Abstract
Rational allocation of limited vaccine resources is one of the key issues in the prevention and control of emerging infectious diseases. An age-structured infectious disease model with limited vaccine resources is proposed to explore the optimal vaccination ages. The effective reproduction number \(R_e\) of the epidemic disease is computed. It is shown that the reproduction number is the threshold value for eradicating disease in the sense that the disease-free steady state is globally stable if \(R_e<1\) and there exists a unique endemic equilibrium if \(R_e>1\). The effective reproduction number is used as an objective to minimize the disease spread risk. Using the epidemic data from the early spread of Wuhan, China and demographic data of Wuhan, we figure out the strategies to distribute the vaccine to the age groups to achieve the optimal vaccination effects. These analyses are helpful to the design of vaccination schedules for emerging infectious diseases.
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References
Brauer F, Castillo-Chavez C, Feng ZL (2019) Mathematical models in epidemiology. Springer, New York
Byrne AW, McEvoy D, Collins AB et al (2020) Inferred duration of infectious period of SARS-CoV-2: rapid scoping review and analysis of available evidence for asymptomatic and symptomatic COVID-19 cases. BMJ Open 10(8):e039856. https://doi.org/10.1136/bmjopen-2020-039856
Bubar KM, Reinholt K, Kissler SM et al (2021) Model-informed COVID-19 vaccine prioritization strategies by age and serostatus. Science 6532:916–921. https://doi.org/10.1126/science.abe6959
Chen QX, Zhang DZ, Wei GQ et al (2010) Fundamentals of real variable functions and functional analysis. Higher Education Press, Beijing
Census Office of The State Council and Division for Population and Employment Statistics of National Bureau of Statistics (2012) China’s 2010 Census. China Statistics Press, Beijing
Cruz-Pacheco G, Esteva L, Vargas C (2014) Vaccination strategies for SIR vector-transmitted diseases. Bull Math Biol 76:2073–2090. https://doi.org/10.1007/s11538-014-9999-6
Dai Y (2022) Rapid epidemic expansion of the SARS-CoV-2 Omicron BA.2 subvariant during China’s largest outbreaks. https://doi.org/10.21203/rs.3.rs-1516063/v4. Accessed 14 June 2023
Datta SD, Tangermann RH, Reef S et al (2017) National, regional and global certification bodies for polio eradication: a framework for verifying measles elimination. J Infect Dis 216:S351–S354. https://doi.org/10.1093/infdis/jiw578
Desch W, Schappacher W (1986) Linearized stability for nonlinear semigroups. In: Favini A, Obrecht E (eds) Differential equations in Banach spaces. Lectures notes in mathematics, vol 1223. Springer, Berlin, pp 61–73
Feng ZJ, Li Q, Zhang YP et al (2020) The epidemiological characteristics of an outbreak of 2019 novel coronavirus diseases (COVID-19)-China, 2020. China CDC Wkly 2:113–122
Franceschetti A, Pugliese A (2008) Threshold behaviour of a SIR epidemic model with age structure and immigration. J Math Biol 57:1–27
Ge JW, Wang WD (2022) Vaccination games in prevention of infectious diseases with application to COVID-19. Chaos Solitons Fractals 161:112294. https://doi.org/10.1016/j.chaos.2022.112294
Hethcote HW (1988) Optimal ages of vaccination for measles. Math Biosci 89:29–52. https://doi.org/10.1016/0025-5564(88)90111-3
Hu Y, Wang KF, Wang WD (2022) Analysis of the geographic transmission differences of COVID-19 in China caused by population movement and population density. Bull Math Biol 84:1–17. https://doi.org/10.1007/s11538-022-01050-2
Iannelli M (1995) Mathematical theory of age-structured population dynamics. Giardini editori e stampatori in Pisa, Pisa
Iannelli M, Milner F (2017) The basic approach to age-structured population dynamics. Models, methods and numerics. Lecture notes on mathematical modelling in the life sciences. Springer, Dordrecht
Inaba H (1990) Threshold and stability results for an age-structured epidemic model. J Math Biol 28:411–434
Inaba H (2017) Age-structured population dynamics in demography and epidemiology. Springer, Singapore
Li XZ, Gupur G, Zhu GT (2001) Threshold and stability results for an age-structured SEIR epidemic model. Comput Math Appl 42:883–907
Li DQ, Liu ZC, Liu QH et al (2020a) Estimating the efficacy of quarantine and traffic blockage for the epidemic caused by 2019-nCoV (COVID-19): a simulation analysis. MedRxiv. https://doi.org/10.1101/2020.02.14.20022913
Li XZ, Yang JY, Martcheva M (2020b) Age structured epidemic modeling. Springer, Cham
Ling Y, Xu SB, Lin YX et al (2020) Persistence and clearance of viral RNA in 2019 novel coronavirus disease rehabilitation patients. Chin Med J 133:1039–1043. https://doi.org/10.1097/CM9.0000000000000774
Liu KH, Lou YJ (2022) Optimizing COVID-19 vaccination programs during vaccine shortages: a review of mathematical models. Infect Dis Model 7:286–298. https://doi.org/10.1016/j.idm.2022.02.002
Martcheva M (2015) An introduction to mathematical epidemiology. Springer, New York
Magal P, Ruan S (2018) Theory and applications of abstract semilinear Cauchy problems. Springer, Cham
Mizumoto K, Kagaya K, Chowell G (2020) Early epidemiological assessment of the transmission potential and virulence of coronavirus disease 2019 (COVID-19) in Wuhan City, China, January-February, 2020. BMC Med 18:217. https://doi.org/10.1186/s12916-020-01691-x
More SJ, McAloon CG, Griffin JM, et al (2020) COVID-19 epidemiological parameters summary document. Department of Health. https://assets.gov.ie/74596/2379b5a28b944140aa3b5d969e2c4beb.pdf. Accessed 14 June 2023
Mohammed A, Tomori O, Nkengasong JN (2021) Lessons from the elimination of poliomyelitis in Africa. Nat Rev Immunol 21:823–828
Qu YM, Kang EM, Cong HY (2020) Positive result of SARS-Cov-2 in sputum from a cured patient with COVID-19. Travel Med Infect Dis 34:101619. https://doi.org/10.1016/j.tmaid.2020.101619
Shi F, Wen HY, Liu R et al (2021) The comparison of epidemiological characteristics between confirmed and clinically diagnosed cases with COVID-19 during the early epidemic in Wuhan, China. Glob Health Res Policy 6:18. https://doi.org/10.1186/s41256-021-00200-8
State Statistics Bureau (2021) China Statistical Yearbook 2021. China Statistics Press, Beijing
Tang B, Wang X, Li Q et al (2020) Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions. J Clin Med 9:462. https://doi.org/10.3390/jcm9020462
Tian X, Wang WD (2020) Dynamical analysis of age-structured pertussis model with covert infection. Math Methods Appl Sci 43:1631–1645. https://doi.org/10.1002/mma.5989
Webb GF (1985) Theory of nonlinear age-dependent population dynamics. Marcel Dekker, New York
World Health Organization (1980) The global eradication of smallpox: final report of the Global Commission for the Certification of Smallpox Eradication, Geneva, December 1979. World Health Organization
Wuhan Statistics Bureau (2022) Wuhan 7th National Census Bulletin (No. 4)—population age composition. http://tjj.wuhan.gov.cn/tjfw/tjgb/202105/t20210528_1707401.shtml
Zhao S, Musa SS, Lin QY et al (2020) Estimating the unreported number of novel coronavirus (2019-nCoV) cases in China in the first half of January 2020: a data-driven modelling analysis of the early outbreak. J Clin Med 9:388. https://doi.org/10.3390/jcm9020388
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We are very grateful to the anonymous reviewers for their valuable suggestions that have helped to improve this paper.
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This study was funded by the NSF of China (12071381).
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In honor of Professor Fred Brauer.
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Ai, M., Wang, W. Optimal vaccination ages for emerging infectious diseases under limited vaccine supply. J. Math. Biol. 88, 13 (2024). https://doi.org/10.1007/s00285-023-02030-3
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DOI: https://doi.org/10.1007/s00285-023-02030-3