Well-mixed Populations Research Articles

Simulation and Analysis Methods for Stochastic Compartmental Epidemic Models

This article considers simulation and analysis of incidence data using stochastic compartmental models in well-mixed populations. Several simulation approaches are described and compared. Thereafter, we provide an overview of likelihood estimation for stochastic models. We apply one such method to a real-life outbreak data set and compare models assuming different kinds of stochasticity. We also give references for other publications where detailed information on this topic can be found.

The impact of travel and timing in eliminating COVID-19

While the spread of communicable diseases such as coronavirus disease 2019 (COVID-19) is often analyzed assuming a well-mixed population, more realistic models distinguish between transmission within and between geographic regions. A disease can be eliminated if the region-to-region reproductive number—i.e., the average number of other regions to which a single infected region will transmit the disease—is reduced to less than one. Here we show that this region-to-region reproductive number is proportional to the travel rate between regions and exponential in the length of the time-delay before region-level control measures are imposed. If, on average, infected regions (including those that become re-infected in the future) impose social distancing measures shortly after experiencing community transmission, the number of infected regions, and thus the number of regions in which such measures are required, will exponentially decrease over time. Elimination will in this case be a stable fixed point even after the social distancing measures have been lifted from most of the regions.

The chemodynamics of prograde and retrograde Milky Way stars

Context. The accretion history of the Milky Way is still unknown, despite the recent discovery of stellar systems that stand out in terms of their energy-angular momentum space, such as Gaia-Enceladus-Sausage. In particular, it is still unclear how these groups are linked and to what extent they are well-mixed. Aims. We investigate the similarities and differences in the properties between the prograde and retrograde (counter-rotating) stars and set those results in context by using the properties of Gaia-Enceladus-Sausage, Thamnos/Sequoia, and other suggested accreted populations. Methods. We used the stellar metallicities of the major large spectroscopic surveys (APOGEE, Gaia-ESO, GALAH, LAMOST, RAVE, SEGUE) in combination with astrometric and photometric data from Gaia’s second data-release. We investigated the presence of radial and vertical metallicity gradients as well as the possible correlations between the azimuthal velocity, vϕ, and metallicity, [M/H], as qualitative indicators of the presence of mixed populations. Results. We find that a handful of super metal-rich stars exist on retrograde orbits at various distances from the Galactic center and the Galactic plane. We also find that the counter-rotating stars appear to be a well-mixed population, exhibiting radial and vertical metallicity gradients on the order of ∼ − 0.04 dex kpc−1 and −0.06 dex kpc−1, respectively, with little (if any) variation when different regions of the Galaxy are probed. The prograde stars show a vϕ − [M/H] relation that flattens – and, perhaps, even reverses as a function of distance from the plane. Retrograde samples selected to roughly probe Thamnos and Gaia-Enceladus-Sausage appear to be different populations yet they also appear to be quite linked, as they follow the same trend in terms of the eccentricity versus metallicity space.

Coupled differentiation and division of embryonic stem cells inferred from clonal snapshots

The deluge of single-cell data obtained by sequencing, imaging and epigenetic markers has led to an increasingly detailed description of cell state. However, it remains challenging to identify how cells transition between different states, in part because data are typically limited to snapshots in time. A prerequisite for inferring cell state transitions from such snapshots is to distinguish whether transitions are coupled to cell divisions. To address this, we present two minimal branching process models of cell division and differentiation in a well-mixed population. These models describe dynamics where differentiation and division are coupled or uncoupled. For each model, we derive analytic expressions for each subpopulation’s mean and variance and for the likelihood, allowing exact Bayesian parameter inference and model selection in the idealised case of fully observed trajectories of differentiation and division events. In the case of snapshots, we present a sample path algorithm and use this to predict optimal temporal spacing of measurements for experimental design. We then apply this methodology to an in vitro dataset assaying the clonal growth of epiblast stem cells in culture conditions promoting self-renewal or differentiation. Here, the larger number of cell states necessitates approximate Bayesian computation. For both culture conditions, our inference supports the model where cell state transitions are coupled to division. For culture conditions promoting differentiation, our analysis indicates a possible shift in dynamics, with these processes becoming more coupled over time.

Aspiration Can Promote Cooperation in Well-Mixed Populations As in Regular Graphs

Classical studies on aspiration-based dynamics suggest that a dissatisfied individual changes strategy without taking into account the success of others. This promotes defection spreading. The imitation-based dynamics allow individuals to imitate successful strategies without taking into account their own-satisfactions. In this article, we propose to study a dynamic based on aspiration which takes into account imitation of successful strategies for dissatisfied individuals. This helps cooperative members to resist. Individuals compare their success to their desired satisfaction level before making a decision to update their strategies. This mechanism helps individuals with a minimum of self-satisfaction to maintain their strategies. If an individual is dissatisfied, it will learn from others by choosing successful strategies. We derive an exact expression of the fixation probability in well-mixed populations as in structured populations in networks. As a result, we show that selection may favor cooperation more than defection in well-mixed populations as in populations ranged over a regular graph. We show that the best scenario is a graph with small connectivity.

Mixing protocols in the public goods game.

If interaction partners in social dilemma games are not selected randomly from the population but are instead determined by a network of contacts, it has far reaching consequences for the evolutionary dynamics. Selecting partners randomly leads to a well-mixed population, where pattern formation is essentially impossible. This rules out important mechanisms that can facilitate cooperation, most notably network reciprocity. In contrast, if interactions are determined by a lattice or a network, then the population is said to be structured, where cooperators can form compact clusters that protect them from invading defectors. Between these two extremes, however, there is ample middle ground that can be brought about by the consideration of temporal networks, mobility, or other coevolutionary processes. The question that we here seek to answer is, when does mixing on a lattice actually lead to well-mixed conditions? To that effect, we use the public goods game on a square lattice, and we consider nearest-neighbor and random mixing with different frequencies, as well as a mix of both mixing protocols. Not surprisingly, we find that nearest-neighbor mixing requires a higher frequency than random mixing to arrive at the well-mixed limit. The differences between the two mixing protocols are most expressed at intermediate mixing frequencies, whilst at very low and very high mixing frequencies the two almost converge. We also find a near universal exponential growth of the average size of cooperator clusters as their fraction increases from zero to one, regardless of whether this increase is due to increasing the multiplication factor of the public goods, decreasing the frequency of mixing, or gradually shifting the mixing from random to nearest neighbors.

Evolution of cooperation in social dilemmas under the coexistence of aspiration and imitation mechanisms.

Imitation and aspiration update rules are frequently observed in human and animal populations. While the imitation process entails payoff comparisons with surroundings, the aspiration process refers to self-evaluation. This work explores the evolution of cooperation in dyadic games under the coexistence of these two dynamics in an infinitely large well-mixed population. Two situations have been explored: (i) individuals adopt either an imitation or aspiration update rule with a certain probability, and (ii) the entire population is divided into two groups where one group only uses imitative rules and the other obeys aspiration updating alone. Both premises have been modeled by taking an infinite approximation of the finite population. In particular, the second mixing principle follows an additive property: the outcome of the whole population is the weighted average of outcomes from imitators and aspiration-driven individuals. Our work progressively investigates several variants of aspiration dynamics under strong selection, encompassing symmetric, asymmetric, and adaptive aspirations, which then coalesce with imitative dynamics. We also demonstrate which of the update rules performs better, under different social dilemmas, by allowing the evolution of the preference of update rules besides strategies. Aspiration dynamics always outperform imitation dynamics in the prisoner's dilemma, however, in the chicken and stag-hunt games the predominance of either update rule depends on the level of aspirations as well as on the extent of greed and fear present in the system. Finally, we examine the coevolution of strategies, aspirations, and update rules which leads to a binary state of obeying either imitation or aspiration dynamics. In such a circumstance, when aspiration dynamics prevail over imitation dynamics, cooperators and defectors coexist to an equal extent.

How to evaluate one's behavior toward ‘bad’ individuals? Exploring good social norms in promoting cooperation in spatial public goods games

Cooperation in modern society is based on mutual trust and reputation information can well reflect people's social image. Good social norms encourage people to cooperate with high-reputation (or called ‘good’) individuals and severely punish those who defect ‘good’ individuals. However, the existing research lacks a unified criterion of how to evaluate the behavior when interacting with low-reputation (or called ‘bad’) people. Based on public goods games, we focus on exploring second-order social norms that can improve the evolution of cooperation in this study. Parameters to control the reward and punishment intensity through increasing or decreasing one’s reputation when interacting with ‘bad’ individuals are introduced. Simulation results show that this reward and punishment mechanism through reputation can effectively promote the evolution of cooperation in both well-mixed and latticed populations. Under certain high-reputation threshold environments, a well-mixed population structure can even promote cooperation more significantly than a lattice network. However, increasing reward intensity for cooperating with ‘bad’ individuals cannot further improve cooperation, but in a high-reputation threshold environment, increasing punishment intensity for defecting ‘bad’ individuals can further improve cooperation. This research extends the use of statistical physics to study the evolution of cooperation from the perspective of reputation-based dynamics.

Structural Heterogeneity and Evolutionary Dynamics on Complex Networks

The study of evolutionary games on networks has revealed the impact of population structure on evolutionary dynamics. Unlike the case in well-mixed population where defection is favored by natural selection, certain types of networks have shown to favor cooperation. However, most previous research work has been focusing on frequency-based analysis, and emphasized on the update strategy adopted by each player, and thus generally considered the group of players with the same strategy as a whole. While it is powerful in deriving analytic results using this approach, the heterogeneity of players within such groups is effectively overlooked. In this paper, we attempt to emphasize more on the heterogeneity of players that comes from the network structure in evolutionary dynamics. Particularly, the prestige of a player is represented by its centrality, and it is reflected in an adapted payoff function. We provide several viable centrality measures that can be calculated using the adjacency matrix of the network. The relation between different centrality measures of the invader and the fixation of cooperation is analyzed via computational simulations. Results show that in the proposed model, compared to other three centrality measures, invaders with maximum betweenness centrality have significant advantage in terms of the fixation probability of cooperation, in both scale-free and small-world networks.

Heterogeneous Mixed Populations of Best-Responders and Imitators: Equilibrium Convergence and Stability

In anticoordination social contexts such as stock selection, resource allocation, and crowd dispersion, an individual earns more if the opponents adopt her opposite strategy. Based on their experience and available information, individuals may either evaluate all available options and decide on the most profitable one, or simply mimic successful others. These two types of decision-makers are known as best-responders and imitators, respectively. Previous studies have shown that in anticoordination social contexts, a population of best-responders reaches an equilibrium state, where every individual is satisfied with her decision, but a population of imitators is quite likely to never settle and undergo perpetual fluctuations. Most real-world populations, however, consist of both types of individuals, and it remains an open problem whether such mixed-populations eventually reach an equilibrium state. We provide a sharp, yet simple answer to this question: the population almost surely reaches an equilibrium if and only if it admits one. More specifically, we study a well-mixed population of both best-responders and imitators playing anticoordination games with two available strategies, cooperation and defection, and earning according to payoff matrices that can be unique to each player, resulting in a heterogeneous population. The individuals update their strategies asynchronously accordingly to their types: best-responders choose the strategy that maximizes their payoffs against the population and imitators copy the strategy of the individual earning the highest payoff. We find the necessary and sufficient condition for the population dynamics to admit an equilibrium, identify all possible equilibria, investigate their stability and perform convergence analysis.

The evolution of trust and trustworthiness.

Trust and trustworthiness form the basis for continued social and economic interactions, and they are also fundamental for cooperation, fairness, honesty, and indeed for many other forms of prosocial and moral behaviour. However, trust entails risks, and building a trustworthy reputation requires effort. So how did trust and trustworthiness evolve, and under which conditions do they thrive? To find answers, we operationalize trust and trustworthiness using the trust game with the trustor's investment and the trustee's return of the investment as the two key parameters. We study this game on different networks, including the complete network, random and scale-free networks, and in the well-mixed limit. We show that in all but one case, the network structure has little effect on the evolution of trust and trustworthiness. Specifically, for well-mixed populations, lattices, random and scale-free networks, we find that trust never evolves, while trustworthiness evolves with some probability depending on the game parameters and the updating dynamics. Only for the scale-free network with degree non-normalized dynamics, we find parameter values for which trust evolves but trustworthiness does not, as well as values for which both trust and trustworthiness evolve. We conclude with a discussion about mechanisms that could lead to the evolution of trust and outline directions for future work.

Mathematical epidemiological analysis of dynamics of delay attacks on pull-based competitive information diffusion

In recent years, competitive information diffusion plays a key role in specific network systems. For example, in the cryptocurrency systems, e.g., Bitcoin, some special nodes, called miners, compete with each other in diffusion races to acquire the system rewards. If the information for diffusion is large, pull-based dissemination is often used to alleviate the communication overhead, where a node notifies a small inventory message to its neighbor(s) before transferring the information itself. On receiving the inventory message, the neighboring node requests the corresponding information from the node. Since the information transfer may require much time due to temporal network/node trouble, the pull-based dissemination generally involves a timeout mechanism. If a timeout occurs, the node retries the information retrieval. A malicious node can conduct a delay attack on the information propagation by exploiting this timeout mechanism, i.e., temporarily interrupting the information transfer after receiving a request. In this paper, a risk of information diffusion with interruption is considered, under which adversaries colluding with a specific source node simultaneously conduct the delay attacks to slow down the information diffusion from competitive source nodes. With the help of the deterministic nonlinear model for the propagation of infectious diseases in mathematical epidemiology, a scalar standby-interrupted-retrieved-attackable (SIRA) model is first developed, which captures the information diffusion with interruption under a well-mixed population. Furthermore, a network SIRA model is proposed by extending the scalar SIRA model to support the information diffusion with interruption over arbitrary networks. Through numerical evaluations, the risk of information diffusion with interruption is quantitatively revealed from the viewpoint of attack scale, attack rate, recovery rate, and network structure.

Convergence of linear threshold decision-making dynamics in finite heterogeneous populations

Linear threshold models have been studied extensively in structured populations; however, less attention is paid to perception differences among the individuals of the population. To focus on this effect, we exclude structure and consider a well-mixed population of heterogeneous agents, each associated with a threshold in the form of a fixed ratio within zero and one that can be unique to this agent. The agents are initialized with a choice of strategy A or B, and at each time step, one agent becomes active to update; if the ratio of agents playing A is higher (resp. lower) than her threshold, she updates to A (resp. B). We show that for any given initial condition, after a finite number of time steps, the population reaches an equilibrium where no agent’s threshold is violated; however, the equilibrium is not necessarily uniquely determined by the initial condition but depends on the agents’ activation sequence. We find all those possible equilibria that the dynamics may reach from a given initial condition and show that, in contrast to the case of homogeneous populations, heterogeneity in the agents’ thresholds gives rise to several equilibria where both A-playing and B-playing agents coexist. Perception heterogeneity can, hence, preserve decision diversity, even in the absence of population structure. We also investigate the asymptotic stability of the equilibria and show how to calculate the contagion probability for a population with two thresholds.

The cancellation effect at the group level.

Group selection models combine selection pressure at the individual level with selection pressure at the group level. Cooperation can be costly for individuals, but beneficial for the group, and therefore, if individuals are sufficiently much assorted, and cooperators find themselves in groups with disproportionately many other cooperators, cooperation can evolve. The existing literature on group selection generally assumes that competition between groups takes place in a well‐mixed population of groups, where any group competes with any other group equally intensely. Competition between groups however might very well occur locally; groups may compete more intensely with nearby than with far‐away groups. We show that if competition between groups is indeed local, then the evolution of cooperation can be hindered significantly by the fact that groups with many cooperators will mostly compete against neighboring groups that are also highly cooperative, and therefore harder to outcompete. The existing empirical method for determining how conducive a group structured population is to the evolution of cooperation also implicitly assumes global between‐group competition, and therefore gives (possibly very) biased estimates.

Space is the Place: Effects of Continuous Spatial Structure on Analysis of Population Genetic Data.

Real geography is continuous, but standard models in population genetics are based on discrete, well-mixed populations. As a result, many methods of analyzing genetic data assume that samples are a random draw from a well-mixed population, but are applied to clustered samples from populations that are structured clinally over space. Here, we use simulations of populations living in continuous geography to study the impacts of dispersal and sampling strategy on population genetic summary statistics, demographic inference, and genome-wide association studies (GWAS). We find that most common summary statistics have distributions that differ substantially from those seen in well-mixed populations, especially when Wright’s neighborhood size is < 100 and sampling is spatially clustered. “Stepping-stone” models reproduce some of these effects, but discretizing the landscape introduces artifacts that in some cases are exacerbated at higher resolutions. The combination of low dispersal and clustered sampling causes demographic inference from the site frequency spectrum to infer more turbulent demographic histories, but averaged results across multiple simulations revealed surprisingly little systematic bias. We also show that the combination of spatially autocorrelated environments and limited dispersal causes GWAS to identify spurious signals of genetic association with purely environmentally determined phenotypes, and that this bias is only partially corrected by regressing out principal components of ancestry. Last, we discuss the relevance of our simulation results for inference from genetic variation in real organisms.

High thresholds encouraging the evolution of cooperation in threshold public-good games

For a well-mixed population, we consider a threshold public good game where group members only obtain benefits from a public good if a sufficiently large number of them cooperates. We investigate the effect of an increase in the threshold on the level of cooperation that evolves. It is shown that for sufficiently large participation costs, the level of cooperation is higher for low and for high thresholds, than it is for intermediate thresholds – where in the latter case cooperation may not evolve at all. The counterintuitive effect where an increase in the threshold from an intermediate to a high one decreases the probability of cooperation, is related to the so-called common-enemy hypothesis of the evolution of cooperation. We further apply our analysis to assess the relative weight of different game types across the parameter space, and show that game types where either a small, or a large fraction of the population evolves as cooperators, receive more weight compared to game types where an intermediate fraction of cooperators evolves.

Costly Participation and The Evolution of Cooperation in the Repeated Public Goods Game

In real life, individuals often need to pay some costs to build and maintain long-termed relationships as well as interactions among them. However, previous studies of the repeated public goods game have focused almost exclusively on the cost-free participation. Here we introduce costly participation to the repeated public goods game in a conditional manner, and study the evolution of cooperation in both deterministic and stochastic dynamics for well-mixed populations. In the limit of an infinite population size, the deterministic dynamics can lead to either a stable coexistence between cooperators and defectors or even a complete dominance of cooperators over defectors if the initial frequency of cooperators is larger than some invasion barrier. In general, defectors are always able to resist invasion by cooperators. However, in finite populations, we show that natural selection can favor the emergence of cooperation in the stochastic dynamics. In the limit of weak selection and large populations, we derive a critical condition required for a cooperator to replace a population of defectors with a selective advantage by using several approximation techniques. Theoretical analysis of the critical condition reveals that participation cost determines the impacts of the number of game round in the emergence of cooperation. Interestingly, there exists an intermediate value of the participation threshold of cooperators leading to the optimal condition for a cooperator to invade and fixate in a population of defectors if the participation cost is smaller than a critical value. Numerical calculations confirm that the validity of the analytical approximations extends to much wider ranges of the selection strength as well as of the population size.

Lying on networks: The role of structure and topology in promoting honesty.

Lies can have a negating impact on governments, companies, and the society as a whole. Understanding the dynamics of lying is therefore of crucial importance across different fields of research. While lying has been studied before in well-mixed populations, it is a fact that real interactions are rarely well-mixed. Indeed, they are usually structured and thus best described by networks. Here we therefore use the Monte Carlo method to study the evolution of lying in the sender-receiver game in a one-parameter family of networks, systematically covering complete networks, small-world networks, and one-dimensional rings. We show that lies that benefit the sender at a cost to the receiver, the so-called black lies, are less likely to proliferate on networks than they do in well-mixed populations. Honesty is thus more likely to evolve, but only when the benefit for the sender is smaller than the cost for the receiver. Moreover, this effect is particularly strong in small-world networks, but less so in the one-dimensional ring. For lies that favor the receiver at a cost to the sender, the so-called altruistic white lies, we show that honesty is also more likely to evolve than it is in well-mixed populations. But contrary to black lies, this effect is more expressed in the one-dimensional ring, whereas in small-world networks it is present only when the cost to the sender is greater than the benefit for the receiver. Last, for lies that benefit both the sender and the receiver, the so-called Pareto white lies, we show that the network structure actually favors the evolution of lying, but this only occurs when the benefit for the sender is slightly greater than the benefit for the receiver. In this case again the small-world topology acts as an amplifier of the effect, while other network topologies fail to do the same. In addition to these main results we discuss several other findings, which together show clearly that the structure of interactions and the overall topology of the network critically determine the dynamics of lying.

The effect of individual attitude on cooperation in social dilemma

Environmentalists spend something such as, time, energy and money to protect the environment, while others in the same environment pay nothing. In this social dilemma associated with the ecological environment, we consider environmental investment and draw some interesting and meaningful conclusions by abstracting a differential equation model, based on the mean field approximation to discuss the impact of various environmental budgets on the dynamics in estimable social states. Firstly, we consider each individual having dual attributes of attitude and behavior, and classify the well-mixed and infinite populations into four categories, namely, indifferent cooperator, passionate cooperator, indifferent defector and passionate defector. The system may be in one of four situations — only the indifferent defectors are left, two types of defectors are survived, two types of cooperators are remained and four types of individuals coexist. A serious of numerical results show that one of causes for environmental tragedy is the indifferent of the public. Increasing investment in environmental protection projects can result in more cooperators if the society is not too hardhearted. When the investment exceeds a threshold, however, more investment has no obvious effect on promoting cooperation further. We should draw public attention to environmental problems and make a reasonable budget for environmental protection so as to promote green and harmonious development to the greatest extent.

A mean-field vaccination game scheme to analyze the effect of a single vaccination strategy on a two-strain epidemic spreading

We propose a mean-field vaccination game framework that combines two distinct processes: the simultaneous spreading of two strains of an influenza-like disease, and the adoption of vaccination based on evolutionary game theory presuming an infinite and well-mixed population. The vaccine is presumed to be imperfect such that it shows better efficacy against the original (resident) strain rather than the new one (mutant). The vaccination-decision takes place at the beginning of an epidemic season and depends upon the vaccine-effectiveness along with the cost. Additionally, we explore a situation if the original strain continuously converts to a new strain through the process of mutation. With the aid of numerical experiments, we explore the impact of vaccinating behavior on a specific strain prevalence. Our results suggest that the emergence of vaccinators can create the possibility of infection-prevalence of the new strain if the vaccine cannot bestow a considerable level of efficiency against that strain. On the other hand, the resident strain can continue to dominate under large-scale vaccine avoidance. Moreover, in the case of continuous mutation, the vaccine efficacy against the new strain plays a pivotal role to control the disease prevalence. We successfully obtain phase diagrams, displaying the infected fraction with each strain, final epidemic size, vaccination coverage, and average social payoff considering two-different strategy-update rules and provide a comprehensive discussion to get an encompassing idea, justifying how the vaccinating behavior can affect the spread of a disease having two strains.Highlights–We build a mean-field vaccination game scheme to analyze the effect of an imperfect vaccine on a two-strain epidemic spreading taking into account individuals’ vaccination behavior.–En masse vaccine avoidance can enhance the possibility of the original strain prevalence.–Propensity for vaccination can create the possibility of infection by the new strain if the vaccine is unable to provide a considerable level of efficiency against that strain.