We quantified the uncertainty associated with exposure-response modeling by using 100 plausible temperature-mortality relationships in addition to the mean relationship (blue and red lines in fig. S1) for each city. We generated these samples using Monte Carlo simulations, assuming a multivariate normal distribution of the spline model coefficients (5, 38). For each decade-long simulation in the 90-member climate model ensemble, we used all 101 temperature-mortality relationships to estimate the number of heat-related deaths.

For Fig. 3, this means that we computed 9090 (90 ensemble members × 101 temperature-mortality relationships) fractions of mortality attributable to heat for each city and climate scenario. We computed the avoidable attributable fractions of heat-related mortality by subtracting the attributable fraction in HAPPI1.5 or HAPPI2.0 from the corresponding attributable fraction in HAPPI3.0. In other words, the avoidable fractions are paired differences. We took the mean avoidable fraction across the 90 ensemble members (considering only the mean temperature-mortality relationship) as the point estimate and the 2.5 to 97.5 percentiles of all 9090 estimates of attributable fractions as the 95% eCI on Fig. 3.

For Fig. 4 and figs. S2 to S4, we used the 900 model years in each experiment (10 years of simulation × 90 ensemble members) to estimate mortality levels of rare events. Using each of the 101 plausible temperature-mortality relationships, we determined the annual numbers of heat-related deaths in the 900-year series and sorted them in descending order. For figs. S2 and S4, we took the additional step of dividing the series by the city’s July 2016 population in 100,000 persons. The return periods of these mortality levels were then 900 divided by their ranks within the sorted series. This means that the highest mortality level is always the rarest (a 1-in-900-year event). Using the 101 plausible temperature-mortality relationships, we generated 101 return period curves for each climate scenario. Each solid line in figs. S3 and S4 represents the mean return period curve across the 101 estimates. We estimated the 95% confidence interval associated with climate variability by bootstrapping this mean curve 1000 times. We added this uncertainty to the 2.5 to 97.5 percentiles of all 101 curves, generating a combined 95% eCI, as indicated by the shading in figs. S3 and S4. We subtracted the 1-in-30-year mortality level in HAPPI1.5 or HAPPI2.0 from HAPPI3.0 for Fig. 4 and fig. S2. Increasing the temperature-mortality relationship sample size from 100 to 500 did not alter the main results.

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