Self-enforcing regional vaccination agreements

Abstract

In a highly interconnected world, immunizing infections are a transboundary problem, and their control and elimination require international cooperation and coordination. In the absence of a global or regional body that can impose a universal vaccination strategy, each individual country sets its own strategy. Mobility of populations across borders can promote free-riding, because a country can benefit from the vaccination efforts of its neighbours, which can result in vaccination coverage lower than the global optimum. Here we explore whether voluntary coalitions that reward countries that join by cooperatively increasing vaccination coverage can solve this problem. We use dynamic epidemiological models embedded in a game-theoretic framework in order to identify conditions in which coalitions are self-enforcing and therefore stable, and thus successful at promoting a cooperative vaccination strategy. We find that countries can achieve significantly greater vaccination coverage at a lower cost by forming coalitions than when acting independently, provided a coalition has the tools to deter free-riding. Furthermore, when economically or epidemiologically asymmetric countries form coalitions, realized coverage is regionally more consistent than in the absence of coalitions.

Keywords: SIR model; epidemic dynamics; metapopulation; regional cooperation; transboundary movement.

Figure 1.

Multi-patch SIR model. (a) Global, fully cooperative optimum (green line) and the independent, non-cooperative optimum (Nash equilibrium), given in red for increasing numbers of identical interconnected countries. Parameters: R₀ = 5, coupling strength = 10μ/(n − 1), a_i = 0.1, c_Ii = 5. Costs (b) and realized coverage (c) for a system of 15 identical interconnected countries, for increasing coalition size (x-axis). Green and red lines show global and independent optima (as in (a)), black lines show the optimum realized by the countries in the coalition, grey lines show optimum for countries that have not joined the coalition. Other parameters as in (a).

Figure 2.

Coalition stability. (a) Externally unstable coalition—benefits of the coalition are greater than the free-riding pay-off, giving non-signatories the incentive to join. (b) Internally unstable coalition—benefits of free-riding are greater than benefits from coalition giving signatories an incentive to defect. (c–e) Effects of travel restrictions on coalition stability for different numbers of interconnected countries n (x-axis) and for increasing numbers of signatories k (y-axis). Blue shading shows internally stable coalitions, externally stable coalitions are shaded orange, and their overlap shows self-enforcing coalitions. Coalitions in yellow are neither externally nor internally stable, and area in white shows unfeasible coalitions. R₀ = 5, coupling strength = 20μ/(n − 1), a_i = 0.1, c_Ii = 5. (c) No travel restrictions, (d) expensive travel restrictions, q = 1000, (e) inexpensive travel restrictions, q = 5000.

Figure 3.

Multi-patch SIR model with 15 coupled countries. Costs (a,d,g), realized coverage (b,e,h) and intensity of travel restrictions (c,f,i) for three control scenarios: no travel restrictions (a–c); expensive travel restrictions, q = 500 (d–f); inexpensive travel restrictions, q = 1000 (g–i). Green lines show vaccination coverage at the global optimum (b,e,h) and the corresponding costs (a,d,g). Red lines show realized coverage and corresponding costs when countries are acting independently (Nash equilibrium). Black and grey lines show optimal coverage and corresponding costs for signatories and non-signatories, respectively. Internally stable coalitions are shaded blue, externally stable coalitions are shaded orange, and their overlap shows self-enforcing coalitions. Coalitions in white are unstable. Parameters: R₀ = 5, coupling strength = 20μ/(n − 1), a_i = 0.1, c_Ii = 5.

Figure 4.

SIR model for a system of eight asymmetric interconnected countries showing summary statistics for costs (a), coverage (b) or prevalence (c). For each country (shown on x-axis in black), the coalition sizes (indicated in grey on x-axis) are ordered from 1 (non-cooperative outcome) to 8 (fully cooperative outcome). There are possible coalitions of size k, and here we show the mean value and range of optimization outcomes for a given country in a coalition of a given size. Circles show mean costs (a), coverage (b) or prevalence (c) for each country and each coalition size when that country is in coalition (black) and outside of coalition (grey). Whiskers show fifth and 95th quantiles. Red and green lines show independent and global optimum for each country, respectively. Cost of infection parameter varies linearly across countries from c_I1 = 1 for country 1, and c_I8 = 15 for country 8. R₀ = 5, coupling strength = 10μ/(n − 1), a_i = 0.1. All 248 optimizations are shown in electronic supplementary material, figure S4.

Figure 5.

Overall coalition savings when a coalition becomes a fully cooperative coalition (all countries participate, k = n), with a country that joins the coalition last indicated on the x-axis. Parameter values as in figure 4.