Alternative Vaccination Politics

Background : The timely implementation of the vaccination campaign and sharp rules for vaccine administration can make a difference. The paper investigates the impact of alternative policies based on scientific vaccination priorities inspired by the extended statistics on Covid-19 fatalities. Methods : In the case of Covid-19 vaccination, a principal role is played by promptly adopting a reverse-order of age approach (to target first the elderly) coupled to covering the high priority categories but postponing the low priority ones. We implemented an in silico vaccination simulator capable of comparing what happened in reality with what might have happened if alternative vaccination policies had been adopted. The immunization profile and the death distribution curve allowed measuring the distance between reality and alternative policies and finally quantifying the expected number of saved lives. Results : The alternative approach to vaccination was applied to Italy and Lombardy that host 60 and 10 million residents respectively. In about 100 days of vaccination based on (a) a reverse-order of age policy (from 90+ to 80-89 to 70-79 year-olds, etc.), (b) vaccination of priority categories, (c) postponement of non-priority categories and reallocation of such doses to (a) and (b), the saved lives would have been 3969 in Italy (of which 799 in Lombardy). In the same period, Italy suffered 30,911 fatalities (of which 5,613 in Lombardy). Of those fatalities, about 12.8% in Italy and 14.2% in Lombardy might have been avoided if a different approach to vaccination had been applied. Even better results would have been achieved if the elderly vaccination had been anticipated a few days (which started only 53 days after the very beginning of the Italian vaccination campaign) or if the vaccination engine had performed better in terms of daily administered doses while respecting the available delivered doses. Conclusions : A different approach to the vaccination politics based on sharp and straight policies based on scientific quantitative data of Covid-19 mortality as a function of age and comorbidities would have accomplished a better quantitative effect on extinguishing the pandemic and containing the fatalities toll.


Background
A recent report from ISS, the Italian Superior Institute of Health, informs that 99% of the deaths since the very beginning of the Covid-19 pandemic involved over 50-year people (Italian Superior Institute of Health, 2021). Table 1 shows that deaths in excess of 85.6% consisted of over 70-year individuals. Table 1: Statistics of the 118,589 Italian fatalities registered from the start of the pandemic (24-Feb-2020) to 28-Apr-2021 (Italian Superior Institute of Health, 2021). Data are organized in 10-year age ranges. F(i) are the fatalities of the i-th age range. F100k are the fatalities of the i-th age range per 100,000 inhabitants. R(i) are the sequential ratios of F100k. P(i) are the cumulative products of R(i), with P(1) = 1. The Covid-19 vaccination campaign started on December 27 th , 2020 in Italy. The first eight weeks saw a vaccination rate of no more than 100,000 doses/day. Indeed, Lombardy features a comparable number of citizens as Sweden, the Czech Republic, Greece, Hungary, and Portugal but is significantly more populated than Israel, Austria, Switzerland, Denmark, Norway, Finland, Slovakia, Slovenia, Serbia, and Croatia. The large number of citizens in Lombardy made the vaccination campaign challenging and the high population density was a key factor together with the towering gross-domestic-product (i.e. industrial activity, travels, and commercial exchange) for the widespread and vast consequences played by Covid-19 in terms of infections, hospitalizations, and deaths (Bignami et al., 2021). Consequently, Lombardy was one of the Italian regions that most adhered to the vaccination campaign.
Italy is a republic and regions do not have significant political independence. Indeed, nationwide rules for vaccine administration characterized the initial phase of vaccination. However, several regions introduced further degrees of freedom as a function of local political pressures, the efficiency of their booking system, and the arrangement of the vaccination sites.
For the sake of clarity, the Italian vaccination campaign was organized into different population categories and age ranges.
Vaccinations were carried out locally, which means that every region according to the number of available vaccines, the vaccination adherence, and the number of citizens belonging to categories and age ranges achieved different vaccination rates and coverages. It is worth adding that Italy, as a member state of the European Community, received over 33 million doses by the end of May 2021 of four different vaccines: Janssen (2.2%), Moderna (9.2%), Pfizer/BioNTech (67.1%), and Vaxzevria (AstraZeneca) (21.5%). Unfortunately, the very rare adverse reactions after AstraZeneca's vaccination played a significant role in the population acceptance of the vaccination campaign, and the residents of some Italian regions showed a noteworthy reluctance against that vaccine. In addition, at the end of March 2021, the Italian health ministry suspended Vaxzevria administrations for four consecutive days, which had repercussions of about 2-3 weeks to restore the scheduled vaccination rate.
For more than 50 days, since the start of the vaccination campaign, Italy chose to cover first some categories instead of prioritizing the population by a reverse-age approach. The main initial reason behind that decision was to immunize hospital personnel. However, besides medical doctors and nurses working in hospitals, the campaign involved also socio-sanitary personnel, technical, maintenance, security, and administrative personnel (both front-and back-office employees), medical students, psychologists, as well as hospital cleaners, bartenders, and gardeners. All the different ages (i.e. from 16 to 90+ years old) of those categories were covered, which induced a fierce debate on the ethical correctness of such prioritization (Craxì et al., 2021;Giubilini et al., 2021;Williams et al., 2021). Two further critical categories were spotlighted: nursing home guests and fragile subjects. In addition, school/university personnel and armed forces received the vaccination before elderly people, thanks to the pressure of a few influential stakeholders. Unevenly, regions vaccinated also some peculiar subcategories that were lumped under the "other" category, for instance, lawyers, scientific informants, social workers, customs and airport personnel, and funeral home staff. Putting non-critical categories before the elderly caused further divisions among the persons in charge of the vaccination policies and highlighted the disparities among regions. A political change in the Italian government, together with the appointment of a new special pandemic commissioner on 1-Mar-2021 (at day #65 of the vaccination campaign), allowed straightening up the vaccination politics with a clear directive towards a reverse-age prioritization.
Since mid-February 2021, people aged over 80 (more than 4.4 million in Italy) became the principal target. The vaccination politics then progressively focused on younger citizens, i.e. seventy yearolds, sixty year-olds, and so on by reverse order of age ranges (ROAR).
Again, regions did not start in unison. Lombardy was one of the slowest to open and implement the vaccination campaign for those over 80 and suffered a significant delay mainly due to a suboptimal booking system, which eventually was reworked. Lombardy took 75 more days to establish the Italian record of vaccinations with more than 116,800 doses administered in a day (vs 546,700 in Italy), well above (21.4%) the proportional percentage of regional vs national population (16.7%).
However, it was not possible to sustain that vaccination rate due to the reduced availability of delivered doses. The following month of May 2021, Lombardy could afford an average of 85-90,000 doses/day, in line with the expected Italian target of half a million daily doses, which was also consistently attained.

An alternative approach to vaccination
The history of implementation of the Covid-19 vaccination in Italy is known and available in detail.
Since the first day, the Italian ministry of health, through the special pandemic commissioner, publishes daily the vaccination data on a GitHub data repository (Italian Special Pandemic Commissioner, 2021) that provides an in-depth vision of both the delivered and administered doses.
The delivered doses are detailed for each region in terms of vaccine brand and delivery date. The administered doses are detailed for each region according to the vaccine brand, gender, age-range, category, and first/second dose. The categories are ten: health workers, non-health hospital workers, nursing home guests, fragile subjects, school/university personnel, armed forces, over-80, 70-79 years old, 60-69 years old, and others. The age ranges are 9: 16-19, 20-29, 30-39, 40-49, 50-59, 60-69, 70-79, 80-89 in preventing the hospitalization of an infected person (Anand and Stahel, 2021;Benenson et al., 2021;Olliaro, 2021;Vasileiou et al., 2021). Since the second wave of Covid-19, which started in Italy at the beginning of October 2020 (as of May 2021, Italy is experiencing the deflation trend of the third wave, which started towards the end of February 2021), Covid-19 patients die only in hospitals (Ciminelli and Garcia-Mandicó, 2020). For the sake of clarity, people do not die anymore at home or in nursing homes. Therefore, if the vaccine zeros the probability of being hospitalized and since patients may die only in hospitals, a reasonable assumption is that the vaccine zeros the probability of passing away. A further consequence of vaccination is that it transforms a deadly pandemic into a more manageable disease.
About point (i), the alternatives might have been to anticipate of some days the vaccination of some specific critical categories, to increase of some percentage points the number of daily vaccinations, to avoid the dispersion/dissipation of doses to less critical individuals as the ones reported in the Introduction.
About point (ii), conceivable priorities might have been (a) to avoid the vaccination of very young people even if they belonged to prioritized categories, (b) to go for a vaccination policy based on ROAR and assume it as the leading one.
Point (iv) calls for a consistent and fair approach to the alternative vaccination politics. We assume that all the simulations of alternative administrations should respect a few constraints: the delivered vaccines in terms of daily and regional availability, and number and brand of doses without any anticipation of what is expected to happen but is not yet fully accessible/regulated. This includes avoiding any a posteriori assumption based on historical data. The doses must also be administered respecting the present rules (that may change at either a regional or a national level during the vaccination campaign) in terms of age range accessibility and the interval between the first and second dose (except Janssen that is a single-dose vaccine). For the sake of clarity, the "age range accessibility" term refers mainly to the Vaxzevria vaccine that underwent a few changes in the Italian administration directives. As the vaccination campaign progressed, the Vaxzevria vaccine was prescribed to under 55, then under 65, then over 60 individuals, then tolerated for under 60, and finally forbidden for the under 60 year-olds (Faranda et al., 2021;Wise, 2021). However, these rules were not strictly respected and few citizens received Vaxzevria even though they were well over 80 years old during the periods when it was prescribed first to under 55 and then to under 65 year-olds (we are not able to explain why it happened). As far as the timespan between two doses is concerned, the Italian vaccination campaign adopted for more than 130 days in a row the following intervals: Pfizer/BioNTech and Moderna 21 days, Vaxzevria 70-84 days. Some regions firmly implemented the 70-day interval for Astra Zeneca (e.g., Lombardy), some others were more flexible and allowed citizens to book the first vaccination day and choose the second one within the 70-84 days timespan. In addition, when it came to "younger" age ranges, some regions (e.g., Lazio) allowed the citizens to book their preferred vaccine. This caused an overbooking of Pfizer/BioNTech and Moderna vaccines due to the biased opinion of a large portion of citizens mostly against the Vaxzevria vaccine and to a lesser degree the Janssen one. In some regions (e.g., Veneto) the selection of the vaccine was "indirect" as, by choosing the place were to be vaccinated, the citizens could become aware in advance of the vaccine brand that would be administered. On the contrary, some regions (e.g., Lombardy) assigned the first (and second) date(s) and the vaccination site once the citizen had expressed their adhesion to the vaccination campaign. In Lombardy, the medical doctors at the vaccination site decided on the vaccine brand after an in-depth analysis of citizens' features and (co)morbidities.

Prioritization
When it comes to proposing an alternative approach to vaccination prioritization, one might be distracted by first paying attention to those who are either more exposed, or more productive, or more contagious with the target of mitigating the virus spreading and enhancing the national productivity (e.g., GDP) (Pardhan and Drydakis, 2021;Sarkodie and Owusu, 2021). However, the most important indicator is the death factor that focuses the attention on simply protecting and saving those who are more fragile and risk their lives. Targeting first those who are more fragile to the virus fulfills the highest ethical values that a democratic and fair society should be inclined to.
Relatedly, in the case of an opposite approach based on either sub-ethical or unethical values determined by a selfish and opportunistic attitude, targeting first those who are more fragile to the virus is still the winning approach. Indeed, the reduction of the death count allows loosening the lockdown measures that are incompatible with the economic targets and the desire to live a fully realized and unconstrained life. Unfortunately, the Covid-19 pandemic has brought out orthogonal values that span from very altruistic principles to very selfish behaviors. Somehow, the preservation and defense of those who are most fragile to the virus are key factors that meet the values and interests of those conflicting categories of people.
Therefore, to preserve and defend the most fragile individuals, one has to minimize their probability of death. People may die if they are fragile because of their compromised physical condition or because of their age (as shown in Table 1). Indeed, the elderly are usually affected by comorbidities that increase the probability of fatality in the case of SARS-CoV-2 infection (Italian Superior Institute of Health, 2021). The older the age, the higher the probability of being affected by a greater number of comorbidities. In addition, the organs' efficiency decreases significantly with age as well as the immune system's capacity to react to external solicitations (Grubeck-Loebenstein, 1997).
These observations lead to identifying the main prioritization criterion simply based on vaccinating first the elderly and the nursing home guests.
Then a dilemma opens about the category of health workers that involves but is not restricted to medical doctors and nurses. One might propose to adopt the same reverse-age approach to that category. However, many reasons urged vaccinating that category first. Among the cogent rationales, it is worth listing (a) the necessity to reduce/avoid infectious outbreaks in hospitals and the contagion of patients by medical staff; (b) to increase the number of available doctors, nurses, and biologists in the key analysis labs, intensive care units, and pneumology departments of hospitals; (c) to test, train, and tune the vaccination engine on a specific category of individuals who are intrinsically ready to face possible side effects and make experience by vaccinating each other; (d) acknowledge the prolonged sacrifice and dedication of both hospital and health workers and reduce the contagion risk that allows enhancing the quality and efficacy of their work. In the long run, the decision of creating a self-contained bubble of protected health workers allowed providing a better hospital service and quickening the progressive return to elective medicine.

Questions and open points
Several questions can be formulated under the umbrella of an optimal vaccination campaign. For instance, what would have happened if: 1. We had strictly administered in reverse-age order the doses that went to non-priority categories?
2. The vaccination campaign to the elderly (i.e. over 80 years old) had started 10 or 20 days before what happened in reality?
3. The daily vaccinations had increased by 10 or 20%?
It is worth underlining that these three points comply with the assumption that the administered doses respect the daily availability and the vaccination capacity of each Italian region.
Point (1) Figure 1 shows how the Italian vaccination campaign was characterized by a weekly periodicity. The reader can observe the so-called frog jump trend that repeats every seven days. The grid of Figure   1 is centered on Sundays and shows how the weekend days (i.e. Saturdays and Sundays) underperform with a decreasing trend in comparison with the maxima on Thursdays and Fridays.
This weekend's underperformance may be justified by the fact that the whole vaccination supply chain should periodically take a breath to deliver a sustainable and prolonged service. However, it is less clear why the working days see a monotonically steep increase from Mondays to Thursdays/Fridays. Specifically, the first working days of the week underperform and it often happens that the administered doses on Mondays, Tuesdays, and Wednesdays are significantly less than those on Thursdays, Fridays, and Saturdays.

The vaccination simulator
The quantification of the differences discussed in Section 2.3 calls for a simulator of the vaccination campaign capable of implementing the real input data recorded in the different regions of a specific nation (e.g., Italy with its 19 regions and 2 autonomous provinces) and accepting dissimilar and flexible hypotheses of doses allocation to different categories and age ranges of the population that adhered to the vaccination campaign.

Recommended features
The recommended features and services of the vaccination simulator are:  Download the input values from a data repository (e.g., GitHub) that is updated periodically (e.g., twice daily in Italy) and reports both administrations and delivered doses;  Filter the input data and check for their consistency;  (a) split the input data into separate regions/provinces, (b) organize the data chronologically,  Import for each region/province and each age range the number of residents;  Import the percentages of adherence to the vaccination campaign for each region and age range;  Implement the possibly different administration rules (in terms of specific vaccines that either have to or do not have to be administered to certain categories or age ranges) enacted either at a national or regional level and that may change in time;  Implement the key dates when the vaccination of specific priority/nonpriority categories took place at a national/regional level;  Consider the vaccine features in terms of single or double administrations with the recall time in the case of two administrations.
The simulator engine loads all the input data reported above and must be able to allocate in different optimal modes the real administered doses that went to non-priority categories according to the ROAR approach discussed in Section 2.
The simulator relies on two further key features: the immunization profile (see Section 2.4.2) and the life-saving assessment (see Section 2.4.3).

Immunization profile
What does it happen to the human body when it receives the first and possibly the second dose of vaccine? We know that the immune system reacts to the exogenous input by developing antibodies that will fight SARS-CoV-2 infection (Mansourabadi et al., 2020). This reaction of the immune system is called immunization and makes a vaccine more or less effective against the virus infection. To our knowledge, the scientific literature does not report a quantitative law describing the dynamic profile of immunization after the Covid-19 vaccination. However, the literature reports some key values, times, and notes about most of the Covid-19 vaccines developed worldwide from which to infer a quantitative dynamic profile of immunization (Lombardi et al., 2021). Some papers report the degree of immunization in terms of the percentage of vaccinated individuals that were infected after they received either the vaccine or a placebo (Creech et al., 2021;Tenforde et al., 2021). These numbers are statistically collected through phase III clinical trials that entail two groups of volunteers that undergo a double-blinded vaccination campaign. Following the first and possibly the second vaccination shot, they lead a conventional life, may get in contact, and be infected by SARS-CoV-2. The ratio of infected volunteers who received the vaccine over those who received the placebo allows determining the vaccine efficiency in decreasing the probability of being infected after contagion (Chagla, 2021;Hall et al., 2021;Vasileiou et al., 2021).
Following the first administration, our body starts reacting and developing an antigen-specific immune response (specifically T and B lymphocytes, which produce antibodies in a rather complex biological cascade of mechanisms and transformations (Tufan et al., 2020)). The first maximum level of immunization is reached a few days later (Chagla, 2021;Knoll and Wonodi, 2021). At that time and in the case of two-dose vaccines, a second dose is administered. A few days later, the immune system reaches the maximum coverage (Kadire et al., 2021). We chose to adopt a conservative approach to implementing a functional dependency of immunization from time (i.e. the immunization profile) (Dan et al., 2013;De Bernardis et al., 2012;Le et al., 2014). Figure 2: Immunization profile. The first administration occurs at t0; t1 is the induction time; at t2 the first level  of immunization is achieved and the second (if any) dose is administered; at t3 the maximum degree of immunization  is reached and preserved for a few months.
As shown in Figure 2, we assumed that for few days (i.e. the induction time, 1 t ) after the first administration (which occurs at 0 t ) the immune system is capable of producing a negligible defense but that in the following days it triggers a suitable response that from zero (based on a conservative approach) increases linearly to the first  maximum value at time 2 t . Usually, at 2 t , the second administration for the two-dose vaccines occurs and a few days later, at 3 t , a new maximum,  , is reached.  (Chagla, 2021;Knoll and Wonodi, 2021). Articles inform that the probability of being hospitalized is almost zeroed after 3 t and that the immunity is supposed to last for 10-12 months (Fergie and Srivastava, 2021;Seow et al., 2020). These bits of information suggest assuming 1   as far as the death probability is concerned. At present, as discussed in Section 2, patients may die only in hospitals if seriously infected by SARS-CoV-2. Therefore, if the vaccine avoids any hospitalization then its efficacy can be assumed thorough ( 1   ) against the risk of death. For the sake of completeness, the immunization period of 10-12 months is much longer than the time horizon of the vaccination campaign and therefore we are authorized to keep  constant for the whole simulation time (after 3 t ).

Life-saving assessment
To To estimate the expected number of deaths in the case of an alternative vaccination politics, it is first necessary to quantify the number of vaccinated citizens, their immunization degree, and their age. The basin of people that may fall seriously sick and eventually die includes those individuals who have not yet reached a full immunization. Statistically, as we are dealing with large numbers of citizens (i.e. national/regional populations), we can consider the degree of immunization of those individuals who are progressively developing their immunity through the administration of the first (and possibly second) dose(s). We can calculate the dynamic percentage of immunization of the population of a region or a nation according to their age ranges and as a function of the administered doses. Figure 3: Cumulated dynamic immunization of a susceptible population belonging to a specific age range of P cardinality. The green curve shows the evolution of the alternative vaccination politics, which prioritizes that specific age range (subject to all the constraints discussed in previous Sections). The blue line reports the degree of immunization achieved through the real vaccination campaign. olds. This vaccination starts at s t as soon as the first withdrawn dose is available and any priorities by the older age range (i.e. 90+ years old) have been complied with. This means that first, all the 90+ years old individuals, who adhered to the vaccination campaign, receive at least the first dose.
If the first dose administration to all the 90+ individuals takes less than 2 t (in Figure 2) then the vaccination campaign to the 80-89 year-olds starts. At the same time, when the 90+ year-olds are ready for the second administration, the priority goes to them and only the remaining withdrawn doses go to the 80-89 year-olds. This priority cascade applies to all the age ranges in reverse order.
Our simulations showed that the second administration priority to the different reverse order age ranges was always feasible (as far as the Italian regions and provinces are concerned).
It is worth spending a few words about point P of Figure 3, which is the number of citizens of that specific age range who live in the nation/region under study. For the sake of precision, P is the number of susceptible citizens i.e. the total number of residents less the individuals who either healed from Covid-19 or died. Indeed, even though Covid-19 plagued the elderly significantly, the to stabilize and equalize their immunization degree (Gobbi et al., 2021).
The immunization degree of each age range depends on the specific adherence to the vaccination campaign, which may vary significantly among regions. For instance, by the end of May 2021, the population fraction that received the first administration was 43% in Molise and only 33% in Sicily.
Equally, the Lombardy citizens in their 90s who received the first administration were 99.97% and 90.88% had received the second shot. According to a more general approach to vaccination, both the blue and green curves of Figure 3 (that depend on the degree of adherence to the real vaccination campaign) reach distinct horizontal asymptotes that are lower than P . Both the blue and green lines at time i t of Figure 3 are the summation of the fractional contributions to immunization of each corresponding vaccinated individual whose immunization degree belongs to the 0,  K interval.
At i t , the number of citizens who are exposed to a possible contagion is either the segment AC in the case of the real vaccination campaign or AB in the case of the alternative vaccination politics.

BC measures the distance between the alternative and the real vaccination campaigns and is
proportional to the number of lives that may be saved. The real number of fatalities ( R i F ) is known at each day ( i t ) and we can rely on the fatalities distribution by age ranges ( k ) reported in Table 1.
Eventually, the number of lives that the alternative vaccination politics would save is: The green line in Figure 3 is not always higher than the blue one for any age range. Actually, the younger age ranges receive a minor knockback due to the doses withdrawn from the non-priority categories and administered according to ROAR. Since the daily doses are assigned according to the real administrations, the withdrawn doses have a minor negative impact on the younger age ranges but produce a major improvement on the older ranges. From a conceptual point of view, the younger categories may experience the following condition: ii AB AC  at some i t . Altogether, an alternative vaccination politics, to be successful, must obey the following condition: 0 LS N  .

Vaccination rules
The vaccination simulator has to comply with and implement a record of rules that are aimed at preserving the most fragile individuals and shorten as far as possible the achievement of their highest immunization degree. The following list reports the main vaccination rules:  In the case of dose shortage, prioritize the second administrations;  Prioritize the vaccines characterized by shorter 2 t times (as of Figure 2) to the older age ranges;  Supersede the  Assume that the fatality distribution with age ranges is constant in time and space (i.e. it does not depend on single regions/provinces as regional data are not publicly available). The time-constant hypothesis was extensively discussed in Section 2 and reported in Table 1).  The citizens who received the first dose before the start of the alternative vaccination politics and are waiting for the second dose will be administered accordingly regardless they belong to non-priority categories.

Program implementation
We chose to implement the vaccination simulator in Matlab 2021a (The MathWorks Inc., Natick, MA, USA). Matlab is a rather practical prototyping tool (although it is not a real programming language) that can download and read remote data on the Internet, work with tables containing different data types (e.g., characters, strings, headings, date and time, integer, and floating-point values) organized in different file formats (e.g., JSON, CSV text files, and raw data). Matlab's most interesting feature for the vaccination simulator is the data filtering option which can prune large tables and efficiently select the data that are necessary to comply with the rules reported in Section 2.4.4. In addition, Matlab can draw good-quality diagrams and figures, and produce reports for the final user. As far as the CPU efficiency is concerned, even though Matlab is an interpreted language and its efficiency is quite low, this is not a limiting feature as a single comparison between the real and the alternative vaccination campaign that covers all the 21 regions and autonomous provinces of Italy takes 50.5 s on an Intel i7-4600U processor running at 2.1 GHz with 16 GB of RAM (i.e. an ordinary Windows 7/10 notebook). In the case of Lombardy, the most populated Italian region, the CPU time is 14.4 s. The Matlab routines that simulate the alternative vaccination politics roughly consist of 2,500 lines of code.

Results and discussion
The basic comparison between the alternative vaccination politics (AVP) and the real vaccination campaign (RVC) refers to Italy with their contemporary start at 18-Feb-2021, when most of the vaccinations to priority categories had occurred. That was day #54 after the very beginning of the Italian vaccination campaign on 27-Dec-2020. Figure 4 and Figure 5 show the comparison for the most exposed and fragile age ranges (over 80 year-olds, and 70-79 year-olds) that cover most of the fatalities due to SARS-CoV-2 infection.  In the over 80 year-olds the AVP first and second administration doses (green lines) are always consistently above the RVC doses (red lines). This shows how the elderly did not receive enough attention and priority, and that a strict ROAR criterion was not respected. Specifically, Figure 5 confirms this statement as in the first nine weeks of AVP the real doses administered to 70-79 yearolds were withdrawn and primarily administered to older citizens (i.e. 90+ and 80-89 year-olds). The dose reallocation of AVP would have covered all the over 80 year-olds with the first administration six weeks earlier than reality. In addition, AVP would have been able to finalize the second administration to over 80 year-olds one/two weeks before the end of May and achieve an almost complete immunization by the end of May instead of the lesser RVC coverage. For the sake of detail, the enhanced vaccination coverage achieved by AVP does not rely only on withdrawing the improperly administered doses to the 70-79 age range but to all the younger age ranges that received any doses before the older ones were fully covered.
Focusing on the 70-79 age range, AVP is capable of crossing and surpassing RVC on 25-Apr as far as the first administration doses are concerned. As of 13-May, the AVP immunization level becomes higher than the RVC one. These results are achieved by a continuous and strict application of the ROAR criterion that withdraws systematically the doses erroneously administered to younger citizens in favor of the older ones.
The rate of administration of the second dose and the achieved immunization degree are further important points of discussion. Figure 4 and Figure 5 show   A parallel discussion can be carried out in the case of a specific region, for instance, Lombardy. For the sake of space, we report only the diagrams of the 70-79 year-olds (see Figure 7) that are the most important and dissimilar when compared with the Italian equivalents. The distance between the green and red lines for the first dose is narrower for Lombardy than for Italy. Similarly, the regional second dose curves and the immunization trends are nearer than the national ones. This means that somehow the vaccination priority towards those above 70 was more preserved in Lombardy than in Italy. Nonetheless, the improvement in terms of saved lives that AVP would have achieved at the regional level is a little better than the national one as the total number of fatalities in Lombardy was proportionally slightly higher than in Italy.  For the sake of detail, the dates reported in Table 2 do not consider the three priority categories that proceed independently and follow the RVC schedule. Conversely, the percentages refer to all the categories split into age ranges and allow grasping the whole picture of the vaccination campaign.    18-Feb, the improvement would have been 11.9% and 40.5% respectively.  at both national and regional levels. How one can achieve an administration increase of 10% or 20%?
Such an increase would interest the whole vaccination engine that builds on the vaccination supply chain. One might work on either improving the efficiency of the engine or increasing the operation of that engine. On the one hand, the efficiency may be tuned and polished in several different ways but it has to deal with the so-called rate-determining step, which is the less efficient step of the engine, i.e. the bottleneck. On the other hand, the operation of the engine may be improved by increasing the time worked. The easiest way would be to introduce overtime. If one assumes that personnel works 8 hours per day an increase of 10% or 20% would call for 48 min and 96 min overtimes. From a practical point of view, personnel might be asked to work either 1 or 1.5 h more.
The improvement in saved lives would be much more significant (as reported in Table 3 and Table   4) and would probably motivate the workers of the vaccination supply chain to embrace overtime with the promise of creating an impact on the number of saved lives. Indeed, the cause of the greater good (coupled with paid overtime) is something that may motivate workers to outperform.
The positive results of AVP are evident and significant (as shown in the Figures and Tables of this   Section). However, is AVP feasible or too optimistic and distant from reality? In our opinion, most of the hypotheses we introduced for AVP to occur are rather realistic and based on pragmatic assumptions. Nonetheless, we see two limitations.
The first limitation is probably the most severe and it deals with the strict ROAR hypothesis that would call for vaccinating first the 90+ and then the 80-89 year-olds and so on. This strict approach to the vaccination sequence of the elderly would call for a very robust engine capable of timely delivering the administrations to citizens who usually have to rely on the help of relatives and/or caregivers and who require a longer time for finalizing a single vaccination. In addition, the reduced time to promptly adhere to the vaccination campaign (to achieve an efficient ROAR approach) is a further and derived limitation of that proposed martial approach. Indeed, a longer time to convince the skeptics would better approximate what in reality happened.
The second limitation is that the algorithm used to determine the expected number of saved lives by AVP uses the daily real fatalities in the region/nation whilst the first and second administrations that contribute to the progressive degree of immunization of the vaccinated population have a delayed effect on the number of deaths. One might introduce a further adaptive parameter in Equation (4) in terms of a time delay to better account for the infection, disease, hospitalization, and death chain. In our opinion, this alternative would introduce a further degree of uncertainty.
That is why we intentionally used the daily fatalities to calculate the expected saved lives as a function of the immunization degree of the different age ranges at the same time of those fatalities.
A final remark should be devoted to the possibility that the fatalities distribution as a function of age might dynamically change its profile according to the unbalanced vaccination coverage of the population. Reality showed that the percentage of elderly fatalities is not significantly affected by the vaccination campaign. This result might seem surprising and counterintuitive. In reality, the vaccination campaign produces the most important results in decreasing significantly the fatalities number. The elderly remain more exposed to life risk than young people and also minor numbers of non-vaccinated elderly still contribute at a large extent to the daily fatalities. This is a further demonstration that it is worth vaccinating the elderly at very high percentages and avoiding any reluctance.

Conclusions
How the real vaccination campaign has been carried out so far in Italy at both national and regional levels inspired and motivated this article. Could it have been better? The answer is a definitive yes.
For the sake of correctness, Italian RVC might have been even worst. Indeed, in March 2021, the new special pandemic commissioner forced RVC to rigorously implement the inverse order of age ranges approach and set the half-million target of daily-administered doses. He stepped up the pressure on every single region of Italy to proportionally reach that target through scheduled weekly and eventually daily increasing values to be scored.
As a matter of fact, this paper does neither want to blame nor to criticize the Italian vaccination campaign that at present is amongst the best in the European Community in terms of both administered doses and population coverage (European Centre for Disease Prevention and Control, 2021). We just wanted to show how a better knowledge and implementation of the vaccination policies together with an earlier start of the vaccination campaign specifically targeted to the elderly would have influenced significantly the fatalities restraint. The initial efficiency of the vaccination engine was rather low regardless of the preliminary and necessary running in. A not negligible amount of doses was administered to non-priority categories leaving the elderly exposed for a longer period to SARS-CoV-2 infection. Some of the over 80 years-olds took more than 100 days before receiving the first dose. After 150 days since the very beginning of the vaccination campaign, quite a few over 80 year-olds were still receiving the first dose. This important delay in the finalization of the vaccination campaign to the elderly played a paramount role in the fatalities toll paid to the Covid-19 pandemic. Besides accelerating the vaccination of the elderly, the higher efficiency of the vaccination engine and overtime would have saved a higher number of lives. These are the lessons learned that may help future decision-makers in the case of new outbreaks towards the implementation of an enlightened vaccination politics. We intentionally used the term politics in this article instead of policy as the setting of a vaccination campaign under a pandemic emergency is more related to strategic political decisions rather than to tactical applied policies.
The availability of an in-silico vaccination simulator allows running a set of parametric simulations based on different scenarios, priority allocations, date shifts, and administration policies. For instance, it can be used to analyze the (in)effectiveness of increasing the time interval between the first and second dose of vaccines (e.g., from 21 to 35 or 42 days for the Pfizer/BioNTech or Moderna vaccines). However, this is another story that goes beyond the scope of this article. Finally, the insilico vaccination simulator can be used for prediction purposes by assigning the expected future administration rates and estimating the upcoming fatalities with suitable predictive models (Manca et al., 2020).

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Availability of data and materials
The data used in the paper are publicly available at: