Endogenous social distancing and its underappreciated impact on the epidemic curve.

Social distancing is an effective strategy to mitigate the impact of infectious diseases. If sick or healthy, or both, predominantly socially distance, the epidemic curve flattens. Contact reductions may occur for different reasons during a pandemic including health-related mobility loss (severity of symptoms), duty of care for a member of a high-risk group, and forced quarantine. Other decisions to reduce contacts are of a more voluntary nature. In particular, sick people reduce contacts consciously to avoid infecting others, and healthy individuals reduce contacts in order to stay healthy. We use game theory to formalize the interaction of voluntary social distancing in a partially infected population. This improves the behavioral micro-foundations of epidemiological models, and predicts differential social distancing rates dependent on health status. The model's key predictions in terms of comparative statics are derived, which concern changes and interactions between social distancing behaviors of sick and healthy. We fit the relevant parameters for endogenous social distancing to an epidemiological model with evidence from influenza waves to provide a benchmark for an epidemic curve with endogenous social distancing. Our results suggest that spreading similar in peak and case numbers to what partial immobilization of the population produces, yet quicker to pass, could occur endogenously. Going forward, eventual social distancing orders and lockdown policies should be benchmarked against more realistic epidemic models that take endogenous social distancing into account, rather than be driven by static, and therefore unrealistic, estimates for social mixing that intrinsically overestimate spreading.

Publisher Correction: Past and future spread of the arbovirus vectors Aedes aegypti and Aedes albopictus.

This Article was mistakenly not made Open Access when originally published; this has now been amended, and information about the Creative Commons Attribution 4.0 International License has been added into the 'Additional information' section.

Publisher Correction: Past and future spread of the arbovirus vectors Aedes aegypti and Aedes albopictus.

In the version of this Article originally published, the affiliation for author Catherine Linard was incorrectly stated as '6Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine, London, UK'. The correct affiliation is '9Spatial Epidemiology Lab (SpELL), Universite Libre de Bruxelles, Brussels, Belgium'. The affiliation for author Hongjie Yu was also incorrectly stated as '11Department of Statistics, Harvard University, Cambridge, MA, USA'. The correct affiliation is '15School of Health, Fudan University, Key Laboratory of Public Health Safety, Ministry of Education, Shanghai, China'. This has now been amended in all versions of the Article.

Past and future spread of the arbovirus vectors Aedes aegypti and Aedes albopictus.

The global population at risk from mosquito-borne diseases-including dengue, yellow fever, chikungunya and Zika-is expanding in concert with changes in the distribution of two key vectors: Aedes aegypti and Aedes albopictus. The distribution of these species is largely driven by both human movement and the presence of suitable climate. Using statistical mapping techniques, we show that human movement patterns explain the spread of both species in Europe and the United States following their introduction. We find that the spread of Ae. aegypti is characterized by long distance importations, while Ae. albopictus has expanded more along the fringes of its distribution. We describe these processes and predict the future distributions of both species in response to accelerating urbanization, connectivity and climate change. Global surveillance and control efforts that aim to mitigate the spread of chikungunya, dengue, yellow fever and Zika viruses must consider the so far unabated spread of these mosquitos. Our maps and predictions offer an opportunity to strategically target surveillance and control programmes and thereby augment efforts to reduce arbovirus burden in human populations globally.

Groups and scores: the decline of cooperation.

Cooperation among unrelated individuals in social-dilemma-type situations is a key topic in social and biological sciences. It has been shown that, without suitable mechanisms, high levels of cooperation/contributions in repeated public goods games are not stable in the long run. Reputation, as a driver of indirect reciprocity, is often proposed as a mechanism that leads to cooperation. A simple and prominent reputation dynamic function through scoring: contributing behaviour increases one's score, non-contributing reduces it. Indeed, many experiments have established that scoring can sustain cooperation in two-player prisoner's dilemmas and donation games. However, these prior studies focused on pairwise interactions, with no experiment studying reputation mechanisms in more general group interactions. In this paper, we focus on groups and scores, proposing and testing several scoring rules that could apply to multi-player prisoners' dilemmas played in groups, which we test in a laboratory experiment. Results are unambiguously negative: we observe a steady decline of cooperation for every tested scoring mechanism. All scoring systems suffer from it in much the same way. We conclude that the positive results obtained by scoring in pairwise interactions do not apply to multi-player prisoner's dilemmas, and that alternative mechanisms are needed.

Assortative Matching with Inequality in Voluntary Contribution Games.

Voluntary contribution games are a classic social dilemma in which the individually dominant strategies result in a poor performance of the population. However, the negative zero-contribution predictions from these types of social dilemma situations give way to more positive (near-)efficient ones when assortativity, instead of random mixing, governs the matching process in the population. Under assortative matching, agents contribute more than what would otherwise be strategically rational in order to be matched with others doing likewise. An open question has been the robustness of such predictions when heterogeneity in budgets amongst individuals is allowed. Here, we show analytically that the consequences of permitting heterogeneity depend crucially on the exact nature of the underlying public-good provision efficacy, but generally are rather devastating. Using computational methods, we quantify the loss resulting from heterogeneity vis-a-vis the homogeneous case as a function of (i) the public-good provision efficacy and (ii) the population inequality.

Adding noise to the institution: an experimental welfare investigation of the contribution-based grouping mechanism.

Real-world institutions dealing with social dilemma situations are based on mechanisms that are rarely implemented without flaw. Usually real-world mechanisms are noisy and imprecise, that is, which we call 'fuzzy'. We therefore conducted a novel type of voluntary contributions experiment where we test a mechanism by varying its fuzziness. We focus on a range of fuzzy mechanisms we call 'meritocratic matching'. These mechanisms generalize the mechanism of 'contribution-based competitive grouping', and their basic function is to group players based on their contribution choices-i.e. high contributors with high contributors, and low contributors with low contributors. Theory predicts the following efficiency-equality tradeoff as a function of the mechanism's inherent fuzziness: high levels of fuzziness should lead to maximal inefficiency, but perfect equality; decreasing fuzziness is predicted to improve efficiency, but at the cost of growing inequality. The main finding of our experimental investigation is that, contrary to tradeoff predictions, less fuzziness increases both efficiency and equality. In fact, these unambiguous welfare gains are partially realized already at levels where the mechanism is too fuzzy for any high-efficiency outcome to even be a Nash equilibrium.

Spread of yellow fever virus outbreak in Angola and the Democratic Republic of the Congo 2015-16: a modelling study.

BACKGROUND: Since late 2015, an epidemic of yellow fever has caused more than 7334 suspected cases in Angola and the Democratic Republic of the Congo, including 393 deaths. We sought to understand the spatial spread of this outbreak to optimise the use of the limited available vaccine stock. METHODS: We jointly analysed datasets describing the epidemic of yellow fever, vector suitability, human demography, and mobility in central Africa to understand and predict the spread of yellow fever virus. We used a standard logistic model to infer the district-specific yellow fever virus infection risk during the course of the epidemic in the region. FINDINGS: The early spread of yellow fever virus was characterised by fast exponential growth (doubling time of 5-7 days) and fast spatial expansion (49 districts reported cases after only 3 months) from Luanda, the capital of Angola. Early invasion was positively correlated with high population density (Pearson's r 0·52, 95% CI 0·34-0·66). The further away locations were from Luanda, the later the date of invasion (Pearson's r 0·60, 95% CI 0·52-0·66). In a Cox model, we noted that districts with higher population densities also had higher risks of sustained transmission (the hazard ratio for cases ceasing was 0·74, 95% CI 0·13-0·92 per log-unit increase in the population size of a district). A model that captured human mobility and vector suitability successfully discriminated districts with high risk of invasion from others with a lower risk (area under the curve 0·94, 95% CI 0·92-0·97). If at the start of the epidemic, sufficient vaccines had been available to target 50 out of 313 districts in the area, our model would have correctly identified 27 (84%) of the 32 districts that were eventually affected. INTERPRETATION: Our findings show the contributions of ecological and demographic factors to the ongoing spread of the yellow fever outbreak and provide estimates of the areas that could be prioritised for vaccination, although other constraints such as vaccine supply and delivery need to be accounted for before such insights can be translated into policy. FUNDING: Wellcome Trust.

Assortativity evolving from social dilemmas.

Assortative mechanisms can overcome tragedies of the commons that otherwise result in dilemma situations. Assortativity criteria include various forms of kin selection, greenbeard genes, and reciprocal behaviors, usually presuming an exogenously fixed matching mechanism. Here, we endogenize the matching process with the aim of investigating how assortativity itself, jointly with cooperation, is driven by evolution. Our main finding is that full-or-null assortativities turn out to be long-run stable in most cases, independent of the relative speeds of both processes. The exact incentive structure of the underlying social dilemma matters crucially. The resulting social loss is evaluated for general classes of dilemma games, thus quantifying to what extent the tragedy of the commons may be endogenously overcome.

Stability of cooperation under image scoring in group interactions.

Image scoring sustains cooperation in the repeated two-player prisoner's dilemma through indirect reciprocity, even though defection is the uniquely dominant selfish behaviour in the one-shot game. Many real-world dilemma situations, however, firstly, take place in groups and, secondly, lack the necessary transparency to inform subjects reliably of others' individual past actions. Instead, there is revelation of information regarding groups, which allows for 'group scoring' but not for image scoring. Here, we study how sensitive the positive results related to image scoring are to information based on group scoring. We combine analytic results and computer simulations to specify the conditions for the emergence of cooperation. We show that under pure group scoring, that is, under the complete absence of image-scoring information, cooperation is unsustainable. Away from this extreme case, however, the necessary degree of image scoring relative to group scoring depends on the population size and is generally very small. We thus conclude that the positive results based on image scoring apply to a much broader range of informational settings that are relevant in the real world than previously assumed.

Payoff-based learning explains the decline in cooperation in public goods games.

Economic games such as the public goods game are increasingly being used to measure social behaviours in humans and non-human primates. The results of such games have been used to argue that people are pro-social, and that humans are uniquely altruistic, willingly sacrificing their own welfare in order to benefit others. However, an alternative explanation for the empirical observations is that individuals are mistaken, but learn, during the game, how to improve their personal payoff. We test between these competing hypotheses, by comparing the explanatory power of different behavioural rules, in public goods games, where individuals are given different amounts of information. We find: (i) that individual behaviour is best explained by a learning rule that is trying to maximize personal income; (ii) that conditional cooperation disappears when the consequences of cooperation are made clearer; and (iii) that social preferences, if they exist, are more anti-social than pro-social.

Directional learning and the provisioning of public goods.

We consider an environment where players are involved in a public goods game and must decide repeatedly whether to make an individual contribution or not. However, players lack strategically relevant information about the game and about the other players in the population. The resulting behavior of players is completely uncoupled from such information, and the individual strategy adjustment dynamics are driven only by reinforcement feedbacks from each player's own past. We show that the resulting "directional learning" is sufficient to explain cooperative deviations away from the Nash equilibrium. We introduce the concept of k-strong equilibria, which nest both the Nash equilibrium and the Aumann-strong equilibrium as two special cases, and we show that, together with the parameters of the learning model, the maximal k-strength of equilibrium determines the stationary distribution. The provisioning of public goods can be secured even under adverse conditions, as long as players are sufficiently responsive to the changes in their own payoffs and adjust their actions accordingly. Substantial levels of public cooperation can thus be explained without arguments involving selflessness or social preferences, solely on the basis of uncoordinated directional (mis)learning.