Monica-Gabriela Cojocaru

Double-layered dynamics: A unified theory of projected dynamical systems and evolutionary variational inequalities

In this paper we continue the study of the unified dynamics resulting from the theory of projected dynamical systems and evolutionary variational inequalities, initiated by Cojocaru, Daniele, and Nagurney. In the process we make explicit the interdependence between the two timeframes used in this new theory. The theoretical results presented here provide a natural context for studying applied problems in disciplines such as operations research, engineering, in particular, transportation science, as well as in economics and finance.

Dynamic equilibrium formulation of the oligopolistic market problem

The aim of this paper is to generalize the oligopolistic market equilibrium problems when the data depend on time. We present results regarding the existence of solutions to such problems. Moreover, we show that continuity regularity results hold, and we use them in order to numerically solve the dynamic equilibrium problem.

Dynamic equilibria of group vaccination strategies in a heterogeneous population

In this paper we present an evolutionary variational inequality model of vaccination strategies games in a population with a known vaccine coverage profile over a certain time interval. The population is considered to be heterogeneous, namely its individuals are divided into a finite number of distinct population groups, where each group has different perceptions of vaccine and disease risks. Previous game theoretical analyses of vaccinating behaviour have studied the strategic interaction between individuals attempting to maximize their health states, in situations where an individual’s health state depends upon the vaccination decisions of others due to the presence of herd immunity. Here we extend such analyses by applying the theory of evolutionary variational inequalities (EVI) to a (one parameter) family of generalized vaccination games. An EVI is used to provide conditions for existence of solutions (generalized Nash equilibria) for the family of vaccination games, while a projected dynamical system is used to compute approximate solutions of the EVI problem. In particular we study a population model with two groups, where the size of one group is strictly larger than the size of the other group (a majority/minority population). The smaller group is considered much less vaccination inclined than the larger group. Under these hypotheses, considering that the vaccine coverage of the entire population is measured during a vaccine scare period, we find that our model reproduces a feature of real populations: the vaccine averse minority will react immediately to a vaccine scare by dropping their strategy to a nonvaccinator one; the vaccine inclined majority does not follow a nonvaccinator strategy during the scare, although vaccination in this group decreases as well. Moreover we find that there is a delay in the majority’s reaction to the scare. This is the first time EVI problems are used in the context of mathematical epidemiology. The results presented emphasize the important role played by social heterogeneity in vaccination behaviour, while also highlighting the valuable role that can be played by EVI in this area of research.

Monotonicity and existence of periodic orbits for projected dynamical systems on Hilbert spaces

We present here results about the existence of periodic orbits for projected dynamical systems (PDS) under Minty-Browder monotonicity conditions. The results are formulated in the general context of a Hilbert space of arbitrary (finite or infinite) dimension. The existence of periodic orbits for such PDS is deduced by means of nonlinear analysis, using a fixed point approach. It is also shown how occurrence of periodic orbits is intimately related to that of critical points (equilibria) of a PDS in certain cases.

Dynamic vaccination games and variational inequalities on time-dependent sets

This paper presents a model of a dynamic vaccination game in a population consisting of a collection of groups, each of which holds distinct perceptions of vaccinating versus non-vaccinating risks. Vaccination is regarded here as a game due to the fact that the payoff to each population group depends on the so-called perceived probability of getting infected given a certain level of the vaccine coverage in the population, a level that is generally obtained by the vaccinating decisions of other members of a population. The novelty of this model resides in the fact that it describes a repeated vaccination game (over a finite time horizon) of population groups whose sizes vary with time. In particular, the dynamic game is proven to have solutions using a parametric variational inequality approach often employed in optimization and network equilibrium problems. Moreover, the model does not make any assumptions upon the level of the vaccine coverage in the population, but rather computes this level as a final result. This model could then be used to compute possible vaccine coverage scenarios in a population, given information about its heterogeneity with respect to perceived vaccine risks. In support of the model, some theoretical results were advanced (presented in the appendix) to ensure that computation of optimal vaccination strategies can take place; this means, the theory states the existence, uniqueness and regularity (in our case piecewise continuity) of the solution curves representing the evolution of optimal vaccination strategies of each population group.

Double-Layer and Hybrid Dynamics of Equilibrium Problems: Applications to Markets of Environmental Products

We present here an original method of tracking the dynamics of an equilibrium problem using an evolutionary variational inequalities and hybrid dynamical systems approach. We apply our method to describe the time evolution of a differentiated product market model under incentive policies with a finite life span. In particular, we describe trajectories of a dynamic game between two producers of a standard product and of an environmental variant of the standard product. We compute and assess the behavior of both the equilibrium (optimal) strategies, as well as the disequilibrium (no-optimal) ones of each producer involved in the oligopolistic market.

Non-equilibrium Solutions of Dynamic Networks: A Hybrid System Approach

Many dynamic networks can be analyzed through the framework of equilibrium problems. While traditionally, the study of equilibrium problems is solely concerned with obtaining or approximating equilibrium solutions, the study of equilibrium problems not in equilibrium provides valuable information into dynamic network behavior. One approach to study such non-equilibrium solutions stems from a connection between equilibrium problems and a class of parametrized projected differential equations. However, there is a drawback of this approach: the requirement of observing distributions of demands and costs. To address this problem we develop a hybrid system framework to model non-equilibrium solutions of dynamic networks, which only requires point observations. We demonstrate stability properties of the hybrid system framework and illustrate the novelty of our approach with a dynamic traffic network example.

Evolution Solutions of Equilibrium Problems: A Computational Approach

This paper proposes a computational method to describe evolution solutions of known classes of time-dependent equilibrium problems (such as time-dependent traffic network, market equilibrium or oligopoly problems, and dynamic noncooperative games). Equilibrium solutions for these classes have been studied extensively from both a theoretical (regularity, stability behaviour) and a computational point of view. In this paper we highlight a method to further study the solution set of such problems from a dynamical systems perspective, namely we study their behaviour when they are not in an (market, traffic, financial, etc.) equilibrium state. To this end, we define what is meant by an evolution solution for a time-dependent equilibrium problem and we introduce a computational method for tracking and visualizing evolution solutions using a projected dynamical system defined on a carefully chosen L2-space. We strengthen our results with various examples.