<figure><img src="image_12.png" /><figcaption>Figure 11: Life expectancy at birth lost in Metropolitan France, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>

steady trend of reduction (in absolute value) in life expectancy loss for both women and men with the gradual increase in temperatures. The 95% uncertainty associated with our life expectancy loss predictions becomes wider from 2050 onwards in this scenario. On the other hand, the RCP8.5 scenario comprises an even greater worsening of life expectancy loss due to extreme heat from the second half of the century, potentially reaching nearly 0.2 years by 2100. The uncertainty associated with these predictions increases significantly over time due to the longer duration, severity, and intensity of heatwave episodes (Figure 4).

This situation at the scale of Metropolitan France varies according to geographical location. Thus, the same figures for temperature-related life expectancy loss are reproduced for Brest, Marseille, Paris, Perpignan and Strasbourg. For that purpose, we use the projections of central death rates, excluding temperature effects, as described in Section 5.2, and then apply the attributable fractions simulated from the temperature trajectories specific to each city using data from DRIAS. Detailed results are presented in Appendix D. For these different cities, the trajectory of temperature-related life expectancy loss clearly depends on the evolution of hot and cold contributions in each area of France. In the northern part of France, including Paris, Brest and Strasbourg, the life expectancy loss related to all temperature effects tends to decrease (in absolute value) in all RCP scenarios due to the reduced contribution of cold to mortality. In the RCP8.5 scenario where average temperatures tend to rise rapidly, the life expectancy loss seems to become lower more quickly. In the southern part of France (Marseille and Perpignan), the trajectory of life expectancy loss reduction is less clear in RCP2.6 and RCP4.5. In the RCP8.5 scenario for Perpignan, the temperature-related life expectancy loss increases (in absolute value) in the second half of the century. For this city, where extreme heat will be more frequent at the end of the century, the median life expectancy loss due to heatwaves alone reaches between 0.3 and 0.4 years.
<figure><img src="image_9.png" /><figcaption>Figure 8: Pearson residuals of the Li-Lee model for the calibration period 1980-2019 and ages between 0-105 for the female and male populations of Metropolitan France. The model is fitted on observed deaths and temperature-adjusted exposures to risk.</figcaption></figure>

of the processes $\hat{\kappa}_t^{(f)}$ and $\hat{\kappa}_t^{(m)}$ must remain stationary, i.e. the absolute value of $\phi^{(g)}$, $g \in \{f, m\}$, must be strictly lower than 1. This constraint is met during the maximum likelihood estimation of the model (21) regarding $\kappa_t^{(m)}$ and $\kappa_t^{(f)}$ for observed deaths. However, it is observed that the values of the coefficients $\phi^{(m)}$ and $\phi^{(f)}$ are very close to 1. Regarding $\phi^{(f)}$ when adjusting exposure to risk to the temperature effect $\hat{T}_{x,t}^{(g)}(D^*)$, the parameter values are slightly above 1. This undesirable situation would thus lead to projecting mortality rates for women diverging from the common trend $K_t$ in the long term. To overcome this issue, Antonio et al. (2020) proposes reducing the calibration period of the model or increasing the lag of the autoregressive processes. Other strategies could be proposed, such as using a mortality model based on mortality increments (Hunt and Villegas, 2023) or employing a data-driven model, as seen in Guibert et al. (2019). However, this topic is beyond the scope of this paper, and we choose to address it more simply by constraining the coefficient $\phi^{(g)}$ to be strictly less than 1 during maximum likelihood maximization.

<table><caption>Table 2: Calibrated parameters in the Li-Lee models (26) and (5) on the period 1980-2019 and age range 0-105.</caption><thead><tr><th>Calibration data</th><th>δ</th><th>c<sup>(m)</sup></th><th>φ<sup>(m)</sup></th><th>c<sup>(f)</sup></th><th>φ<sup>(f)</sup></th></tr></thead><tbody><tr><td>Original Li-Lee model</td><td>-0.2338</td><td>-0.0083</td><td>0.9681</td><td>-0.0264</td><td>0.9978</td></tr><tr><td>Adjusted exposure to risk</td><td>-0.2339</td><td>-0.0055</td><td>0.9681</td><td>-0.0254</td><td>0.9999</td></tr></tbody></table>

<table><caption>Table 3: Correlation matrix of residuals in the Li-Lee models (26) and (5) on the period 1980-2019 and age range 0-105.</caption><thead><tr><th rowspan="2"></th><th colspan="3">Original Li-Lee model</th><th colspan="3">Adjusted exposure to risk</th></tr><tr><th>e<sub>t</sub></th><th>r<sub>t</sub><sup>(m)</sup></th><th>r<sub>t</sub><sup>(f)</sup></th><th>e<sub>t</sub></th><th>r<sub>t</sub><sup>(m)</sup></th><th>r<sub>t</sub><sup>(f)</sup></th></tr></thead><tbody><tr><td>e<sub>t</sub></td><td>1.0000</td><td>-0.7658</td><td>0.9437</td><td>1.0000</td><td>-0.5826</td><td>0.9330</td></tr><tr><td>r<sub>t</sub><sup>(m)</sup></td><td>-0.7658</td><td>1.0000</td><td>-0.7451</td><td>-0.5826</td><td>1.0000</td><td>-0.5683</td></tr><tr><td>r<sub>t</sub><sup>(f)</sup></td><td>0.9437</td><td>-0.7451</td><td>1.0000</td><td>0.9330</td><td>-0.5683</td><td>1.0000</td></tr></tbody></table>

Figure 9 finally presents the projection of parameters $\hat{K}_t$, $\hat{\kappa}_t^{(f)}$, and $\hat{\kappa}_t^{(m)}$ over the period 2020-2100, along with the 95% prediction intervals obtained by 1,000 Monte-Carlo simulation of innovation errors. These projections are shown for both models, with and without adjusting exposure to risk for temperature effects. More precisely, the trends of model (5) correspond to the dynamic of central death rates not attributable to temperature effects. We first observe that the common trend $K_t$ is virtually identical in terms of median for both parameter estimations. However, as expected, the projected uncertainty for the model (26) is wider than the trend for central death rates not attributable to temperatures $\tilde{m}_{x,t}^{(g)}$. This is explained by adjustments in exposure to risk related to mortality peaks
# 7 Conclusion

Temperature changes linked to climate change create disturbances that will affect the frequency of observed deaths throughout the year. In this paper, we design a multi-population mortality model incorporating the effect of temperature changes on mortality. This framework includes a classic climate epidemiology model, known as the Distributed Lag Non-Linear Model (DLNM), which measures deaths attributable to hot and cold temperatures over different periods of the year. By coupling these two modeling frameworks and integrating temperature trajectories from various climate models, we project central death rates impacted by temperature effects and measure these impacts in terms of gains or losses in life expectancy.

Our results describe a situation dependent on both climate scenarios and the geographical location of populations. Specifically, the simulated gains or losses in life expectancy until the end of the century, particularly from 2050 in Metropolitan France, show a downward trend in mortality related to cold and an upward trend in mortality related to extreme heat. These two effects evolve in opposite directions and reveal a source of divergence between certain regions of France, especially under the RCP8.5 scenario. For all results, we incorporate multiple sources of uncertainty, including climate scenarios, prediction errors of the mortality model, and estimation errors of the DLNM model. The modeling framework established here and illustrated with French data offers an adaptable approach for other countries or regions, whether for demographic needs or for insurers, actuaries, or supervisory authorities interested in designing climate scenarios. Further research perspectives are opened by extending the multi-population mortality model to other countries or by refining the geographical resolution of the considered territory. For instance, some regions will be more affected than others by climate change, and more pronounced impacts of heat-related mortality are expected in Southern Europe or the MENA region (Hajat et al., 2023). Otherwise, the study of daily mortality or on finer time scales may encompass other environmental components such as air pollution or humidity, which we do not consider in this study. These factors introduce additional sources of uncertainty that should be addressed in future research. In this regard, the DLNM could be calibrated on multiple sites and additional environmental variables. Thus, a possible extension would be to model the simultaneous occurrence of mortality shocks related to temperature extremes and other environmental variables.

In this paper, we make the strong assumption that populations do not adapt to their local environment, which is debated in recent literature (Wu et al., 2024). Indeed, the number of deaths related to temperatures could decrease thanks to acclimatization. This can result both from technology, e.g. improvement of house insulation, development of air conditioning, or by a physiological process, see Heiden et al. (2020) among others. Furthermore, prevention and protection plans could be set up to alert and guide the population on how to behave in the face of heatwaves. Concrete actions like changing working hours, opening air-conditioned reception areas, distributing water in public transport, monitoring the most vulnerable at home, and limiting or prohibiting outdoor sports activities could also affect heat-related mortality. Considering these acclimatization factors offers research perspectives to be explored in the construction of prospective scenarios.

**Supplementary material:** The results in this paper is obtained using R. Supplementary material related to this paper can be found at https://github.com/qguibert/mortalityheatwaves.

**Author Contributions:** Q.G., G.P. and F.P. design the study. Q.G. and G.P. collect, cure and verify the data. Q.G. designs the models and the methodology. Q.G. implements methods, produces results and visualizations. Q.G. and G.P. write the manuscript. F.P. edit the manuscript.

**Funding:** This research received no external funding.

**Conflicts of interest:** The authors declare no conflict of interest.
induced by temperature variations. Regarding the parameters $\hat{\kappa}_t^{(f)}$ and $\hat{\kappa}_t^{(m)}$, we notice the similarity between the forecasted series, and the same phenomenon in terms of uncertainty.

<figure><img src="image_10.png" /><figcaption>Figure 9: Projection of trend parameters $\hat{K}_t$, $\hat{\kappa}_t^{(f)}$, and $\hat{\kappa}_t^{(m)}$ over the period 2020-2100 for the Li-Lee models with original exposure to risk (26) and adjusted exposure to risk (5) for the calibration period 1980-2019 and ages between 0-105 for the entire population of Metropolitan France. The solid lines represent from 2020 the median of the projections. The 95% prediction intervals are constructed based on 1,000 Monte-Carlo simulations.</figcaption></figure>

In Appendix D, we further verify the consistency of projections obtained from the Li-Lee model using both the original exposure to risk (26) and the temperature-adjusted exposure to risk (5), as explained in Section 5. We also compare their forecasting performance, noting that the Li-Lee model combined with the DLNM slightly outperforms the Li-Lee model with the original exposure to risk.

## 6.4 Simulating temperatures effects

We now focus on the simulation of total attributable fractions to temperatures $\hat{AF}_{x,t}^{(g)}(\mathcal{D}_t)$, as defined in Equation (24). In this equation, we consider weights, estimated based on daily deaths not attributable to temperatures from the last five years of our historical data.<fnref n="4" /> These attributable fractions depend on a period $\mathcal{D}_t$ for each year $t$ that allows for distinguishing the effects of heat and cold over the year. These quantities are also calculated for each age group, sex and calendar year. To simplify notation, we do not include the index corresponding to the Monte-Carlo simulation numbers.

To facilitate visual analysis, we calculate an aggregated attributable fraction to temperatures using the distribution of deaths known at the end of 2019, as follows

$$ \hat{AF}_t(\mathcal{D}_t) = \sum_{g \in \mathcal{G}} \sum_{x \in \mathcal{X}} \hat{AF}_{x,t}^{(g)}(\mathcal{D}_t) \frac{D_{x,2019}^{(g)}}{D_{2019}}, $$

where $D_{2019} = \sum_{g \in \mathcal{G}} \sum_{x \in \mathcal{X}} D_{x,2019}^{(g)}$. The details of the attributable fractions to temperatures $\hat{AF}_{x,t}^{(g)}(\mathcal{D}_t)$ per age bucket and sex are provided in Appendix E.1.

Figure 10 breaks down the evolution of attributable fractions related to hot and cold temperatures according to the RCP scenarios for the period 2020-2100 and all population. The presented 95% confidence intervals combine, for each RCP scenario, the estimation error associated with the coefficients $\theta_k^{(g)}$, measured by 1,000 bootstrap replications, and the uncertainty related to different climate models listed in Table 1. Similar to Figure 6, we examine the evolution of components associated with moderate cold and hot days and extreme cold and hot days. Firstly, we observe a relatively stable overall effect of mean temperatures until the end of the century in the RCP2.6 scenario and a slightly decreasing effect in the RCP4.5 scenario for women and men. The RCP4.5 scenario is also

<footnote n="4">⁴Alternatively, the use of constant weights empirically leads to a very slight increase in the temperature-attributable fractions on these data.</footnote>
# A Heatwaves in France

Heatwaves represent a particular risk of acute mortality and their importance has increased over the past 20 years. To understand this phenomenon for the studied population, this appendix provides a synthetic overview of historical recent heatwaves in metropolitan France since 1950. Additional details are provided by Pincemin (2021).

## A.1 Definition and characterization of heatwaves

Heatwaves do not have a single definition. Depending on the temperature distribution and the adaptation capacity of populations in the region concerned, the definitions and tolerance thresholds adopted may vary. In particular, the identification of a heatwave may depend on the climate, urbanization of the area concerned, as well as indicators measuring its intensity (average temperature, daytime or nighttime temperature, duration of the reference period, humidity rate, etc.).

In the context of this study on French population, we define a heatwave according to the criteria used for heatwave alerts by Météo France, the French national meteorological service. These criteria are also used by Santé Publique France (Santé Publique France, 2019b). Météo France defines a heatwave period as a period where the average of daily temperatures during the day and at night exceeds the 99.5% percentile of observations made between 1981 and 2010.

We introduce for $d \in D^*$

$$ \bar{\vartheta}_d^{\max} = \frac{1}{3} \sum_{j=1}^{3} \vartheta_{d-j}^{\max}, \quad \bar{\vartheta}_d^{\min} = \frac{1}{3} \sum_{j=1}^{3} \vartheta_{d-j}^{\min} $$

where $\vartheta_d^{\max}$ and $\vartheta_d^{\min}$ are the aggregated maximum and minimum temperatures in metropolitan France for day $d$. This 3-day moving average is sometimes referred to as a biometeorological indicator. Thus, a day is marked by a heatwave if the minimum $\bar{\vartheta}_d^{\min}$ and maximum $\bar{\vartheta}_d^{\max}$ 3-day moving averages of temperatures exceed respective minimum $r^{\min}$ and maximum $r^{\max}$ thresholds.

Furthermore, we characterize each heatwave period, i.e. the set of days $D \subset D^*$, using the following indicators:

* the duration corresponds to the number of consecutive days above the alert thresholds, i.e., $|D|$,

* the cumulative severity corresponds to the sum of the daily severities $sev_d$ over the heatwave period $D$

$$ sev_d = |\vartheta_d^{\min} - r^{\min}| + |\vartheta_d^{\max} - r^{\max}|, $$

* the intensity of a heatwave corresponds to

$$ \text{int} = \max_{d \in D} \{|\vartheta_d^{\min} - r^{\min}|\} + \max_{d \in D} \{|\vartheta_d^{\max} - r^{\max}|\}. $$

These indicators allow for triggering the appropriate level of alert in line with the danger associated with the heatwave, and for evaluating and ranking heatwaves among themselves.

## A.2 History of recent heatwaves in France

Gradually, Western European countries such as France have experienced a decrease in the intensity and frequency of cold spells (Météo France, 2024). Conversely, heatwaves have become more frequent, and summers are increasingly hotter. In France, the major heatwaves of 1976 and 2003 resulted in an excess of 4,500 and 15,000 deaths, respectively. In recent years, heatwaves have occurred almost every single year with varying intensities. Some of them have had temperatures similar to the historical heatwave of 2003. In 2019 and 2020, they even reached the highest level of the alert plan for the first time in history.
characterized by a decrease in the effect of moderate cold and a progressive increase in the effect of extreme heat. It is noteworthy that there is an increase in the uncertainty of the effect of extreme heat over time.

The RCP8.5 scenario exhibits a different behavior. Specifically, it shows a more pronounced decrease over time than the RCP4.5 scenario in the effect of moderate cold temperatures, which is the main component of temperature-attributable mortality for both women and men. Concurrently, a marked increase in the effect of extreme heat is observed starting from the 2050s. However, as highlighted in Appendix E.1, this increase is not sufficient to reverse the overall declining trend in the effect of temperatures over the century for the males, except for those aged between 0 and 64 years. For the female population, we observe a slight resurgence in the effect of temperatures towards the end of the century, as well as greater uncertainty in temperature-attributable mortality.

<figure><img src="image_11.png" /><figcaption>Figure 10: Temperature attributable fraction in Metropolitan France, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>

It is also noteworthy that the effects of temperatures vary according to geographical locations (Martínez-Solanas et al., 2021). To illustrate this pronounced effect between the north and south of France, we simulate the trajectory of attributable fractions for different French cities across the country, namely Brest, Marseille, Paris, Perpignan, and Strasbourg, using data from DRIAS. Detailed results are presented in Appendix E.2. For this purpose, we employ the DLNM model estimated on the daily mortality data of Metropolitan France as presented in Section 6.2. This is an approximation that could be improved by having access to time-series of death counts per city. Indeed, each of the sub-populations residing there may have specific adaptation capacities to hot and cold temperatures, e.g. air conditioning or thermal insulation of buildings, which we do not account for.

These simulations reveal different sensitivities of each of these cities to extreme hot temperatures and moderate cold temperatures. For the city of Brest for example, the contribution of extreme hot temperatures to mortality remains very limited in the latter part of the century, even under the RCP8.5 scenario. On the other hand, the contribution of cold temperatures decreases gradually, which could tend to decrease the contribution of temperatures to mortality in this region. In contrast, the city of Perpignan appears significantly more exposed to extreme hot temperatures in the RCP4.5 scenario and in the RCP8.5 scenario.

## 6.5 Impacts on life-years lost due to temperature

Finally, we examine the effect of temperatures on the loss of life expectancy at birth. Figure 11 shows the loss of life expectancy at birth, as defined in Equation (25), for women and men between 2020 and
Their consequences in terms of mortality are listed in Table 4, which contains data compiled from *Santé Publique France* and the Centre for Research on the Epidemiology of Disasters (CRED) Database. It is noteworthy that the dates and durations of each heatwave are calculated on average by department and are extracted from the annual reports of *Santé Publique France*. Considering the average for the entirety of France would result in a shorter duration of heatwaves. For comparative purposes, the CRED data is also included in this table. Significant differences in excess mortality attributed to these various events are observed depending on the data source, due to variations in the methodologies employed.

<table><caption>Table 4: Main heatwaves in metropolitan France since 1976. This table presents for each year the cumulative number of heatwave days and the attributed number of deaths, calculated on average per department by *Santé Publique France*. These data are extracted from the *Bulletins de Santé Publique France* (reports analyzing the departmental, regional, and national situation) on the relevant summers (Santé Publique France, 2018; Santé Publique France, 2019a; Santé Publique France, 2020) and from the publication of the Institut de Veille Sanitaire (2015). Data from the earlier years (1976, 1983, 2003, 2006) are sourced from a study on heatwaves since 1970 (Pascal et al., 2019). The CRED data is drawn from the EM'DAT database (Centre for Research on the Epidemiology of Disasters, 2021). This open and free database is a project of the Integrated Research on Disaster Risk. It is one of the three largest databases for disasters, along with MunichRe's NatCatSERVICE and SwissRe's Sigma. It is fed by numerous governmental and non-governmental sources, insurers and reinsurers, press agencies, specialized entities and various global organizations (United Nations, SwissRe, MunichRe, NOAA, etc.).</caption><thead><tr><th rowspan="2">Key information</th><th colspan="2">Excess Death</th></tr><tr><th>Santé Publique France</th><th>Cred Em'Dat</th></tr></thead><tbody><tr><td><strong>1976</strong><br>Late June - mid-July</td><td>4,540</td><td>-</td></tr><tr><td><strong>1983</strong><br>July 9 - July 31 (22 days)</td><td>2,900</td><td>-</td></tr><tr><td><strong>2003</strong><br>The summer of 2003 was the hottest in France and Europe in over 50 years, with exceptional intensity during the first half of August. This episode was associated with significant ozone pollution.</td><td>14,800</td><td>19,490</td></tr><tr><td><strong>2006</strong><br>July 2006 witnessed a very intense heatwave, although less severe than in 2003, but with a longer duration.</td><td>1,442</td><td>1,388</td></tr><tr><td><strong>2015</strong><br>Three waves unevenly distributed over the territory. The first was quite intense, affecting the north for 10 days. The second, lasting 11 days, was less intense and concentrated in the south-east. The last one, shorter, was very localized in some eastern and southern departments.</td><td>2,040</td><td>3,275</td></tr><tr><td><strong>2018</strong><br>A two-week heatwave peaking in early August, exposing 70% of the French population.</td><td>1,480</td><td>-</td></tr><tr><td><strong>2019</strong><br>Two very extensive and intense heatwaves. For the first time, several departments were under red alert during both waves.</td><td>2,034</td><td>1,435</td></tr><tr><td><strong>2020</strong><br>Three heatwaves, one particularly severe in the north, resulting in a red alert for the second consecutive year, over a cumulative period of around ten days.</td><td>1,924</td><td>1,924</td></tr></tbody></table>
2100 due to temperatures in Metropolitan France. We specifically distinguish the overall temperature effects, which account for both hot and cold temperature-related mortality, and the sole effects of extreme hot temperatures. The different trajectories presented correspond to RCP2.6, RCP4.5 and RCP8.5 scenarios. These assessments are conducted under the assumption that populations and healthcare systems do not adapt to temperature changes throughout the century and evolve similarly at national level. Hence, the potential evolution of adaptation to both heat and cold, and the differences between regions are not analyzed in this study. This is a strong hypothesis but difficult to address in a prospective approach as it may evolve according to numerous factors (physiological, technological, immunological, etc.). It is noteworthy that recent literature suggest that populations can adapt to their local environment (Wu et al., 2024).

<figure><img src="image_12.png" /><figcaption>Figure 11: Life expectancy at birth lost in Metropolitan France, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>

For the overall temperature effects, we first note that the loss of life expectancy for men is greater (in absolute value) than that for women. This difference is explained by men's higher sensitivity to cold temperatures compared to women (Figure 5), especially in the age groups between 65 and 84 years. In the three RCP scenarios presented, rising temperatures would limit the contribution of cold to mortality. This would conduct to a general reduction of temperature-related life expectancy loss. We also observe in the RCP8.5 scenario a small reduction of the gap between women and men from 2050 onward due to increasing temperatures.

At the same time, attention must be paid to the evolution of life expectancy loss related to extreme hot temperatures. These excesses lead to reductions in life expectancy loss that generally do not exceed 0.1 years, except in a few exceptional years, under the RCP2.6 scenario. This loss does not show any particular trend throughout the century. The RCP4.5 scenario shows a slight but steady trend of reduction (in absolute value) in life expectancy loss for both women and men with the gradual increase in temperatures. The 95% uncertainty associated with our life expectancy loss predictions becomes wider from 2050 onwards in this scenario. On the other hand, the RCP8.5 scenario
In 2022, we observed an excess mortality of 73 deaths per million (26, 124), which corresponds to 4,807 (1,739, 8,123)<fnref n="2" /> deaths (Ballester et al., 2023).

# B Validation of the DLNM model

## B.1 Goodness of fit for the DLNM model

The figures 12 and 13 compare the monthly distributions of observed and modeled death counts over the observation period for females and males. We observe a close similarity in the monthly distributions, notably during summers and winters. However, our models appear to fit less well to the left tails of the distribution, i.e. to the low values of the number of deaths, as well as to some extreme values of deaths in the spring and autumn. These variations in the number of deaths could possibly be explained by other factors not included in our model, e.g. air pollution, flu epidemics, etc.

<figure><img src="image_13.png" /><figcaption>Figure 12: Monthly distribution of observed (blue) and predicted (green) numbers of deaths based on the DLNM model (17) per year for women in metropolitan France for the years between 1980 and 2019. The distributions are grouped by decade.</figcaption></figure>

Figures 14 and 15 depict the analysis of deviance residuals by age group and indicate a relatively good fit to the data for women and men. It is worth noting a few extreme points that were not well captured by the model, such as the 2003 heatwave for 85+, which was an event of extreme magnitude.

<footnote n="2">²The values in parentheses represent the 95% confidence intervals.</footnote>
comprises an even greater worsening of life expectancy loss due to extreme heat from the second half of the century, potentially reaching nearly 0.2 years by 2100. The uncertainty associated with these predictions increases significantly over time due to the longer duration, severity, and intensity of heatwave episodes (Figure 4).

While many studies quantify the impact of temperature in terms of excess mortality, fewer examine its effect on life expectancy to our knowledge. Among these rare studies, Hauer and Santos-Lozada (2021) estimates the impact of climate change under a business-as-usual scenario, equivalent to our RCP8.5 scenario. Specifically for France, they find a reduction in life expectancy due to climate change of approximately 0.65 years by 2080, which is more pessimistic than our results. However, it is important to note that their study only accounts for the extreme effects of temperature (heatwaves and cold spells) and ignores the compensatory mechanisms between the effects of cold and heat, particularly the gain in life expectancy resulting from the reduction in the attributable effect of moderate cold. Therefore, their estimates should be more closely compared to the approximate 0.2-year loss in life expectancy that we observe for heatwaves by the end of the century, with a large uncertainty depending on the climate model used.

This situation at the scale of Metropolitan France varies according to geographical location. Thus, the same figures for temperature-related life expectancy loss are reproduced for Brest, Marseille, Paris, Perpignan and Strasbourg. For that purpose, we use the projections of central death rates not attributable to temperature effects, as described in Section 5.2, and then apply the attributable fractions simulated from the temperature trajectories specific to each city using data from DRIAS. Detailed results are presented in Appendix F. For these different cities, the trajectory of temperature-related life expectancy loss clearly depends on the evolution of hot and cold contributions in each area of France. In the northern part of France, including Paris, Brest and Strasbourg, the life expectancy loss related to all temperature effects tends to decrease (in absolute value) in all RCP scenarios due to the reduced contribution of cold to mortality. In the RCP8.5 scenario where average temperatures tend to rise rapidly, the life expectancy loss seems to become lower more quickly. In the southern part of France (Marseille and Perpignan), the trajectory of life expectancy loss reduction is less clear in RCP2.6 and RCP4.5. In the RCP8.5 scenario for Perpignan, the temperature-related life expectancy loss increases (in absolute value) in the second half of the century. For this city, where extreme heat will be more frequent at the end of the century, the median life expectancy loss due to heatwaves alone reaches between 0.3 and 0.4 years.

# 7 Conclusion

Temperature changes linked to climate change create disturbances that will affect the frequency of observed deaths throughout the year. In this paper, we design a multi-population mortality model incorporating the effect of temperature changes on mortality. This framework includes a classic climate epidemiology model, known as the Distributed Lag Non-Linear Model (DLNM), which measures deaths attributable to hot and cold temperatures over different periods of the year. By coupling these two modeling frameworks and integrating temperature trajectories from various climate models, we project central death rates impacted by temperature effects and measure these impacts in terms of gains or losses in life expectancy.

Our results describe a situation dependent on both climate scenarios and the geographical location of populations. Specifically, the simulated gains or losses in life expectancy until the end of the century, particularly from 2050 in Metropolitan France, show a downward trend in mortality related to cold and an upward trend in mortality related to extreme heat. These two effects evolve in opposite directions and reveal a source of divergence between certain regions of France, especially under the RCP8.5 scenario. For all results, we incorporate multiple sources of uncertainty, including climate scenarios, prediction errors of the mortality model, and estimation errors of the DLNM model. The modeling framework established here and illustrated with French data offers an adaptable approach for other countries or regions, whether for demographic needs or for insurers, actuaries, or supervisory
<figure><img src="image_14.png" /><figcaption>Figure 13: Monthly distribution of observed (blue) and predicted (green) numbers of deaths based on the DLNM model (17) per year for men in metropolitan France for the years between 1980 and 2019. The distributions are grouped by decade.</figcaption></figure>
authorities interested in designing climate scenarios. Further research perspectives are opened by
extending the multi-population mortality model to other countries or by refining the geographical
resolution of the considered territory. For instance, some regions will be more affected than others
by climate change, and more pronounced impacts of heat-related mortality are expected in Southern
Europe or the MENA region (Hajat et al., 2023). Otherwise, the study of daily mortality or on finer
time scales may encompass other environmental components such as air pollution or humidity, which
we do not consider in this study. These factors introduce additional sources of uncertainty that should
be addressed in future research. In this regard, the DLNM model could be calibrated on multiple sites
and additional environmental variables. Thus, a possible extension would be to model the simultaneous
occurrence of mortality shocks related to temperature extremes and other environmental variables.

In this paper, we make the strong assumption that populations do not adapt to their local envi-
ronment, which is debated in recent literature (Wu et al., 2024). Indeed, the number of deaths related
to temperatures could decrease thanks acclimatization. This can result both from technology, e.g.
improvement of house insulation, development of air conditioning, or by a physiological process, see
Heiden et al. (2020) among others. The vulnerability of populations to low temperatures, influenced
in particular by respiratory virus infections, also needs to be questioned in light of changes in acquired
immunity and the duration of infection periods across different regions (Walkowiak et al., 2024). Fur-
thermore, prevention and protection plans could be set up to alert and guide the population on how
to behave in the face of heatwaves. Concrete actions like raising awareness of population about heat-
waves, changing working hours, opening air-conditioned reception areas, distributing water in public
transport, monitoring the most vulnerable at home, and limiting or prohibiting outdoor sports activ-
ities could also affect heat-related mortality. Considering these acclimatization factors offers research
perspectives to be explored in the construction of prospective scenarios.

**Data and code availability:** The results in this paper is obtained using R. The numerical results presented in this manuscript were reproduced by the CASCaD on 8 July 2025. All code used for the analyses are publicly available in the following repository: https://github.com/qguibert/mortalityheatwaves.

**Author Contributions:** Q.G., G.P. and F.P. design the study. Q.G. and G.P. collect, cure and verify the data. Q.G. designs the models and the methodology. Q.G. implements methods, produces results and visualizations. Q.G. and G.P. write the manuscript. Q.G., G.P and F.P. edit the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors are very grateful for the useful suggestions of the two anonymous referees, which led to significant improvements of this article. This paper also benefits from fruitful discussions with members of the *Réseau thématique MATRISK* during the Inaugural Conference of this group at Le Mans (France) in June 2024. The authors would like to thank the International Actuarial Association for awarding this paper the Best Paper Award from the LIFE Section at the 2024 Joint Colloquium of all IAA Sections (JOCO), held in Brussels, Belgium, on 22-25 September 2024. For the purpose of Open Access, a CC-BY public copyright licence has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.
<figure><img src="image_15.png" /><figcaption>Figure 14: Representation of deviance residuals for DLNM models associated with age groups 0-64, 65-74, 75-84, and 85+ for women in metropolitan France for the years between 1980 and 2019.</figcaption></figure>
# A Heatwaves in France

Heatwaves represent a particular risk of acute mortality and their importance has increased over the past 20 years. To understand this phenomenon for the studied population, this appendix provides a synthetic overview of historical recent heatwaves in metropolitan France since 1950. Additional details are provided by Pincemin (2021).

Heatwaves do not have a single definition. Depending on the temperature distribution and the adaptation capacity of populations in the region concerned, the definitions and tolerance thresholds adopted may vary. In particular, the identification of a heatwave may depend on the climate, urbanization of the area concerned, as well as indicators measuring its intensity (average temperature, daytime or nighttime temperature, duration of the reference period, humidity rate, etc.). In the context of this study on French population, we define a heatwave according to the criteria used for heatwave alerts by Météo France, the French national meteorological service. These criteria are also used by Santé Publique France (Santé Publique France, 2019b). Météo France defines a heatwave period as a period where the average of daily temperatures during the day and at night exceeds the 99.5% percentile of observations made between 1981 and 2010.

Gradually, Western European countries such as France have experienced a decrease in the intensity and frequency of cold spells (Météo France, 2024). Conversely, heatwaves have become more frequent, and summers are increasingly hotter. In France, the major heatwaves of 1976 and 2003 resulted in an excess of 4,500 and 15,000 deaths, respectively. In recent years, heatwaves have occurred almost every single year with varying intensities. Some of them have had temperatures similar to the historical heatwave of 2003. In 2019 and 2020, they even reached the highest level of the alert plan for the first time in history.

Their consequences in terms of mortality are listed in Table 4, which contains data compiled from *Santé Publique France* and the Centre for Research on the Epidemiology of Disasters (CRED) Database. It is noteworthy that the dates and duration of each heatwave are calculated on average by department and are extracted from the annual reports of *Santé Publique France*. Considering the average for the entirety of France would result in a shorter duration of heatwaves. For comparative purposes, the CRED data is also included in this table. Significant differences in excess mortality attributed to these various events are observed depending on the data source, due to variations in the methodologies employed.

In 2022, we observed an excess mortality of 73 deaths per million (26, 124), which corresponds to 4,807 (1,739, 8,123)<fnref n="5" /> deaths (Ballester et al., 2023).

# B Monthly and annual mortality data

In this appendix, we reconcile monthly and annual mortality data. These data originate from different sources and we compare in Figure 12 annual discrepancies in death counts by age group and sex. For the period from 1980 to 1998, we observe slight differences between the two data sources. Beyond this period, the two sources provide equivalent death counts.

Regarding the DLNM model, these minor gaps are not expected to influence the estimation conducted in Section 4.1, as the differences are not specifically causes by temperature or seasonal mortality variations. Given the small magnitude of the observed differences, it is reasonable to proportionally adjust the daily temperature-attributable death counts in Equation (19) using the annual death counts provided by the HMD database. This adjustment does not affect the estimation of the total attributable fraction given in Equation (20), which is used to adjust the exposure to risk for the Li-Lee model estimation in Section 4.3. For this estimation, we choose to rely on the annual HMD data rather than the aggregated daily data, since the former is publicly accessible which facilitates the reproducibility of our results.

<footnote n="5">⁵The values in parentheses represent the 95% confidence intervals.</footnote>
<figure><img src="image_16.png" /><figcaption>Figure 15: Representation of deviance residuals for DLNM models associated with age groups 0-64, 65-74, 75-84, and 85+ for men in metropolitan France for the years between 1980 and 2019.</figcaption></figure>
<table><caption>Table 4: Main heatwaves in Metropolitan France since 1976. This table presents for each year the cumulative number of heatwave days and the attributed number of deaths, calculated on average per department by *Santé Publique France*. These data are extracted from the *Bulletins de Santé Publique France* (reports analyzing the departmental, regional, and national situation) on the relevant summers (Santé Publique France, 2018; Santé Publique France, 2019a; Santé Publique France, 2020) and from the publication of the Institut de Veille Sanitaire (2015). Data from the earlier years (1976, 1983, 2003, 2006) are sourced from a study on heatwaves since 1970 (Pascal et al., 2019). The CRED data is drawn from the EM'DAT database (Centre for Research on the Epidemiology of Disasters, 2021). This open and free database is a project of the Integrated Research on Disaster Risk. It is one of the three largest databases for disasters, along with MunichRe's NatCatSERVICE and SwissRe's Sigma. It is fed by numerous governmental and non-governmental sources, insurers and reinsurers, press agencies, specialized entities and various global organizations (United Nations, SwissRe, MunichRe, NOAA, etc.).</caption><thead><tr><th rowspan="2">Key information</th><th colspan="2">Excess Death</th></tr><tr><th>Santé Publique France</th><th>Cred Em'Dat</th></tr></thead><tbody><tr><td><strong>1976</strong><br>Late June - mid-July</td><td>4,540</td><td>-</td></tr><tr><td><strong>1983</strong><br>July 9 - July 31 (22 days)</td><td>2,900</td><td>-</td></tr><tr><td><strong>2003</strong><br>The summer of 2003 was the hottest in France and Europe in over 50 years, with exceptional intensity during the first half of August. This episode was associated with significant ozone pollution.</td><td>14,800</td><td>19,490</td></tr><tr><td><strong>2006</strong><br>July 2006 witnessed a very intense heatwave, although less severe than in 2003, but with a longer duration.</td><td>1,442</td><td>1,388</td></tr><tr><td><strong>2015</strong><br>Three waves unevenly distributed over the territory. The first was quite intense, affecting the north for 10 days. The second, lasting 11 days, was less intense and concentrated in the south-east. The last one, shorter, was very localized in some eastern and southern departments.</td><td>2,040</td><td>3,275</td></tr><tr><td><strong>2018</strong><br>A two-week heatwave peaking in early August, exposing 70% of the French population.</td><td>1,480</td><td>-</td></tr><tr><td><strong>2019</strong><br>Two very extensive and intense heatwaves. For the first time, several departments were under red alert during both waves.</td><td>2,034</td><td>1,435</td></tr><tr><td><strong>2020</strong><br>Three heatwaves, one particularly severe in the north, resulting in a red alert for the second consecutive year, over a cumulative period of around ten days.</td><td>1,924</td><td>1,924</td></tr></tbody></table>

# C The DLNM model

## C.1 Visual examination of predicted death counts

In this appendix, we present the predicted all-cause death counts obtained from the DLNM model, as given by Equation (16), and visually assess its ability to reproduce the seasonality of daily deaths across age groups and genders. Figures 13 and 14 respectively display the observed daily deaths (grey points) and those predicted by the DLNM model (black line) for women and men by age group. The results show the model's ability to reproduce the daily death patterns across all-cause mortality for each age group. Notably, the model effectively captures seasonality, which intensifies with age, and
## B.2 Sensitivity analysis

In order to present the sensitivity to the number of days used to estimate the cumulative effects of the temperature-mortality association, we present in Figure 16 the response curves in terms of RR with a lag of 14 days. Compared to Figure 5, the Relative Risk (RR) of mortality over a 21-day period does not capture the longer-term effects related to periods of heat and cold, such as the harvesting effect associated with certain extreme heatwaves. This results in underestimated effects, mainly for the extremes of heat and cold periods.

<figure><img src="image_17.png" /></figure>

Figure 16: Cumulative relative risk of mortality over a 14-day period in Metropolitan France calculated for the years 1980-2019 for women (red) and men (blue) across age groups 0-64, 65-74, 75-84, and 85+, with their 95% confidence intervals (shaded areas). The dashed vertical lines represent the quantiles of daily mean temperatures at 1% (blue), 5% (cyan), 95% (orange), and 99% (red) calculated over the period 1980-2019. Daily average temperatures are calculated for each city, and then an average of 14 cities is used to derive the daily average temperatures for Metropolitan France. Temperatures are measured in degrees Celsius. Based on GHCN data.
<figure><img src="image_13.png" /><figcaption>Figure 12: Comparison of annual death counts from daily data (INSEE, 2020) and annual data (HMD, 2024) for both women and men with age groups 0-64, 65-74, 75-84, and 85+. For daily data, deaths are aggregated annually. Observation period: 1980-2019, based on INSEE and HMD data.</figcaption></figure>

reproduces long-term trends driven primarily by changes in population age structure and aging.

<figure><img src="image_14.png" /><figcaption>Figure 13: Comparison of observed daily deaths (grey points) and predicted deaths (black line) for women. Observation period: 1980-2019, based on INSEE data</figcaption></figure>

Figure 15 decomposes daily death counts for women into temperature-attributable deaths, as given by Equation (19), and non-temperature-attributable deaths, obtained by difference with the all-cause predicted death counts in Equation (16). These visualizations align with our model specifications in Equation (15), where temperature-attributable deaths exhibit seasonal dynamics linked to temperature patterns (Figure 17), and non-temperature-attributable deaths display residual seasonality and distinct trends. Consistent with Figure 13, the seasonal effect attributable to temperatures becomes more
# C Additional figures on projected temperature-attributable fractions

## C.1 Attributable fraction per age bucket and sex

Figures 17, 18, 19 and 20 describe the projected temperature-attributable fractions simulated by sex and age group.

<figure><img src="image_18.png" /></figure>

Figure 17: Temperature attributable fraction in Metropolitan France, simulated for the years 2020-2100 for both women and men across all age groups. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.
<figure><img src="image_15.png" /><figcaption>Figure 14: Comparison of observed daily deaths (grey points) and predicted deaths (black line) for men. Observation period: 1980-2019, based on INSEE data</figcaption></figure>

pronounced with age. Figure 16 illustrates a similar decomposition for the male population.
<figure><img src="image_19.png" /><figcaption>Figure 18: Temperature attributable fraction in Metropolitan France, simulated for the years 2020-2100 for both women and men across for the age group 65-74. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure 15: Decomposition of daily death counts for women predicted by the DLNM model into those attributable to temperatures (in red) and those not attributable to temperatures (in black). Observation period: 1980-2019, based on INSEE data</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 16: Decomposition of daily death counts for men predicted by the DLNM model into those attributable to temperatures (in red) and those not attributable to temperatures (in black). Observation period: 1980-2019, based on INSEE data.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Figure 19: Temperature attributable fraction in Metropolitan France, simulated for the years 2020-2100 for both women and men across the age group 75-84. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure 17: Evolution of daily average temperatures. Daily average temperatures are calculated for each city, and then an average of 14 cities is used to derive the daily average temperatures for Metropolitan France. Temperatures are measured in degrees Celsius. Observation period: 1980-2019, based on GHCN data.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>Figure 20: Temperature attributable fraction in Metropolitan France, simulated for the years 2020-2100 for both women and men across for the age group 85+. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>
## C.2 Goodness of fit analysis and model checking

In this appendix, we conduct a statistical analysis to evaluate the goodness of fit of the DLNM model. Since, the attributable and non-attributable death counts related to temperature are unobservable, they cannot be used to evaluate the model's performance. Therefore, we compare the observed daily death counts $D_{k,t,d}^{(g)}$ with those predicted by the DLNM model. To ensure that the model accurately replicates both the seasonality of daily deaths and long-term trends, we examine the monthly distribution of deaths over the four decades comprising the period 1980–2019.

Figures 18 and 19 compare the monthly distributions of observed and modeled death counts over the observation period for females and males. We observe a close similarity in the monthly distributions. For females, the model replicates the distribution of deaths accurately, except for some extreme values. For males, the model fits less well in the central part of the distribution, but it still provides a good overall fit, especially in the last decade.

<figure><img src="image_19.png" /><figcaption>Figure 18: Monthly distribution of observed (blue) and predicted (green) numbers of deaths based on the DLNM model (15) per year for women in Metropolitan France for the years between 1980 and 2019. The distributions are grouped by decade.</figcaption></figure>

As suggested by Bhaskaran et al. (2013), we examine the deviance residuals to check for any residual trends not captured by the DLNM model. Figures 20 and 21 present the analysis of deviance residuals by age group for women and men. Since seasonality and long-term trends are controlled in Equation (15), the results indicate a relatively good fit to the data for both women and men, as well as the absence of observable trends in deviance residuals. It is worth noting a few extreme points that the model did not capture well, such as the 2003 heatwave for the 85+ age group, which was an event of extreme magnitude. Extreme variations in the number of deaths could potentially be explained by other factors not included in our model, such as air pollution, flu epidemics, etc.
## C.2 Attributable fraction per city

In this appendix, Figures 21, 22, 23, 24 and 25 respectively present the projected fractions of deaths attributable to temperatures for the cities of Brest, Marseille, Paris, Perpignan and Strasbourg.

<figure><img src="image_22.png" /></figure>

Figure 21: Temperature attributable fraction in Brest, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.

<figure><img src="image_23.png" /></figure>

Figure 22: Temperature attributable fraction in Marseille, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.
<figure><img src="image_20.png" /><figcaption>Figure 19: Monthly distribution of observed (blue) and predicted (green) numbers of deaths based on the DLNM model (15) per year for men in Metropolitan France for the years between 1980 and 2019. The distributions are grouped by decade.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Figure 23: Temperature attributable fraction in Paris, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Figure 24: Temperature attributable fraction in Perpignan, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>Figure 20: Representation of deviance residuals for DLNM models associated with age groups 0-64, 65-74, 75-84, and 85+ for women in Metropolitan France from 1980 to 2019. The blue curve represents the smoothed residuals, while the dashed lines indicate the +2 and -2 thresholds.</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Figure 25: Temperature attributable fraction in Strasbourg, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Figure 21: Representation of deviance residuals for DLNM models associated with age groups 0-64, 65-74, 75-84, and 85+ for men in Metropolitan France from 1980 to 2019. The blue curve represents the smoothed residuals, while the dashed lines indicate the +2 and -2 thresholds.</figcaption></figure>
D Additional figures on projected life expectancy losses by city

In this appendix, Figures 26, 27, 28, 29 and 30 respectively present the projected Life expectancy at birth losses attributable to overall temperatures and extreme hot temperatures for the cities of Brest, Marseille, Paris, Perpignan and Strasbourg.

<figure><img src="image_27.png" /><figcaption>Figure 26: Life expectancy at birth lost in Brest, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>
## C.3 Sensitivity analysis

As explained in Section 4.2, the proportion of deaths attributable to temperatures is not directly observable but is instead predicted by a model. Following the recommendations of Bhaskaran et al. (2013), we compare in this appendix various specifications of the DLNM model to assess the robustness of our estimates.

### C.3.1 Degrees of freedom and knots hyperparameters

First, we compare different specifications of the DLNM model and assess the model's goodness of fit across varying hyperparameters used in Equation (15). There is no definitive consensus in the empirical literature regarding these choices. For instance, Bhaskaran et al. (2013) recommend using 7 knots per year for the natural cubic B-spline in the lag-response curve $w(\cdot)$. They strikes that this is "a balance between providing adequate control for seasonality and other confounding by trends in time, while leaving sufficient information from which to estimate exposure effects." Other studies, such as Gasparrini and al. (2015), Gasparrini et al. (2017), Vicedo-Cabrera et al. (2018), Lee et al. (2020), Martínez-Solanas et al. (2021), and Ballester et al. (2023), use eight knots, while Lee et al. (2018) employ seven and nine knots. For modeling warm periods only, Guo et al. (2018) and Vicedo-Cabrera et al. (2021) use four knots. We thus investigate the robustness of the choices made in Section 4.

We examine the sensitivity of the DLNM model to the number of internal knots for the natural cubic B-spline in the lag-response curve $w(\cdot)$, as well as to the degrees of freedom for $h_p(\cdot)$ which controls the residual seasonality. We have also tested the impact of the placement of internal knots for the natural cubic B-spline of the exposure-response curve $f(\cdot)$. Different specifications were considered, such as three internal knots placed at the 25th, 50th, and 75th percentiles of the daily average temperature distribution, or four internal knots placed at the 20th, 40th, 60th, and 80th percentiles. Although these specifications can sometimes slightly improve model fit, we ultimately retain the initial configuration with three internal knots placed at the 10th, 75th, and 90th percentiles of the daily average temperature distribution. Indeed, this choice minimizes overfitting and facilitates the extrapolation of results to extreme temperatures. Therefore, we only present the results of the sensitivity analysis for the first two hyperparameters.

Here, we vary the degrees of freedom per year between 4 and 12 for $h_p(\cdot)$ functions, corresponding to one degree of freedom per quarter or per month. Additionally, we evaluate the impact of the number of internal knots between 1 and 10 used for the natural cubic B-spline in the lag-response curve $w(\cdot)$. The combination of these two hyperparameters leads us to compare 90 specifications of the DLNM model. Following the notation from Section 3.3.2, we compute the modified Akaike and Bayesian information criteria for GLMs with overdispersed responses, defined as follows for $k \in \{1, \dots, K\}$ and $g \in \mathcal{G}$

$$ \text{QAIC}_{k}^{(g)} = -2\mathcal{L}(\hat{\eta}_{k}^{(g)}, \hat{\theta}_{k}^{(g)}, \hat{\zeta}_{k,1}^{(g)}, \hat{\zeta}_{k,y_{\min}}^{(g)}, \dots, \hat{\zeta}_{k,y_{\max}}^{(g)}) + 2\hat{\varphi}_{k}^{(g)}\xi_{k}^{(g)}, $$

$$ \text{QBIC}_{k}^{(g)} = -2\mathcal{L}(\hat{\eta}_{k}^{(g)}, \hat{\theta}_{k}^{(g)}, \hat{\zeta}_{k,1}^{(g)}, \hat{\zeta}_{k,y_{\min}}^{(g)}, \dots, \hat{\zeta}_{k,y_{\max}}^{(g)}) + \ln(n_{k}^{(g)})\hat{\varphi}_{k}^{(g)}\xi_{k}^{(g)}, $$

where $\xi_k^{(g)}$ is the number of parameters, $n_k^{(g)}$ is the number of observations, $\mathcal{L}$ represents the log-likelihood of the DLNM fitted with parameters $\hat{\eta}_k^{(g)}, \hat{\theta}_k^{(g)}, \hat{\zeta}_{k,1}^{(g)}, \hat{\zeta}_{k,y_{\min}}^{(g)}, \dots, \hat{\zeta}_{k,y_{\max}}^{(g)}$, and $\hat{\varphi}_k^{(g)}$ is the estimated dispersion parameter. The QAIC and QBIC values are then summed across each subgroup $k$ and for each sex $g$.

Table 5 compares the QAIC and QBIC values of the ten best models, ranked by QBIC, to the model selected in Section 6.2. The model with 6 degrees of freedom and 2 knots achieves the lowest QBIC statistic. Our model ranks 8th, but its QBIC is close to those of the top models. However, it outperforms them according to the QAIC statistic.

Finally, we examine the sensitivity of the curve representing the cumulative temperature-mortality association estimated for each of these 10 DLNM models. Figures 22 and 23 respectively compare
<figure><img src="image_28.png" /><figcaption>Figure 27: Life expectancy at birth lost in Marseille, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>
<table><caption>Table 5: Comparison of QAIC and QBIC statistics for the 10 best DLNM models estimated over the period 1980-2019, varying the number of knots for the lag-response curve and the degrees of freedom associated with residual seasonality. These top ten models are ranked according to the QBIC statistic. The results of the model used in Section 6.2 are presented in bold.</caption><thead><tr><th>Model</th><th>Number of knots (lag-response)</th><th>Seasonality degree</th><th>QAIC</th><th>QBIC</th></tr></thead><tbody><tr><td>Model 1</td><td>2</td><td>6</td><td>952354.8</td><td>971552.8</td></tr><tr><td>Model 2</td><td>2</td><td>5</td><td>954926.7</td><td>971594.7</td></tr><tr><td>Model 3</td><td>3</td><td>6</td><td>952213.0</td><td>971743.6</td></tr><tr><td>Model 4</td><td>2</td><td>8</td><td>947781.3</td><td>971793.2</td></tr><tr><td>Model 5</td><td>3</td><td>5</td><td>954789.9</td><td>971802.5</td></tr><tr><td>Model 6</td><td>4</td><td>6</td><td>951960.9</td><td>971807.6</td></tr><tr><td>Model 7</td><td>4</td><td>5</td><td>954542.9</td><td>971885.8</td></tr><tr><td><strong>Model 8</strong></td><td><strong>3</strong></td><td><strong>8</strong></td><td><strong>947653.4</strong></td><td><strong>971979.6</strong></td></tr><tr><td>Model 9</td><td>2</td><td>7</td><td>950322.1</td><td>972019.5</td></tr><tr><td>Model 10</td><td>4</td><td>8</td><td>947410.4</td><td>972028.9</td></tr></tbody></table>

the cumulative relative risk of mortality over a 21-day period for women and men. These figures demonstrate the robustness of our model (Model 8) to variations in the hyperparameters listed in Table 5.
<figure><img src="image_29.png" /><figcaption>Figure 28: Life expectancy at birth lost in Paris, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Figure 22: Cumulative relative risk of mortality over a 21-day period in Metropolitan France calculated for the years 1980-2019 for women across age groups 0-64, 65-74, 75-84, and 85+ for the ten best DLNM models. Daily average temperatures are calculated for each city, and then an average of 14 cities is used to derive the daily average temperatures for Metropolitan France. Temperatures are measured in degrees Celsius. Based on GHCN and INSEE data.</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>Figure 29: Life expectancy at birth lost in Perpignan, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Figure 23: Cumulative relative risk of mortality over a 21-day period in Metropolitan France calculated for the years 1980-2019 for men across age groups 0-64, 65-74, 75-84, and 85+ for the ten best DLNM models. Daily average temperatures are calculated for each city, and then an average of 14 cities is used to derive the daily average temperatures for Metropolitan France. Temperatures are measured in degrees Celsius. Based on GHCN and INSEE data.</figcaption></figure>
<figure><img src="image_31.png" /><figcaption>Figure 30: Life expectancy at birth lost in Strasbourg, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>
### C.3.2 Lag hyperparameter

The choice of lag for the function $s(\cdot, \cdot)$ in Equation (13) influences the short-term seasonality attributable to temperature. In studies focusing exclusively on the effects of heatwaves (Guo et al., 2018; Vicedo-Cabrera et al., 2021), the impact of temperature on mortality typically dissipates after approximately ten days. However, when accounting for the effects of cold, a longer lag period is necessary. For instance, Gasparrini et al. (2017) use 21 days. If the lag is set too long, the $s(\cdot, \cdot)$ component of the DLNM model (15) might inadvertently capture seasonal mortality effects not related to temperature. To remain consistent with the literature, we limit the lag to 21 days and we analyze in this appendix the impact of shorter lags on the cumulative effects of the temperature-mortality association.

To illustrate the sensitivity to the number of lagged days, Figures 24 and 25 display the response curves in terms of cumulative relative risk (RR) for women and men, respectively, using lags of 7, 14, and 21 days. Differences in lag selection primarily affect the excess mortality attributable to cold, while the impact on heat-related excess mortality is less pronounced. Specifically, the RR of mortality for 7- and 14-day periods fails to capture the longer-term effects of cold and to a lesser extent the delayed effects of heat, such as the harvesting effect associated with certain extreme heatwaves.
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### C.3.3 Calibration period length

The length of the period used for calibrating the DLNM model may influence the estimated short-term mortality response to temperature. In this appendix, we assess the sensitivity of this response with respect to the calibration period length. Figures 26 and 27 display the response curves in terms of cumulative relative risk (RR) for women and men, respectively, using calibration periods ending in 2011, 2013, 2015, 2017 and 2019. We observe that the model continues to capture the temperature-mortality association accurately, suggesting that the estimates are not materially affected by the length of the calibration period. Minor differences in the RR curves are only observed for extreme heat levels.

<figure><img src="image_27.png" /><figcaption>Figure 26: Cumulative relative risk of mortality over a 21-day period in Metropolitan France, estimated for women across the age groups 0-64, 65-74, 75-84, and 85+. We compare five scenarios in which the DLNM is trained on different historical periods: 1980-2011, 1980-2013, 1980-2015, 1980-2017, and 1980-2019. Daily average temperatures are calculated for each city, and then an average of 14 cities is used to derive the daily average temperatures for Metropolitan France. Temperatures are measured in degrees Celsius. Based on GHCN and INSEE data.</figcaption></figure>
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# D Consistency of projected central death rates and out-of-sample forecasting analysis

## D.1 Consistency of projected central death rates

In this appendix, we verify the consistency of the projections obtained from the original Li-Lee model (26) and the Li-Lee model with adjusted exposures to risk (5). This verification is carried out for the different methods of calculating the temperature-attributable fraction used in this paper. Indeed, the adjustment of death rates can be performed in two ways:

1. For historical data, the annual attributable fractions $AF_{x,t}^{(g)}$ are estimated using Equation (20), which relies on daily deaths. The central death rates are then obtained via Equation (3).

2. During the projection phase, daily deaths are unknown. In this case, we use Equation (23), using both constant weights $\omega_{x,t,d}^{(g)}$ and weights calibrated on historical data.

Figures 28 and 29 illustrate the trajectories of the crude central death rates at ages 45, 65, 75, and 85 for females and males, comparing them with the central death rates predicted by the two approaches described above. We specifically distinguish two sub-cases when central death rates are calculated using Equation (23): one considering constant daily weights and the other using weights estimated based on daily deaths unaffected by temperature from the last five years of our historical data. First, we note that the death rates obtained from the original Li-Lee model and those predicted using the attributable fractions from Equation (20) are strictly equivalent. This is expected, as we apply Equation (3), while the Li-Lee model is estimated based on adjusted exposure to risk. When death rates are derived from Equation (23), a slight bias may appear compared to predictions from the original Li-Lee model. Empirically, we observe however that this bias is small and decreases when non-constant weights $\omega_{x,t,d}^{(g)}$ are applied in Equation (23).
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<figure><img src="image_29.png" /><figcaption>Figure 28: Comparison of observed and predicted central death rates at ages 45, 65, 75, and 85 for females over the observation period. The red curves represent the crude death rates. The blue curves correspond to the central death rates predicted by the original Li-Lee model. The green curves represent the central death rates predicted using Equation (4), after estimating the total attributable fractions via Equation (20). The purple curves utilize Equation (23) with constant weights, while the orange curves use weights estimated based on daily deaths unaffected by temperature from the last five years of our historical data.</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>Figure 29: Comparison of observed and predicted central death rates at ages 45, 65, 75, and 85 for males over the observation period. The red curves represent the crude death rates. The blue curves correspond to the central death rates predicted by the original Li-Lee model. The green curves represent the central death rates predicted using Equation (4), after estimating the total attributable fractions via Equation (20). The purple curves utilize Equation (23) with constant weights, while the orange curves use weights estimated based on daily deaths unaffected by temperature from the last five years of our historical data.</figcaption></figure>
## D.2 Out-of-sample forecasting analysis

In this appendix, we analyze the out-of-sample forecasting performance of our Li-Lee model with adjusted exposure to risk and compare it with that of a standard Li-Lee model estimated using original exposure to risk. To this end, both models are calibrated on data from 1980 to 2011, as described in Section 4, and evaluated over the out-of-sample period $T_y^{ofs} = \{2012, \dots, 2019\}$. During this testing period, we project central death rates using the approach outlined in Section 5. The DLNM predictions are obtained using the observed average temperatures from 2012 to 2019. Indeed, we do not simulate a temperature scenario, as temperature is treated as an exogenous input in our framework.

Figure 30 first illustrates the log-mortality predictions for both Li-Lee models in 2019, showing their ability to reproduce observed death rates across all ages. For both men and women, the Li-Lee model with adjusted exposure to risk slightly outperforms the original Li-Lee model. We then assess the forecasting performance of the two models across the full out-of-sample period, both by age and by year. For that, we define the Root Mean Squared Error (RMSE) for each year $t \in T_y^{ofs}$ and for each age $x \in \mathcal{X}$ as follows

$$ \text{RMSE}_{t}^{(g)} = \sqrt{\frac{1}{|\mathcal{X}|} \sum_{x \in \mathcal{X}} \left( \ln(\hat{\hat{m}}_{x,t}^{(g)}) - \ln(\hat{m}_{x,t}^{(g)}) \right)^2}, $$

$$ \text{RMSE}_{x}^{(g)} = \sqrt{\frac{1}{|T_y^{ofs}|} \sum_{t \in T_y^{ofs}} \left( \ln(\hat{\hat{m}}_{x,t}^{(g)}) - \ln(\hat{m}_{x,t}^{(g)}) \right)^2}. $$

<figure><img src="image_31.png" /><figcaption>Figure 30: Comparison of observed and predicted log-mortality rates for men and women in 2019. These values correspond to the median of predictions obtained from 1,000 Monte Carlo simulations for the both Li-Lee models.</figcaption></figure>

The RMSE by year and by age are compared in Figures 31 and 32 respectively. Both models yield similar predictions for short-term mortality trends. However, the Li-Lee model with adjusted exposure to risk appears to perform better over time. This improvement is likely due to the additional information incorporated through the observed temperatures. The age-specific analysis further suggests that our Li-Lee model provides more accurate predictions at ages above 50, particularly for women. Finally, Table 6 reports the RMSE and Mean Absolute Error (MAE) computed over all ages
across the full out-of-sample period $\mathcal{T}_y^{ofs}$

$$ \text{RMSE}^{(g)} = \sqrt{\frac{1}{|\mathcal{X}| |\mathcal{T}_y^{ofs}|} \sum_{x \in \mathcal{X}} \sum_{t \in \mathcal{T}_y^{ofs}} \left( \ln(\hat{m}_{x,t}^{(g)}) - \ln(\hat{m}_{x,t}^{(g)}) \right)^2}, $$

$$ \text{MAE}^{(g)} = \frac{1}{|\mathcal{X}| |\mathcal{T}_y^{ofs}|} \sum_{x \in \mathcal{X}} \sum_{t \in \mathcal{T}_y^{ofs}} |\ln(\hat{m}_{x,t}^{(g)}) - \ln(\hat{m}_{x,t}^{(g)})|. $$

Overall, it shows that the Li-Lee model with adjusted exposure to risk performs slightly better than the original Li-Lee model.

<figure><img src="image_32.png" /><figcaption>Figure 31: Comparison of RMSE of predicted log-mortality rates by year for the original Li-Lee model and those with adjusted exposure over the out-of-sample period. The median of predictions is obtained from 1,000 Monte Carlo simulations for both Li-Lee models.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>Figure 32: Comparison of RMSE of predicted log-mortality rates by age for the original Li-Lee model and those with adjusted exposure over the out-of-sample period. The median of predictions is obtained from 1,000 Monte Carlo simulations for both Li-Lee models.</figcaption></figure>
<table><caption>Table 6: The overall RMSE and MAE for the original Li-Lee model and the model with adjusted exposure over the out-of-sample period</caption><thead><tr><th>Sex</th><th>Model</th><th>RMSE</th><th>MAE</th></tr></thead><tbody><tr><td>Females</td><td>Original Li-Lee model</td><td>0.1682</td><td>0.1270</td></tr><tr><td>Females</td><td>Adjusted exposure to risk</td><td>0.1612</td><td>0.1158</td></tr><tr><td>Males</td><td>Original Li-Lee model</td><td>0.1460</td><td>0.0982</td></tr><tr><td>Males</td><td>Adjusted exposure to risk</td><td>0.1401</td><td>0.0932</td></tr></tbody></table>
# E Additional figures on projected temperature-attributable fractions

## E.1 Attributable fraction per age bucket and sex

Figures 33, 34, 35 and 36 describe the projected temperature-attributable fractions simulated by sex and age group.

<figure><img src="image_34.png" /></figure>

Figure 33: Temperature attributable fraction in Metropolitan France, simulated for the years 2020-2100 for both women and men across for the age group 0-64. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.
<figure><img src="image_35.png" /><figcaption>Figure 34: Temperature attributable fraction in Metropolitan France, simulated for the years 2020-2100 for both women and men across all age groups. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>
<figure><img src="image_36.png" /><figcaption>Figure 35: Temperature attributable fraction in Metropolitan France, simulated for the years 2020-2100 for both women and men across for the age group 75-84. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>
<figure><img src="image_37.png" /><figcaption>Figure 36: Temperature attributable fraction in Metropolitan France, simulated for the years 2020-2100 for both women and men across for the age group 85+. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>
## E.2 Attributable fraction per city

In this appendix, Figures 37, 38, 39, 40 and 41 respectively present the projected fractions of deaths attributable to temperatures for the cities of Brest, Marseille, Paris, Perpignan and Strasbourg.

<figure><img src="image_38.png" /></figure>

Figure 37: Temperature attributable fraction in Brest, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.

<figure><img src="image_39.png" /></figure>

Figure 38: Temperature attributable fraction in Marseille, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.
<figure><img src="image_40.png" /><figcaption>Figure 39: Temperature attributable fraction in Paris, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>

<figure><img src="image_41.png" /><figcaption>Figure 40: Temperature attributable fraction in Perpignan, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>
<figure><img src="image_42.png" /><figcaption>Figure 41: Temperature attributable fraction in Strasbourg, simulated for the years 2020-2100 for both women and men across. These values, along with their 95% confidence intervals (shaded areas) are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients and the ensemble of climate models. Each line corresponds to the average of these simulations. Attributable fraction is expressed in % for all effect, decomposed into moderate cold, moderate hot, extreme cold and extreme hot.</figcaption></figure>
F Additional figures on projected life expectancy losses by city

In this appendix, Figures 42, 43, 44, 45 and 46 respectively present the projected life expectancy at birth losses attributable to overall temperatures and extreme hot temperatures for the cities of Brest, Marseille, Paris, Perpignan and Strasbourg.

<figure><img src="image_43.png" /><figcaption>Figure 42: Life expectancy at birth lost in Brest, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>
<figure><img src="image_44.png" /><figcaption>Figure 43: Life expectancy at birth lost in Marseille, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>
<figure><img src="image_45.png" /><figcaption>Figure 44: Life expectancy at birth lost in Paris, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>
<figure><img src="image_46.png" /><figcaption>Figure 45: Life expectancy at birth lost in Perpignan, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>
<figure><img src="image_47.png" /><figcaption>Figure 46: Life expectancy at birth lost in Strasbourg, simulated for the years 2020-2100 for both women and men. We present both the loss related to all temperature effects and extreme hot effects only. These values, along with their 95% confidence intervals (shaded areas), are calculated through 1,000 Monte Carlo simulations of the DLNM coefficients, the Li-Lee model and the ensemble of climate models. Each line corresponds to the median of these simulations.</figcaption></figure>
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