
probability or else simply drop the edge if ), so-called
preventive rewiring. The models are denoted SIR- and SEIR-, and
we focus attention on the early stages of an outbreak, where we derive
expression for the basic reproduction number and the expected degree of
the infectious nodes using two different approximation approaches. The
first approach approximates the early spread of an epidemic by a branching
process, whereas the second one uses pair approximation. The expressions are
compared with the corresponding empirical means obtained from stochastic
simulations of SIR- and SEIR- epidemics on Poisson and
scale-free networks. Without rewiring of exposed nodes, the two approaches
predict the same epidemic threshold and the same for both types of
epidemics, the latter being very close to the mean degree obtained from
simulated epidemics over Poisson networks. Above the epidemic threshold,
pairwise models overestimate the value of computed from simulations,
which turns out to be very close to the one predicted by the branching process
approximation. When exposed individuals also rewire with (perhaps
unaware of being infected), the two approaches give different epidemic
thresholds, with the branching process approximation being more in agreement
with simulations.Comment: 25 pages, 7 figure
Robustness of behaviourally-induced oscillations in epidemic models under a low rate of imported cases
This paper is concerned with the robustness of the sustained oscillations
predicted by an epidemic ODE model defined on contact networks. The model
incorporates the spread of awareness among individuals and, moreover, a small
inflow of imported cases. These cases prevent stochastic extinctions when we
simulate the epidemics and, hence, they allow to check whether the average
dynamics for the fraction of infected individuals are accurately predicted by
the ODE model. Stochastic simulations confirm the existence of sustained
oscillations for different types of random networks, with a sharp transition
from a non-oscillatory asymptotic regime to a periodic one as the alerting rate
of susceptible individuals increases from very small values. This abrupt
transition to periodic epidemics of high amplitude is quite accurately
predicted by the Hopf-bifurcation curve computed from the ODE model using the
alerting rate and the infection transmission rate for aware individuals as
tuning parameters.Comment: 17 pages, 11 figure
Saddle-node bifurcation of limit cycles in an epidemic model with two levels of awareness
In this paper we study the appearance of bifurcations of limit cycles in an
epidemic model with two types of aware individuals. All the transition rates
are constant except for the alerting decay rate of the most aware individuals
and the rate of creation of the less aware individuals, which depend on the
disease prevalence in a non-linear way. For the ODE model, the numerical
computation of the limit cycles and the study of their stability are made by
means of the Poincar\'e map. Moreover, sufficient conditions for the existence
of an endemic equilibrium are also obtained. These conditions involve a rather
natural relationship between the transmissibility of the disease and that of
awareness. Finally, stochastic simulations of the model under a very low rate
of imported cases are used to confirm the scenarios of bistability (endemic
equilibrium and limit cycle) observed in the solutions of the ODE model.Comment: 18 page
Characterization of the tree cycles with minimum positive entropy for any period
Consider, for any integer , the set of all -periodic
tree patterns with positive topological entropy and the set
of all -periodic irreducible tree
patterns. The aim of this paper is to determine the elements of minimum entropy
in the families , and
. Let be the unique real root of
the polynomial in . We explicitly construct an
irreducible -periodic tree pattern whose entropy is
. We prove that this entropy is minimum in .
Since the pattern is irreducible, also
minimizes the entropy in the family . We also prove that the
minimum positive entropy in the set (which
is nonempty only for composite integers ) is ,
where is the least prime factor of .Comment: 39 pages, 21 figure
Erratum to: A Network Epidemic Model with Preventive Rewiring: Comparative Analysis of the Initial Phase
Erratum de l'article: https://doi.org/10.1007/s11538-016-0227-
A multilayer temporal network model for STD spreading accounting for permanent and casual partners
Sexually transmitted diseases (STD) modeling has used contact networks to study the spreading of pathogens. Recent findings have stressed the increasing role of casual partners, often enabled by online dating applications. We study the Susceptible-Infected-Susceptible (SIS) epidemic model -appropriate for STDs- over a two-layer network aimed to account for the effect of casual partners in the spreading of STDs. In this novel model, individuals have a set of steady partnerships (links in layer 1). At certain rates, every individual can switch between active and inactive states and, while active, it establishes casual partnerships with some probability with active neighbors in layer 2 (whose links can be thought as potential casual partnerships). Individuals that are not engaged in casual partnerships are classified as inactive, and the transitions between active and inactive states are independent of their infectious state. We use mean-field equations as well as stochastic simulations to derive the epidemic threshold, which decreases substantially with the addition of the second layer. Interestingly, for a given expected number of casual partnerships, which depends on the probabilities of being active, this threshold turns out to depend on the duration of casual partnerships: the longer they are, the lower the threshold
On the minimum positive entropy for cycles on trees
Consider, for any n ∈ N, the set Posn of all n-periodic tree patterns with positive topological entropy and the set Irrn ( Posn of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families Posn and Irrn. Let λn be the unique real root of the polynomial xn − 2x − 1 in (1, +∞). We explicitly construct an irreducible n-periodic tree pattern Qn whose entropy is log(λn). For n = mk, where m is a prime, we prove that this entropy is minimum in the set Posn. Since the pattern Qn is irreducible, Qn also minimizes the entropy in the family Irrn
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