# A Modeling Study on Vaccination and Spread of SARS-CoV-2 Variants in Italy

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. SEIRL-V Compartmental Model

- S: susceptible individuals,
- ${E}_{\nu}$: exposed individuals, where $\nu =0,1,2$ denotes the numbers of vaccines doses received,
- ${P}_{S,\nu}$: pre-symptomatic individuals, where $\nu =0,1,2$ denotes the numbers of vaccines doses received,
- A: asymptomatic individuals,
- M: people with mild infection,
- H: people in hospital with severe symptoms,
- $ICU$: people with critical infection which requires ICU level care,
- R: recovered individuals,
- D: dead people,
- ${V}_{1}$: people vaccinated with the first dose of vaccine,
- ${V}_{2}$: people vaccinated with the second dose of vaccine,
- $Im$: immune individuals.
- ${S}_{\nu}$ with $\nu =0,1,2$ as the number of doses received. In more detail, ${S}_{0}=S$, ${S}_{1}=(1-{\rho}_{1}){V}_{1}$ and ${S}_{2}=(1-{\rho}_{2}){V}_{2}$.

- (1)
- ${a}_{1}=PresymPerio{d}^{-1}$
- (2)
- ${a}_{0}={(IncubPeriod-PresymPeriod)}^{-1}$
- (3)
- ${g}_{1}=DurMildIn{f}^{-1}\xb7(1-FracSevere-FracCritical)$
- (4)
- ${p}_{1}=DurMildIn{f}^{-1}-{g}_{1}$
- (5)
- ${p}_{2}=DurHos{p}^{-1}\xb7\frac{FracCritical}{(FracSevere+FracCritical)}$
- (6)
- ${u}_{1}=DurHos{p}^{-1}\xb7(\frac{(ProbDeathH\xb7\frac{FracSevere}{100})}{FracSevere}$
- (7)
- ${g}_{2}=DurHos{p}^{-1}-{p}_{2}-{u}_{1}$
- (8)
- $u=TimeICUDeat{h}^{-1}\xb7(\frac{(ProbDeath\xb7\frac{FracCritical}{100})}{FracCritical}$
- (9)
- ${g}_{3}=TimeICUDeat{h}^{-1}-u$
- (10)
- $f=FracAsym$
- (11)
- ${g}_{0}=DurAsy{m}^{-1}$,

- parameter $\eta $ is the rate of injection of the first dose. It is modeled as a piecewise constant function;
- parameter $\tau $ is the time between the first and second dose of vaccine and it is set to 21 days [11];
- parameter ${\tau}_{imm}$ is the number of days between the second dose and the acquired immunity and it is set to 14 days [11];
- parameter ${\rho}_{1}$ is the efficacy of the first shot of vaccine and it is set to 0.8 [27];
- parameter ${\rho}_{2}$ is the efficacy of the second shot of vaccine and it is set to 0.95 [11];
- parameter ${z}_{\nu}$ with $\nu =0,1,2$ is introduced to represent vaccine efficacy against disease. Thus, ${z}_{0}=0$, ${z}_{1}={\rho}_{3}$ and ${z}_{2}={\rho}_{4}$.

#### 2.2. Data

#### 2.3. Conditional Robust Calibration (CRC) for Parameter Estimation

**P**. It returns in output an approximation of the parameter posterior distribution conditioned to the available data ${f}_{\mathbf{P}|{\mathbf{y}}^{*}}\left(\mathbf{p}\right)$, where ${\mathbf{y}}^{*}$ is the dataset. At each iteration, CRC generates a matrix ${P}_{O}$ of ${N}_{S}$ parameter vectors through Latin Hypercube Sampling (LHS). Each parameter vector $\mathbf{p}$ is sampled inside an interval between a lower and upper boundary, ${L}^{1}$ and ${U}^{1}$, chosen by the user. Parameters are assumed to be uniformly or logarithmically distributed in their intervals. Then, for each sample $\mathbf{p}\in {P}_{O}$, the ODE model is integrated to compute the in silico vector of observables

**y**. The fitting between the simulated vector

**y**and the dataset ${\mathbf{y}}^{*}$ is measured through the Absolute Distance Function (ADF):

**p**$\in {P}_{O}$, each $AD{F}_{i}\phantom{\rule{1.em}{0ex}}i=1,...,m$ is computed, and we select only those distance functions under a user defined threshold ${\u03f5}_{i}\ge 0$. Thus, we obtain different parameter sets ${P}_{O,{\u03f5}_{i}}$, one for each output variable. Each set contains only those parameters that yield the values of a specific distance function under the corresponding threshold. All these sets are intersected to obtain ${P}_{O,\u03f5}=\left\{{\bigcap}_{i=1}^{m}{P}_{O,{\u03f5}_{i}}\right\}$, where $\u03f5=\{{\u03f5}_{1},...,{\u03f5}_{i},...,{\u03f5}_{m}\}$. Using a kernel density approach, the approximate posterior distribution ${f}_{\mathbf{P}|{P}_{O,\u03f5}}$ is estimated. This procedure is repeated for multiple iterations, updating the sampling interval on the basis of the posterior distribution of the previous iteration. The final output of the algorithm is ${f}_{\mathbf{P}|{P}_{O,\u03f5}}$, where $\u03f5$ is the set of thresholds chosen in the final CRC iteration. The code for running CRC and the SEIRL-V model of COVID-19 is available at https://github.com/fortunatobianconi/CRC (accessed on 7 June 2021).

## 3. Results

#### 3.1. Spread of Sars-CoV-2 Lineages in Umbria and Italy

#### 3.2. Umbria Case Study

- (1)
- 14 September 2020 (${T}_{lock,1}$), school reopening;
- (2)
- 19 October 2020 (${T}_{lock,2}$), the Regional Government adopted some preventative measures such as remote teaching for part of the students, limited capacity of public transportation and closure of shopping malls during the weekend [37];
- (3)
- 11 November 2020 (${T}_{lock,3}$), Umbria region is classified as “orange”, i.e., as a medium-risk contagion zone;
- (4)
- 6 December 2020 (${T}_{lock,4}$), Umbria goes back to “yellow” zone, i.e., with moderate risk of virus spread;
- (5)
- 7 January 2021 (${T}_{lock,5}$), school reopening and easing of some restrictions after the country-wide red area;
- (6)
- 8 February 2021 (${T}_{lock,6}$), “red” area for the entire Province of Perugia, i.e., the highest level of restrictions, following an improvement in the contagion data and the identification of variants;
- (7)
- 22 March 2021 (${T}_{lock,7}$), back to ’orange’ zone with reopening of schools for the youngest.

- $FracCritical=FracCritica{l}_{1}$ from day 0 (1 September 2020) to day 35 (5 October 2020);
- $FracCritical=FracCritica{l}_{2}$ from day 36 to day 83 (22 November 2020);
- $FracCritical=FracCritica{l}_{3}$ from day 84 to day 152 (30 January 2021);
- $FracCritical=FracCritica{l}_{4}$ from day 153 to day 200 (19 March 2021);
- $FracCritical=FracCritica{l}_{5}$ from day 201 onward.

- 26 April 2021: reintroduction of the low-risk “yellow” zone;
- 24 May 2021: curfew extension, gym reopening and restaurants with indoor seating;
- 21 June 2021: curfew lifted and holiday season.

#### 3.3. Italy Case Study

- (1)
- 14 September 2020 (${T}_{lock,1}$), school reopening;
- (2)
- 6 November 2020 (${T}_{lock,2}$), introduction of a three-tier color coded system of restrictive measures, based on the risk profile of each region;
- (3)
- 24 December 2020 (${T}_{lock,3}$), country-wide lockdown for Christmas holidays;
- (4)
- 7 January 2021 (${T}_{lock,4}$), school reopening and easing of some restrictions after the country-wide red area;
- (5)
- 15 March 2021 (${T}_{lock,5}$), removal of ’yellow’ zone in the color-coded system, leaving only medium and high risk zones.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**CRC results for Umbria. The columns show the 60th, 70th, and 90th percentile of the probability density function (pdf) of the parameter vector returned by CRC in the last iteration, for one of the realizations.

Parameter | 60th Percentile | 70th Percentile | 90th Percentile |
---|---|---|---|

${b}_{e,1}$ | [0.1483–0.1521] | [0.1462–0.154] | [0.1422–0.1581] |

${b}_{0,1}$ | [0.3084–0.3173] | [0.3032–0.3232] | [0.2895–0.3331] |

${b}_{e,2}$ | [0.2044–0.2377] | [0.1896–0.2609] | [0.1621–0.313] |

${b}_{0,2}$ | [0.3392–0.3589] | [0.3299–0.3673] | [0.31–0.3883] |

${b}_{1}$ | [0.0314–0.0365] | [0.0291–0.0387] | [0.0223–0.0441] |

${b}_{2}$ | [0.0158–0.0195] | [0.0142–0.0218] | [0.0114–0.0272] |

${b}_{3}$ | [0.0286–0.0342] | [0.0263–0.0379] | [0.0217–0.0461] |

$FracSevere$ | [0.0166–0.0175] | [0.0161–0.018] | [0.0148–0.0192] |

$FracCritica{l}_{1}$ | [0.0044–0.0046] | [0.0043–0.0047] | [0.0041–0.0049] |

$FracCritica{l}_{2}$ | [0.0051–0.0054] | [0.005–0.0056] | [0.0047–0.0059] |

$FracCritica{l}_{3}$ | [0.0039–0.0041] | [0.0038–0.0042] | [.0036–0.0044] |

$FracCritica{l}_{4}$ | [0.0048–0.0051] | [0.0047–0.0052] | [0.0046–0.0054] |

$FracCritica{l}_{5}$ | [0.0028–0.003] | [0.0027–0.0031] | [0.0026–0.0034] |

$FracAsym$ | [0.4738–0.4834] | [0.4684–0.4878] | [0.4567–0.4962] |

$IncubPeriod$ | [4.4704–4.5473] | [4.429–4.591] | [4.3421–4.662] |

$DurMildInf$ | [15.8483–16.2392] | [15.63–16.4292] | [15.2157–16.809] |

$DurAsym$ | [10.9094–11.3187] | [10.7329–11.4892] | [10.3026–11.844] |

$DurHosp$ | [12.9659–13.3531] | [12.8103–13.5196] | [12.33–13.8242] |

$TimeICUDeath$ | [11.3651–11.5532] | [11.2718–11.6631] | [11.0916–11.8755] |

$ProbDeath$ | [28.5227–29.8384] | [27.5732–30.6969] | [25.865–32.129] |

$ProbDeat{h}_{H}$ | [21.9761–22.7784] | [21.5211–23.2579] | [20.6255–24.3682] |

$PresymPeriod$ | [0.5967–0.6039] | [0.5928–0.6085] | [0.5844–0.6163] |

n | [48.1783–51.8111] | [45.8733–53.8871] | [41.8768–57.764] |

K | [$2.8576\times {10}^{4}$–$3.8125\times {10}^{4}$] | [$2.3307\times {10}^{4}$–$4.3381\times {10}^{4}$] | [$1.2627\times {10}^{4}$–$5.4783\times {10}^{4}$] |

${s}_{01}$ | [1.0424–1.0614] | [1.0331–1.0727] | [1.0110–1.0911] |

${s}_{02}$ | [0.2722–0.2807] | [0.2663–0.2857] | [0.2562–0.2949] |

${s}_{03}$ | [0.263–0.2746] | [0.2578–0.2802] | [0.2456–0.2929] |

${s}_{04}$ | [0.4205–0.4297] | [0.4146–0.4346] | [0.4049–0.4447] |

${s}_{05}$ | [0.5608–0.5818] | [0.5506–0.5922] | [0.5307–0.6165] |

${s}_{06}$ | [0.2678–03279] | [0.2634–0.2839] | [0.2542–0.2944] |

${s}_{07}$ | [0.3181–0.3279] | [0.3126–0.334] | [0.3044–0.344] |

**Table A2.**CRC results for Italy. The columns show the 60th, 70th, and 90th percentile of the probability density function (pdf) of the parameter vector returned by CRC in the last iteration, for one of the realizations.

Parameter | 60th Percentile | 70th Percentile | 90th Percentile |
---|---|---|---|

${b}_{e,1}$ | [0.1467–0.1559] | [0.1418–0.1609] | [0.1318–0.1721] |

${b}_{0,1}$ | [0.1933–0.2016] | [0.1901–0.2054] | [0.1833–0.214] |

${b}_{e,2}$ | [0.2686–0.2912] | [0.2587–0.3063] | [0.2393–0.3451] |

${b}_{0,2}$ | [0.1564–0.1711] | [0.1493–0.1792] | [0.1366–0.205] |

${b}_{1}$ | [0.0151–0.0189] | [0.0135–0.0208] | [0.0112–0.0261] |

${b}_{2}$ | [0.057–0.0608] | [0.0552–0.0627] | [0.0517–0.0677] |

${b}_{3}$ | [0.0157–0.0196] | [0.0141–0.0218] | [0.0112–0.0267] |

$FracSevere$ | [0.0246–0.0258] | [0.024–0.0264] | [0.0221–0.0283] |

$FracCritica{l}_{1}$ | [0.0053–0.0054] | [0.0052–0.0055] | [0.0051–0.0057] |

$FracCritica{l}_{2}$ | [0.0066–0.0069] | [0.0064–0.007] | [0.0061–0.0073] |

$FracCritica{l}_{3}$ | [0.0033–0.0035] | [0.0033–0.0036] | [0.0031–0.0039] |

$FracCritica{l}_{4}$ | [0.0044–0.0045] | [0.0043–0.0046] | [0.0041–0.0049] |

$FracAsym$ | [0.4622–0.4683] | [0.4592–0.4712] | [0.4529–0.4772] |

$IncubPeriod$ | [5.1812–5.2188] | [5.16–5.2379] | [5.1188–5.2792] |

$DurMildInf$ | [9.3392–9.9241] | [9.0192–10.2157] | [8.366–10.7008] |

$DurAsym$ | [14.6655–15.0305] | [14.4845–15.2173] | [14.1736–15.7103] |

$DurHosp$ | [15.3648–15.5912] | [15.2701–15.7051] | [15.0902–15.8961] |

$TimeICUDeath$ | [12.3828–12.5781] | [12.2907–12.673] | [12.0992–12.8822] |

$ProbDeath$ | [34.7139–35.4889] | [34.3299–35.9058] | [33.4267–36.5895] |

$ProbDeat{h}_{H}$ | [23.6944–24.3566] | [23.2392–24.8239] | [22.1059–25.5674] |

$PresymPeriod$ | [0.615–0.6227] | [0.6114–0.6274] | [0.6033–0.6363] |

n | [50.9573–53.9461] | [49.4666–55.479] | [46.4324–58.8729] |

K | [$4.321\times {10}^{4}$–$4.9343\times {10}^{4}$] | [$3.9531\times {10}^{4}$–$5.2406\times {10}^{4}$] | [$3.3381\times {10}^{4}$–$5.7296\times {10}^{4}$] |

${s}_{01}$ | [1.0205–1.0297] | [1.0153–1.0359] | [1.0055–1.0442] |

${s}_{02}$ | [0.3–0.3233] | [0.289–0.3369] | [0.2653–0.369] |

${s}_{03}$ | [0.242–0.2708] | [0.2305–0.2842] | [0.2099–03254] |

${s}_{04}$ | [0.6434–0.6667] | [0.6337–0.6829] | [0.612–0.7307] |

${s}_{05}$ | [0.1744–0.2092] | [0.1614–0.2262] | [0.1344–0.2704] |

## References

- WHO. COVID-19 Dashboard; World Health Organization: Geneva, Switzerland, 2020; Available online: https://covid19.who.int/ (accessed on 15 June 2021).
- Vinceti, M.; Filippini, T.; Rothman, K.J.; Di Federico, S.; Orsini, N. SARS-CoV-2 infection incidence during the first and second COVID-19 waves in Italy. Environ. Res.
**2021**, 197, 111097. [Google Scholar] [CrossRef] - Trasmissione Documento “Prevenzione e Risposta a Covid-19: Evoluzione Della Strategia e Pianificazione Nella Fase di Transizione Per il Periodo Autunno-Invernale. Available online: https://www.trovanorme.salute.gov.it/norme/renderNormsanPdf?anno=2020&codLeg=76597&parte=1%20&serie=null (accessed on 15 October 2020).
- Giordano, G.; Colaneri, M.; Di Filippo, A.; Blanchini, F.; Bolzern, P.; De Nicolao, G.; Sacchi, P.; Colaneri, P.; Bruno, R. Modeling vaccination rollouts, SARS-CoV-2 variants and the requirement for non-pharmaceutical interventions in Italy. Nat. Med.
**2021**, 27, 993–998. [Google Scholar] [CrossRef] - Di Giallonardo, F.; Puglia, I.; Curini, V.; Cammà, C.; Mangone, I.; Calistri, P.; Cobbin, J.C.; Holmes, E.C.; Lorusso, A. Emergence and spread of SARS-CoV-2 lineages B. 1.1. 7 and P. 1 in Italy. Viruses
**2021**, 13, 794. [Google Scholar] [CrossRef] - Davies, N.G.; Abbott, S.; Barnard, R.C.; Jarvis, C.I.; Kucharski, A.J.; Munday, J.D.; Pearson, C.A.; Russell, T.W.; Tully, D.C.; Washburne, A.D.; et al. Estimated transmissibility and impact of SARS-CoV-2 lineage B. 1.1. 7 in England. Science
**2021**, 372, 6538. [Google Scholar] [CrossRef] - Faria, N.R.; Mellan, T.A.; Whittaker, C.; Claro, I.M.; Candido, D.d.S.; Mishra, S.; Crispim, M.A.; Sales, F.C.; Hawryluk, I.; McCrone, J.T.; et al. Genomics and epidemiology of the P. 1 SARS-CoV-2 lineage in Manaus, Brazil. Science
**2021**, 372, 815–821. [Google Scholar] [CrossRef] - Prevalenza delle VOC (Variant of Concern) del virus SARS-CoV-2 in Italia: Blineage B.1.1.7, P.1 e B.1.351, e Altre Varianti (Variant of Interest, VOI) Tra Cui Lineage P.2 e Lineage B.1.525 (Indagine del 20 April 2021). Available online: https://www.iss.it/documents/20126/0/Relazione+tecnica+indagine+rapida+varianti+SARS-COV-2.pdf/f425e647-efdb-3f8c-2f86-87379d56ce8d?t=1620232350272 (accessed on 26 April 2021).
- COVID: Bolzano, Umbria Likely to Become Red Zones. Available online: https://www.ansa.it/english/news/politics/2021/02/19/covid-bolzano-umbria-likely-to-become-red-zones_b92c110b-a505-4100-a241-7ee2d459e8a0.html (accessed on 20 February 2021).
- Le, T.T.; Andreadakis, Z.; Kumar, A.; Román, R.G.; Tollefsen, S.; Saville, M.; Mayhew, S. The COVID-19 vaccine development landscape. Nat. Rev. Drug Discov.
**2020**, 19, 305–306. [Google Scholar] [CrossRef] - Polack, F.P.; Thomas, S.J.; Kitchin, N.; Absalon, J.; Gurtman, A.; Lockhart, S.; Perez, J.L.; Pérez Marc, G.; Moreira, E.D.; Zerbini, C.; et al. Safety and efficacy of the BNT162b2 mRNA Covid-19 vaccine. N. Engl. J. Med.
**2020**, 383, 2603–2615. [Google Scholar] [CrossRef] - Baden, L.R.; El Sahly, H.M.; Essink, B.; Kotloff, K.; Frey, S.; Novak, R.; Diemert, D.; Spector, S.A.; Rouphael, N.; Creech, C.B.; et al. Efficacy and safety of the mRNA-1273 SARS-CoV-2 vaccine. N. Engl. J. Med.
**2021**, 384, 403–416. [Google Scholar] [CrossRef] [PubMed] - Voysey, M.; Clemens, S.A.C.; Madhi, S.A.; Weckx, L.Y.; Folegatti, P.M.; Aley, P.K.; Angus, B.; Baillie, V.L.; Barnabas, S.L.; Bhorat, Q.E.; et al. Single-dose administration and the influence of the timing of the booster dose on immunogenicity and efficacy of ChAdOx1 nCoV-19 (AZD1222) vaccine: A pooled analysis of four randomised trials. Lancet
**2021**, 397, 881–891. [Google Scholar] [CrossRef] - Sadoff, J.; Gray, G.; Vandebosch, A.; Cárdenas, V.; Shukarev, G.; Grinsztejn, B.; Goepfert, P.A.; Truyers, C.; Fennema, H.; Spiessens, B.; et al. Safety and efficacy of single-dose Ad26. COV2. S vaccine against Covid-19. N. Engl. J. Med.
**2021**, 384, 2187–2201. [Google Scholar] [CrossRef] [PubMed] - Piano Vaccini Anti Covid-19. Available online: https://www.salute.gov.it/portale/nuovocoronavirus/dettaglioContenutiNuovo-Coronavirus.jsp?lingua=italiano&id=5452&area=nuovoCoronavirus&menu=vuoto (accessed on 20 December 2020).
- Moore, S.; Hill, E.M.; Tildesley, M.J.; Dyson, L.; Keeling, M.J. Vaccination and non-pharmaceutical interventions for COVID-19: A mathematical modelling study. Lancet Infect. Dis.
**2021**, 21, 793–802. [Google Scholar] [CrossRef] - Bubar, K.M.; Reinholt, K.; Kissler, S.M.; Lipsitch, M.; Cobey, S.; Grad, Y.H.; Larremore, D.B. Model-informed COVID-19 vaccine prioritization strategies by age and serostatus. Science
**2021**, 371, 916–921. [Google Scholar] [CrossRef] [PubMed] - Saad-Roy, C.M.; Wagner, C.E.; Baker, R.E.; Morris, S.E.; Farrar, J.; Graham, A.L.; Levin, S.A.; Mina, M.J.; Metcalf, C.J.E.; Grenfell, B.T. Immune life history, vaccination, and the dynamics of SARS-CoV-2 over the next 5 years. Science
**2020**, 370, 811–818. [Google Scholar] [CrossRef] [PubMed] - Antonini, C.; Calandrini, S.; Stracci, F.; Dario, C.; Bianconi, F. Mathematical Modeling and Robustness Analysis to Unravel COVID-19 Transmission Dynamics: The Italy Case. Biology
**2020**, 9, 394. [Google Scholar] [CrossRef] [PubMed] - Bianconi, F.; Antonini, C.; Tomassoni, L.; Valigi, P. Robust calibration of high dimension nonlinear dynamical models for omics data: An application in cancer systems biology. IEEE Trans. Control Syst. Technol.
**2018**, 28, 196–207. [Google Scholar] [CrossRef] - Bianconi, F.; Antonini, C.; Tomassoni, L.; Valigi, P. Application of conditional robust calibration to ordinary differential equations models in computational systems biology: A comparison of two sampling strategies. IET Syst. Biol.
**2020**, 14, 107–119. [Google Scholar] [CrossRef] [PubMed] - Bianconi, F.; Tomassoni, L.; Antonini, C.; Valigi, P. A New Bayesian Methodology for Nonlinear Model Calibration in Computational Systems Biology. Front. Appl. Math. Stat.
**2020**, 6, 25. [Google Scholar] [CrossRef] - López, L.; Rodo, X. A modified SEIR model to predict the COVID-19 outbreak in Spain and Italy: Simulating control scenarios and multi-scale epidemics. Results Phys.
**2020**, 21, 103746. [Google Scholar] [CrossRef] - Covid-19 Opendata Vaccini, GitHub Repository. Available online: https://github.com/italia/covid19-opendata-vaccini (accessed on 1 June 2021).
- Dispinseri, S.; Secchi, M.; Pirillo, M.F.; Tolazzi, M.; Borghi, M.; Brigatti, C.; De Angelis, M.L.; Baratella, M.; Bazzigaluppi, E.; Venturi, G.; et al. Neutralizing antibody responses to SARS-CoV-2 in symptomatic COVID-19 is persistent and critical for survival. Nat. Commun.
**2021**, 12, 1–12. [Google Scholar] [CrossRef] - Pfizer and BioNTech Initiate a Study as Part of Broad Development Plant o Evaluate Covid-19 Booster and New Variants. Available online: https://www.pfizer.com/news/press-release/press-release-detail/pfizer-and-biontech-initiate-study-part-broad-development (accessed on 27 February 2021).
- Thompson, M.G.; Burgess, J.L.; Naleway, A.L.; Tyner, H.L.; Yoon, S.K.; Meece, J.; Olsho, L.E.; Caban-Martinez, A.J.; Fowlkes, A.; Lutrick, K.; et al. Interim estimates of vaccine effectiveness of BNT162b2 and mRNA-1273 COVID-19 vaccines in preventing SARS-CoV-2 infection among health care personnel, first responders, and other essential and frontline workers—Eight US locations, December 2020–March 2021. Morb. Mortal. Wkly. Rep.
**2021**, 70, 495. [Google Scholar] [CrossRef] - Task force COVID-19 del Dipartimento Malattie Infettive e Servizio di Informatica, Istituto Superiore di Sanità. Epidemia COVID-19. Aggiornamento Nazionale: 14 Luglio 2021. Available online: https://www.epicentro.iss.it/coronavirus/bollettino/Bollettino-sorveglianza-integrata-COVID-19_14-luglio-2021.pdf (accessed on 15 July 2021).
- Mateo-Urdiales, A.; Alegiani, S.S.; Fabiani, M.; Pezzotti, P.; Filia, A.; Massari, M.; Riccardo, F.; Tallon, M.; Proietti, V.; Del Manso, M.; et al. Risk of SARS-CoV-2 infection and subsequent hospital admission and death at different time intervals since first dose of COVID-19 vaccine administration, Italy, 27 December 2020 to mid-April 2021. Eurosurveillance
**2021**, 26, 2100507. [Google Scholar] [CrossRef] - Nguyen, L.K.; Kulasiri, D. On the functional diversity of dynamical behaviour in genetic and metabolic feedback systems. BMC Syst. Biol.
**2009**, 3, 51. [Google Scholar] [CrossRef] [Green Version] - Fosu, G.O.; Akweittey, E.; Adu-Sackey, A. Next-generation matrices and basic reproductive numbers for all phases of the Coronavirus disease. Open J. Math. Sci.
**2020**, 4, 261–272. [Google Scholar] [CrossRef] - Dipartimento Protezione Civile, GitHub Repository. Available online: https://github.com/pcm-dpc/COVID-19 (accessed on 2 May 2021).
- SARS-CoV-2 Variant Classifications and Definitions. Available online: https://www.cdc.gov/coronavirus/2019-ncov/cases-updates/variant-surveillance/variant-info.html#Concern (accessed on 15 May 2021).
- Pearson, C.A.; Russell, T.W.; Davies, N.; Kucharski, A.J.; CMMID COVID-19 working group; Edmunds, W.J.; Eggo, R.M. Estimates of Severity and Transmissibility of Novel South Africa SARS-CoV-2 Variant 501Y. V2. Available online: https://cmmid.github.io/topics/covid19/sa-novel-variant.html (accessed on 1 May 2021).
- Prevalenza Della Variante VOC 202012/01, Lineage B.1.1.7 in Italia, Studio di Prevalenza 4–5 Febbraio 2021. Available online: https://www.iss.it/documents/20126/0/Relazione+tecnica+prima+indagine+flash+VOC+UK+12.2.2021_rev+finale.pdf/636b3651-a358-468e-24d5-65364a004994?t=1613400983576 (accessed on 10 February 2021).
- Prevalenza Delle Varianti VOC 202012/01 (Lineage B.1.1.7), P.1, e 501.V2 (Lineage B.1.351) in Italia Indagine del 18 Febbraio 2021. Available online: https://www.iss.it/documents/20126/0/Relazione+tecnica+terza+indagine+flash+per+le+varianti+del+virus+SARS-CoV-2+%282%29.pdf/a03f33e6-d775-9ab0-b0ce-9cdd289c711d?t=1614707205598 (accessed on 20 February 2021).
- Ordinanza della Presidente Della Giunta Regionale, 19 Ottobre 2020, n. 65. Available online: https://www.regione.umbria.it/documents/18/24917656/ORDINANZA_65.pdf/80514ae4-6ae5-43f0-a0d7-43e23dae6194 (accessed on 20 October 2020).
- Riaperture Italia, il Calendario del Governo. Available online: https://www.adnkronos.com/riaperture-italia-il-calendario-del-governo_4GbnBUMmxOnrDY2YEKt5Jv (accessed on 17 May 2021).
- Anderson, R.M.; Vegvari, C.; Truscott, J.; Collyer, B.S. Challenges in creating herd immunity to SARS-CoV-2 infection by mass vaccination. Lancet
**2020**, 396, 1614–1616. [Google Scholar] [CrossRef] - D’Souza, G.; Dowdy, D. What’s herd immunity and how can we achieve it with COVID-19. Johns Hopkins Sch. Public Health Expert Insights. 2020, 10. Available online: https://www.jhsph.edu/covid-19/articles/achieving-herd-immunity-with-covid19.html (accessed on 1 June 2021).
- Certificazioni Verdi Covid-19. Available online: https://www.salute.gov.it/portale/nuovocoronavirus/dettaglioFaqNuovo-Coronavirus.jsp?lingua=italiano&id=264 (accessed on 1 June 2021).
- COVID-19: What You Have to Know about It. Available online: https://www.inmp.it/eng/CoViD-19-what-you-have-to-know-about-it/Covid-19-Red-zone (accessed on 15 January 2021).
- Epidemia COVID-19 Aggiornamento Nazionale 27 Ottobre 2020—Ore 11:00. Available online: https://www.epicentro.iss.it/coronavirus/bollettino/Bollettino-sorveglianza-integrata-COVID-19_27-ottobre-2020.pdf (accessed on 31 October 2020).
- Epidemia COVID-19 Aggiornamento Nazionale 18 Novembre 2020—Ore 11:00. Available online: https://www.epicentro.iss.it/coronavirus/bollettino/Bollettino-sorveglianza-integrata-COVID-19_18-novembre-2020.pdf (accessed on 21 November 2020).
- Ledford, H. COVID reinfections are unusual-but could still help the virus to spread. Nature
**2021**. [Google Scholar] [CrossRef] - Britton, T.; Ball, F.; Trapman, P. A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2. Science
**2020**, 369, 846–849. [Google Scholar] [CrossRef] - Fontanet, A.; Cauchemez, S. COVID-19 herd immunity: Where are we? Nat. Rev. Immunol.
**2020**, 20, 583–584. [Google Scholar] [CrossRef]

**Figure 1.**Graphic representation of the SEIRL-V model. Clinical stages for the population are as follows: Susceptible (S), Exposed (${E}_{\nu}$), Presymptomatic (${P}_{S,\nu}$), Asymptomatic (A), Recovered (R), Mild infection (M), Severe infection (H), Critical infection ($ICU$), Dead (D), Vaccinated 1st dose (${V}_{1})$, Vaccinated 2nd dose (${V}_{2})$, and Immunized ($Im$). The intervention measures are represented by L.

**Figure 2.**Comparison of hospitalization and ICU admissions in Umbria (light blue line) and Italy (red line) from 1 September until 1 May 2021. (

**a**) Daily hospitalizations normalized over the whole population (∼$\mathrm{882,000}$ for Umbria and ∼60 million for Italy) and multiplied by ${10}^{5}$. (

**b**) Daily ICU admissions normalized over the whole population and multiplied by ${10}^{5}$.

**Figure 3.**Vaccination campaign in Umbria. (

**a**) Number of first (light blue) and second (blue) doses injected every day. (

**b**) Profile of the evolution of the vaccination rate $\eta $ chosen for model calibration.

**Figure 4.**Umbria. Time behavior of H, ICU and D variables using as parameter vector the final mode vector computed by CRC (black line) (see Table 1); dots are the public data available at [32]. Both data and simulations are in log-scale, normalized over the population of Umbria (∼$\mathrm{882,000}$) and multiplied by ${10}^{5}$. The colored area represents the variation of the temporal behavior when the parameter vector varies between the 60th, 70th, and 90th percentile of its conditional probability density function (pdf) (see Table A1).

**Figure 5.**Projections of the epidemic evolution with a vaccination schedule of 8000 first doses of vaccines per day in Umbria (slow vaccination). Each line corresponds to a different reopening strategy, i.e., different values of transmission rate parameters ${b}_{e}$ and ${b}_{0}$ (yellow for high values, blue for medium values, and pink for low values).

**Figure 6.**Projections of the epidemic evolution with a vaccination schedule of $\mathrm{12,500}$ first doses of vaccines per day in Umbria (medium vaccination). Each line corresponds to a different reopening strategy, i.e., different values of transmission rate parameters ${b}_{e}$ and ${b}_{0}$ (yellow for high values, blue for medium values, and pink for low values).

**Figure 7.**Projections of the epidemic evolution with a vaccination schedule of $\mathrm{18,000}$ first doses of vaccines per day in Umbria (fast vaccination). Each line corresponds to a different reopening strategy, i.e., different values of transmission rate parameters ${b}_{e}$ and ${b}_{0}$ (yellow for high values, blue for medium values, and pink for low values).

**Figure 8.**Area plots of model variables to compare the long-term evolution of the epidemic dynamics in Umbria. (

**a**) 8000 first doses every day and low values for ${b}_{e}$ and ${b}_{0}$ (Low/Low). (

**b**) 8000 first doses every day and high values for ${b}_{e}$ and ${b}_{0}$ (High/Low). (

**c**) $\mathrm{18,000}$ first doses every day and low values for ${b}_{e}$ and ${b}_{0}$ (High/Low). (

**d**) $\mathrm{18,000}$ first doses every day and high values for ${b}_{e}$ and ${b}_{0}$ (High/High).

**Figure 9.**Vaccination campaign in Italy. (

**a**) Number of first (light blue) and second (blue) doses injected every day. (

**b**) Profile of the evolution of the vaccination rate $\eta $ chosen for model calibration.

**Figure 10.**Italy. Time behavior of H, ICU, and D variables using as parameter vector the final mode vector computed by CRC (black line) (see Table 1); dots are the public data available in [32]. Both data and simulations are in log-scale, normalized over the population of Italy (∼60 million) and multiplied by $\mathrm{100,000}$. The colored area represents the variation of the temporal behavior when the parameter vector varies between the 60th, 70th and 90th percentile of its conditional pdf (see Table A2).

**Figure 11.**Projections of the epidemic evolution with a vaccination schedule of $\mathrm{500,000}$ first doses of vaccines per day in Italy (slow vaccination). Each line corresponds to a different reopening strategy, i.e., different values of transmission rate parameters ${b}_{e}$ and ${b}_{0}$ (yellow for high values, blue for medium values, and pink for low values).

**Figure 12.**Projections of the epidemic evolution with a vaccination schedule of $\mathrm{800,000}$ first doses of vaccines per day in Italy (medium vaccination). Each line corresponds to a different reopening strategy, i.e., different values of transmission rate parameters ${b}_{e}$ and ${b}_{0}$ (yellow for high values, blue for medium values, and pink for low values).

**Figure 13.**Projections of the epidemic evolution with a vaccination schedule of $\mathrm{1,000,000}$ first doses of vaccines per day in Italy (fast vaccination). Each line corresponds to a different reopening strategy, i.e., different values of transmission rate parameters ${b}_{e}$ and ${b}_{0}$ (yellow for high values, blue for medium values, and pink for low values).

**Figure 14.**Area plots of model variables to compare the long-term evolution of the epidemic dynamics in Italy. (

**a**) $\mathrm{500,000}$ first doses every day and low values for ${b}_{e}$ and ${b}_{0}$ (Low/Low). (

**b**) $\mathrm{500,000}$ first doses every day and high values for ${b}_{e}$ and ${b}_{0}$ (Low/High). (

**c**) $\mathrm{1,000,000}$ first doses every day and low values for ${b}_{e}$ and ${b}_{0}$ (High/Low). (

**d**) $\mathrm{1,000,000}$ first doses every day and high values for ${b}_{e}$ and ${b}_{0}$ (High/High).

**Table 1.**CRC results for Umbria and Italy. The second column shows the prior distribution of parameters, set at the beginning of the first CRC iteration. The third and fourth columns show, respectively, the mode vector of ${f}_{\mathbf{P}|{P}_{O,\u03f5}}$ in one of the 10 final realizations for Umbria and Italy. Note that the pre-symptomatic period (PresymPeriod) is supposed to be a percentage of the incubation period (IncubPeriod).

Parameter | Prior | Umbria ${\mathit{p}}_{\mathbf{mode}}$ | Italy ${\mathit{p}}_{\mathbf{mode}}$ |
---|---|---|---|

${b}_{e,1}$ | log-U(0.01,1) | 0.1442 | 0.1342 |

${b}_{0,1}$ | log-U(0.01,1) | 0.3178 | 0.2109 |

${b}_{e,2}$ | log-U(0.01,1) | 0.2172 | 0.2512 |

${b}_{0,2}$ | log-U(0.01,1) | 0.3672 | 0.19 |

${b}_{1}$ | log-U(0.001,1) | 0.0269 | 0.0120 |

${b}_{2}$ | log-U(0.001,1) | 0.0119 | 0.0516 |

${b}_{3}$ | log-U(0.001,1) | 0.0260 | 0.0145 |

$FracSevere$ | log-U(0.01,0.08) | 0.0182 | 0.0253 |

$FracCritica{l}_{1}$ | log-U(0.001,0.02) | 0.0041 | 0.005 |

$FracCritica{l}_{2}$ | log-U(0.001,0.02) | 0.0055 | 0.007 |

$FracCritica{l}_{3}$ | log-U(0.001,0.02) | 0.0045 | 0.0035 |

$FracCritica{l}_{4}$ | log-U(0.001,0.02) | 0.0053 | 0.0048 |

$FracCritica{l}_{5}$ | log-U(0.001,0.02) | 0.0033 | - |

$FracAsym$ | U(0.2,0.7) | 0.488 | 0.4618 |

$IncubPeriod$ | U(4,6) | 4.6389 | 5.2668 |

$DurMildInf$ | U(5,30) | 15.6157 | 9.8982 |

$DurAsym$ | U(5,20) | 10.7014 | 14.5149 |

$DurHosp$ | U(4,30) | 13.2214 | 15.6204 |

$TimeICUDeath$ | U(4,30) | 11.1608 | 12.6742 |

$ProbDeath$ | U(10,90) | 31.0621 | 34.7989 |

$ProbDeat{h}_{H}$ | U(10,90) | 22.2085 | 25.3705 |

$PresymPeriod$ | log-U(0.5,0.9) | 0.6075 | 0.637 |

n | U(1,100) | 47.1727 | 47.1717 |

K | U(1,${10}^{5}$) | $2.48\times {10}^{4}$ | $4.44\times {10}^{4}$ |

${s}_{01}$ | log-U(0.4,1.5) | 1.0729 | 1.0409 |

${s}_{02}$ | log-U(0.1,0.9) | 0.2705 | 0.3095 |

${s}_{03}$ | log-U(0.1,0.9) | 0.2815 | 0.2634 |

${s}_{04}$ | log-U(0.4,1.5) | 0.432 | 0.6842 |

${s}_{05}$ | log-U(0.4,1.5) | 0.5805 | 0.1837 |

${s}_{06}$ | log-U(0.1,0.9) | 0.2952 | - |

${s}_{07}$ | log-U(0.1,0.9) | 0.3077 | - |

**Table 2.**Future scenarios simulated for Umbria and Italy through variation of the vaccination rate and of intervention parameters. The three values on the first column represent the increase of transmission parameters on 26 April, 24 May, and 21 June 2021, respectively. The first row indicates the number of first doses of vaccine per day.

Vaccination Rate | |||||||
---|---|---|---|---|---|---|---|

Umbria | Italy | ||||||

$8\times {10}^{3}$ | $12.5\times {10}^{3}$ | $18\times {10}^{3}$ | $5\times {10}^{5}$ | $8\times {10}^{5}$ | ${10}^{6}$ | ||

Intervention | [0.4–0.6–0.8] | Low/Low | Low/Medium | Low/High | Low/Low | Low/Medium | Low/High |

[0.6–0.8–1] | Medium/Low | Medium/Medium | Medium/High | Medium/Low | Medium/Medium | Medium/High | |

[0.8–1–1.2] | High/Low | High/Medium | High/High | High/Low | High/Medium | High/High |

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**MDPI and ACS Style**

Antonini, C.; Calandrini, S.; Bianconi, F.
A Modeling Study on Vaccination and Spread of SARS-CoV-2 Variants in Italy. *Vaccines* **2021**, *9*, 915.
https://doi.org/10.3390/vaccines9080915

**AMA Style**

Antonini C, Calandrini S, Bianconi F.
A Modeling Study on Vaccination and Spread of SARS-CoV-2 Variants in Italy. *Vaccines*. 2021; 9(8):915.
https://doi.org/10.3390/vaccines9080915

**Chicago/Turabian Style**

Antonini, Chiara, Sara Calandrini, and Fortunato Bianconi.
2021. "A Modeling Study on Vaccination and Spread of SARS-CoV-2 Variants in Italy" *Vaccines* 9, no. 8: 915.
https://doi.org/10.3390/vaccines9080915