An SIS epidemiology game with two subpopulations

There is significant current interest in the application of game theory to problems in epidemiology. Most mathematical analyses of epidemiology games have studied populations where all individuals have the same risks and interests. This paper analyses the rational-expectation equilibria in an epidemiology game with two interacting subpopulations of equal size where decisions change the prevalence and transmission patterns of an infectious disease. The transmission dynamics are described by an SIS model and individuals are only allowed to invest in daily prevention measures like hygiene. The analysis shows that disassortative mixing may lead to multiple Nash equilibria when there are two interacting subpopulations affecting disease prevalence. The dynamic stability of these equilibria is analysed under the assumption that strategies change slowly in the direction of self-interest. When mixing is disassortative, interior Nash equilibria are always unstable. When mixing is positively assortative, there is a unique Nash equilibrium that is globally stable.     

Games of age-dependent prevention of chronic infections by social distancing

Epidemiological games combine epidemic modelling with game theory to assess strategic choices in response to risks from infectious diseases. In most epidemiological games studied thus-far, the strategies of an individual are represented with a single choice parameter. There are many natural situations where strategies can not be represented by a single dimension, including situations where individuals can change their behavior as they age. To better understand how age-dependent variations in behavior can help individuals deal with infection risks, we study an epidemiological game in an SI model with two life-history stages where social distancing behaviors that reduce exposure rates are age-dependent. When considering a special case of the general model, we show that there is a unique Nash equilibrium when the infection pressure is a monotone function of aggregate exposure rates, but non-monotone effects can appear even in our special case. The non-monotone effects sometimes result in three Nash equilibria, two of which have local invasion potential simultaneously. Returning to a general case, we also describe a game with continuous age-structure using partial-differential equations, numerically identify some Nash equilibria, and conjecture about uniqueness.     

Population structure and the spread of disease

A common assumption of many mathematical models for the spread of disease is that there is random mixing among all individuals in the host population. This paper analyzes and develops a model for the spread of disease in a population consisting of several interacting subpopulations. The model considers 2 different types of interactions between individuals: 1) within a subpopulation because of geographic proximity, and 2) of the same or different subpopulations because of attendance at common social functions. A stability analysis performed on the equilibria of the model shows 2 stable states: 1) a population composed solely of susceptible individuals with no disease present, and 2) an interior point where there are susceptible, infective, and recovered individuals present at all times. The analysis shows that the threshold for disease maintenance is more easily exceed in centers that are members of a small local cluster than in randomly mixing centers, but that the spread of the disease throughout the population occurs more rapidly when the initial case attends a randomly mixing center. The conditions under which a disease will become established are dependent upon the transmission rate for the disease, the birth and death rate in each neighborhood, the recovery rate from the disease in each neighborhood, and the movement patterns of the individuals in the population. The study of the spread of disease in a population by means of mathematical models provides a valuable addition to the statistical data analyzed by epidemiologists. This model is relevant any time there is a division of the population into several interacting groups in which the probability of disease spread is a function both of neighborhood contact because of geographic proximity and of social interactions between groups.     

Dynamics of Vaccination Strategies via Projected Dynamical Systems

Previous game theoretical analyses of vaccinating behaviour have underscored the strategic interaction between individuals attempting to maximise 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 theories of variational inequalities (VI) and projected dynamical systems (PDS) to vaccination games. A PDS provides a dynamics that gives the conditions for existence, uniqueness and stability properties of Nash equilibria. In this paper, it is used to analyse the dynamics of vaccinating behaviour in a population consisting of distinct social groups, where each group has different perceptions of vaccine and disease risks. In particular, we study populations with two groups, where the size of one group is strictly larger than the size of the other group (a majority/minority population). We find that a population with a vaccine-inclined majority group and a vaccine-averse minority group exhibits higher average vaccine coverage than the corresponding homogeneous population, when the vaccine is perceived as being risky relative to the disease. Our model also reproduces a feature of real populations: In certain parameter regimes, it is possible to have a majority group adopting high vaccination rates and simultaneously a vaccine-averse minority group adopting low vaccination rates. Moreover, we find that minority groups will tend to exhibit more extreme changes in vaccinating behaviour for a given change in risk perception, in comparison to majority groups. These results emphasise the important role played by social heterogeneity in vaccination behaviour, while also highlighting the valuable role that can be played by PDS and VI in mathematical epidemiology.     

The effects of social structure and sex-biased transmission on macroparasite infection

Mathematical models of disease dynamics tend to assume that individuals within a population mix at random and so transmission is random, and yet, in reality social structure creates heterogeneous contact patterns. We investigated the effect of heterogeneity in host contact patterns on potential macroparasite transmission by first quantifying the level of assortativity in a socially structured wild rodent population (Apodemus flavicollis) with respect to the directly-transmitted macroparasitic helminth, Heligmosomoides polygyrus. We found the population to be disassortatively mixed (i.e. male mice mixing with female mice more often than same sex mixing) at a constant level over time. The macroparasite H. polygyrus has previously been shown to exhibit male-biased transmission so we used a Susceptible-Infected (SI) mathematical model to simulate the effect of increasing strengths of male-biased transmission on the prevalence of the macroparasite using empirically-derived transmission networks. When transmission was equal between the sexes the model predicted macroparasite prevalence to be 73% and infection was male biased (82% of infection in the male mice). With a male-bias in transmission ten times that of the females, the expected macroparasite prevalence was 50% and was equally prevalent in both sexes, results that both most closely resembled empirical dynamics. As such, disassortative mixing alone did not produce macroparasite dynamics analogous to those from empirical observations; a strong male-bias in transmission was also required. We discuss the relevance of our results in the context of network models for transmission dynamics and control.     

Mating strategies in primates: a game theoretical approach to infanticide

Infanticide by newly immigrated or newly dominant males is reported among a variety of taxa, such as birds, rodents, carnivores and primates. Here we present a game theoretical model to explain the presence and prevalence of infanticide in primate groups. We have formulated a three-player game involving two males and one female and show that the strategies of infanticide on the males' part and polyandrous mating on the females' part emerge as Nash equilibria that are stable under certain conditions. Moreover, we have identified all the Nash equilibria of the game and arranged them in a novel hierarchical scheme. Only in the subspace spanned by the males are the Nash equilibria found to be strict, and hence evolutionarily stable. We have therefore proposed a selection mechanism informed by adaptive dynamics to permit the females to transition to, and remain in, optimal equilibria after successive generations. Our model concludes that polyandrous mating by females is an optimal strategy for the females that minimizes infanticide and that infanticide confers advantage to the males only in certain regions of parameter space. We have shown that infanticide occurs during turbulent changes accompanying male immigration into the group. For changes in the dominance hierarchy within the group, we have shown that infanticide occurs only in primate groups where the chance for the killer to sire the next infant is high. These conclusions are confirmed by observations in the wild. This model thus has enabled us to pinpoint the fundamental processes behind the reproductive decisions of the players involved, which was not possible using earlier theoretical studies.     

Nash equilibria for an evolutionary language game

We study an evolutionary language game that describes how signals become associated with meaning. In our context, a language, L, is described by two matrices: the P matrix contains the probabilities that for a speaker certain objects are associated with certain signals, while the Q matrix contains the probabilities that for a listener certain signals are associated with certain objects. We define the payoff in our evolutionary language game as the total amount of information exchanged between two individuals. We give a formal classification of all languages, L(P, Q), describing the conditions for Nash equilibria and evolutionarily stable strategies (ESS). We describe an algorithm for generating all languages that are Nash equilibria. Finally, we show that starting from any random language, there exists an evolutionary trajectory using selection and neutral drift that ends up with a strategy that is a strict Nash equilibrium (or very close to a strict Nash equilibrium). Received: 1 March 2000 / Published online: 3 August 2000     

Nash Equilibria in Multi-Agent Motor Interactions

Social interactions in classic cognitive games like the ultimatum game or the prisoner''s dilemma typically lead to Nash equilibria when multiple competitive decision makers with perfect knowledge select optimal strategies. However, in evolutionary game theory it has been shown that Nash equilibria can also arise as attractors in dynamical systems that can describe, for example, the population dynamics of microorganisms. Similar to such evolutionary dynamics, we find that Nash equilibria arise naturally in motor interactions in which players vie for control and try to minimize effort. When confronted with sensorimotor interaction tasks that correspond to the classical prisoner''s dilemma and the rope-pulling game, two-player motor interactions led predominantly to Nash solutions. In contrast, when a single player took both roles, playing the sensorimotor game bimanually, cooperative solutions were found. Our methodology opens up a new avenue for the study of human motor interactions within a game theoretic framework, suggesting that the coupling of motor systems can lead to game theoretic solutions.     

Imitation dynamics predict vaccinating behaviour

There exists an interplay between vaccine coverage, disease prevalence and the vaccinating behaviour of individuals. Moreover, because of herd immunity, there is also a strategic interaction between individuals when they are deciding whether or not to vaccinate, because the probability that an individual becomes infected depends upon how many other individuals are vaccinated. To understand this potentially complex interplay, a game dynamic model is developed in which individuals adopt strategies according to an imitation dynamic (a learning process), and base vaccination decisions on disease prevalence and perceived risks of vaccines and disease. The model predicts that oscillations in vaccine uptake are more likely in populations where individuals imitate others more readily or where vaccinating behaviour is more sensitive to changes in disease prevalence. Oscillations are also more likely when the perceived risk of vaccines is high. The model reproduces salient features of the time evolution of vaccine uptake and disease prevalence during the whole-cell pertussis vaccine scare in England and Wales during the 1970s. This suggests that using game theoretical models to predict, and even manage, the population dynamics of vaccinating behaviour may be feasible.     

The donation game with roles played between relatives

We consider a social game with two choices, played between two relatives, where roles are assigned to individuals so that the interaction is asymmetric. Behaviour in each of the two roles is determined by a separate genetic locus. Such asymmetric interactions between relatives, in which individuals occupy different behavioural contexts, may occur in nature, for example between adult parents and juvenile offspring. The social game considered is known to be equivalent to a donation game with non-additive payoffs, and has previously been analysed for the single locus case, both for discrete and continuous strategy traits. We present an inclusive fitness analysis of the discrete trait game with roles and recover equilibrium conditions including fixation of selfish or altruistic behaviour under both behavioural contexts, or fixation of selfish behaviour under one context and altruistic behaviour under the other context. These equilibrium solutions assume that the payoff matrices under each behavioural context are identical. The equilibria possible do depend crucially, however, on the deviation from payoff additivity that occurs when both interacting individuals act altruistically.     

The spread and persistence of infectious diseases in structured populations

A basic assumption of many epidemic models is that populations are composed of a homogeneous group of randomly mixing individuals. This is not a realistic assumption. Most actual populations are divided into a number of subpopulations, within which there may be relatively random mixing, but among which there is nonrandom mixing. As a consequence of the structuring of the population, there are several sources of heterogeneity within populations that can affect the course of an infection through the population. Two of these sources of heterogeneity are differences in contact number between subpopulations, and differences in the patterns of contact among subpopulations. A model for the spread of a disease in such a population is described. The model considers two levels of interaction: interactions between individuals within a subpopulation because of geographic proximity, and interactions between individuals of the same or different subpopulations because of attendance at common social functions. Because of this structure, it is possible to analyze with the model both heterogeneity in contact number and variation in the patterns of contact. A stability analysis of the model is presented which shows that there is a unique threshold for disease maintenance. Below the threshold the disease goes extinct, and the equilibrium is globally asymptotically stable. Above the threshold, the extinction equilibrium is unstable, and there is a unique endemic equilibrium. The analysis presents a sufficient condition for disease maintenance, which determines critical subpopulation sizes above which the disease cannot go extinct. The condition is a simple inequality relating the removal rate of infectives to the infection rate of susceptibles. In addition, bounds on the actual threshold and the effect of symmetry in the interaction matrix on the threshold are presented.     

The use of case-parent triads to study joint effects of genotype and exposure

Most noninfectious disease is caused by low-penetrance alleles interacting with other genes and environmental factors. Consider the simple setting where a diallelic autosomal candidate gene and a binary exposure together affect disease susceptibility. Suppose that one has genotyped affected probands and their parents and has determined each proband's exposure status. One proposed method for assessment of etiologic interaction of genotype and exposure, an extension of the transmission/disequilibrium test, tests for differences in transmission of the variant allele from heterozygous parents to exposed versus unexposed probands. We show that this test is not generally valid. An alternative approach compares the conditional genotype distribution of unexposed cases, given parental genotypes, versus that of exposed cases. This approach provides maximum-likelihood estimators for genetic relative-risk parameters and genotype-exposure-interaction parameters, as well as a likelihood-ratio test (LRT) of the no-interaction null hypothesis. We show how to apply this approach, using log-linear models. When a genotype-exposure association arises solely through incomplete mixing of subpopulations that differ in both exposure prevalence and allele frequency, the LRT remains valid. The LRT becomes invalid, however, if offspring genotypes do not follow Mendelian proportions in each parental mating type-for example, because of genotypic differences in survival-or if a genotype-exposure association reflects an influence of genotype on propensity for exposure-for example, through behavioral mechanisms. Because the needed assumptions likely hold in many situations, the likelihood-based approach should be broadly applicable for diseases in which probands commonly have living parents.     

The Worst-Case Weighted Multi-Objective Game with an Application to Supply Chain Competitions

In this paper, we propose a worst-case weighted approach to the multi-objective n-person non-zero sum game model where each player has more than one competing objective. Our “worst-case weighted multi-objective game” model supposes that each player has a set of weights to its objectives and wishes to minimize its maximum weighted sum objectives where the maximization is with respect to the set of weights. This new model gives rise to a new Pareto Nash equilibrium concept, which we call “robust-weighted Nash equilibrium”. We prove that the robust-weighted Nash equilibria are guaranteed to exist even when the weight sets are unbounded. For the worst-case weighted multi-objective game with the weight sets of players all given as polytope, we show that a robust-weighted Nash equilibrium can be obtained by solving a mathematical program with equilibrium constraints (MPEC). For an application, we illustrate the usefulness of the worst-case weighted multi-objective game to a supply chain risk management problem under demand uncertainty. By the comparison with the existed weighted approach, we show that our method is more robust and can be more efficiently used for the real-world applications.     

Nash equilibrium and evolutionary stability in large- and finite-population "playing the field" models

This paper studies the correspondence between Nash equilibrium and evolutionary stability in large- and finite-population "playing the field" models. Whenever the fitness function is sufficiently continuous, any large-population ESS corresponds to a symmetric Nash equilibrium in the game that describes the simultaneous interaction of the individuals in the population, and any strict, symmetric Nash equilibrium in that game corresponds to a large-population ESS. This correspondence continues to hold, approximately, in finite populations; and it holds exactly for strict pure-strategy equilibria in sufficiently large finite populations. By contrast, a sequence of (mixed-strategy) finite-population ESSs can converge, as the population grows, to a limit that is not a large-population ESS, and a large-population ESS need not be the limit of any sequence of finite-population ESSs.     

A game dynamic model for delayer strategies in vaccinating behaviour for pediatric infectious diseases

Several studies have found that some parents delay the age at which their children receive pediatric vaccines due to perception of higher vaccine risk at the recommended age of vaccination. This has been particularly apparently during the Measles-Mumps-Rubella scare in the United Kingdom. Under a voluntary vaccination policy, vaccine coverage in certain age groups is a potentially complex interplay between vaccinating behaviour, disease dynamics, and age-specific risk factors. Here, we construct an age-structured game dynamic model, where individuals decide whether to vaccinate according to imitation dynamics depending on age-dependent disease prevalence and perceived risk of vaccination. Individuals may be timely vaccinators, delayers, or non-vaccinators. The model exhibits multiple equilibria and a broad range of possible dynamics. For certain parameter regimes, the proportion of timely vaccinators and delayers oscillate in an anti-phase fashion in response to oscillations in infection prevalence. Under an exogenous change to the perceived risk of vaccination as might occur during a vaccine scare, the model can also capture an increase in delayer strategists similar in magnitude to that observed during the Measles-Mumps-Rubella vaccine scare in the United Kingdom. Our model also shows that number of delayers steadily increases with increasing severity of the scare, whereas it saturates to specific value with increases in duration of the scare. Finally, by comparing the model dynamics with and without the option of a delayer strategy, we show that adding a third delayer strategy can have a stabilizing effect on model dynamics. In an era where individual choice—rather than accessibility—is becoming an increasingly important determinant of vaccine uptake, more infectious disease models may need to use game theory or related techniques to determine vaccine uptake.     

The effects of population structure on the spread of the HIV infection

A model for the spread of human immunodeficiency virus (HIV) in a population of male homosexuals is presented. The population is divided into five groups on the basis of degree of sexual activity. Within each group, the individuals are classified as 1) susceptible; 2) infective; or 3) removed because of a lack of sexual activity associated with advanced acquired immunodeficiency disease (AIDS). The infective individuals are further subdivided into four stages of infection. Analyses of the model address two questions with regard to the spread of HIV: (1) What is the effect of level of sexual activity on an individual's risk for infection, and (2) What is the effect that assumptions about mixing between groups have on both individual risk and transmission throughout a population? Results from analyses using a number of different parameter estimates show that increased levels of sexual activity increase the likelihood that an individual will become infected. In addition, the initial spread of the disease is markedly affected by variation in the amount of contact among individuals from different subpopulations. The steady-state incidence of the disease is not markedly affected by variation in the contact patterns, but the size of the steady-state population and therefore the proportion of infected individuals in the population does vary significantly with changes in the degree of mixing among subpopulations. These results show clearly the sensitivity of model outcomes to variation in the patterns of contact among individuals and the need for better data on such interactions to aid in understanding and predicting the spread of HIV.     

On Nash Equilibrium and Evolutionarily Stable States That Are Not Characterised by the Folk Theorem

In evolutionary game theory, evolutionarily stable states are characterised by the folk theorem because exact solutions to the replicator equation are difficult to obtain. It is generally assumed that the folk theorem, which is the fundamental theory for non-cooperative games, defines all Nash equilibria in infinitely repeated games. Here, we prove that Nash equilibria that are not characterised by the folk theorem do exist. By adopting specific reactive strategies, a group of players can be better off by coordinating their actions in repeated games. We call it a type-k equilibrium when a group of k players coordinate their actions and they have no incentive to deviate from their strategies simultaneously. The existence and stability of the type-k equilibrium in general games is discussed. This study shows that the sets of Nash equilibria and evolutionarily stable states have greater cardinality than classic game theory has predicted in many repeated games.     

When do host–parasite interactions drive the evolution of non‐random mating?

Interactions with parasites may promote the evolution of disassortative mating in host populations as a mechanism through which genetically diverse offspring can be produced. This possibility has been confirmed through simulation studies and suggested for some empirical systems in which disassortative mating by disease resistance genotype has been documented. The generality of this phenomenon is unclear, however, because existing theory has considered only a subset of possible genetic and mating scenarios. Here we present results from analytical models that consider a broader range of genetic and mating scenarios and allow the evolution of non-random mating in the parasite as well. Our results confirm results of previous simulation studies, demonstrating that coevolutionary interactions with parasites can indeed lead to the evolution of host disassortative mating. However, our results also show that the conditions under which this occurs are significantly more fickle than previously thought, requiring specific forms of infection genetics and modes of non-random mating that do not generate substantial sexual selection. In cases where such conditions are not met, hosts may evolve random or assortative mating. Our analyses also reveal that coevolutionary interactions with hosts cause the evolution of non-random mating in parasites as well. In some cases, particularly those where mating occurs within groups, we find that assortative mating evolves sufficiently to catalyze sympatric speciation in the interacting species.     

Dynamics to Equilibrium in Network Games: Individual Behavior and Global Response

Various social contexts can be depicted as games of strategic interactions on networks, where an individual’s welfare depends on both her and her partners’ actions. Whereas much attention has been devoted to Bayes-Nash equilibria in such games, here we look at strategic interactions from an evolutionary perspective. To this end, we present the results of a numerical simulations program for these games, which allows us to find out whether Nash equilibria are accessible by adaptation of player strategies, and in general to identify the attractors of the evolution. Simulations allow us to go beyond a global characterization of the cooperativeness at equilibrium and probe into individual behavior. We find that when players imitate each other, evolution does not reach Nash equilibria and, worse, leads to very unfavorable states in terms of welfare. On the contrary, when players update their behavior rationally, they self-organize into a rich variety of Nash equilibria, where individual behavior and payoffs are shaped by the nature of the game, the social network’s structure and the players’ position within the network. Our results allow to assess the validity of mean-field approaches we use to describe the dynamics of these games. Interestingly, our dynamically-found equilibria generally do not coincide with (but show qualitatively the same features of) those resulting from theoretical predictions in the context of one-shot games under incomplete information.     

Core recruitment effects in SIS models with constant total populations

We consider a set of SIS models for a heterosexually transmitted disease in which there is recruitment between core and non-core subpopulations as a function of prevalence of the disease. Behavior diverges from the traditional R0 threshold behavior and yields an extra pair of endemic equilibria in one case and a limit cycle in the other. Total at-risk population is constant.