A game theoretic approach identifies conditions that foster vaccine-rich to vaccine-poor country donation of surplus vaccines

The COVID-19 pandemic highlighted the need to understand drivers of policy solutions for global infectious disease crises. By November 2021, over 5.23 million deaths and severe economic losses had occurred. Lockdowns and social distancing were necessary before effective vaccines and treatments were available. The development of safe and effective vaccines was a remarkable scientific achievement, but their equitable global distribution remains a major challenge. Unequal vaccine allocation presents the most formidable challenge to maximizing vaccine benefits. While allocating doses proportionally to population size is estimated as near-optimal, the reality is extremely unequal distribution, with many in the world's poorest countries lacking access until at least mid-2023. This exposes a large portion of humanity to continued outbreaks and increases the risk of new virus variants emerging, potentially undermining existing vaccine efficacy. COVAX, a pooled procurement initiative, aimed to prioritize global vaccination, but has fallen short due to vaccine nationalism. Wealthier countries prioritized their own populations over healthcare workers and high-risk populations in low-income countries. This led to initiatives focusing on country-level and ad hoc intergovernmental approaches, such as donations of surplus vaccines from high-income to low- and middle-income countries. The G7 pledged 1 billion doses by June 2022, representing an effort to redistribute excess stocks. This study uses a game-theoretical approach to determine under what conditions self-interested vaccine-rich countries would donate surplus vaccines to vaccine-poor nations, rather than keeping them for domestic use. Vaccine-rich countries have self-interested reasons for sharing surpluses: large open economies dependent on international trade and the risk of new variants undermining national pandemic control efforts. The probability of a variant of concern (VOC) is crucial; it can either increase short-term national self-interest or create awareness that long-term pandemic control necessitates effective global vaccination. The challenge is that donation benefits all countries by reducing the probability of future variants but incurs a cost to the donor by leaving it without available surplus in case of a VOC outbreak.

Previous game-theoretic models on epidemics focused on individual-level vaccination decisions, assuming vaccine availability. This study addresses the significant issue of international vaccine inequality, where vaccines are scarce in some countries and surplus in others. The focus is on country-level decisions, particularly those made by governments controlling surplus stocks. The study differs from previous work on international subsidies (e.g., smallpox vaccine), which lacked the urgency and scarcity of vaccine supplies that characterize the COVID-19 situation. Other relevant work examined international subsidies for invasive species control and the influence of national policies on pandemic spread (weakest-link games), but these models didn't directly address the vaccine donation problem with its unique aspects of surplus and potential variant emergence.

The study models the vaccine donation game involving *N* vaccine-rich countries. Each country has surplus vaccine doses sufficient to vaccinate its population twice beyond initial vaccination needs. Each country can choose to donate 0, 1, or 2 extra doses per capita. The fraction of the global unvaccinated population that could be vaccinated due to donations is denoted as *v*, ranging from 0 (no donation) to *v_max* (full donation). The expected annual rate of VOCs is *λ*, reduced to (1-*v*) if *v* ≤ 1 and 0 if *v* > 1, following a Poisson distribution. The probability of *k* variants is calculated using the Poisson coefficient formula. The expected cost of future outbreaks (*C_i*) for country *i* depends on its donation strategy (*s_i*) and the fraction (*α*) of the total cost of a new VOC outbreak that remains unavoidable even with surplus doses. The formula for *C_i* varies depending on *s_i* (0, 1, or 2) and incorporates probabilities *P₁*, *P₂*, and *P₃* (probabilities of 1, 2, or 3 or more variants emerging). The study identifies optimal solutions from a social planner's perspective (minimizing total expected cost for all vaccine-rich countries) and assesses the stability of these solutions using Nash equilibrium and self-enforcing international agreement (SEA) concepts. Numerical methods were used to calculate optimal solutions by evaluating total cost across all possible strategy combinations and finding the minimum. Stability was checked by examining if a single country could benefit from deviating from the optimal solution. For SEA, the model analyzed scenarios with different numbers of signatories (countries contributing) and assessed if any signatory would benefit from opting out or a non-signatory from opting in. All calculations are fully reproducible using the described methods and available code.

The optimal solution always involves equal donation by all countries (0, 1, or 2 doses), depending on parameters *λ*, *α*, *N*, and *v_max*. Zero donation is optimal when *λ* is high and *v_max* and *α* are low. Full donation is optimal when *v_max* is close to 1 and/or *α* is high. Partial donation (one dose) is optimal for intermediate values of *λ*, *v_max*, and *α*. The *v_max* threshold where the optimal strategy becomes full donation isn't highly sensitive to *λ* above a certain *v_max* level. Below this threshold, *λ* significantly influences whether one dose or no donation is optimal. The effect of *α* (unavoidable outbreak cost fraction) is also examined, with higher *α* values leading to more donations. When *α* = 1 (stocking is worthless), full donation is optimal. As *α* decreases, holding doses domestically becomes more appealing. The number of vaccine-rich countries (*N*) doesn't affect the optimal solution but impacts its stability. Smaller *N* leads to broader parameter ranges where full or partial donation is a stable solution (Nash equilibrium or SEA). In regions where full donation is optimal but not a Nash equilibrium, it can still be an SEA, suggesting potential for stable donation despite individual self-interest. Similar stability properties are found for partial donation. The study also identified regions where optimal solutions (full or partial donation) are unstable, resulting in suboptimal outcomes where only some countries donate. Figures illustrate these findings, mapping optimal solutions and their stability across various parameter combinations.

This study shows that strictly self-interested vaccine-rich countries may voluntarily donate surplus vaccines under specific conditions, without relying on altruism or international solidarity. Full donation is optimal when the potential impact is high (*v_max* is large) and when stocking surplus doses is less effective in averting outbreak costs (*α* is large). Above a certain threshold of *v_max*, full donation is optimal and relatively insensitive to *λ*. Below this threshold, *λ* significantly influences the optimal strategy. The results highlight the importance of considering both the potential global impact of donations and the effectiveness of domestic stockpiling when designing international cooperation strategies. The findings suggest that coordinated efforts to increase *v_max* (e.g., through increased vaccine production and equitable allocation mechanisms) can significantly improve the likelihood of substantial vaccine donations even from self-interested countries. The model’s stability analysis provides insights into the potential for international agreements to support vaccine donation. The model suggests that smaller numbers of vaccine-rich countries could facilitate more stable agreements for international vaccine donations. The study also demonstrates the importance of considering the avoidable and unavoidable costs of a future outbreak when deciding whether to donate vaccines or not.

This study presents a game-theoretic model showing that under certain conditions, notably when a large fraction of the global unvaccinated population can be covered by surplus vaccines from rich countries, full or partial donations of surplus vaccines are optimal even for strictly self-interested countries. These optimal strategies are stable under more restricted but still relevant parameter ranges. The results provide valuable guidance for policymakers seeking to facilitate international vaccine donations, emphasizing the importance of increasing the global impact of donations and considering the effectiveness of domestic stockpiling strategies in managing the risk of new variants. Future research could explore the impact of relaxing simplifying assumptions, such as unequal country sizes and multiple reasons for vaccine stockpiling.

The model incorporates several simplifying assumptions. The focus is on the risk of VOCs as the primary reason for stockpiling, neglecting other factors like uncertainty regarding vaccine efficacy duration or diversification of vaccine sources. Countries stock vaccines for reasons beyond VOC risks, which could reduce the likelihood of donations. The model also assumes equal-sized vaccine-rich countries; relaxing this assumption would likely alter the results. Additionally, the model assumes that variants occur independently of one another, which may not perfectly capture the complexity of viral evolution. The model's sensitivity to changes in parameter values might vary depending on the specific context and data used.