Guillermo Abramson

Small world effect in an epidemiological model.

A model for the spread of an infection is analyzed for different population structures. The interactions within the population are described by small world networks, ranging from ordered lattices to random graphs. For the more ordered systems, there is a fluctuating endemic state of low infection. At a finite value of the disorder of the network, we find a transition to self-sustained oscillations in the size of the infected subpopulation.

Traveling waves of infection in the hantavirus epidemics.

Traveling waves are analyzed in a model of the hantavirus infection in deer mice. The existence of two kinds of wave phenomena is predicted. An environmental parameter governs a transition between two regimes of propagation. In one of them the front of infection lags behind at a constant rate. In the other, fronts of susceptible and infected mice travel at the same speed, separated by a constant delay. The dependence of the delay on system parameters is analyzed numerically and through a piecewise linearization.

Spatiotemporal patterns in the Hantavirus infection

We present a model of the infection of Hantavirus in deer mouse, Peromyscus maniculatus, based on biological observations of the system in the North American Southwest. The results of the analysis shed light on relevant observations of the biological system, such as the sporadical disappearance of the infection, and the existence of foci or refugia that perform as reservoirs of the virus when environmental conditions are less than optimal.

Analytic solutions for nonlinear waves in coupled reacting systems

We analyze a system of reacting elements harmonically coupled to nearest neighbors in the continuum limit. An analytic solution is found for traveling waves. The procedure is used to find oscillatory as well as solitary waves. A comparison is made between exact solutions and solutions of the piecewise linearized system, showing how the linearization affects the amplitude and frequency of the solutions.

THE EFFECT OF BIODIVERSITY ON THE HANTAVIRUS EPIZOOTIC

We analyze a mathematical model of the epizootic of Hantavirus in mice populations, including the effect of species that compete with the host. We show that the existence of the second species has an important consequence for the prevalence of the infectious agent in the host. When the two mice species survive in the ecosystem, the competitive pressure of the second species may lead to reduction or complete elimination of the prevalence of infection. The transition between the disappearance of the infection and its presence occurs at a critical value of the competitors population, resembling a second-order phase transition in a statistical system. The results provide a rigorous framework for the study of the impact of biodiversity in the propagation of infectious diseases, and further lends itself to future experimental verification.

Effects of transport memory and nonlinear damping in a generalized Fisher's equation.

Memory effects in transport require, for their incorporation into reaction-diffusion investigations, a generalization of traditional equations. The well-known Fishers equation, which combines diffusion with a logistic nonlinearity, is generalized to include memory effects, and traveling wave solutions of the equation are found. Comparison is made with alternative generalization procedures.

Vector opinion dynamics in a model for social influence

We present numerical simulations of a model of social influence, where the opinion of each agent is represented by a binary vector. Agents adjust their opinions as a result of random encounters, whenever the difference between opinions is below a given threshold. Evolution leads to a steady state, which highly depends on the threshold and a convergence parameter of the model. We analyze the transition between clustered and homogeneous steady states. Results of the cases of complete mixing and small-world networks are compared.

Statistics of extinction and survival in Lotka-Volterra systems

We analyze purely competitive many-species Lotka-Volterra systems with random interaction matrices, focusing the attention on statistical properties of their asymptotic states. Generic features of the evolution are outlined from a semiquantitative analysis of the phase-space structure and extensive numerical simulations are performed to study the statistics of the extinctions. We find that the number of surviving species depends strongly on the statistical properties of the interaction matrix and that the probability of survival is weakly correlated to specific initial conditions. @S1063-651X~98!03204-8#

Diffusion and home range parameters for rodents: Peromyscus maniculatus in New Mexico

We analyze data from a long-term field project in New Mexico, consisting of repeated sessions of mark-recaptures of Peromyscus maniculatus (Rodentia: Muridae), the host and reservoir of Sin Nombre virus (Bunyaviridae: Hantavirus). The displacements of the recaptured animals provide a means to study their movement from a statistical point of view. We extract two parameters from the data with the help of a simple model: the diffusion constant of the rodents, and the size of their home range. The short time behavior shows the motion to be approximately diffusive and the diffusion constant to be 470 � 50 m 2 /day. The long time behavior provides an estimation of the diameter of the rodent home ranges, with an average value of 100 � 25 m. As in previous investigations directed at Zygodontomys brevicauda observations in Panama, we use a box model for home range estimation. We also use a harmonic model in the present investigation to study the sensitivity of the conclusions to the model used and find that both models lead to similar estimates.

The dynamics of opinion in hierarchical organizations

We study the mutual influence of authority and persuasion in the flow of opinions. We describe a simple model with no social mobility, where each agent belongs to a class in the hierarchy and has also a persuasion capability. Agents use the force of its persuasion to propagate their opinions; however a high-rank agent can also use its authority to impose its opinion on other ones. The model is studied analytically within a mean field approximation and by means of numerical simulations. In the case of a three authority level hierarchy the agreement between the two approaches is excellent. We obtain a phase diagram identifying the relative frequency of the prevailing opinions, and find that the stratum where the dominant opinion arises from is strongly dependent on the relative population of each hierarchy level. The time evolution shows that conflicting opinions polarize after a short transient.