Population-Level Immunity for Transient Suppression of COVID-19 Waves in Japan from April 2021 to September 2022.

Multiple COVID-19 waves have been observed worldwide, with varying numbers of positive cases. Population-level immunity can partly explain a transient suppression of epidemic waves, including immunity acquired after vaccination strategies. In this study, we aimed to estimate population-level immunity in 47 Japanese prefectures during the three waves from April 2021 to September 2022. For each wave, characterized by the predominant variants, namely, Delta, Omicron, and BA.5, the estimated rates of population-level immunity in the 10-64-years age group, wherein the most positive cases were observed, were 20%, 35%, and 45%, respectively. The number of infected cases in the BA.5 wave was inversely associated with the vaccination rates for the second and third injections. We employed machine learning to replicate positive cases in three Japanese prefectures to validate the reliability of our model for population-level immunity. Using interpolation based on machine learning, we estimated the impact of behavioral factors and vaccination on the fifth wave of new positive cases that occurred during the Tokyo 2020 Olympic Games. Our computational results highlighted the critical role of population-level immunity, such as vaccination, in infection suppression. These findings underscore the importance of estimating and monitoring population-level immunity to predict the number of infected cases in future waves. Such estimations that combine numerical derivation and machine learning are of utmost significance for effective management of medical resources, including the vaccination strategy.

Replicator dynamics in public goods games with reward funds.

Which punishment or rewards are most effective at maintaining cooperation in public goods interactions and deterring defectors who are willing to freeload on others' contribution? The sanction system is itself a public good and can cause problematic "second-order free riders" who do not contribute to the provisions of the sanctions and thus may subvert the cooperation supported by sanctioning. Recent studies have shown that public goods games with punishment can lead to a coercion-based regime if participation in the game is optional. Here, we reveal that even with compulsory participation, rewards can maintain cooperation within an infinitely large population. We consider three strategies for players in a standard public goods game: to be a cooperator or a defector in a standard public goods game, or to be a rewarder who contributes to the public good and to a fund that rewards players who contribute during the game. Cooperators do not contribute to the reward fund and are therefore classified as second-order free riders. The replicator dynamics for the three strategies exhibit a rock-scissors-paper cycle, and can be analyzed fully, despite the fact that the expected payoffs are nonlinear. The model does not require repeated interaction, spatial structure, group selection, or reputation. We also discuss a simple method for second-order sanctions, which can lead to a globally stable state where 100% of the population are rewarders.

Probabilistic participation in public goods games.

Voluntary participation in public goods games (PGGs) has turned out to be a simple but effective mechanism for promoting cooperation under full anonymity. Voluntary participation allows individuals to adopt a risk-aversion strategy, termed loner. A loner refuses to participate in unpromising public enterprises and instead relies on a small but fixed pay-off. This system leads to a cyclic dominance of three pure strategies, cooperators, defectors and loners, but at the same time, there remain two considerable restrictions: the addition of loners cannot stabilize the dynamics and the time average pay-off for each strategy remains equal to the pay-off of loners. Here, we introduce probabilistic participation in PGGs from the standpoint of diversification of risk, namely simple mixed strategies with loners, and prove the existence of a dynamical regime in which the restrictions ono longer hold. Considering two kinds of mixed strategies associated with participants (cooperators or defectors) and non-participants (loners), we can recover all basic evolutionary dynamics of the two strategies: dominance; coexistence; bistability; and neutrality, as special cases depending on pairs of probabilities. Of special interest is that the expected pay-off of each mixed strategy exceeds the pay-off of loners at some interior equilibrium in the coexistence region.