Luis F. Lafuerza

Simulating non-Markovian stochastic processes.

We present a simple and general framework to simulate statistically correct realizations of a system of non-Markovian discrete stochastic processes. We give the exact analytical solution and a practical and efficient algorithm like the Gillespie algorithm for Markovian processes, with the difference being that now the occurrence rates of the events depend on the time elapsed since the event last took place. We use our non-Markovian generalized Gillespie stochastic simulation methodology to investigate the effects of nonexponential interevent time distributions in the susceptible-infected-susceptible model of epidemic spreading. Strikingly, our results unveil the drastic effects that very subtle differences in the modeling of non-Markovian processes have on the global behavior of complex systems, with important implications for their understanding and prediction. We also assess our generalized Gillespie algorithm on a system of biochemical reactions with time delays. As compared to other existing methods, we find that the generalized Gillespie algorithm is the most general because it can be implemented very easily in cases (such as for delays coupled to the evolution of the system) in which other algorithms do not work or need adapted versions that are less efficient in computational terms.

Nonuniversal Results Induced by Diversity Distribution in Coupled Excitable Systems

We consider a system of globally coupled active rotators near the excitable regime. The system displays a transition to a state of collective firing induced by disorder. We show that this transition is found generically for any diversity distribution with well-defined moments. Singularly, for the Lorentzian distribution (widely used in Kuramoto-like systems) the transition is not present. This warns about the use of Lorentzian distributions to understand the generic properties of coupled oscillators.

On the effect of heterogeneity in stochastic interacting-particle systems.

We study stochastic particle systems made up of heterogeneous units. We introduce a general framework suitable to analytically study this kind of systems and apply it to two particular models of interest in economy and epidemiology. We show that particle heterogeneity can enhance or decrease the size of the collective fluctuations depending on the system, and that it is possible to infer the degree and the form of the heterogeneity distribution in the system by measuring only global variables and their fluctuations. Our work shows that, in some cases, heterogeneity among the units composing a system can be fully taken into account without losing analytical tractability.

Residential segregation and cultural dissemination: An Axelrod-Schelling model

In the Axelrods model of cultural dissemination, we consider the mobility of cultural agents through the introduction of a density of empty sites and the possibility that agents in a dissimilar neighborhood can move to them if their mean cultural similarity with the neighborhood is below some threshold. While for low values of the density of empty sites, the mobility enhances the convergence to a global culture, for high enough values of it, the dynamics can lead to the coexistence of disconnected domains of different cultures. In this regime, the increase in initial cultural diversity paradoxically increases the convergence to a dominant culture. Further increase in diversity leads to the fragmentation of the dominant culture into domains, forever changing in shape and number, as an effect of the never ending eroding activity of cultural minorities.

Exact solution of a stochastic protein dynamics model with delayed degradation.

We study a stochastic model of protein dynamics that explicitly includes delay in the degradation. We rigorously derive the master equation for the processes and solve it exactly. We show that the equations for the mean values obtained differ from others intuitively proposed and that oscillatory behavior is not possible in this system. We discuss the calculation of correlation functions in stochastic systems with delay, stressing the differences with Markovian processes. The exact results allow us to clarify the interplay between stochasticity and delay.

Role of delay in the stochastic creation process

We develop an approximate theoretical method to study discrete stochastic birth and death models that include a delay time. We analyze the effect of the delay in the fluctuations of the system and obtain that it can qualitatively alter them. We also study the effect of distributed delay. We apply the method to a protein-dynamics model that explicitly includes transcription and translation delays. The theoretical model allows us to understand in a general way the interplay between stochasticity and delay.

Staged Models for Interdisciplinary Research

Modellers of complex biological or social systems are often faced with an invidious choice: to use simple models with few mechanisms that can be fully analysed, or to construct complicated models that include all the features which are thought relevant. The former ensures rigour, the latter relevance. We discuss a method that combines these two approaches, beginning with a complex model and then modelling the complicated model with simpler models. The resulting “chain” of models ensures some rigour and relevance. We illustrate this process on a complex model of voting intentions, constructing a reduced model which agrees well with the predictions of the full model. Experiments with variations of the simpler model yield additional insights which are hidden by the complexity of the full model. This approach facilitated collaboration between social scientists and physicists—the complex model was specified based on the social science literature, and the simpler model constrained to agree (in core aspects) with the complicated model.

Stochastic description of delayed systems

We study general stochastic birth and death processes including delay. We develop several approaches for the analytical treatment of these non-Markovian systems, valid, not only for constant delays, but also for stochastic delays with arbitrary probability distributions. The interplay between stochasticity and delay and, in particular, the effects of delay in the fluctuations and time correlations are discussed.

Evolution of surname distribution under gender-equality measures.

We consider a model for the evolution of surname distribution under a gender-equality measure currently being discussed by the Spanish Parliament (whereby children would adopt their mothers and fathers surnames in alphabetical order). We quantify how this would bias the alphabetical distribution of surnames, and analyze its effect on the present distribution of surnames in Spain.

Role of demographic stochasticity in a speciation model with sexual reproduction

Recent theoretical studies have shown that demographic stochasticity can greatly increase the tendency of asexually reproducing phenotypically diverse organisms to spontaneously evolve into localized clusters, suggesting a simple mechanism for sympatric speciation. Here we study the role of demographic stochasticity in a model of competing organisms subject to assortative mating. We find that in models with sexual reproduction, noise can also lead to the formation of phenotypic clusters in parameter ranges where deterministic models would lead to a homogeneous distribution. In some cases, noise can have a sizable effect, rendering the deterministic modeling insufficient to understand the phenotypic distribution.