Monica Cojocaru

The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

Projected Dynamical Systems, Evolutionary Variational Inequalities, Applications, and a Computational Procedure

In this paper, we establish the equivalence between the solutions to an evolutionary variational inequality and the critical points of a projected dynamical system in infinite–dimensional spaces. We then present an algorithm, with convergence results, for the computation of solutions to evolutionary variational inequalities based on a discretization method and with the aid of projected dynamical systems theory. A numerical traffic network example is given for illustrative purposes.

Coevolution of risk perception, sexual behaviour, and HIV transmission in an agent-based model.

Risk perception shapes individual behaviour, and is in turn shaped by the consequences of that behaviour. Here we explore this dynamics in the context of human immunodeficiency virus (HIV) spread. We construct a simplified agent-based model based on a partner selection game, where individuals are paired with others in the population, and through a decision tree, agree on unprotected sex, protected sex, or no sex. An individuals choice is conditioned on their HIV status, their perceived population-level HIV prevalence, and the preferences expressed by the individual with whom they are paired. HIV is transmitted during unprotected sex with a certain probability. As expected, in model simulations, the perceived population-level HIV prevalence climbs along with actual HIV prevalence. During this time, HIV- individuals increasingly switch from unprotected sex to protected sex, HIV+ individuals continue practicing unprotected sex whenever possible, and unprotected sex between HIV+ and HIV- individuals eventually becomes rare. We also find that the perceived population-level HIV prevalence diverges according to HIV status: HIV- individuals develop a higher perceived HIV prevalence than HIV+ individuals, although this result is sensitive to how much information is derived from global versus local sources. This research illustrates a potential mechanism by which distinct groups, as defined by their sexual behaviour, HIV status, and risk perceptions, can emerge through coevolution of HIV transmission and risk perception dynamics.

Sexual behavior, risk perception, and HIV transmission can respond to HIV antiviral drugs and vaccines through multiple pathways

There has been growing use of highly active antiretroviral treatment (HAART) for HIV and significant progress in developing prophylactic HIV vaccines. The simplest theories of counterproductive behavioral responses to such interventions tend to focus on single feedback mechanisms: for instance, HAART optimism makes infection less scary and thus promotes risky sexual behavior. Here, we develop an agent based, age-structured model of HIV transmission, risk perception, and partner selection in a core group to explore behavioral responses to interventions. We find that interventions can activate not one, but several feedback mechanisms that could potentially influence decision-making and HIV prevalence. In the model, HAART increases the attractiveness of unprotected sex, but it also increases perceived risk of infection and, on longer timescales, causes demographic impacts that partially counteract HAART optimism. Both HAART and vaccination usually lead to lower rates of unprotected sex on the whole, but intervention effectiveness depends strongly on whether individuals over- or under-estimate intervention coverage. Age-specific effects cause sexual behavior and HIV prevalence to change in opposite ways in old and young age groups. For complex infections like HIV—where interventions influence transmission, demography, sexual behavior and risk perception—we conclude that evaluations of behavioral responses should consider multiple feedback mechanisms.

Nonpivot and Implicit Projected Dynamical Systems on Hilbert Spaces

This paper presents a generalization of the concept and uses of projected dynamical systems to the case of nonpivot Hilbert spaces. These are Hilbert spaces in which the topological dual space is not identified with the base space. The generalization consists of showing the existence of such systems and their relation to variational problems, such as variational inequalities. In the case of usual Hilbert spaces these systems have been extensively studied, and, as in previous works, this new generalization has been motivated by applications, as shown below.

Shopkeeper strategies in the iterated prisoner's dilemma

Many studies have evolved agents to play the iterated prisoners dilemma. This study models a different situation, called the Shopkeeper model of interaction, in which a state conditioned agent interacts with a series of other agents without resetting its internal state. This is intended to simulate the situation in which a shopkeeper interacts with a series of customers. In a majority of other studies agents either reset their internal state information before each new encounter or have relatively little internal state information. This means they cannot model situations such as being the customer that meets the shopkeeper after an obnoxious customer. We train shopkeeper prisoners dilemma agents against a variety of distributions of possible customers. The shopkeepers specialize their behavior to their customers but sometimes fail to discover maximally exploitative behaviors. The evolved shopkeeper agents are subject to fingerprint analysis and are shown to differ substantially from agents evolved with a round-robin fitness functions. Evaluation of the behavior of the shopkeeper agents with customers they did not encounter during evolution provides additional evidence that shopkeepers specialized to the customers, but did so incompletely for the more complex sets of customers.

Time-Dependent Casual Encounters Games and HIV Spread

In Tully et al. (Math Biosci Eng AIMS, 2015, to submitted) the authors model and investigate casual sexual encounters between two members of a population with two possible HIV states: positive and negative, using a Nash game framework in which players try to maximize their expected payoff resulting out of a possible encounter. Each player knows their own HIV status, but do not know the HIV status of a potential partner. They do however have a personal assessment of the risk that the potential partner may be HIV positive. Last but not least, each player has a ranked list of preferences of potential types of sexual outcomes: unprotected, protected, or no sexual outcome. In Tully et al. (Math Biosci Eng AIMS, 2015, to submitted), the game model is studied via 1- and 2-dimensional sensitivity analyses on parameters such as the utility values of unprotected sex of an HIV negative individual with an HIV positive, and values of personal risk (of encountering an HIV positive partner) perception. In this work, we introduce time as a variable which affects players’ risk perceptions, and thus their strategies. Given that HIV transmission happens when an HIV positive player has a non-zero probability (strategy) of having unprotected sex with a HIV negative player, we are also able to keep track of the time evolution of the overall fraction of HIV positive individuals in the population, as reflected as an outcome of repeated casual encounters. We model a continuous time dynamic game (as in Cojocaru et al. (J Optim Theory Appl 127(3):549–563, 2005)) where we compute the stable strategies of each player based on a dynamical system defined on a set of functions. We observe that with change in choices the HIV prevalence in the population increases.

Absenteeism Impact on Local Economy During a Pandemic via Hybrid SIR Dynamics

In this paper we study the cost of absenteeism and presenteeism (going to work while sick) during a pandemic in a local economy with several geographically distinct locations, and with work force populations consisting of individuals who live and work in the same city, and individuals who live and work in different locations (daily commuters).We run simulations to study the effects of the fear factor and of the severity of disease on the number of missed work days in the region, which we translate into loss of productivity costs. We find that higher values of the fear parameter lead to high absenteeism and lower infection levels. However, we also show that for severe pandemics (such as the number of secondary infections is higher) there are scenarios where there exists a unique value of the fear parameter which leads to minimum economic costs for the regional economy. This indicates that “staying at home” policies during an epidemic could be implemented for the work force, without reaching a state of emergency.

Adoption of New Products with Global and Local Social Influence in a 2D Characteristics Space

We present here an agent-based model (ABM) of adoption of new products including: dynamic consumer preferences and product demands, a 2D characteristics space where products are placed, global and local (nearest neighbours) social influence. The ABM model is built from a continuous time model of the market (Cojocaru et al., Environ Model Softw, 2013), driven by agents’ heterogeneity and their local connections. We find that consumer populations where a large fraction of population is sensitive to product popularity displays higher adoption levels of a new product, especially when local social connections are taken into account.

Replicator Dynamics of Axelrod’s Norms Games

We create pure strategy versions of Robert Axelrod’s well-known norms and metanorms games. Our findings show that the only evolutionarily stable strategy (ESS) in the norms game is one in which a player defects and is lenient. This result is derived using classic game theoretical tools, and we conclude that Axelrod’s original statement that the norms game always collapses holds. The metanorms game, however, has two evolutionarily stable strategies. The first is a repeat from the norms game, while the other is one in which a player follows the norm and punishes those who are lenient and those who defect.