Dynamic equilibria of group vaccination strategies in a heterogeneous population

Abstract

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.

Evolution Solutions of Equilibrium Problems: A Computational Approach

“…There is a wide array of TDEPs (in the sense defined in this section) whose constraint sets amount to a subset of L p .OE0; T; R q / of the form (2) (models of time-dependent traffic network problems, spatial equilibrium problems, financial equilibrium problems, as well as oligopolies and dynamic games: [4,10,11,16,21] and the references therein).…”

“…Most papers in the literature to date are concerned with the equilibrium states of these problems, where equilibrium is defined depending on the context of the problem: Wardrop [5,15,16,27], Nash-Cournot [10,11,21,27], market [4,20,27], physical/mathematical equilibrium [1,2,23,25] and many others. The classes of equilibrium problems have been shown to be equivalent to different types of variational inequality problems both finite-and infinite-dimensional, as is the case in all cited papers above.…”

“…For instance, in a traffic network problem, the flows on routes will be timedependent, and thus the demands on those routes can be thought of varying with time. In a game setting (see [10,11]), strategies, as well as players' utilities or payoffs can be time-dependent. Infinite-dimensional variational inequalities provide a mathematical framework for studying equilibrium states of TDEP which are described as a succession of equilibrium states over a time interval of interest, generically denoted by t 2 OE0; T.…”

“…[8][9][10][11]14]. Consequently, any non-stationary solution of the projected equation represents a time-dependent, evolving state of the underlying problem.…”

Simulation models for socioeconomic inequalities in health: A systematic review

“…An artificial space was simulated in the four agent-based models (grid space) and in two other studies (Banach space) [44,75].…”