Inductive Reasoning Games as Influenza Vaccination Models: Mean Field Analysis

We define and analyze an inductive reasoning game of voluntary yearly vaccination in order to establish whether or not a population of individuals acting in their own self-interest would be able to prevent influenza epidemics. We find that epidemics are rarely prevented. We also find that severe epidemics may occur without the introduction of pandemic strains. We further address the situation where market incentives are introduced to help ameliorating epidemics. Surprisingly, we find that vaccinating families exacerbates epidemics. However, a public health program requesting prepayment of vaccinations may significantly ameliorate influenza epidemics.Comment: 20 pages, 7 figure

On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders.

We develop a financial market model with heterogeneous interacting agents: market makers adjust prices with respect to excess demand, chartists believe in the persistence of bull and bear markets and fundamentalists bet on mean reversion. Moreover, speculators trade asymmetrically in over and undervalued markets and while some of them determine the size of their orders via linear trading rules others always trade the same amount of assets. The dynamics of our model is driven by a one-dimensional discontinuous map. Despite the simplicity of our model, analytical, graphical and numerical analysis reveals a surprisingly rich set of interesting dynamical behaviors.Financial markets, heterogeneous agents, technical and fundamental analysis, nonlinear dynamics, discontinuous map, bifurcation analysis.

The dynamics of the NAIRU model with two switching regimes

We consider a model of inflation and unemployment proposed in Ferri et al. (JEBO, 2001), in which the dynamics are described by a discontinuous piecewise linear map, made up of two branches. We shall show that the bounded dynamics may be classified in two cases: we may have either regular dynamics with stable cycles of any period or quasiperiodic trajectories, or only chaotic dynamics (pure chaos in which a unique absolutely continuous invariant ergodic measure exists, and structurally stable),in a rich variety of cyclical chaotic intervals. The main results are the analytical formulation of the border collision bifurcation curves, through which we give a complete picture of the possible outcomes of the model.Phillips curve, Regime switching, NAIRU, Nonlinearities, Discontinuous maps.

Bifurcation Curves in Discontinuous Maps

Several discrete-time dynamic models are ultimately expressed in the form of iterated piecewise linear functions, in one or two-dimensional spaces. In this paper we study a one-dimensional map made up of three linear pieces which are separated by two discontinuity points, motivated by a dynamic model arising in social sciences. Starting from the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modied by the introduction of a second discontinuity point, and we give the analytic expressions of the bifurcation curves of the principal tongues (or tongues of first degree), for the family of maps considered, that depends on five parameters.iterated piecewise linear functions, discrete-time dynamic models, bifurcation curves.

A unified view on bipartite species-reaction and interaction graphs for chemical reaction networks

The Jacobian matrix of a dynamic system and its principal minors play a prominent role in the study of qualitative dynamics and bifurcation analysis. When interpreting the Jacobian as an adjacency matrix of an interaction graph, its principal minors correspond to sets of disjoint cycles in this graph and conditions for various dynamic behaviors can be inferred from its cycle structure. For chemical reaction systems, more fine-grained analyses are possible by studying a bipartite species-reaction graph. Several results on injectivity, multistationarity, and bifurcations of a chemical reaction system have been derived by using various definitions of such bipartite graph. Here, we present a new definition of the species-reaction graph that more directly connects the cycle structure with determinant expansion terms, principal minors, and the coefficients of the characteristic polynomial and encompasses previous graph constructions as special cases. This graph has a direct relation to the interaction graph, and properties of cycles and sub-graphs can be translated in both directions. A simple equivalence relation enables to decompose determinant expansions more directly and allows simpler and more direct proofs of previous results.Comment: 27 pages. submitted to J. Math. Bio

Optimal Management and Differential Games in the Presence of Threshold Effects - The Shallow Lake Model

Abstract: In this article we analyze how the presence of thresholds influences multi agent decision making situations. We introduce a class of discounted autonomous optimal control problems with threshold effects and discuss tools to analyze these problems. Later, using these results we investigate two types of threshold effects; namely, simple and hysteresis switching, in the canonical model of the shallow lake. We solve the optimal management and open loop Nash equilibrium solutions for the shallow lake model with threshold effects. We establish a bifurcation analysis of the optimal vector field. Further, we observe that modeling with threshold effects simplifies this analysis. To be precise, the bifurcation scenarios rely on simple rules (inequalities) which can be verified easily. However, the qualitative behavior of the switching vector field is similar to the smooth case.Optimal control;Differential games;Threshold effects;Discontinuous dynamics;Shallow lake