We consider two countries, A and B, each confronting a COVID-19 outbreak in which one country (A) has a vaccine stockpile and the other country (B) does not. The outbreak is modelled using SEIRV (Susceptible - Exposed - Infected - Recovered - Vaccinated) dynamics. The system dynamics of the countries are coupled and active infections in one country can cause infections in the other country. Further, country A has the option of donating a part of its vaccine stock to country B. This vaccine sharing can only be done once at the start of the outbreak - between A (the donor country) and B (the recipient country).
The objective of the selfish, optimal vaccine sharing policy is to minimize fatalities in the donor country. The epidemic dynamics between countries are coupled. Here, we explore optimal policies as a function of epidemiological parameters as well as two features of the two-nation problem: (i) the epidemic coupling constant; (ii) the rate of vaccine uptake in the donor country. Increase in the epidemic coupling constant makes it more likely that infections in the recipient country lead to new cases in the donor country (and vice-versa). The vaccine uptake rate controls the rate at which a donor country can potentially use its vaccine stockpile; and we focus on limits in which the rate of vaccine uptake (on the order of a year) is slower than that of typical outbreak dynamics (i.e., on the order of months). In all cases the objective of the donor country is to minimize fatalities in its own country.
Code written in MATLAB 2021a
Run main.m for generating .mat files (~5-10 minutes execution time) Run plot_all.m for generating figures used in manuscript (~1 minute execution time)
A1. Figures - Main Text
Run “plot_all.m” for generating all figures, if mat files are present. To generate mat files, run "main.m"
Figure 1: Dependence of the optimal fraction of vaccines to be donated to κ (coupling coefficient) and the vaccination rate.
Figure 2: Variation of fatalities per 10^7 over 1 year in countries A and B for different values of µ. The value of µ which minimizes the fatalities in country A is termed the ‘optimal policy’. The simulation is run for low, medium and high coupling coefficients, κ (with κ ∈ {10^−6 , 10^−4, 10^−2}) and the daily vaccination rate is 0.28% of the total population. For plotting, change line 50 in plot_all.m with appropriate value of κ, as needed ({10^−6 , 10^−4, 10^−2})
Figure 3: Plots of the fatalities per 107 over 1 year for the no-sharing, optimal and hybrid policies in countries A and B. The analysis is done for different coupling coefficients (κ ∈ [10^−8 , 10^−1]) and three vaccination rates (50%, 75% and 100% of the population vaccinated in 1 year, assuming vaccine stock lasts).
A2. Figures - Supplementary Information
Figure S1: Fatalities in countries A and B when the (a) no-sharing policy, (b) optimal policy and (c) hybrid policy is implemented, over different κ ∈[10^−8,10^−1] and vaccination rates (from 0.14% to 0.42% of the population daily).
Figure S2: Change in fatalities in countries A and B when comparing the hybrid policy with the optimal policy, over different κ ∈ [10−8,10−1] and vaccination rates (from 0.14% to 0.42% of the population daily).
B. Description of key scripts
- force_of_infection.m: implementation of force of infection for 2 coupled systems
- state_f.m: state dynamical system
- state_solver.m: numerical integration of state system based on Euler method (forward direction)
- main.m: code to find optimal sharing fraction for different vaccination rates and coupling factor; creates data files (.mat) for plotting
- plot_all.m: plot all graphs using saved data