A data-validated temporary immunity model of COVID-19 spread in Michigan

Bruce Pell; Matthew D. Johnston; Patrick Nelson; Bruce Pell; Matthew D. Johnston; Patrick Nelson

doi:10.3934/mbe.2022474

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 10: 10122-10142. doi: 10.3934/mbe.2022474

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A data-validated temporary immunity model of COVID-19 spread in Michigan

Department of Mathematics & Computer Science, Lawrence Technological University, 21000 W 10 Mile Rd, Southfield, MI 48075, USA

Academic Editor: Quentin Griette

Received: 10 May 2022 Revised: 23 June 2022 Accepted: 12 July 2022 Published: 18 July 2022

We introduce a distributed-delay differential equation disease spread model for COVID-19 spread. The model explicitly incorporates the population's time-dependent vaccine uptake and incorporates a gamma-distributed temporary immunity period for both vaccination and previous infection. We validate the model on COVID-19 cases and deaths data from the state of Michigan and use the calibrated model to forecast the spread and impact of the disease under a variety of realistic booster vaccine strategies. The model suggests that the mean immunity duration for individuals after vaccination is $ 350 $ days and after a prior infection is $ 242 $ days. Simulations suggest that both high population-wide adherence to vaccination mandates and a more-than-annually frequency of booster doses will be required to contain outbreaks in the future.
- delay differential equation,
- distributed delay differential equation,
- temporary immunity,
- COVID-19,
- vaccination,
- linear chain trick
Citation: Bruce Pell, Matthew D. Johnston, Patrick Nelson. A data-validated temporary immunity model of COVID-19 spread in Michigan[J]. Mathematical Biosciences and Engineering, 2022, 19(10): 10122-10142. doi: 10.3934/mbe.2022474

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Abstract

We introduce a distributed-delay differential equation disease spread model for COVID-19 spread. The model explicitly incorporates the population's time-dependent vaccine uptake and incorporates a gamma-distributed temporary immunity period for both vaccination and previous infection. We validate the model on COVID-19 cases and deaths data from the state of Michigan and use the calibrated model to forecast the spread and impact of the disease under a variety of realistic booster vaccine strategies. The model suggests that the mean immunity duration for individuals after vaccination is $ 350 $ days and after a prior infection is $ 242 $ days. Simulations suggest that both high population-wide adherence to vaccination mandates and a more-than-annually frequency of booster doses will be required to contain outbreaks in the future.

References

[1]	N. Anggriani, M. Z. Ndii, R. Amelia, W. Suryaningrat, M. A. A. Pratama, A mathematical COVID-19 model considering asymptomatic and symptomatic classes with waning immunity, Alexandria Eng. J., 61 (2022), 113–124, https://doi.org/10.1016/j.aej.2021.04.104 doi: 10.1016/j.aej.2021.04.104
[2]	A. Vespignani, H. Tian, C. Dye, J. O. Lloyd-Smith, R. M. Eggo, M. Shrestha, et al., Modelling COVID-19, Nat. Rev. Phys., 2 (2020), 279–281. https://doi.org/10.1038/s42254-020-0178-4
[3]	W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700–721.
[4]	A. J. Kucharski, T. W. Russell, C. Diamond, Y. Liu, J. Edmunds, S. Funk, et al., Early dynamics of transmission and control of COVID-19: a mathematical modelling study, Lancet Infect. Dis., 20 (2020), 553–558. https://doi.org/10.1016/S1473-3099(20)30144-4 doi: 10.1016/S1473-3099(20)30144-4
[5]	F. A. Rihan, H. J. Alsakaji, Dynamics of a stochastic delay differential model for COVID-19 infection with asymptomatic infected and interacting people: Case study in the UAE, Results Phys., 28 (2021). https://doi.org/10.1016/j.rinp.2021.104658
[6]	S. Zhanga, M. Diao, W. Yuc, L. Pei, Z. Lind, D. Chena, Estimation of the reproductive number of novel coronavirus (COVID-19) and the probable outbreak size on the diamond princess cruise ship: A data-driven analysis, Int. J. Infect. Dis., 93 (2020), 201–204. https://doi.org/10.1016/j.ijid.2020.02.033 doi: 10.1016/j.ijid.2020.02.033
[7]	Z. Zhuang, S. Zhao, Q. Lin, P. Cao, Y. Lou, L. Yang, et al., Preliminary estimates of the reproduction number of the coronavirus disease (COVID-19) outbreak in republic of Korea and Italy by 5 March 2020, Int. J. Infect. Dis., 95 (2020), 308–310. https://doi.org/10.1016/j.ijid.2020.04.044 doi: 10.1016/j.ijid.2020.04.044
[8]	M. V. Barbarossa, J. Fuhrmann, J. H. Meinke, S. Krieg, H. V. Varma, N. Castelletti, et al., Modeling the spread of COVID-19 in {G}ermany: Early assessment and possible scenarios, PLOS ONE, 15 (2020), 1–22. https://doi.org/10.1371/journal.pone.0238559 doi: 10.1371/journal.pone.0238559
[9]	A. Bouchnita, A. Jebrane, A hybrid multi-scale model of COVID-19 transmission dynamics to assess the potential of non-pharmaceutical interventions, Chaos, Solitons Fractals, 138 (2020). https://doi.org/10.1016/j.chaos.2020.109941
[10]	S. Chang, N. Harding, C. Zachreson, O. Cliff, M. Prokopenko, Modelling transmission and control of the COVID-19 pandemic in Australia, Nat. Commun., 11 (2020), 5710. https://doi.org/10.1038/s41467-020-19393-6 doi: 10.1038/s41467-020-19393-6
[11]	S. E. Eikenberry, M. Mancuso, E. Iboi, T. Phan, K. Eikenberry, Y. Kuang, et al., To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic, Infect. Dis. Model., 5 (2020), 293–308. https://doi.org/10.1016/j.idm.2020.04.001 doi: 10.1016/j.idm.2020.04.001
[12]	K. M. Bubar, K. Reinholt, S. M. Kissler, M. Lipsitch, S. Cobey, Y. H. Grad, et al., Model-informed COVID-19 vaccine prioritization strategies by age and serostatus, Science, 371 (2021), 916–921. https://doi.org/10.1126/science.abe6959 doi: 10.1126/science.abe6959
[13]	B. H. Foy, B. Wahl, K. Mehta, A. Shet, G. I. Menon, C. Britto, Comparing COVID-19 vaccine allocation strategies in india: A mathematical modelling study, Int. J. Infect. Dis., 103 (2021), 431–438. https://doi.org/10.1016/j.ijid.2020.12.075 doi: 10.1016/j.ijid.2020.12.075
[14]	M. Johnston, B. Pell, P. Nelson, A mathematical study of COVID-19 spread by vaccination status in Virginia, Appl. Sci., 12 (2022), 1723. https://doi.org/10.3390/app12031723 doi: 10.3390/app12031723
[15]	N. Guglielmi, E. Iacomini, A. Viguerie, Delay differential equations for the spatially resolved simulation of epidemics with specific application to COVID-19, Math. Meth. Appl. Sci., 45 (2022), 4752–4771. https://doi.org/10.1002/mma.8068 doi: 10.1002/mma.8068
[16]	A. Viguerie, G. Lorenzo, F. Auricchio, D. Baroli, T. Hughes, A. Patton, et al., Simulating the spread of COVID-19 via a spatially-resolved susceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion, Appl. Math. Lett., 111 (2021), 106617. https://doi.org/10.1016/j.aml.2020.106617 doi: 10.1016/j.aml.2020.106617
[17]	N. Yamamoto, B. Jiang, H. Wang, Quantifying compliance with COVID-19 mitigation policies in the US: A mathematical modeling study, Infect. Dis. Modell., 6 (2021), 503–513. https://doi.org/10.1016/j.idm.2021.02.004 doi: 10.1016/j.idm.2021.02.004
[18]	Y. Goldberg, M. Mandel, Y. M. Bar-On, O. Bodenheimer, L. Freedman, E. J. Haas, et al., Waning immunity after the BNT162b2 vaccine in Israel, N. Engl. J. Med., 385 (2021), e85. https://doi.org/10.1056/NEJMoa2114228 doi: 10.1056/NEJMoa2114228
[19]	E. G. Levin, Y. Lustig, C. Cohen, R. Fluss, V. Indenbaum, S. Amit, et al., Waning immune humoral response to BNT162b2 COVID-19 vaccine over 6 months, N. Engl. J. Med., 385 (2021), e84. https://doi.org/10.1056/NEJMoa2114583 doi: 10.1056/NEJMoa2114583
[20]	F. Inayaturohmat, R. N. Zikkah, A. K. Supriatna, N. Anggriani, Mathematical model of COVID-19 transmission in the presence of waning immunity, J. Phys. Conf. Ser., 1722 (2021), 012038, https://doi.org/10.1088/1742-6596/1722/1/012038 doi: 10.1088/1742-6596/1722/1/012038
[21]	M. Q. Shakhany, K. Salimifard, Predicting the dynamical behavior of COVID-19 epidemic and the effect of control strategies, Chaos Solitons Fractals, 146 (2021), 110823. https://doi.org/10.1016/j.chaos.2021.110823 doi: 10.1016/j.chaos.2021.110823
[22]	F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, 2rd edition, Springer, New York, 2012.
[23]	R. Carlsson, L. M. Childs, Z. Feng, J. W. Glasser, J. M. Heffernan, J. Li, et al., Modeling the waning and boosting of immunity from infection or vaccination, J. Theor. Biol., 497 (2020), 110265. https://doi.org/10.1016/j.jtbi.2020.110265 doi: 10.1016/j.jtbi.2020.110265
[24]	D. Hamami, R. Cameron, K. G. Pollock, C. Shankland, Waning immunity is associated with periodic large outbreaks of mumps: A mathematical modeling study of Scottish data, Front. Physiol., 8 (2017), 233. https://doi.org/10.3389/fphys.2017.00233 doi: 10.3389/fphys.2017.00233
[25]	M. Barbarossa, G. Röst, Immuno-epidemiology of a population structured by immune status: a mathematical study of waning immunity and immune system boosting, Math. Biol., 71 (2015), 1737–1770. https://doi.org/10.1007/s00285-015-0880-5 doi: 10.1007/s00285-015-0880-5
[26]	M. L. Taylor, T. W. Carr, An SIR epidemic model with partial temporary immunity modeled with delay, J. Math. Biol., 59 (2009), 841–880. https://doi.org/10.1007/s00285-009-0256-9 doi: 10.1007/s00285-009-0256-9
[27]	A. Feng, U. Obolski, L. Stone, D. He, Modelling COVID-19 vaccine breakthrough infections in highly vaccinated Israel-the effects of waning immunity and third vaccination dose, medRxiv, 2022. https://doi.org/10.1101/2022.01.08.22268950
[28]	F. Richards, A flexible growth function for empirical use, J. Exp. Bot., 10 (1959), 290–301.
[29]	P. J. Hurtado, A. S. Kirosingh, Generalizations of the 'linear chain trick' : incorporating more flexible dwell time distributions into mean field ode models, J. Math. Biol., 79 (2019), 1831–1883. https://doi.org/10.1007/s00285-019-01412-w doi: 10.1007/s00285-019-01412-w
[30]	Y. Kuang, Delay Differential Equations: with Applications in Population Dynamics, Academic Press, 1993. https://doi.org/10.1016/s0076-5392(08)x6164-8
[31]	H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer Science & Business Media, 2010. https://doi.org/10.1007/978-1-4419-7646-8
[32]	O. Diekmann, J. Heesterbeek, M. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2009), 873–885.
[33]	P. van den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Model., 2 (2017), 288–303. https://doi.org/10.1016/j.idm.2017.06.002 doi: 10.1016/j.idm.2017.06.002
[34]	RStudio Team, Rstudio: Integrated Development Environment for r, 2016. Available from: http://www.rstudio.com/.
[35]	E. Mathieu, H. Ritchie, E. Ortiz-Ospina, M. Roser, J. Hasell, C. Appel, et al., A global database of COVID-19 vaccinations, Nat. Hum. Behav., 5 (2021), 947–953. https://doi.org/10.1038/s41562-021-01122-8 doi: 10.1038/s41562-021-01122-8
[36]	S. of Michigan, Cases and Deaths by County and by Date of Symptom Onset or by Date of Death 2022-01-12 745533 7 (1), https://www.michigan.gov/coronavirus/-/media/Project/Websites/coronavirus/Michigan-Data/07-12-2022/Cases-and-Deaths-by-County-2022-07-12.xlsx?rev=0b8b993775f841a18aa6cd9c7ce6d0a0&hash=48104B6EDCFCD25280F8E794F098929E.
[37]	MATLAB, version 9.6.0 (r2019a), 2019.
[38]	F. A. Rihan, Parameter estimation with delay differential equations, in Delay Differential Equations and Applications to Biology, Springer, Singapore, (2021), 87–102. https://doi.org/10.1007/978-981-16-0626-7_5
[39]	A. Saltelli, K. Chan, E. M. Scott, Sensitivity Analysis, Wiley, New York, 2000.
[40]	F. A. Rihan, Sensitivity analysis for dynamic systems with time-lags, J. Comp. App. Math., 28 (2003). https://doi.org/10.1016/S0377-0427(02)00659-3
[41]	State Population by Characteristics: 2010–2019, Available from: https://www.census.gov/data/datasets/time-series/demo/popest/2010s-state-detail.html. Accessed date: 2022-01-11.
[42]	D. M. Fargue, Reducibilite' des systemes dynamiues, C. R. Acad. Sci. Paris, Set. B., 277 (1973), 471–473.
[43]	O. Diekmann, M. Gyllenberg, J. A. Metz, Finite dimensional state representation of physiologically structured populations, J. Math. Biol., 80 (2020), 205–273. https://doi.org/10.1007/s00285-019-01454-0 doi: 10.1007/s00285-019-01454-0
[44]	O. Diekmann, M. Gyllenberg, J. A. Metz, On models of physiologically structured populations and their reduction to ordinary differential equations, J. Math. Biol., 80 (2020), 189–204. https://doi.org/10.1007/s00285-019-01431-7 doi: 10.1007/s00285-019-01431-7
[45]	S. Al-Beltagi, L. V. Goulding, D. K. Chang, K. H. Mellits, C. J. Hayes, P. Gershkovich, et al., Emergent SARS-CoV-2 variants: comparative replication dynamics and high sensitivity to thapsigargin, Virulence, 12 (2021), 2946–2956. https://doi.org/10.1080/21505594.2021.2006960 doi: 10.1080/21505594.2021.2006960
[46]	F. J. Ibarrondo, J. A. Fulcher, D. Goodman-Meza, J. Elliott, C. Hofmann, M. A. Hausner, et al., Rapid decay of anti-SARS-CoV-2 antibodies in persons with mild COVID-19, N. Engl. J. Med., 383 (2020), 1085–1087. https://doi.org/10.1056/NEJMc2025179 doi: 10.1056/NEJMc2025179
[47]	G. Chowell, C. Viboud, J. M. Hyman, L. Simonsen, The western africa ebola virus disease epidemic exhibits both global exponential and local polynomial growth rates, PLoS Curr., 2015. https://doi.org/10.1371/currents.outbreaks.8b55f4bad99ac5c5db3663e916803261
[48]	C. Viboud, L. Simonsen, G. Chowell, A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks, Epidemics, 15 (2016), 27–37. https://doi.org/10.1016/j.epidem.2016.01.002 doi: 10.1016/j.epidem.2016.01.002
[49]	E. B. Hodcroft, Covariants: SARS-CoV-2 Mutations and Variants of Interest, 2021, Available from: https://covariants.org/.
[50]	D. Hutchinson, Michigan to Lift All COVID Restrictions on Capacity, Masks, Gatherings June 22, June 2021, Available from: https://www.clickondetroit.com/news/michigan/2021/06/17/michigan-to-lift-all-covid-restrictions-on-capacity-masks-gatherings-june-22/.

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