Modeling vaccination rollouts, SARS-CoV-2 variants and the requirement for non-pharmaceutical interventions in Italy

Despite progress in clinical care for patients with coronavirus disease 2019 (COVID-19)1, population-wide interventions are still crucial to manage the pandemic, which has been aggravated by the emergence of new, highly transmissible variants. In this study, we combined the SIDARTHE model2, which predicts the spread of SARS-CoV-2 infections, with a new data-based model that projects new cases onto casualties and healthcare system costs. Based on the Italian case study, we outline several scenarios: mass vaccination campaigns with different paces, different transmission rates due to new variants and different enforced countermeasures, including the alternation of opening and closure phases. Our results demonstrate that non-pharmaceutical interventions (NPIs) have a higher effect on the epidemic evolution than vaccination alone, advocating for the need to keep NPIs in place during the first phase of the vaccination campaign. Our model predicts that, from April 2021 to January 2022, in a scenario with no vaccine rollout and weak NPIs (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{R}}_0$$\end{document}R0 = 1.27), as many as 298,000 deaths associated with COVID-19 could occur. However, fast vaccination rollouts could reduce mortality to as few as 51,000 deaths. Implementation of restrictive NPIs (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{R}}_0$$\end{document}R0 = 0.9) could reduce COVID-19 deaths to 30,000 without vaccinating the population and to 18,000 with a fast rollout of vaccines. We also show that, if intermittent open–close strategies are adopted, implementing a closing phase first could reduce deaths (from 47,000 to 27,000 with slow vaccine rollout) and healthcare system costs, without substantive aggravation of socioeconomic losses.

The infeasibility of long-term lockdowns (due to detrimental socioeconomic consequences and behavioural fatigue 3,4 ) and of effective contact tracing at high case numbers led many countries to invest in mass vaccination to control the COVID-19 pandemic. Since the SARS-CoV-2 genome was sequenced in January 2020 5 , researchers rushed to develop a vaccine suitable for mass distribution 6,7 . As of February 1 st , 2021, 2 vaccines (Moderna and Pfizer/BioNTech) were approved by FDA for full use 8,9 and 8 more for limited use 10 . The registration clinical trials report 94% and 95% efficacy rates respectively for Moderna and Pfizer/BioNTech vaccines 11,12 , with a favourable safety profile. Italy began its vaccination campaign on December 27 th , 2020 with the delivery of the first In this complex layout, where vaccines and new variants are potential game-changers, models to forecast epidemic scenarios and assess the associated healthcare costs are essential. Our proposed integrated model ( Figure 1A) feeds the predicted evolution of new positive cases, provided by the compartmental model SIDARTHE 2 , extended to include the effect of vaccination, into a new data-based dynamic model, derived from Italian field data, that, once fed by the new cases, computes the time profile of the resulting healthcare system costs (hospital and ICU occupancy and deaths). To capture the progressive vaccination of the older population, the model takes into account the specific aggravation and death probability for different age classes (Extended Data Figure 5). Details are provided in the Methods.
We compare different scenarios to assess the effect of mass vaccination campaigns with different paces, in the presence of varying profiles of the reproduction number ℛ 0 over time, due to specific SARS-CoV-2 variants and/or restrictions. We consider four effective vaccination schedules (Extended Data Figure 4), obtained by modulating linearly the speed of the four phases T1-T4 of Italy's vaccination plan 26 so as to yield a different fraction of successfully immunised people within one year: absent (0%); slow (46%); medium (64%); fast (90%). We also consider five different profiles of ℛ 0 as a function of time: constant ℛ 0 = 1.27 (high transmission); Open-Close periodic ℛ 0 with average value 1.1, alternating first Openings (ℛ 0 = 1.3), and then Closures (ℛ 0 = 0.9); constant ℛ 0 = 1.1; Close-Open periodic ℛ 0 with average value 1.1, alternating first Closures and then Openings; constant ℛ 0 = 0.9 (eradication).
Our main findings and implications for policy-makers are outlined in Table 1 and summarised by the deaths vs. speed curves in Figure 1B, showing the death toll as a function of the vaccination speed for each ℛ 0 profile. The combination of the four vaccination schedules with the five ℛ 0 profiles leads to twenty distinct scenarios, associated with the dots in the deaths vs. speed curves ( Figure   1B). Leaving aside eradication, associated with an almost constant curve (green), with all the other ℛ 0 profiles, vaccination saves at least 44% of lives compared to no vaccination, and even 50% and 57% with medium and fast schedules. The deaths vs. speed curves are flatter when ℛ 0 is kept smaller: stringent NPIs drastically reduce sensitivity to vaccination delays. Containment strategies have an impact on human losses that is 5-fold: depending on the ℛ 0 profile (not considering eradication), deaths in the period from February 2021 to January 2022 vary in the range of 39,000-196,000 (slow vaccination), 35,000-174,000 (medium vaccination) and 31,000-146,000 (fast vaccination). Therefore, NPIs have a much larger effect than vaccination speed. In the planning of mid-term interventions, pre-emption reduces life and healthcare system costs at no socio-economic cost: intermittent containment strategies with the same average ℛ 0 involve the same amount of socioeconomic restrictions, but starting with a Closing phase (Close-Open, purple curve) improves on constant containment (yellow), which is in turn better than starting with an Opening phase (Open-Close, orange). For all vaccination schedules, the Close-Open strategy saves no less than 23,000 lives compared to the Open-Close strategy. Hospital and ICU occupancy as a function of the vaccination speed follow a similar pattern (Extended Data Figure 8).
Considering a medium vaccination speed, Figure 2 shows the epidemic evolution for different constant values of ℛ 0 (the scenarios in the absence of vaccination are in Extended Data Figure 2).
In spite of vaccination and of containment measures, a higher transmissibility due to the spread of new variants would cause a dramatic surge in infection cases, leading to a peak of some 9,000 ICU beds needed and more than 1,600 daily deaths before sufficient population immunity can be reached. A significant reduction in hospital occupancy and death toll can be obtained by enforcing, throughout the vaccination campaign, NPIs to reduce ℛ 0 , which need to be even more stringent in the presence of highly transmissible variants.
The need to enforce new restrictions is likely to trigger intermittent containment measures, with the alternation of higher-ℛ 0 and lower-ℛ 0 phases 27,28 . In Open-Close strategies, closures are delayed and only enforced when the pressure on the healthcare system becomes unbearable. Each intermittent Open-Close strategy can be associated with a Close-Open strategy that alternates the same opening and closing phases, with the only difference of starting with a closure. Figure 3 compares the two different intermittent strategies, with average ℛ 0 equal to 1.1, under mediumspeed vaccination (the comparison in the absence of vaccination is in Extended Data Figure 3). It appears that opening first (Open-Close) or closing first (Close-Open) strongly affects healthcare system costs (which depend on case numbers), while socioeconomic costs (which depend on the duration and stringency of restrictions) are substantially unchanged. Without aggravation of social and economic losses, a pre-emptive Close-Open strategy drastically reduces forthcoming infection numbers (decreasing the peak of daily new cases from more than 35,000 to 16,000), hospital and ICU occupancy, and deaths (decreasing the peak of daily deaths from 650 to 280), with a sustained reduction of the epidemic phenomenon over time, which facilitates the vaccination campaign and increases the efficacy and feasibility of testing and tracing 18 . Even though the average ℛ 0 is above 1, the effective reproduction number ℛ = ℛ 0 ( ) goes below 1 due to the decreasing susceptible fraction ( ), hence the epidemic is eventually suppressed (see Methods).
Finally, we comparatively assess the effect of mass vaccination with different paces. We assume that no reinfections occur in a one-year horizon. Figure 4 compares the effect of slow vs. fast vaccination under the intermittent Open-Close strategy. Although vaccination leads to a net reduction in deaths and hospital and ICU occupancy compared to the corresponding scenario without vaccination, the difference in impact between slow and fast vaccination is modest: by the time the impact of a faster vaccination could become evident, the epidemic is already contained.
The difference is more visible at a higher ℛ 0 , at the price of much higher losses. In Extended Data Figures 6 and 7, we also consider an adaptive vaccination scenario, where an increase in the number of current infection cases leads to a reduction of the vaccination rate, due to the augmented strain on the healthcare system: both death toll and healthcare system costs increase, reinforcing conclusions about the greater importance of containment measures over vaccination rates.
Our findings confirm that physical distancing, testing and contact tracing are now more crucial than ever due to the circulation of highly transmissible variants of SARS-CoV-2 and the risk of the emergence of vaccine-resistant mutations. We have shown the role of non-pharmaceutical interventions throughout a mass vaccination campaign in keeping the reproduction number low until a sufficient population immunity is achieved. In order to contain the epidemic, restrictions appear more effective than a fast vaccination campaign, in line with US-based studies 29 . Casualties and healthcare system costs predicted by our data-based model also show the importance of preemptive action when enforcing intermittent close-open strategies: without any aggravation of socioeconomic costs, early closures can drastically lower healthcare system costs with respect to delayed closures of the same duration and stringency.
In our scenarios, vaccination has been assumed effective against SARS-CoV-2 variants. However, several concerns are raised regarding variants and their potential for vaccine-induced immunity escape 30,31 ; preliminary reports suggest that COVID-19 vaccines likely retain efficacy against variants 32 , although it might be attenuated 33 , but data are currently limited. Long-term and largescale monitoring is required to prove these assumptions; in the meanwhile, enforcing NPIs to keep case numbers low is particularly important. 17

Methods
Our overall model (see Figure 1A) combines the flexibility and insight of compartmental models with the intrinsic robustness of a black-box healthcare system cost model based on observed data.
The SIDARTHE-V model, including the compartment of vaccinated individuals (first block in Figure   1A), generates the predicted evolution of new positive cases, which feeds the data-based model (second block in Figure 1A) that captures hospitalisation flows and quantifies healthcare system costs in terms of deaths and of hospital and ICU occupancy.

SIDARTHE-V Compartmental Model
The SIDARTHE-V compartmental model shown in Figure  The dynamic interaction between these nine clusters of the population is described by the following nine ordinary differential equations, describing how the fraction of the population in each cluster evolves over time: The uppercase Latin letters (state variables) represent the fraction of population in each stage, while all the considered parameters, denoted by Greek letters, are positive numbers and have the following meaning.

•
The contagion parameters , , , respectively denote the transmission rate (defined as the probability of disease transmission in a single contact multiplied by the average number of contacts per person) due to contacts between a Susceptible subject and an Infected, a Diagnosed, an Ailing, a Recognised subject. These parameters can be modified by social distancing policies (e.g., closing schools, remote working, lockdown), as well as physical distancing, adoption of proper hygiene behaviours and use of personal protective equipment. The risk of contagion due to Threatened subjects, treated in proper ICUs, is assumed negligible.

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The diagnosis parameters and respectively denote the probability rate of detection, relative to asymptomatic and symptomatic cases. These parameters, also modifiable, reflect the level of attention on the disease and the number of tests performed over the population: they can be increased by enforcing a massive contact-tracing and testing campaign.

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The symptom-onset parameters and represent the probability rate at which an infected subject, respectively undetected and detected, develops clinically relevant symptoms. Although disease-dependent, they may be partially reduced by improved therapies and acquisition of immunity against the virus.

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The critical/aggravation parameters µ and respectively denote the rate at which undetected and detected infected symptomatic subjects develop life-threatening symptoms.
They parameters can be reduced by means of improved therapies and acquisition of immunity against the virus.

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The mortality parameters 1 and 2 respectively denote the mortality rate for infected subjects with symptoms (presumably in hospital wards) and with acute symptoms (presumably in intensive care units) and can be reduced by means of improved therapies.

•
The healing parameters , , , and denote the rate of recovery for the five classes of infected subjects and can be increased thanks to improved treatments and acquisition of immunity against the virus. To better understand the system behaviour, we partition it into three subsystems: the first includes just variable (corresponding to susceptible individuals); the second, which we denote as the subsystem, includes , , , and (the infected individuals); and the third includes variables , and (representing healed, defuncts and vaccinated/immunised).
As a consequence, epidemic suppression is achieved when the inequality ℛ = ℛ 0 ( ) < 1 is always verified from a certain moment onwards.

Fit of the SIDARTHE-V Model for the COVID-19 Epidemic in Italy
We infer the parameters for model (1) to 47 (fast speed), while they would be 100 without vaccination.

Data-Driven Model of Healthcare System Costs
In order to predict the evolution of deaths from the time series of reported cases, a field estimate of the apparent CFR (Case Fatality Rate) is needed. This parameter is affected by the testing protocol, the healthcare system reaction, and the age distribution of vaccinated people. For these reasons, a specific model should be derived for each country resorting to a data-based approach. We where the weight ( ) denotes the fraction of subjects that became infected at day − that eventually die at day . The CFR for the unvaccinated population is then given by In order to estimate the weights, an exponential model with delay was assumed: where is the delay and and are unknown parameters. Since both the new cases and the daily deaths exhibit an apparent weekly seasonality, the original series were replaced by their 7 Then, recalling the formula for the sum of the harmonic series, the second-wave CFR for the unvaccinated population is given by In the second step, the effect of vaccination on lethality was modelled by estimating a time-varying ( ) that depends on the vaccination schedule, described by the fraction ( ) of vaccineimmunized subjects at time . Order of vaccination follows the reverse of the age. Slower or faster vaccination speeds correspond to different curves ( ), whose rate of increase may be less or more rapid. Three schedules were considered: fast, medium, and slow. The fast schedule assumes that each the four phases T1-T4 of the Italian vaccination plan 37 is completed in one trimester. In the medium and slow schedule the time was linearly extended by a factor 1.2 and 1.4, respectively. The three schedules are graphically displayed in Extended Data Figure 4. If we assume that the number of deaths does not affect significantly the overall age distribution, the probability ( = ) can be directly inferred by ISTAT statistical tables 39 . The distribution of population by age is displayed in Extended Data Figure 5, Panel C.
Recalling that ( ) is the fraction of vaccine-immunized subjects and vaccination order follows the reverse of the age, the probability of death for a subject of age a that becomes infected at time t is Then, the time varying case fatality rate is obtained by the total probability theorem: where ( = | , ) denotes the probability that the age of a subject is , knowing that it became infected at time . Currently, the age distribution of the infected subjects 38  Assuming that gravity reduction parallels the lethality one, the effect of vaccination on hospital and ICU occupancies was described by modulating the input through the time-varying coefficient ( ).
As seen in Extended Data Figure 9, the three data-based dynamic models provide a very good fitting of deaths, hospital and ICU occupancies.

Reporting Summary. Further information on research design is available in the Life Sciences
Reporting Summary linked to this article.
Data are also included in the Extended Data Figures 1 and 9.

Code availability
The codes will be made available online.     Our scenarios are outlined based on reasonable assumptions, but the actual epidemic evolution will depend on the adopted measures and the possible emergence of other variants.