key: cord-0181588-b84i4oju
authors: Schulenburg, Arthur; Cota, Wesley; Costa, Guilherme S.; Ferreira, Silvio C.
title: Effects of infection fatality ratio and social contact matrices on vaccine prioritization strategies
date: 2022-01-08
journal: nan
DOI: nan
sha: 8da9354aa821806acc6c39eb8c209508ddf19197
doc_id: 181588
cord_uid: b84i4oju

Effective strategies of vaccine prioritization are essential to mitigate the impacts of severe infectious diseases. We investigate the role of infection fatality ratio (IFR) and social contact matrices on vaccination prioritization using a compartmental epidemic model fueled by real-world data of different diseases (COVID-19 and influenza) and countries (Brazil, Germany, and Uganda). Our study confirms that massive and early vaccination is extremely effective to reduce the disease fatality if the contagion is mitigated, but the effectiveness is increasingly reduced as vaccination beginning delays in an uncontrolled epidemiological scenario. The optimal and least effective prioritization strategies depend non-linearly on epidemiological variables. Regions of the epidemiological parameter space, in which prioritizing the most vulnerable is more effective than the most contagious individuals, are substantially broader for COVID-19's IFR in comparison with influenza. Demographics and social contact matrices deform the phase diagrams but do not alter their qualitative shapes.

Effective strategies of vaccine prioritization are essential to mitigate the impacts of severe infectious diseases. We investigate the role of infection fatality ratio (IFR) and social contact matrices on vaccination prioritization using a compartmental epidemic model fueled by real-world data of different diseases (COVID-19 and influenza) and countries (Brazil, Germany, and Uganda). Our study confirms that massive and early vaccination is extremely effective to reduce the disease fatality if the contagion is mitigated, but the effectiveness is increasingly reduced as vaccination beginning delays in an uncontrolled epidemiological scenario. The optimal and least effective prioritization strategies depend non-linearly on epidemiological variables. Regions of the epidemiological parameter space, in which prioritizing the most vulnerable is more effective than the most contagious individuals, are substantially broader for COVID-19's IFR in comparison with influenza. Demographics and social contact matrices deform the phase diagrams but do not alter their qualitative shapes.

Densely connected and unequal societies impose enormous challenges for combating emerging infections diseases and their potentially catastrophic consequences. The pandemic of COVID-19, identified initially in Wuhan, China at the end of 2019 and exported to essentially all populated regions around the world in a few months [1, 2] , is an example that has reshaped the form of how people interact. Several and variable nonpharmacological interventions (NPIs) have been adopted across different places, as wearing of face masks, physical and social distancing, as well as more extreme ones such as lockdown, school closure, and traveling restrictions [3] [4] [5] [6] [7] . While being clearly efficient to momentarily reduce the transmission and unburden heath systems [6, 7] , they are insufficient to restore the pre-COVID-19 lifestyle and avoid economic crashes caused by the disease [8] . Natural emergence of virus variants [9] [10] [11] [12] [13] [14] [15] , fueled by negligent human behaviors, means that natural herd immunity by infections, in which individuals are immune and the susceptible pool is insufficient to a sustained transmission [16] , may never be reached [17] . Since efficient pharmacological therapies have not been made available yet for COVID-19, the only viable alternative is to reach herd immunity through massive vaccination.

Vaccines for COVID-19 were developed with unprecedented speed due to immense collaborative efforts, resources, and accumulated expertise from other viral infectious diseases [18, 19] . In December of 2020, a number of efficient vaccines by different pharmaceutical companies were available for emergency use [20] [21] [22] [23] [24] [25] such as Pfizer-BioNTech, Vaxzevria, CoronaVac, and Janssen. While herd immunity is an important aim of massive * wlcota@gmail.com † silviojr@ufv.br vaccination, its main function is to prevent severe cases which can result in deaths and serious sequelae [26] [27] [28] [29] [30] .

All vaccines approved for emergency use have a high potential to prevent severe cases after complete vaccination and immunization whose time depend on the type of vaccine. CoronaVac, Pfizer-BioNTech, and Vaxzevria demand at least two shots with different intervals (21 to 90 days) for maximum protection [31, 32] while a late third dose has been currently recommended, especially for the most vulnerable groups that present weak antibody responses [33] [34] [35] . However, as usual for anti-viral vaccines, the capacity of COVID-19 vaccines to impede infections and mild symptoms is lower than their efficiency to reduce death and severe cases [20, 21, [23] [24] [25] . The development of efficient vaccines is only the first challenge preceding massive immunization. Large-scale production and timely distribution, particularly to lowincome economies, remain a major barrier to drastically reduce severe cases and to reach herd immunity [36] [37] [38] . Another great matter of concern, especially for highincome economies, is the low demand for vaccines by the populations, which not rarely refuse to get their shots or to complete the immunization schedule. Therefore, given the finite capacity of vaccination, the logistic must be engineered to minimize damages [26, 27, 29, 30, [39] [40] [41] [42] [43] [44] . Infection fatality ratio (IFR) is age-and illnessdependent due to, among other factors, cumulative comorbidities [45] [46] [47] [48] [49] [50] [51] . While on the one hand, a remarkable characteristics of COVID-19, when compared with influenza, is the very high IFR on elderly in contrast with much lower values among kids and newborns [50] [51] [52] [53] , on the other hand, younger population are socially more active and exposed to infections, being potentially the key vectors for contagion of the most vulnerable population [54] [55] [56] . Finally, but not least, the level of sustained transmission is also decisive for the degree of success in vaccination campaigns [38, 41-43, 56, 57] .

In the case of COVID-19, NPIs remain indispensable as the vaccination coverage progresses since the arXiv:2201.02869v1 [q-bio.PE] 8 Jan 2022 excellent but imperfect protection conferred by vaccines will be much less effective in an environment with a high risk of contagion.

In the present work, we investigate the role of vaccination on an age-structured compartmental model [16] , following a susceptible-exposed-asymptomatic-infected-recovereddeceased (SEAIRD) dynamics, using social contact matrices [58] obtained for countries with very distinct population age distribution, namely Brazil, Germany, and Uganda. Outcomes for age-dependent fatality ratios estimated for COVID-19 [50] [51] [52] are compared with seasonal influenza [53] . The interplay between vaccine efficacy to prevent deaths after full immunization and the time taken to acquire protection were also addressed. Epidemic scenarios representing different levels of NPIs were studied. We have found that massive vaccination, even with modest protection against infections, is effective to reduce the disease fatality if adopted early and concomitantly with the contagion mitigation, but losses effectiveness as the epidemic transmission becomes uncontrolled. Comparing different prioritization strategies, vaccinating the most vulnerable first is the optimal strategy to reduce deaths for high transmission scenarios while distributing first shots to the most exposed ones can be better in lower transmission regimes. The region of the epidemiological parameters' space, in which prioritization of the vulnerable population is the most effective strategy, is substantially broader for the COVID-19's IFR than for seasonal influenza. Demographics quantitatively change the phase diagrams of the optimal strategy but preserve their qualitative aspects. Finally, we also analyzed the least effective strategies in a pool of four proposals and found out a dependence on epidemiological parameters more complex than the case of the optimal strategy, in which the phase diagrams indicating the least effective strategy depends strongly on both IFR and demograph-ics.

The remaining of the paper is organized as follows. We describe the model and parameters used in the datadriven approach in section II. Results dealing with four prioritization strategies are shown in section III. We conclude the paper discussing the main results in section IV.

A. SEAIRD compartmental model

We start by considering a SEAIRD, age-structured, compartmental model [16] to simulate the epidemic spreading without vaccination. The schematic representation of this dynamics is shown in Fig. 1 . Individuals can belong to the following broad compartments: S (susceptible), E (exposed), A (asymptomatic or presymptomatic), I (infected), R (recovered), and D (deceased). Every compartment is subdivided according to the age group i = 1, . . . , N g , defined in Sec. II C. Susceptible individuals of age group i become exposed with rate Π (i) upon contact with infectious individuals (labeled by in Fig. 1 ) of all age groups according to the contact matrix; see subsection II C. The other transitions are spontaneous: E → A happens with rate µ 

The transmission dynamics of vaccinated individuals belonging to Z V are the same as Z with their respective transition rates. Additionally, those in the S V compartment can turn into P with rate ν (i) P , granting protection against deaths with high probability. The inverse of this rate is associated with the time between vaccine shots, an important epidemiological parameter. The protected individuals P can evolve to either susceptible S P or directly to recovered R P states with rates ν inf , respectively: S P individuals can still be infected, transmit the pathogen, and eventually die, whereas R P individuals are fully protected against both infections and deaths. The probability that a protected individual acquires full protection, given by

is directly obtained from the model and used to parameterize the rates in terms of the vaccine efficacy against infection and a characteristic time 1/ ν

SP . Both P and S P individuals can be infected with rate Π (i) P . For sake of simplicity, the infections of protected individuals follow a SIR-like dynamics with a single infectious compartment I P which can be recovered (move to R P ) or die (move to D P ) with rates α (i) RP or α (i) DP , respectively. The last two rates can also be parameterized in terms of the efficacy to prevent deaths which is the reduction of probability that an infected protected individual dies in comparison with an unprotected population. In terms of the model rates, it becomes

in which Θ (i) is the IFR of group age i; see Sec. II E. The dynamics of vaccinated individuals is highlighted in the shaded area of Fig. 1 .

We consider individuals divided into N g = 16 age groups, starting from 0-4, 5-9, up to 70-74 and ≥ 75 years. We refer to young population as those with age 0 to 19, adults for 20 to 59, and elderly with age equal or above 60 years. The relative size of each group is given in Table I for three countries investigated in the present work. The estimates for the Brazilian demographics of 2020 [60] are shown in Fig. 2 (a). The infection rate of a susceptible individual within a group i is given by

where the corresponding prevalence in each population Z ∈ {A, I, A V , I V , I P } is denoted by the italics symbols while the corresponding infection rate per contact is represented by λ Z . Also, k i is the number of contacts made by individuals in age group i while C ij is the contact matrix that gives the fraction of the contacts of group i done with j. The parameters k i and C ij were estimated from Ref. [58] considering contacts made at work, school, home, and other locations; see Appendix SI-II A in SM [59] . For sake of simplicity, we consider the same infection rate for protected and non-vaccinated susceptible individuals, Π (i) P = Π (i) , assuming the former is less contagions (λ IP < λ Z ) due to a reduced viral load [62] .

We emulate a hypothetical epidemic scenario of social distancing, denoted by {S}, where 100%, 50%, 50%, and 30% of contacts are allowed in home, work, school, and other places, respectively. The social distancing will be compared with the unmitigated scenario, denoted by {U}, where no reduction of contacts is implemented. The respective average number of contacts in different age groups for Brazil considering each scenario is shown in Fig. 2(a) . A symmetrization procedure described in Appendix. SI-II A of the SM [59] was adopted to compute the contact matrices of Brazil and shown in Fig. 2(b) . For instance, the average number of contact decays from k ≈ 15 in the unmitigated scenario to 8.3 when the social distancing is adopted. Equivalent figures for Uganda and Germany are provided in Appendix SI-II B of the SM [59].

The basic reproduction number in the absence of vaccines is given by Finally, we integrate the dynamical system given in Appendix SI-I of the SM [59] using initial conditions where a single exposed individual is introduced in a single age group s in a total population n = 10 5 individuals (results are insensitive to this parameter given it is large enough). The averages were computed over initial conditions s = 1, . . . , N g using as weight the total number of contacts made by each group s, in which the epidemic process is initiated.

The infection fatality ratio (IFR) 1 Θ (i) , defined as the fraction of infected individuals that evolve to death, can [50] [51] [52] and influenza [53] . As a baseline we also consider a uniform value 0.68%, that is the average for COVID-19's IFR weighted by the population of each age group.

be straightforwardly computed in terms of the model's rates ( Fig. 1) as

which can be used to determine the rate α Here we used the IFR estimates for COVID-19 reported by Verity et al. [52] , that follows an exponential increase with age. Similar values have been reported elsewhere [50, 51] . To investigate the role of IFR, data for influenza from the Centers for Disease Control and Prevention [53] and a hypothetical uniform IFR, given by the averaged COVID-19's IFR weighted by the population size of each age group, were also considered while the remaining parameters were the same estimated for COVID-19. Influenza's IFR also increases exponentially with age, but it is lower than for COVID-19 and influences the choice of the vaccination strategies [27, 41] . The IFR age dependence is shown in Fig. 3 .

The rate ξ(t) is defined as the per capita number of daily first shots of vaccines. For sake of generality, it is assumed to be time-dependent even though we performed simulations assuming ξ constant in the present work. Let us define Ω (i) (t) = 1 if age group i is being vaccinated at time t and Ω (i) (t) = 0 otherwise. All non-vaccinated individuals belonging to the compartments S (i) , E (i) , A (i) , or R (i) can receive their first shots with equal chance if 

The prioritization of vaccine shots across different age groups over time can be modeled with Ω (i) (t). The vaccination begins at a time t v such that Ω (i) (t < t v ) = 0 ∀ i. Once 80% of a priority group has been vaccinated, the vaccination of the next priority groups starts concomitantly with all other groups where vaccination had started previously. Four prioritization strategies are investigated in the present work. In decreasing age priority (DAP) strategy, one starts in the oldest age group and proceeds progressively down to the youngest one as adopted in many countries for the general population. In highly-vulnerable priority (HVP) strategy, only the elderly are prioritized according to the age, then all adults (age 20-59) and later all young individuals (0-19) are vaccinated without age prioritization. This could represent the economically active population being vaccinated altogether after the most vulnerable individuals were protected. The decreasing contact priority (DCP) strategy starts with the age group of the higher number of contacts and proceeds progressively down to the one less connected. This strategy corresponds to vaccinating the most exposed first. Finally, in the no priority (NP) strategy all age groups are vaccinated concomitantly 2 .

For sake of simplicity, the individuals are labeled as vaccinated after the first shot, but acquire protection and are moved to the compartment P (i) only after the second shot. The average interval between shots, which corresponds to the transition S V → P is given by 1/ν (i) SP , assumed uniform across age groups. With respect to the efficacy against infection and death, both an ideal case with moderate protection (ψ . In all cases, after an average time of 1/ (ν RP + ν SP ) = 7 days, uniform across ages, she or he either become fully immunized (P → R P ) or remain susceptible to the disease (P → S P ). All remaining epi-demiological parameters and references are provided in Appendix SI-II C of the SM [59].

Considering = 1.3, social distancing scenario, modest vaccination with constant rate ξ = 0.15% of the population per day, age-dependent efficacy, short delay t v = 30 days, and DAP vaccination strategy, we computed the fractions ρ inf of infectious (I, I V , A, A V , and I P ) and ρ d of deceased (D, D V , and D P ) individuals to compare with the case without vaccines. Figures 4(a,b,d,e) present the temporal evolution of ρ inf and ρ d in the form of stack plots split according to the age profiles of young, adult, and elderly populations of Brazil. Even a modest vaccination rate, if started early, substantially reduces the total amount of deaths while the reduction of infections is not expressive. The age profiles for infections and deaths are shown in Fig. 4 (c,f). Since the percentages of young, adults, and elderly in the Brazilian populations are 27.9%, 57.4%, and 14.7% [60] , respectively, the age profile of infections without vaccines is highly correlated with the demography, Fig. 4(c) , while the deaths' profile is determined by the COVID-19's IFR used in this simulations, as shown in Figs. 3 and 4(f) where deaths are highly concentrated in elderly, even they corresponding to the minor part of the population. DAP strategy moderately alters the age profile of infected individuals. Beyond reducing deaths in all age groups, the fatality age profile is highly affected, presenting a big drop in death among the elderly and a fractional rise in the adult population that now concentrates most of the deaths. Not surprisingly, this inversion was observed during the first semester of 2021 in Brazil that adopted DAP after vaccination of healthcare workers and persons with morbid conditions. Remark that the age profile of deaths changes substantially after the lifespan of immune response 1/ν (i) P = (21 days) −1 , highlighting the importance of a complete immunization scheme. Similar results are found using uniform values of efficacy against death and infection, as shown in Appendix SI-III A of the SM [59], with quantitative changes on the age profile due to the absence of the drop of efficacy in the elderly individuals.

The interplay between vaccination rate ξ and effective infection rate parameterized by is investigated considering the DAP strategy and age-dependent values of efficacy with a delay of t v = 30 days to start vaccination. Heatmaps for the reduction of deaths in the space parameter ξ versus under unmitigated and social distance contact scenarios are presented in Fig. 5 for Brazilian contact matrices. The corresponding heatmaps for total recovered population is given in Fig. SI-6 in Appendix SI-III B of the SM [59]. As expected, the reduction of deaths is much more expressive than of infections for both scenarios. In the scenario of social distancing shown in Fig. 5(a) , DAP vaccination performs very well to reduce deaths if the immunization rate is not too low and dissemination rate is not too high (reduced ); the latter is feasible through simple NPIs such as face mask-wearing. For the case of unmitigated contacts shown in Fig. 5(b) , the vaccination can significantly reduce the number of deaths only at a high vaccination rate of ξ = 0.5% per day (approximately seven months for the total population to be immunized), only if infection rate is kept near to the lower bound of . The unmitigated scenario is not able to reduce the transmission by more than 10% in the whole parameter space, while we can still see a significant effect in the social distancing scenario; see Fig. 5(c) , in which one sees that the death reduction drops significantly for t v > ∼ 80 days, the more for lower vaccination rate. Delayed vaccination becomes ineffective in the scenarios without mitigation even at a high vaccination rate, as shown in Fig. 5(d) . 

Given the limited availability of vaccine shots, an essential problem is to determine which prioritization strategy can be more effective. Aiming at saving the maximum number of lives, we compared different strategies in the parameter space ξ × shown in Fig. 6 for Brazilian demography (Fig. 2) and uniform values of vaccine efficacy against death and infection. To isolate the effects of different IFR age profiles (Fig. 3) , the same epidemiological parameters of COVID-19, except the IFR itself, were considered. For the unreal case of age-independent IFR, the optimal strategy is to prioritize those who are more exposed, i.e. make more contacts, using DCP followed by no priority; the latter is better than the remaining ones since the adults constitute simultaneously the largest and most connected populations in Brazil. For influenza's IFR, we observe that prioritizing the most exposed individuals is more advantageous than the most vulnerable for a broader region of the space parameter, especially if the vaccination rate is high. However, for uncontrolled transmission (high ), it is still more effective to vaccinate according to a decreasing age criterion. Finally, the simulations with COVID-19's IFR yield that the most effective strategy is DAP in most of the investigated parameter space. Only in a narrow region, prioritizing the most exposed through DCP is the most effective. Notice that prioritizing only the highly vulnerable (elderly) by adopting the HVP strategy is not the most effective strategy in the investigated diagrams.

We also compute the least effective among the four investigated strategies, as shown in the right column of Fig. 6 . For the uniform IFR age profile, the DAP strategy reduces deaths least while for realistic IFR of influenza and COVID-19, a complex pattern emerges in the diagrams. HVP can perform worst if the infection is moderately uncontrolled while DAP is the least effective only for almost controlled spreading ( = R {S} 0 ≈ 1). A remarkable result is that prioritization of the most exposed, which has been frequently debated in pandemics of COVID-19 [27, 28] , is the least effective if the epidemic is out of control. The respective plots for Uganda and Germany are shown in Fig. SI-7 of Appendix SI-III C of the SM [59].

The role of time delay in the strategy effectiveness is presented in Fig. 7 with fixed infection parameter = R {S} 0 = 1.3 in a social distancing scenario, COVID-19's IFR age profile, and age-dependent values of efficiency. As shown in Fig. 7(a) , DAP is the most effective for large delays while DCP is for earlier interventions. Moreover, the re-entrant behavior for intermediate delays (t v ∼ 100 days) reveals a complex interplay between epidemiological parameters and optimal strategies. No prioritization has the worst performance in almost the entire parameter diagram; see Fig. 7(b) . Despite the unavoidable ethics concerns in prioritization strategies, a remarkable feature of these diagrams is that the opti- . Four vaccination strategies are considered: decreasing age (DAP), highly vulnerable (HVP), no (NP), and decreasing contact (DCP) priorities. A time delay of tv = 30 days, vaccination with uniform values of efficacy against death and infection, Brazilian demography, and social distancing contact scenario were considered. The gradient colors refer to the respective reduction of deaths, the darker the higher. Differences between the most and least effective strategies smaller than 5% are depicted in gray. mal strategy depends, in a very nonlinear fashion, on the level of epidemic transmission, immunization rates, and timeliness of starting the vaccination.

We have so far observed the nonlinear interplay between epidemiological parameters to determine the optimal strategy to reduce deaths. It depends nontrivially on infection scenarios, vaccination rates, efficacy, and IFR age profiles. Now we explore the effect of demography and contact structures considering two other countries: Uganda, with a higher number of young individuals, and Germany, with a higher number of elderly individuals in comparison with Brazil. Table I summarizes . The plot for optimal strategies has the same qualitative patterns for the three countries with age prioritization (DAP) being the most effective for higher infection regimes and contact prioritization (DCP) for lower infection and high vaccination rates. Quantitatively, Brazil and Germany present very similar diagrams despite the substantially higher fraction of elderly individuals in Germany. For Uganda, with a mostly young population, the diagram region where DAP outperforms DCP is larger than in Brazil and Germany. The last result seems counter-intuitive at a first glance since prioritizing the most exposed is expected to be more effective in a population with very few elderly (3.3%; see Table I ). However, the most exposed population in Uganda are the young while in Germany and Brazil, adults perform more contacts; see Figs. 2, SI-2, and SI-3 of the SM [59]. Since the COVID-19's IFR for the adult population is still much higher than for the young, DAP is also the best strategy for Uganda's contact pattern. The least effective strategy provides more complex diagrams, depending strongly on the contact pattern and demography of the country. No priority strategy dominates the phase diagram for demographics of Brazil and Uganda. Prioritizing contacts is the least effective option for uncontrolled dissemination, more evidently for Germany's demography. Prioritizing age is the least effective option in a small region of the diagrams consisting of concomitantly low transmission and vaccination rate in the case of Brazil, with a broader region for Germany; See Figs. 8(b,f) , respectively. Note, however, that even the least effective strategy strategies can still reduce significantly the number of deaths. The diagrams for optimal and least effective strategies for young, adults, and elderly individuals also present nonlinear effects. The diagrams for the whole population are ruled by deaths of elderly; See Figs. SI-8 and SI-9 in Appendix SI-III D of the SM [59].

The rise of a new, highly transmissible, and lethal infectious disease implies an enormous logistic challenge to minimize the damages and, especially in the case of viral pathogens, quick development and massive distribution of vaccines to the entire population is the most, maybe the only, viable option to mitigate the impacts of the disease. Moreover, the capability of the viruses in mutating and thus evading the protection conferred by either previous infections or vaccination imposes a constant concern about the optimal prioritization strategy to be adopted in a realistic scenario of a limited supply of vaccine shots. We are nowadays witnessing a remarkable success of massive vaccination to reduce the severe cases of COVID-19, wherever it has been adopted. However, unequal and delayed supply of vaccines across the globe still puts COVID-19 as a worldwide threat.

The choice of the optimal prioritization strategy aiming at reducing the number of severe cases and consequently of deaths is far from being trivial due to the wide pool of relevant epidemiological parameters involved in the analysis. To tackle this problem, we investigated the role of social contact patterns (Fig. 2) and infection fatality ratio (Fig. 3) on vaccine prioritization strategies using an age-structured compartmental model and a datadriven approach, in which real epidemiological data are used as inputs to the numerical analysis. Prioritization of the most vulnerable (high risk of death) and of the most exposed (perform more contact) were compared with no prioritization. We report that vaccines, even with modest protection against infections, are very effective to reduce fatality irrespective of the strategy. For age prioritization, which corresponds to the most vulnerable in the case of COVID-19, the age profiles of deaths are significantly altered while the infection profile changes little. Another important outcome of the simulations is that the effectiveness of the vaccination depends strongly on the contagion mitigation through, for example, nonpharmacological interventions. In the case of COVID-19, delays in starting vaccination imply the ineffectiveness of vaccines if the contagion is uncontrolled.

The optimal and least effective strategy to reduce deaths also depends on the epidemic scenario and IFR age profile. Vaccination of the most exposed first is more effective than of the most vulnerable when the epidemic is highly controlled with a low transmission rate. The prioritization of the most vulnerable becomes the optimal approach for highly contagious scenarios. However, nonlinear dependence on the vaccination and contagion rates, depending on the IFR profile, is observed. Comparing COVID-19 and seasonal influenza IFRs, we report that the region in the epidemiological parameter space, where prioritizing vulnerable persons is the most effective strategy, is broader for the former than the latter, despite the qualitative similarity between them. Finally, the diagrams in the epidemiological space parameter reporting the optimal strategy depend little on demography and so-cial contact profile when comparing data for Brazil, Germany, and Uganda, which present very distinct patterns. However, the least effective strategy is very sensitive to demography and contact matrices, revealing a complex dependence on epidemiological parameters.

The data-driven analysis developed in this work is not a forecast of the number of deaths or infections in the pandemics of COVID-19. Actually, it raises important issues, from the perspective of nonlinear dynamical systems, that can be underestimated in applied mathematical or statistical epidemiological modeling. This kind of modeling can help decision-makers to select the vaccination prioritization strategy according to the current scenario, but our central aim is to quantitatively address the importance of epidemiological parameters on the outcomes of the theoretical analysis using a mechanistic approach. Some simple, but still essential messages were presented. Beyond the obvious ones reporting that the faster and earlier the vaccination, the better its result is, we also show that the outcomes depend nonlinearly on the epidemiological situation and particularities of the infectious disease. We expect that our more mechanistic approach can join statistical inference methods to provide more accurate responses to vaccination prioritization strategies.

Author contributions: W.C. and S.C.F. conceptualized the study and jointly wrote the first version of the manuscript. A.S. wrote the codes and performed the numerical analysis. W.C. gathered the data for IFR and contact matrices, verified the codes, and produced the figures. G.S.C. and S.C.F. performed the analytical treatment of the model. All authors edited the manuscript and analyzed the results.

Competing interest: The authors declare no competing interests.

Data and code availability: Data and codes used in the present work are available in a public repository at https://github.com/wcota/covid19-vac-st/. The codes for simulations were written in Fortran and compiled with the Intel Fortran Compiler. Data analysis and figures were done Python 3.10 and the following open source libraries: pandas [68] , matplotlib [69] , and seaborn [70] . 

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