Interrupting vaccination policies can greatly spread SARS-CoV-2 and enhance mortality from COVID-19 disease: The AstraZeneca case for France and Italy

As of March 2021, the spread of the SARS-CoV-2 virus¹1. J. Wu, W. Cai, D. Watkins, and J. Glanz, How The Virus Got Out (The New York Times, 2020). has caused more than $120\times {10}^6$ infections worldwide with a total death toll of more than $2\times {10}^6$. Up to the end of 2020, the only effective measures to contain the spread of the virus were based on social distancing, wearing face masks, and more/less stringent lockdown.^2–42. R. M. Anderson, H. Heesterbeek, D. Klinkenberg, and T. D. Hollingsworth, “How will country-based mitigation measures influence the course of the covid-19 epidemic?,” Lancet 395, 931 (2020). https://doi.org/10.1016/S0140-6736(20)30567-53. M. Chinazzi, J. T. Davis, M. Ajelli, C. Gioannini, M. Litvinova, S. Merler, A. Pastore y Piontti, K. Mu, L. Rossi, K. Sun, C. Viboud, X. Xiong, H. Yu, M. E. Halloran, I. M. Longini, and A. Vespignani, “The effect of travel restrictions on the spread of the 2019 novel coronavirus (covid-19) outbreak,” Science 368, 395 (2020). https://doi.org/10.1126/science.aba97574. H.-Y. Yuan, G. Han, H. Yuan, S. Pfeiffer, A. Mao, L. Wu, and D. Pfeiffer, “The importance of the timing of quarantine measures before symptom onset to prevent covid-19 outbreaks—illustrated by Hong Kong’s intervention model,” medRxiv https://doi.org/10.1101/2020.05.03.20089482 (2020). Later on, a massive vaccination campaign kicked off in several countries thanks to the availability of a variety of vaccines (e.g., AstraZeneca, Johnson&Johnson, Moderna, Pfizer/BionTech, and Sputnik V, among others). Such vaccines differ substantially in terms of efficacy, legal status, availability, and logistics needed for their delivery to patients. According to various estimates,⁵5. D. Sridhar and D. Gurdasani, “Herd immunity by infection is not an option,” Science 371, 230 (2021). https://doi.org/10.1126/science.abf7921 vaccinations would produce a substantial reduction in infections, and eventually yield to “herd immunity” when $\approx 70\mathrm{\%}$ of the population gets fully vaccinated. When such a large fraction of the population becomes immune to the disease, its spread from person to person becomes very unlikely, and the whole community becomes protected. By allowing for an earlier easing of non-medical measures against the SARS-CoV-2 virus, vaccination is also expected to significantly reduce the economical, social, and psychological impacts of lockdown measures.⁶6. N. Fernandes, “Economic effects of coronavirus outbreak (covid-19) on the world economy,” available at SSRN 3557504 (2020). Those estimates assume that there is no break in the supply of vaccines or any other suspension in the procedure due to side effects from vaccination. On March 15, 2021, several European countries suspended the use of the AstraZeneca COVID-19 vaccine as a precaution in order to investigate the death of a few dozens of patients developing blood clots—associated with deep vein thrombosis (DVT)⁷7. M. Cushman, “Epidemiology and risk factors for venous thrombosis,” Semin. Hematol. 44, 62 (2007). https://doi.org/10.1053/j.seminhematol.2007.02.004—after such a vaccine. Health personnel who inoculated the vaccine to those who died as a result of DVT are being investigated in Italy for manslaughter.⁹9. ANSA, March 2021, see https://tinyurl.com/4ywxt5kp. The contingent situation with the widespread COVID-19 pandemic naturally raises the question of whether a prolonged stop in vaccinations coming from adopting the precautionary principle¹⁰10. K. Steele, “The precautionary principle: A new approach to public decision-making?,” Law Probab. Risk 5, 19 (2006). https://doi.org/10.1093/lpr/mgl010 could cause an excess mortality beyond that caused by side effects of the vaccines. In this work, we aim at exploring this issue by computing future COVID-19 epidemic scenarios by comparing (i) the excess mortality caused by reducing the vaccinations using the stochastic susceptible-exposed-infected-recovered (SEIR) model¹¹11. D. Faranda and T. Alberti, “Modeling the second wave of COVID-19 infections in France and Italy via a stochastic SEIR model,” Chaos 30, 111101 (2020). https://doi.org/10.1063/5.0015943 and (ii) the estimates of the possible casualties caused by side effects of a vaccine, namely, those associated with DVT. We remark that the additional, longer-term effect of the presence of higher infection rates, e.g., the increased risk of virus mutations leading to possibly more malignant and/or more infectious variants, is not included in our treatment. Our analysis focuses on France and Italy, which have been among the countries that have been most severely impacted by the COVID-19 pandemic.¹²12. T. Alberti and D. Faranda, “On the uncertainty of real-time predictions of epidemic growths: A covid-19 case study for china and italy,” Commun. Nonlinear Sci. Numer. Simul. 90, 105372 (2020). https://doi.org/10.1016/j.cnsns.2020.105372 An important remark follows. Our goal is not to provide an exact estimate of both (i) and (ii) but rather to perform an order-of-magnitude comparison between excess deaths resulting from different scenarios of vaccination policy. We proceed in the spirit of complexity science, where simple models are useful for elucidating the main mechanisms behind complex behavior and provide useful inputs for the deployment of more advanced modeling suites and data collection strategies.^13–1713. I. M. Held, “The gap between simulation and understanding in climate modeling,” Bull. Am. Meteorol. Soc. 86, 1609 (2005). https://doi.org/10.1175/BAMS-86-11-160914. M. Pascual, M. Roy, and K. Laneri, “Simple models for complex systems: Exploiting the relationship between local and global densities,” Theor. Ecol. 4, 211 (2011). https://doi.org/10.1007/s12080-011-0116-215. U. Gähde, S. Hartmann, and J. Wolf, Models, Simulations, and the Reduction of Complexity, Abhandlungen der Akademie der Wissenschaften in Hamburg (De Gruyter, 2013).16. P. Almaraz, “Simple models, complex models, useful models: Can we tell them from the flap of a butterfly’s wings?,” Front. Ecol. Evol. 2, 54 (2014). https://doi.org/10.3389/fevo.2014.0005417. M. Ghil and V. Lucarini, “The physics of climate variability and climate change,” Rev. Mod. Phys. 92, 035002 (2020). https://doi.org/10.1103/RevModPhys.92.035002 In other words, we will approach the problem by performing Fermi estimates¹⁸18. S. Mahajan, The Art of Insight in Science and Engineering: Mastering Complexity (The MIT Press, Cambridge, 2014). where the classical back-of-the-envelope calculations are performed via the SEIR model, allowing to take into account the uncertainties in both model parameters and data. In nuce, we perform a counterfactual analysis based on a story-line approach, which has become a powerful investigation method for assessing risks coming from extreme events.¹⁹19. T. G. Shepherd, “A common framework for approaches to extreme event attribution,” Curr. Clim. Change Rep. 2, 28 (2016). https://doi.org/10.1007/s40641-016-0033-y While the quantitative consolidation of our results clearly requires extensive data analysis and modeling, our findings show with great confidence that excess deaths due to the interruption of the vaccination campaign largely override those due to DVT even in the worst case scenarios of frequency and gravity of the vaccine side effects. Fermi estimates can provide valuable inputs for an efficient and pragmatic application of the precautionary principle able to reduce the negative impacts of hazards of various natures, as done in economics.²⁰20. P. M. Anderson and C. A. Sherman, “Applying the fermi estimation technique to business problems,” J. Appl. Bus. Econ. 10, 33–42 (2010).

The model²¹21. D. Faranda, I. P. Castillo, O. Hulme, A. Jezequel, J. S. W. Lamb, Y. Sato, and E. L. Thompson, “Asymptotic estimates of sars-cov-2 infection counts and their sensitivity to stochastic perturbation,” Chaos 30, 051107 (2020). https://doi.org/10.1063/5.0008834 with time-dependent control parameters can mimic the dependence on additional/external factors such as variability in the detected cases, different physiological response to the virus, release, or reinforcement of distancing measures.¹¹11. D. Faranda and T. Alberti, “Modeling the second wave of COVID-19 infections in France and Italy via a stochastic SEIR model,” Chaos 30, 111101 (2020). https://doi.org/10.1063/5.0015943 Our compartmental model²²22. F. Brauer, “Compartmental models in epidemiology,” in Mathematical Epidemiology (Springer, 2008), pp. 19–79. divides the population into four groups, namely, susceptible (S), exposed (E), infected (I), and recovered (R) individuals, according to the following discrete-time evolution equations:

$$\begin{array}{rl}{S}_{t+1}& =-\lambda (1-\alpha ){\displaystyle \frac{I_t{S}_t}{N_t}}-\lambda \alpha (1-\sigma ){\displaystyle \frac{I_t{S}_t}{N_t}}+(1-\sigma \alpha )S_t,\end{array}$$

(1)

$$\begin{array}{rl}{E}_{t+1}& =+\lambda (1-\alpha ){\displaystyle \frac{I_t{S}_t}{N_t}}+\lambda \alpha (1-\sigma ){\displaystyle \frac{I_t{S}_t}{N_t}}+(1-ϵ)E_t,\end{array}$$

(2)

$$\begin{array}{rl}{I}_{t+1}& =+ϵE_t+(1-\alpha -\beta )I_t,\end{array}$$

(3)

$$\begin{array}{rl}{R}_{t+1}& =R_t+\sigma \alpha S_t+\beta I_t.\end{array}$$

(4)

In the SEIR model above, the classical parameters are the recovery rate ($\beta$), the inverse of the incubation period ($ϵ$), and the infection rate ($\lambda$). Here, we have generalized the model presented in Faranda and Alberti¹¹11. D. Faranda and T. Alberti, “Modeling the second wave of COVID-19 infections in France and Italy via a stochastic SEIR model,” Chaos 30, 111101 (2020). https://doi.org/10.1063/5.0015943 by introducing two additional parameters able to succinctly mimic the strategies of a vaccination campaign, namely, the vaccination rate per capita $\alpha$ and the vaccine efficacy $\sigma$, see Sun and Hsieh.²³23. C. Sun and Y.-H. Hsieh, “Global analysis of an SEIR model with varying population size and vaccination,” Appl. Math. Model. 34, 2685 (2010). https://doi.org/10.1016/j.apm.2009.12.005 In order to consider uncertainties in long-term extrapolations and time-dependent control parameters, a stochastic approach is used through which the control parameters $\kappa \in \{\alpha ,\beta ,ϵ,\lambda ,\sigma \}$ are described by an Ornstein–Uhlenbeck process²⁴24. G. E. Uhlenbeck and L. S. Ornstein, “On the theory of the Brownian motion,” Phys. Rev. 36, 823 (1930). https://doi.org/10.1103/PhysRev.36.823 with drift as follows:

$$\mathrm{d}\kappa =-\kappa (t)\mathrm{d}t+{\kappa }_0\mathrm{d}t+{\varsigma }_{\kappa }d{W}_t,$$

(5)

where ${\kappa }_0\in \{{\alpha }_0,{\beta }_0,{ϵ}_0,{\lambda }_0,{\sigma }_0\}$, $d{W}_t$ is the increment of a Wiener process. We remind that the basic reproduction number²⁵25. P. L. Delamater et al., “Complexity of the basic reproduction number (r0),” Emerg. Infect. Dis. 25(1), 1–4 (2019). https://doi.org/10.3201/eid2501.171901 is written as $R_0={\beta }_0/{\lambda }_0$. In Eqs. (1)–(5) we set $dt=1$ day, which is the highest time resolution available for official COVID-19–related counts and is relatively small compared to the characteristic times associated with COVID-19 infection, incubation, and recovery/death.

Initializing parameters with their associated reference are shown in Table I. The mortality rate $m_0$ is also shown, set to $0.015$.²⁹29. D. Fanelli and F. Piazza, “Analysis and forecast of COVID-19 spreading in China, Italy and France,” Chaos Solitons Fractals 134, 109761 (2020). https://doi.org/10.1016/j.chaos.2020.109761 While ${\beta }_0$ and ${ϵ}_0$ and the associated $\varsigma$ are the same as in Ref. 1111. D. Faranda and T. Alberti, “Modeling the second wave of COVID-19 infections in France and Italy via a stochastic SEIR model,” Chaos 30, 111101 (2020). https://doi.org/10.1063/5.0015943, the values of ${\sigma }_0$ and respective $\varsigma$ are derived from the range given for the AstraZeneca vaccine phase 3 tests for the first dose,²⁸28. M. Voysey et al., “Safety and efficacy of the chadox1 ncov-19 vaccine (azd1222) against sars-cov-2: An interim analysis of four randomised controlled trials in Brazil, South Africa, and the UK,” Lancet 397, 99 (2021). https://doi.org/10.1016/S0140-6736(20)32661-1 and ${\alpha }_0$ and ${\varsigma }_{\alpha }$ are given supposing that both Italy and France keep vaccinating ${10}^5$ individuals per day with a 20% daily fluctuation.²⁶26. L. Salvioli, “Il vaccino anti covid in italia in tempo reale: Il sole 24 ore” (2021). As in Ref. 1111. D. Faranda and T. Alberti, “Modeling the second wave of COVID-19 infections in France and Italy via a stochastic SEIR model,” Chaos 30, 111101 (2020). https://doi.org/10.1063/5.0015943, we also set ${\varsigma }_{\lambda }=0.2$, allowing for 20% daily fluctuations in the infection rate. Note that here we restrict to Gaussian fluctuations: as shown in Ref. 1111. D. Faranda and T. Alberti, “Modeling the second wave of COVID-19 infections in France and Italy via a stochastic SEIR model,” Chaos 30, 111101 (2020). https://doi.org/10.1063/5.0015943, allowing for log-normal fluctuations of the parameters does not change the average results but slightly enhance their dispersion. See the supplementary material for the numerical code.

Figure 1 reports the daily number of deaths $m_0\times I_t$ as a function of time for Italy (a) and France (b). Initial conditions are set for both countries to the values reported on March 15 as follows: for Italy, we set $N=60\times {10}^6$ for the population, $E_{t=1}=I_{t=1}=20\times {10}^4$ as the infected and exposed populations, $R_{t=1}=11\times {10}^6$ as the sum of $9\times {10}^6$ recovered estimated from serologic tests and $2\times {10}^6$ immunized from 2 doses of either Pfizer/BioNTech, Moderna or AstraZeneca vaccines and $R_0=1.16$. For France, we set $N=67\times {10}^6$, $E_{t=1}=I_{t=1}=25\times {10}^4$, $R_{t=1}=13.2\times {10}^6$ as the sum of $11\times {10}^6$ recovered estimated from serologic tests and $2.2\times {10}^6$ immunized from vaccines and $R_0=1.02$. For both France and Italy, we assume that the virus, after the second wave, has infected 15% of the population. This estimates are based on Pullano et al.³⁰30. G. Pullano, L. Di Domenico, C. E. Sabbatini, E. Valdano, C. Turbelin, M. Debin, C. Guerrisi, C. Kengne-Kuetche, C. Souty, T. Hanslik et al., “Underdetection of cases of covid-19 in France threatens epidemic control,” Nature 590, 134 (2021). https://doi.org/10.1038/s41586-020-03095-6 who reported a $7\mathrm{\%}\pm 3\mathrm{\%}$ total infections for France after the first wave, assuming that the second wave had a similar magnitude for both countries. We remark however, that our results are basically insensitive to oscillation of $S(1)$ of the order of 5 millions individuals (cf. supplementary material Fig. S1). Rather than integrating the Fokker–Planck equation³¹31. H. Risken, The Fokker-Planck Equation (Springer, Berlin, 1996). corresponding to the system of equations given above, neglecting the difference between discrete and continuum time, we follow a Monte Carlo approach and we perform two sets of $N_r=1000$ realizations (see supplementary material Fig. S2 for a justification of this value): stopping (red) and continuing (blue) the vaccination campaign at the same rate. The model is integrated for 500 days, that is approximately the time it would take to vaccine the rest of the susceptible population with AstraZeneca at the rate of ${10}^5$ individuals per day.

First, we observe a monotonic decrease in the daily deaths for all scenarios considered from the initial date $t=t_0$ corresponding to March 15, 2021. This is in agreement with early March estimates that for Italy and France the so-called third wave should reach its peak in the second half of March 2021.³²32. The Guardian, March 2021, see https://tinyurl.com/vvwhcydz. Moreover, we observe that the cumulative number of deaths significantly (we take the width of the error bars as level of significance) reduces if vaccinations are continued at 100 000 doses per day with respect to the scenario where vaccination is stopped. For Italy (France) completely halting the vaccination, at the actual epidemic rate, the number of excess deaths from COVID-19 would amount to $9\pm 3\times {10}^3$ ($1.2\pm 0.4\times {10}^3$) excess deaths from COVID-19. The difference between the two countries is largely due to the value of $R_0$, which is larger for Italy. This suggests that halting vaccination in a growing epidemics phase (Italy) has more dramatic consequences than in a more controlled scenario of $R_0\approx 1$ (France).

Our previous analysis is based on a total stop of AstraZeneca vaccination. However, a more realistic scenario is to assume that AstraZeneca vaccination will resume after a limited number of days used for verification. We investigate this effect in Fig. 2. There, we consider the average excess deaths as a function of the interruption length in number of days (x axis) and $R_0$ (y axis) for Italy (a) and France (b). The excess deaths are computed with respect to a base scenario where vaccine injections are never interrupted and they are averaged over 1000 realizations of the SEIR model. Figure 2 shows that the longer is the vaccine injections disruption, the higher is the number of excess deaths. The impact is stronger for higher values of $R_0$. While waiting the advice of EMA about AstraZeneca safety, many national health agencies also announced that, when allowed, they would resume the vaccination at a higher rate than before, in order to to override the effects of the temporary stop. Hence, in the supplementary Fig. S3, we present a set of simulation where, for a number of days equal to those of the vaccination interruption, injections are performed at a double rate than originally planned, i.e., $2\times {10}^5$ individuals/day, in order to compensate for the lost vaccinations. Although reduced, the number of excess deaths is still high and of the same order of magnitude as the one estimated in Fig. 2, as a result of the nonlinear cascade effect of the extra infections occurred in the period when vaccinations were interrupted. A focus on the actual values of $R_0$ for Italy and France is reported in Fig. 3. Here, we compare the two countries and we also show the effect of doubling vaccination rates. This shows that excess deaths scale down by a factor of two but they remain of the same order of magnitude as for the case of a business-as-usual vaccination rate, namely, ${10}^5$ vaccinations/day.

The final step in our investigation is to compare the previous estimates of excess deaths with an order of magnitude estimate of deaths due to DVT resulting from side effects of the AstraZeneca vaccine. In order to make a meaningful comparison, in a case where uncertainties are very large and hard to quantify, we will consider a worst case scenario for the impacts of the side effects. This scenario relies on the unrealistic hypothesis that the totality of susceptible population to DVT suffers from DVT shortly after being vaccinated, and the lethality rate is similar to the one observed in the overall population.

As of March 15, 2021, few dozens suspect cases of DVT have been reported over a number of $5\times {10}^6$ vaccinated people with AstraZeneca in Europe.³³33. Here, we use European data accessible via the website of the European Medicines Agency at https://tinyurl.com/ht8y98kr to average out the large spread of national data. By suspect cases we mean people who have developed DVT in the few days following the vaccination. This leads us to an estimate of a frequency of 6 cases per million of vaccines. Let us call this rate $r_{AZ}^{DVT}$. Let us also consider that, in the case of France, the incidence of DVT has been estimated to 1800 people per 1 million inhabitants per year,³⁴34. S. Bouée, C. Emery, A. Samson, J. Gourmelen, C. Bailly, and F.-E. Cotté, “Incidence of venous thromboembolism in France: A retrospective analysis of a national insurance claims database,” Thromb. J. 14, 4 (2016). https://doi.org/10.1186/s12959-016-0078-0 with a lethality rate after three months of 5%,³⁵35. J. A. Heit, “Epidemiology of venous thromboembolism,” Nat. Rev. Cardiol. 12, 464 (2015). https://doi.org/10.1038/nrcardio.2015.83 raising to 30% when a period of 5 years is considered.⁷7. M. Cushman, “Epidemiology and risk factors for venous thrombosis,” Semin. Hematol. 44, 62 (2007). https://doi.org/10.1053/j.seminhematol.2007.02.004 This leads to estimating a total of the order of 10 000 deaths per year as a result of DVT. Even assuming that all DVT cases following the inoculation of the AstraZeneca vaccine would have not manifested themselves in the absence of the injection, we have that $N$ vaccinations would lead to an extra $N\times r_{AZ}^{DVT}$ DVT cases. Let us assume that all of these cases result into death.³⁶36. Current data suggest that this is manifestly a gross worst case approximation. We then have that ${10}^5$ daily vaccinations would result into a maximum of 0.6 daily deaths. In 500 days, which is the time needed to cover the entirety of the French population, this leads to an upper bound of 300 deaths. Considering a death rate of $30\mathrm{\%}$, the number scales down to approximately 100, while considering a death rate of $5\mathrm{\%}$ the number scales down to approximately 15. Similar figures apply for Italy.

Decision-making in the presence of strong uncertainties associated with health and environmental risks is an extremely complex process, resulting from the interplay between science, politics, stakeholders, activists, lobbies, media, and society at large.^37–3937. Anonymous, “AGU statement: Investigation of scientists and officials in l’aquila, Italy, is unfounded,” Eos Trans. Am. Geophys. Union 91, 245–252 (2010). https://doi.org/10.1029/2010EO28000538. A. Benessia and B. De Marchi, “When the earth shakes$\dots$ and science with it. The management and communication of uncertainty in the l’aquila earthquake,” Futures 91, 35 (2017), Post-normal science in practice. https://doi.org/10.1016/j.futures.2016.11.01139. J. Reis and P. S. Spencer, “Decision-making under uncertainty in environmental health policy: New approaches,” Environ. Health Prev. Med. 24, 7176 (2019). https://doi.org/10.1186/s12199-019-0813-9 In this letter, we have aimed at contributing to the debate on revolving around the definition of strategies for combating, in conditions of great uncertainties in terms of health and social response, pandemic like the current one caused by the SARS-CoV-2 virus. We have focused on the case of the AstraZeneca COVID-19 vaccine and on the locales of Italy and France, for the period starting on March 15, 2021. The goal is providing a semi-quantitative comparison, based on Fermi estimates informed by a simple yet robust stochastic model, between the excess deaths due to temporal restriction in the deployment of a still experimental vaccine and the excess deaths due to its possible side effects. Given the many uncertainties on the (possible) side effects of the vaccine, we have resorted to making worst case scenario calculations in order to provide a robust upper bound to the related excess deaths. Our results are preliminary and should be supplemented by more detailed modeling and data collection exercises. Indeed: (i) we assume a single vaccine with the nominal AstraZeneca efficacy, neglecting the other available vaccines, (ii) we consider a fixed vaccination rate, (iii) for AstraZeneca DVT side effects we consider French data and rescale them for the Italian populations, (iv) we focused our analysis on DVT side effects, but other pathologies could be considered with the same approach. Yet, these results clearly suggest—see a useful summary in Table II—that the benefits of deploying the vaccine greatly outweigh the associated risks, and that the relative benefits are wider in situations where the reproduction number is larger, and they increase with the temporal duration of the vaccine ban. We have also analyzed the case of resuming the vaccinations at a double rate ($2\times {10}^5$ vaccinations/day) for an amount of days equal to vaccine interruption period (Figs. 3 and S3). This analysis has pointed out that excess deaths are still of the same order of magnitude as those observed by resuming vaccinations with ${10}^5$ vaccinations/day injection rate but scale down by a factor 2. This is a clear outcome of the nonlinear effects of epidemiological dynamics: those who have not been vaccinated can contaminate other individuals before vaccination resumes, as a result of a cascade mechanism also observed in turbulent flows: there, energy injected in large scales vortex is transferred to small scales via nonlinear interactions between scales.⁴⁰40. A. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers,” Akad. Nauk SSSR Dokl. 30, 301 (1941). https://doi.org/10.1098/rspa.1991.0075 Here, in analogy, a few non-vaccinated individuals can produce a large number of infected individuals. The process can only stop if a huge number of daily vaccinations (much larger than a factor 2) is performed. Nevertheless, this still requires a characteristic recovery timescale $T$ that is larger than the typical immunization scale $\eta$ (e.g., a few months for AstraZeneca²⁸28. M. Voysey et al., “Safety and efficacy of the chadox1 ncov-19 vaccine (azd1222) against sars-cov-2: An interim analysis of four randomised controlled trials in Brazil, South Africa, and the UK,” Lancet 397, 99 (2021). https://doi.org/10.1016/S0140-6736(20)32661-1). Finally, even if several countries have resumed, or are going to resume, AstraZeneca vaccinations, the effect of the interruption is hard to counterbalance and require vaccination efforts difficult to set up in due times. Furthermore, at least for large countries where AstraZeneca vaccination could resume, the confidence of the population in the vaccines is reduced by a non-negligible percentage.⁴¹41. The Economist, March 2021, see https://tinyurl.com/83cbr4d3. In this sense, our estimates are likely to be conservative and might possibly underestimate the excess deaths deriving from the disbelief in the vaccination policies observed in the largest European countries. The analysis presented here has been performed with a parsimonious but well-posed and tested model and we hope that the results we obtain might be the starting point for more detailed, more advanced, and more mature investigations with sophisticated models and data collection exercises.