June  2022, 17(3): 443-466. doi: 10.3934/nhm.2022016

Optimization of vaccination for COVID-19 in the midst of a pandemic

1. 

Department of Industrial Engineering, Clemson University, Clemson, SC, USA

2. 

Center for Computational and Integrative Biology, Rutgers Camden, Camden NJ, USA

3. 

Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USA

4. 

Sorbonne Université, CNRS, Université Paris Cité, Inria, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France

5. 

Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN, USA

6. 

School of Civil and Environmental Engineering, Cornell University, Ithaca, NY, USA

7. 

Joseph and Loretta Lopez chair professor of Mathematics, Center for Computational and Integrative Biology, Rutgers Camden, Camden NJ, USA

Received  September 2021 Revised  January 2022 Published  June 2022 Early access  March 2022

Fund Project: The authors acknowledge the support of the NSF CMMI project # 2033580 "Managing pandemic by managing mobility". R.W., S.T.M. and B.P. acknowledge the support of the Joseph and Loretta Lopez Chair endowment

During the Covid-19 pandemic a key role is played by vaccination to combat the virus. There are many possible policies for prioritizing vaccines, and different criteria for optimization: minimize death, time to herd immunity, functioning of the health system. Using an age-structured population compartmental finite-dimensional optimal control model, our results suggest that the eldest to youngest vaccination policy is optimal to minimize deaths. Our model includes the possible infection of vaccinated populations. We apply our model to real-life data from the US Census for New Jersey and Florida, which have a significantly different population structure. We also provide various estimates of the number of lives saved by optimizing the vaccine schedule and compared to no vaccination.

Citation: Qi Luo, Ryan Weightman, Sean T. McQuade, Mateo Díaz, Emmanuel Trélat, William Barbour, Dan Work, Samitha Samaranayake, Benedetto Piccoli. Optimization of vaccination for COVID-19 in the midst of a pandemic. Networks and Heterogeneous Media, 2022, 17 (3) : 443-466. doi: 10.3934/nhm.2022016
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show all references

References:
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D. Acemoglu, V. Chernozhukov, I. Werning and M. D. Whinston, Optimal Targeted Lockdowns in a Multi-group SIR Model, Volume 27102., National Bureau of Economic Research, 2020.

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F. E. Alvarez, D. Argente and F. Lippi, A Simple Planning Problem for Covid-19 Lockdown, Technical report, National Bureau of Economic Research, 2020.

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J. A. E. AnderssonJ. GillisG. HornJ. B. Rawlings and M. Diehl, Casadi–A software framework for nonlinear optimization and optimal control, Mathematical Programming Computation, 11 (2019), 1-36.  doi: 10.1007/s12532-018-0139-4.

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M. S. AronnaR. Guglielmi and L. M. Moschen, A model for covid-19 with isolation, quarantine and testing as control measures, Epidemics, 34 (2021), 100437.  doi: 10.1016/j.epidem.2021.100437.

[6]

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B. BonnardJ.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM: Control, Optimisation and Calculus of Variations, 13 (2007), 207-236.  doi: 10.1051/cocv:2007012.

[10]

R. K. BorcheringC. ViboudE. HowertonC. P. SmithS. TrueloveM. C. RungeN. G. ReichL. ContaminJ. Levander and J. Salerno, Modeling of future covid-19 cases, hospitalizations, and deaths, by vaccination rates and nonpharmaceutical intervention scenarios-united states, april–september 2021,, Morbidity and Mortality Weekly Report, 70 (2021), 719-724.  doi: 10.15585/mmwr.mm7019e3.

[11]

S. Bowong and J. J. Tewa, Mathematical analysis of a tuberculosis model with differential infectivity, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 4010-4021.  doi: 10.1016/j.cnsns.2009.02.017.

[12]

C. C. Branas, A. Rundle, S. Pei, W. Yang, B. G. Carr, S. Sims, A. Zebrowski, R. Doorley, N. Schluger, J. W. Quinn and J. Shaman, Flattening the curve before it flattens us: Hospital critical care capacity limits and mortality from novel coronavirus (sars-cov2) cases in us counties, medRxiv, 2020, 20049759. doi: 10.1101/2020.04.01.20049759.

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[15]

T. Brosh-NissimovE. Orenbuch-HarrochM. ChowersM. ElbazL. NesherM. SteinY. MaorR. CohenK. Hussein and M. Weinberger, Bnt162b2 vaccine breakthrough: Clinical characteristics of 152 fully vaccinated hospitalized covid-19 patients in israel, Clinical Microbiology and Infection, 27 (2021), 1652-1657.  doi: 10.1016/j.cmi.2021.06.036.

[16]

V. L. Brown and K. A. Jane White, The role of optimal control in assessing the most cost-effective implementation of a vaccination programme: HPV as a case study, Math. Biosci., 231 (2011), 126-134.  doi: 10.1016/j.mbs.2011.02.009.

[17]

K. M. BubarK. ReinholtS. M. KisslerM. LipsitchS. CobeyY. H. Grad and D. B Larremore, Model-informed covid-19 vaccine prioritization strategies by age and serostatus, Science, 371 (2021), 916-921.  doi: 10.1126/science.abe6959.

[18]

F. Casella, Can the covid-19 epidemic be managed on the basis of daily test reports?, IEEE Control Syst. Lett., 5 (2021), 1079–1084, arXiv: 2003.06967. doi: 10.1109/LCSYS.2020.3009912.

[19]

Y.-C. ChenP.-E. LuC.-S. Chang and T.-H. Liu, A time-dependent SIR model for COVID-19 with undetectable infected persons, IEEE Trans. Network Sci. Eng., 7 (2020), 3279-3294.  doi: 10.1109/TNSE.2020.3024723.

[20]

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Figure 1.  All possible paths through which populations may flow into other populations
Figure 2.  Sample tests for New Jersey with a choice of $ R0 = 1.0 $ (mildest case) while varying percent of essential worker and beta
Figure 3.  Results using New Jersey data-set plotted by initial replication rate
Figure 4.  Results using Florida data-set plotted by initial replication rate
Figure 6.  Population dynamics for the unvaccinated compartments: Susceptible, Exposed, Infected, and Recovered
Figure 5.  Optimal vaccination strategy for Reproduction number 1.2, Percent of workers considered essential 44
Figure 7.  Population dynamics of the vaccinated compartments: Susceptible, Vaccinated, Exposed vaccinated, Infected vaccinated, and Recovered vaccinated
Table 1.  Groups by Age
Name Description
Group 1 Age 0-4 population
Group 2 Age 5-14 population
Group 3 Age 15-19 population with no job or non-essential
Group 4 Age 20-39 population with no job or non-essential
Group 5 Age 40-59 population with no job or non-essential
Group 6 Age 60-69 population with no job or non-essential
Group 7 Age 70+ population
Group 8 Age 15-19 population who are essential workers
Group 9 Age 20-39 population who are essential workers
Group 10 Age 40-59 population who are essential workers
Group 11 Age 60-69 population who are essential workers
Name Description
Group 1 Age 0-4 population
Group 2 Age 5-14 population
Group 3 Age 15-19 population with no job or non-essential
Group 4 Age 20-39 population with no job or non-essential
Group 5 Age 40-59 population with no job or non-essential
Group 6 Age 60-69 population with no job or non-essential
Group 7 Age 70+ population
Group 8 Age 15-19 population who are essential workers
Group 9 Age 20-39 population who are essential workers
Group 10 Age 40-59 population who are essential workers
Group 11 Age 60-69 population who are essential workers
Table 2.  Description of Variables
Name Description Estimate Units
$ R_0 $ Rate of infection 1.0-1.2
$ D_I $ Infectious period 5-14 days
$ D_E $ Latent period 4-7 days
Name Description Estimate Units
$ R_0 $ Rate of infection 1.0-1.2
$ D_I $ Infectious period 5-14 days
$ D_E $ Latent period 4-7 days
Table 3.  Deaths Projected with Varying $ R0 $
State $ R0 $ Projected Deaths With No Vaccine Projected Deaths With Vaccine
New Jersey $ 1.0 $ 9316 6710
New Jersey $ 1.1 $ 15609 6906
New Jersey $ 1.2 $ 31681 7289
Florida $ 1.0 $ 28467 21678
Florida $ 1.1 $ 44657 22287
Florida $ 1.2 $ 87349 23298
State $ R0 $ Projected Deaths With No Vaccine Projected Deaths With Vaccine
New Jersey $ 1.0 $ 9316 6710
New Jersey $ 1.1 $ 15609 6906
New Jersey $ 1.2 $ 31681 7289
Florida $ 1.0 $ 28467 21678
Florida $ 1.1 $ 44657 22287
Florida $ 1.2 $ 87349 23298
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