1.1 Modeling framework overview

Our modeling framework consists of a vaccine-producing “Country A” choosing what fraction of its vaccine production to share with a vaccine receiving “Country B,” based on the dynamics of a two-country disease model coupled with viral evolution and an associated infection cost which Country A is trying to minimize. The framework thus connects an epidemiological model, an evolutionary model, and an economic model (Fig 1). In the epidemiological model, the two countries’ dynamics are coupled by vaccine sharing and a small degree of between-country transmission. The evolutionary model uses population-level phylodynamic principles [30,31] to calculate the probability that an immune-escape variant with a 10% transmission increase will emerge in each country at each time point, given each country’s prevalence of the wild-type strain and immune landscape at that time. Crucially, this framework uniquely allows us to mechanistically calculate variant emergence probabilities at all points in continuous time while simultaneously tracking all of the different epidemic trajectories that would arise from variant emergence at every point in time.

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Fig 1. Overview of our modeling framework and motivation.

(A) An epidemiological model of wild-type (blue) and immune-escape variant (red) strains of a respiratory virus spreading in two countries, coupled by between-country transmission and the vaccine-producing Country A sharing a fraction of its vaccines with the vaccine receiving Country B. Vaccine sharing increases Country A’s wild-type infections while reducing later infections from variants due to Country B’s reduced infections. (B) The evolutionary model extends the concepts of the population-scale phylodynamic curve (introduced by [30]) to predict the timing and probability of an escape variant emerging, which may be reduced by vaccine sharing. (C) The economic model defines an infection cost for Country A which discounts future infections and that Country A seeks to minimize through optimal vaccine allocation. Though vaccine sharing is favored under low discounting, vaccine hoarding is preferred if future variant waves are highly discounted. We also explore how prosociality and costs to vaccine distribution in Country B shape both countries’ vaccine allocation decisions in a game-theoretic framework. Small blue or red arrows show the projected impacts of vaccine sharing on wild-type or variant infections, respectively. More details in S1 Appendix.

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The economic model then describes how each country calculates the total infection costs that result from a policy as its cumulative (wild-type and variant) incidence over time, discounted at an annual rate . In order to account for all the different possible times at which an escape variant emerges, their cost functions sum variant infections across all scenarios depending on variant emergence times, weighted by the probabilities of emergence at each time. Prosociality acts on a country’s infection cost by additionally incorporating a fraction of the other country’s infections into the cost. And in later sections, we allow Country B agency in deciding how much it will invest in a costly new vaccine-distribution infrastructure, quantified by a cap on the share of vaccines it is able to accept. Consequently, Country A shares a fraction of its vaccines, but Country B ends up distributing only a fraction to its populace, which may or may not be less than depending on the values of and that the two countries choose. We then explore how the high vaccine-distribution cost function to Country B must be to force a Nash equilibrium vaccine-sharing strategy in which Country A donates fewer vaccines than its preference, in order to avoid doses being wasted on an underfunded distribution infrastructure. We thus assume that, while vaccine distribution may be costly for the lower-income Country B, these costs are negligible to the higher-income Country A.

2.1 Temporal discounting and prosociality

Fig 2 shows a typical model output for a viral infection with a wild-type basic reproduction number R₀ of 2 (roughly the infectiousness of the original SARS-CoV-2 strains [32]), exemplifying the competing pressures that arise for a vaccine-producing country. Comparing Fig 2A, where Country A hoards all of its vaccines, to Fig 2B, where the vaccine-producing Country A shares 40% of its vaccines with the receiving Country B, we see that vaccine sharing produces a short-term setback for Country A with increased wild-type infections (black curve) but a long-term benefit with reduced later waves of variant infections (colored curves, each one representing a different scenario of variant emergence with a different emergence week). Fig 2C then shows the accumulated infection costs over time for Country A either sharing (red) or hoarding (green) vaccines and at an annual discount rate of 1% (solid) or 10% (dashed), with variant infections averaged over all possible variant trajectories weighted by their respective emerge probabilities (black curves inside color bars).

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Fig 2. Example model output.

(A)–(B) Wild-type (black) and variant (colors) prevalence in Country A, which either (A) hoards all vaccines or (B) shares 40% of vaccines with Country B. Each variant prevalence curve represents one of many parallel scenarios, stratified and colored by the week of variant emergence. Color bars map each color to its corresponding variant emergence week on the x-axis above it, with the curves inside showing the probability of a variant emerging in each week. (C) Country A’s cumulative infection cost, with infections discounted at annual rate of 1% (solid) or 10% (dashed), and with Country A either hoarding all vaccines (brown) or sharing 40% of its vaccines (green). The infection cost accounts for both wild-type infections and a weighted average of variant infections across all possible variant emergence scenarios. In this figure, to better demonstrate the model’s motivation, population-scaled daily mutation rate and between-country coupling , while the default in other figures is and . And for this and all other figures unless otherwise noted: wild-type R₀ = 2, annual per-capita vaccine production rate , and the escape variant has a 10% transmission increase.

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This example illustrates the intertemporal conflict that a vaccine-producing country faces and the importance of the discount rate. For both the low annual discount rate of 1% and the high discount rate of 10%, vaccine sharing is more costly in the first two years as having fewer vaccines exacerbates Country A’s initial wave of infections. These initial infections are mostly due to the wild-type virus, but early variant emergence is also possible albeit with very low probability (dark blue curves in Fig 2). However, vaccine sharing greatly reduces costs from variant infections in later years once the population’s initial wild-type immunity has waned (Fig D in S1 Appendix), ultimately resulting in vaccine hoarding being the costlier option if discounting is low, while high discounting effectively ignores the later benefits of vaccine sharing and maintains hoarding as the less costly option.

The pattern that high temporal discounting favors vaccine hoarding appears universally from this model. Fig 3A, which visualizes the infection cost to Country A across all possible vaccine-sharing options and for a range of annual discount rates, shows that there is a clear single cost-minimizing vaccine-sharing fraction (black curve) which decreases with higher discount rates. The optimal decreasing with higher discount rates remains true for a wide variety of other parameter regimes, as heatmaps plotting for discount rates varying along with a host of other parameters show (Figs 3B, 4A, 4C, and 5). The impact of varying the discount rate is quite significant on the optimal vaccine-sharing fraction: in the absence of any prosociality, Country A can go from optimally hoarding all vaccines () at 20% annual discounting to optimally sharing about 30% of vaccines () at 1% annual discounting. Fig 3B also shows this effect of discounting on in conjunction with the effect of prosociality. We incorporate prosocial behavior by having Country A’s cost function include a fraction of the infection cost from Country B. As expected, increasing this prosociality factor boosts the proportion of vaccines Country A will share. Mechanistically, prosociality has this effect because hoarding vaccines universally hurts Country B in both wild-type and variant infections (Fig D in S1 Appendix); thus, accounting for Country B’s welfare deters against vaccine hoarding.

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Fig 3. Effects of discounting and prosociality on optimal vaccine sharing.

(A) Heatmap shows how the cumulative infection cost to the vaccine-producing Country A varies across the whole range of vaccine-sharing possibilities on the x-axis and with the y-axis varying the infection cost’s annual discount rate . The black curve shows the fraction of vaccines Country A shares that minimizes its infection cost . (B) The optimal vaccine-sharing fraction for simultaneously varying discount rate and prosociality factor (the fraction of Country B’s infection cost that Country A considers).

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Fig 4. Effects of varying the wild-type basic reproduction number R₀ (top row) or the annual per-capita vaccine production rate (bottom row), on the cost-minimizing fraction of vaccines for Country A to share (left column), and the probability of an immune-escape variant emerging in either country over the course of a simulation (right column).

(A) and (C) also vary the infection cost’s annual discount rate ; (B) and (D) also vary the fraction of vaccines Country A shares. In (B) and (D), the variant R₀ scales with the wild-type R₀ so that the variant always has a 10% increase in transmissibility against primary susceptibles, while the variant’s reproduction number against secondary susceptibles is scaled down to 1 to standardize the model’s long-term behavior. Benefits from vaccine sharing are either primarily evolutionary (reducing variant emergence probability) when wild-type R₀ is low or vaccine production is high, or primarily epidemiological (reducing variant cases given emergence) when wild-type R₀ is high or vaccine production is low.

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Fig 5. Effects of varying the virus’s population-scaled daily mutation rate .

(A) or the degree of coupling between the two countries (B, measured as the fraction of interactions that occur between people of different countries) on the cost-minimizing fraction of vaccines for Country A to share. Both (A) and (B) also vary the infection cost’s annual discount rate .

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Taken together, these results demonstrate that vaccine-hoarding tendencies can be promoted by temporal discounting and a lack of prosociality—in other words, by disregarding infections (and viral evolution) in the future and in other countries. Times of crisis and increased nationalism such as pandemics may trigger greater temporal discounting in decision-making and declines in outgroup prosociality, as it seemingly becomes more important to invest resources into managing the crisis at the present time and place [2,22]. Uncertainty in the future can also discourage consideration of future outcomes [26,27,33]; this can be seen from the COVID-19 pandemic as the potential for destructive variant evolution was widely underestimated in the pandemic’s early stages [34]. Thus, high temporal discounting and low levels of international prosociality are likely contributors to vaccine nationalism during pandemics, specifically in the cases of the 2009 H1N1 and COVID-19 pandemics. Furthermore, these two factors may be closely linked, as experiments have uncovered correlations between individuals’ propensities for discounting future outcomes and eschewing prosocial behavior [35].

It is worth noting that the discount rate we model may in practice be a combination of government-appointed economic discount rates as well as psychologically-induced time preferences, which are known to increase with level of uncertainty in future outcomes [33,36]. Similarly with prosociality, our model greatly simplifies how prosociality functions, making no distinction between policy-mandated and psychologically-based prosociality in their influences on policy decisions. Also, people typically discount the future “hyperbolically,” discounting the far future similarly to the near future [37,38], but we do not explore that possibility here.

2.2 Vaccine sharing’s epidemiological and evolutionary benefits

As we have seen, vaccine sharing provides long-term benefits in the form of reduced future costs from escape variant infections, and these benefits can outweigh the extra wild-type infections that reduced vaccination causes in the short-term if temporal discounting is sufficiently low. But what are the mechanisms by which vaccine sharing reduces variant infections?

Costs due to variant infections can be reduced in two ways. Since we model infection costs due to variants as the sum of all possible variant trajectories weighted by the probabilities that each trajectory will occur (based on their emergence time and our evolutionary model), variant costs can be reduced by either controlling variant infections given emergence—an “epidemiological benefit”—or by controlling the probability of variant emergence in the first place—an “evolutionary benefit.” In general, we see that vaccine sharing consistently lowers the overall probability of variant emergence—despite increasing emergence probability in Country A (Fig G in S1 Appendix)—when Country A shares fewer than half of its vaccines. However, the vaccine sharing’s effectiveness in lowering emergence probability and providing an evolutionary benefit is largely dependent on other parameters. We examine how the wild-type’s basic reproduction number and the total vaccine production rate impact whether vaccine sharing’s benefits are primarily epidemiological or evolutionary.

For a wild-type basic reproduction number of R₀ = 2, 40% vaccine sharing between countries () as compared to complete vaccine hoarding () only reduces the overall probability of variant emergence from 40% to 37% (Fig 4B, or areas under the emergence probability curves in Fig 2A and 2B). The benefit of vaccine sharing is thus mainly epidemiological rather than evolutionary, drawing from the substantial decrease in variant infections given variant emergence (as seen in Fig 2A and 2B). This decrease in variant infections given emergence is a product of the reduced population-level susceptibility in Country B that access to vaccination provides them; however, this decrease in susceptibility does not happen soon enough to have a large effect on dampening the probability of variant emergence (Fig D in S1 Appendix).

By contrast, with a lower wild-type R₀ of 1.5, equitable sharing results in a more significant decrease in emergence probability from 18% to 2% (Fig 4B), while Country A’s variant infections given emergence are not substantially reduced (Figs E and F in S1 Appendix). Thus, the benefit from vaccine-sharing is primarily evolutionary when wild-type R₀ is low. The difference lies in the fact that when the basic reproduction number is sufficiently low, vaccine sharing is able to lower the reproduction number in both countries below 1, curbing the outbreaks and evolutionary potential in both countries. So when wild-type R₀ is low, vaccine sharing actually reduces Country A’s wild-type infections (contrary to when wild-type R₀ is higher as in Fig 2), though the cost this saves is small compared to the cost reduction resulting from the lower emergence probability (Fig E3 in S1 Appendix).

When wild-type R₀ is low, the benefit from vaccine sharing also occurs much earlier. Consequently, the temporal discount rate has little effect on Country A’s optimal vaccine-sharing strategy for small wild-type R₀ (Fig 4A). Raising the vaccine production rate has a similar effect as lowering wild-type R₀, in that both render vaccine sharing more effective at rapidly controlling the pandemic before variant emergence can occur. We thus see that higher vaccine production rates also result in vaccine sharing having a greater evolutionary benefit (Fig 4D) and discount rate having a lesser impact on Country A’s preferred vaccine-sharing strategy (Fig 4C).

Unless otherwise noted, we focus on the case of higher wild-type R₀ = 2 and lower , corresponding to the regime where vaccine sharing has a primarily epidemiological benefit, as this is the more likely case for an emerging epidemic and the more interesting one to study with regards to temporal discounting. However, we more thoroughly examine the case of the lower wild-type R₀ = 1.5 with Figs E and F in S1 Appendix.

2.3 Impacts of epidemiological and evolutionary parameters on vaccine sharing

The optimal vaccine-sharing strategy for the producing country responds in interesting ways to some of the model’s key epidemiological and evolutionary parameters, yielding important insights into the biological factors influencing vaccine nationalism. Most interestingly, the optimal fraction that Country A shares is a bimodal function of the wild-type basic reproduction number R₀ (Fig 4A), each mode corresponding to one of the two regimes discussed in the previous section for how vaccine sharing is beneficial. The first peak for lower wild-type R₀ of 1.2–1.7 corresponds to the “evolutionary benefit” regime, where vaccine sharing’s greatest impact is in preventing the early rise of escape variants by bringing both countries’ effective reproductive numbers below 1; and the second peak for higher wild-type R₀ of 2–3 corresponds to the “epidemiological benefit” regime, where vaccine sharing’s impact occurs later on in reducing variant infections given emergence without appreciably changing emergence probability. In between these two regimes there is a drop in the effectiveness of vaccine sharing when wild-type R₀ is around the 1.7–2 range. And on either extreme, for very high wild-type R₀ or R₀ very close to 1, vaccine sharing is also ineffective because, respectively, the virus is too strong to be controlled by the total available vaccine production or the virus is too weak to warrant vaccine-related expenditures.

The vaccine production rate similarly follows the general trend that the regime in which vaccine sharing provides evolutionary benefits (high ) incentivizes greater vaccine sharing than the regime in which vaccine sharing provides epidemiological benefits (low ). This pattern can be more simply understood by the fact that when Country A has more vaccine production, it becomes less costly to give some of that vaccine production away as it will still have enough vaccination to control its own infections. However, the response of the optimal vaccine-sharing fraction to is simply monotonic (unlike its bimodal response to wild-type R₀), under this parameterization where variants have a 10% transmission advantage.

Vaccine sharing can still be an optimal strategy even when the vaccine has a 10% transmission decrease (as opposed to the 10% increase we explore in our main results), as we show with Fig I in S1 Appendix across a wide range of parameters. Thus, variant transmission increases (the status quo in vaccine nationalism models) are not necessary to incentivize vaccine sharing: we see that vaccine sharing can still be optimal to reduce the long-term threat of variant infections even when the variant is less transmissible than the wild-type, though the optimal vaccine-sharing fraction is generally lower in this case. Intriguingly, does have a bimodal response to vaccine production rate in this weaker-variant parameterization (Fig I3 in S1 Appendix).

The degree of between-country coupling and the virus’s immune-escape mutation rate also factor into the vaccine producer’s optimal sharing strategy (Fig 5). The optimal monotonically increases with the strength of coupling, measured by the fraction of interactions a person has with another country, as the damages to Country B incurred by vaccine hoarding are more detrimental to Country A in the long-term when the two countries are tightly linked (Fig 5A). Similarly, higher mutation rates also exacerbate the danger from variant infections to further incentivize vaccine sharing, but only up to a point. Above a threshold (10–15), variant emergence becomes so likely that vaccine sharing loses value as a tool to control variant emergence, leading to a unimodal response of to mutation rate (Fig 5B). We also find that the pattern of vaccine sharing having a stronger evolutionary impact when wild-type R₀ is low or vaccine production rate is high still holds true under a range of different mutation rates (Fig H in S1 Appendix).

2.4 High costs for vaccine-distribution infrastructure encourage vaccine hoarding

We now extend the economic model to incorporate costs to developing novel vaccine-distribution infrastructure for the receiving country, allowing it decision-making agency in choosing how much infrastructure it will invest in. Utilizing a game-theoretic framework, Country B now decides on a maximum fraction of the total vaccine production it is willing to accept, while Country A simultaneously and independently decides what fraction of vaccines it will share. Effectively, Country B vaccinates its populace at rate , with acting as a cap to the amount of vaccines Country B can accept due to its limited vaccine-distribution infrastructure (while Country A still vaccinates itself at rate ). The up-front cost to developing vaccine-distribution infrastructure is added to Country B’s infection cost in the form of the vaccine-distribution cost function .

We analyze the joint decision-making processes of the two countries through heatmaps plotting each country’s total costs (infection costs for Country A and infection plus vaccine-distribution costs for Country B) for all possible values of and , along with curves showing the cost-minimizing choice of for each value of (black) and the cost-minimizing choice of for each value of (gray; Fig 6). Wherever these two curves overlap represents a Nash equilibrium, meaning neither country would choose to change its strategy given the other country stays constant. Given perfect information about each others’ costs, the two countries should decide on a strategy set that is a Nash equilibrium, at least one of which is guaranteed to exist [39].

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Fig 6. Infection costs to Country A (, top row) and Country B (, bottom row) with x-axes varying (fraction of its vaccines that Country A shares with Country B) and y-axes varying (the maximum fraction of vaccine production that Country B is able to accept), and with different vaccine-distribution costs to Country B.

In the left column (A,B), there is a low vaccine-distribution cost function , where distributing one vaccine is as costly as one average infection; and in the middle column (C,D), there is a high cost function , where one vaccine is as costly as two infections. Black curves show the values of that minimize Country A’s infection cost for each value of , gray curves show the values of that minimize Country B’s infection cost for each value of , and their overlaps (black-gray dashed) represent regions of Nash equilibria. (E) Qualitative outcomes for each country in space: Country A and Country B both balance between sharing and accepting, respectively, more or fewer vaccines than is optimal for them, with the possibility that some vaccines go to waste (when ). This figure uses population-scaled daily mutation rate to better illustrate these concepts.

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No matter the cost function, all equilibria occur on the line , since Country B would prefer to not invest an unnecessarily large amount in vaccine infrastructure () and Country A would prefer not to have vaccines go to waste by sharing more than Country B can distribute (). Furthermore, all equilibria occur for (with Country A’s optimal vaccine-sharing fraction assuming Country B distributes all vaccines), since Country A will not share more vaccines than is optimal for them even if Country B is able to accept more. When vaccine-distribution costs are on the lower end relative to infection costs (, Fig 6A and 6B) or there are no vaccine-distribution costs at all (, Fig J1 and J2 in S1 Appendix), there are equilibria for all possible ; in particular, Country A is able to safely share its optimal vaccine allocation without worries that any vaccines will be wasted on an underfunded vaccine-distribution infrastructure in Country B.

However, these worries become realized when Country B’s distribution costs are sufficiently high (, Fig 6C and 6D). In this case, Country B is still able to invest in some vaccine-distribution infrastructure, but only a very limited amount, as the benefits of accepting more vaccines become outweighed by the costs. At extreme distribution costs (, Fig J3 and J4 in S1 Appendix), it becomes infeasible for Country B to accept any vaccines at all and becomes the only equilibrium. In both instances, Country A is unable to share its preferred fraction due to Country B’s lack of distribution infrastructure. We note that both and represent situations where infection cost is on a similar order of magnitude to vaccination costs, which may be true for a disease with a low infection fatality rate (such as COVID-19), but for a disease with much higher infection fatality rate it may represent a situation closer to as vaccination costs may be negligible relative to very high infection costs.

If we only consider linear distribution cost functions of the form , with representing the cost of distributing one vaccine relative to one infection, then we can predict the exact value of for which Country B’s distribution costs become too high for Country A to share its optimal (Fig B in S1 Appendix). Specifically, we prove in S1 Appendix that Country A sharing its optimal is a Nash equilibrium if and only if is less than Country B’s marginal increase in infection cost from declining one vaccine when is shared. In general, Country A will share its desired amount of vaccines only when the infection cost for Country B to accept them all outweighs the distribution cost. For the parameter regime we present here, this (Fig B in S1 Appendix; it is apparent that this must be between 1 and 2 from visually interpolating between the high and low distribution cost cases in Fig 6).

Thus, high costs to developing novel vaccine-distribution infrastructure in other countries can dissuade a vaccine-producing country from sharing its stock, to avoid doses going unused. SARS-CoV-2 vaccine wastage was in fact observed during the pandemic as some low-income countries were unable to fund the appropriate cold-chain storage facilities across their more rural areas, while vaccine access to other countries was thwarted by natural disasters [40]. And any uncertainty in other countries’ ability to successfully distribute vaccines would further dissuade producers from sharing doses, even to countries that may have the necessary infrastructure. As a result, economic hardships and uncertainties in low-income countries’ distribution infrastructures may contribute to stockpiling by vaccine-producing countries.

If we further modify the model to allow the producing country to help shoulder some of the lower-income country’s infrastructure costs, this may allow the lower-income country to invest more in infrastructure to distribute more vaccines, leading to a scenario better for both countries. However, such a strategy is infeasible for the producing country from a game-theoretic standpoint, as we discuss in more detail in Fig K in S1 Appendix. Faced with a decision to help pay some portion of Country B’s infrastructure development costs, Country A at equilibrium will deny helping with infrastructure development, since it would receive no direct benefit from doing so without Country B simultaneously accepting more vaccines. But the simultaneous combination of Country A helping with infrastructure costs and Country B accepting more vaccines can result in a more optimal strategy for both countries, revealing a discrepancy between the realized Nash equilibrium and what is actually optimal for both countries. This emphasizes the need for international organizations to act as central planners in problems of vaccine allocation and, especially, development of vaccine-distribution infrastructure. Indeed, much of the help with creating cold-chain infrastructure to distribute SARS-CoV-2 vaccines in low-income countries has come from international humanitarian organizations [41].

This work only considers two countries, one a vaccine producer and the other a vaccine receiver. Future work could consider more countries interacting in a network of vaccine production and sharing. Just as free-riding can emerge among a group of individuals deciding whether to vaccinate or take advantage of the population-level immunity that arises from others vaccinating [42,43], free-riding may emerge among a group of vaccine-producing countries deciding whether to share vaccines with lower-income countries or take advantage of the benefits from other producing countries sharing their vaccines. Exploring the conditions in multi-country systems for which free-riding emerges is a salient avenue for future research.

Future work may also benefit from drawing parallels to climate modeling literature focusing on how to foster multi-national collaboration to abate climate change. Just as mutual compliance with international climate accords will be crucial for effective climate change mitigation [44,45], we have shown how global vaccine distribution can act as a common pool resource to limit infectious disease spread and harmful variant evolution. A fundamental difference between pandemics and climate change, however, is that of time scale: pandemic scenarios unfold over the course of just several years, which may make mitigation more challenging due to the need for immediate action, but possibly easier to spur countries to action due to the highly visible and rapidly evolving nature of emerging pandemics.

5.1 Epidemiological model

We consider two countries operating under SIR(S) dynamics, modifying the original SIRS model to have post-infection return to susceptibility marked by a transmissibility decrease for “secondary susceptibles” [49,59]. Specifically, we assume secondary susceptibles can be infected only by variant strains of the virus, not the wild-type, so that the wild-type strain effectively follows SIR dynamics with fully protective immunity, while variants follow SIRS dynamics with imperfect immunity (either due to strain replacement or reinfection by the same strain). In order to standardize the model’s behavior while varying the wild-type transmission rate , the variant’s transmission rate against primary susceptibles is always increased (or decreased) 10% from and its transmission rate against secondary susceptibles is reduced by factor so that its reproduction number is scaled down to 1. This last assumption can be interpreted as individuals adopting interventions at just the right level to ensure the variants eventually die out [60]. The two countries’ dynamics are coupled by a small fraction of interactions which happen between people of different countries and may lead to between-country transmissions, and by Country A sharing a fraction of its vaccine production rate with Country B.

5.2 Evolutionary model

We mechanistically measure the rate of variant emergence in each country based off the evolutionary potential formulae derived by [30]:

(1)

(2)

with S_1,X, S_2,X, and representing primary susceptibles, secondary susceptibles, and wild-type infections, respectively, in Country X (A or B); is the per-transmission mutation rate from wild-type to variant; and is the infection recovery rate. From these, the probability that the variant has not yet emerged follows , and the instantaneous probability that the variant emerges in Country X (A or B) at time t is .

5.3 Economic model

We calculate the instantaneous infection costs for Country X (A or B) at time t, , as the weighted average of the wild-type/variant incidence in Country X at time t in all possible scenarios of variant emergence: no variant has emerged (weighted with probability ), the variant emerged in Country A at time (probability ), or the variant emerged in Country B at time (probability ). The total infection cost for Country A from present time t = 0 until some maximum time t = T, with temporal discounting at rate and prosociality factor , is then

(3)

and a similar equation with A and B switched gives the total infection cost for Country B. Country A is trying to minimize its infection costs by sharing the optimal fraction of its vaccines. In later sections, we allow Country B to simultaneously decide the maximum fraction of vaccines it is able to accept based on an added cost to developing a sufficient vaccine-distribution infrastructure. In this case, Country B’s vaccination rate becomes , while Country A’s vaccination rate remains .