Abstract
For some infectious diseases such as mumps, HBV, there is evidence showing that vaccinated individuals always lose their immunity at different rates depending on the inoculation time. In this paper, we propose an age-structured epidemic model using a step function to describe the rate at which vaccinated individuals lose immunity and reduce the age-structured epidemic model to the delay differential model. For the age-structured model, we consider the positivity, boundedness, and compactness of the semiflow and study global stability of equilibria by constructing appropriate Lyapunov functionals. Moreover, for the reduced delay differential equation model, we study the existence of the endemic equilibrium and prove the global stability of equilibria. Finally, some numerical simulations are provided to support our theoretical results and a brief discussion is given.
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Arunachalam, P.S., Walls, A.C., Golden, N. et al. Adjuvanting a subunit COVID-19 vaccine to induce protective immunity. Nature., 594: 253–258 (2021)
Capasso, V., Serio, G. A generalization of the Kermack-Mackendric deterministic model. Math. Biosci., 42: 43–61 (1978)
Chaves, S., Gargiullo, P., Zhang, J. et al. Loss of vaccine induced immunity to varicella over time. Engl. J. Med., 356: 1121–1129 (2007)
Chen, Y., Zou, S., Yang, J. Global analysis of an SIR epidemic model with infection age and saturated incidence. Nonlinear Anal. Real World Appl., 30: 16–31 (2016)
Cui, F., Wang, X., Cao, L., et al. Progress in hepatitis B prevention through universal infant vaccination-China 1997–2006. MMWR-Morbid. Mortal. W., 56: 441–445 (2007)
Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol., 28: 365–382 (1990)
Ding, P., Qiu, Z., Li, X. The population-level impact of HBV and its vaccination on HIV transmission dynamics. Math. Method. Appl. Sci., 39: 5539–5556 (2016)
Duan, X., Yuan, S., Li, X. Global stability of an SVIR model with age of vaccination. Appl. Math. Comput., 226: 528–540 (2014)
d’Onofrio, A. Vaccination policies and nonlinear force of infection: generalization of an observation by Alexander and Moghadas (2004). Appl. Math. Comput., 168: 613–622 (2005)
Edmunds, W. J., Medley, G.F., Nokes, D. J. Vaccination agains the hepatitis B virus in highly endemic area: waning vaccine-induced immunity and the need for boosterdoses. Trans. R. Soc. Trop. Med. Hyg., 90: 436–440 (1996)
Edmunds, W., Medley, G., Nokes, D. The transmission dynamics and control of hepatitis B virus in the Gambia. Stat. Med., 15: 2215–2233 (1996)
Hale, J.K., Lunel, S.M.V. Introduction to Functional Differential Equations. Appl., Math., Sci., 99, Springer, New York, 1993
Hale, J.K., Waltman, P. Persistence in infinite-dimensional systems. SIAM J. Math. Anal., 20: 388–395 (1989)
Huang, G., Liu, X., Takeuchi, Y. Lyapunov functions and global stability for age structured HIV infection model. SIAM J. Appl. Math., 72: 25–38 (2012)
Huang, M., Luo, J., Hu, L., Zheng, B., Yu, J. Assessing the efficiency of Wolbachia driven Aedes mosquito suppression by delay differential equations. J. Theoret. Biol., 440: 1–11 (2018)
Iannelli, M. Mathematical Theory of Age-Structured Population Dynamics. Giardini Editori E Stampatori, Pisa, 1995
Iannelli, M., Martcheva, M., Li, X.Z. Strain replacement in an epidemic model with super-infection and perfect vaccination. Math. Biosci., 195: 23–46 (2005)
Kermack, W., Mckendrick, A.G. Contributions to the mathematical theory of epidemics-II. The problem of endemicity. Proc. R. Soc. Ser. A., 115: 55–83 (1932)
Kermack, W., Mckendrick, A.G. Contributions to the mathematical theory of epidemics-III. Further studies of the problem of endemicity. Proc. R. Soc. Ser. A., 141: 94–122 (1933)
Korobeinikov, A. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol., 30: 615–626 (2006)
Li, J., Song, X., Gao, F. Global stability of a viral infection model with delays and two types of target cells. J. Appl. Anal. Comput., 2: 281–292 (2012)
Li, X., Wang, J., Ghosh, M. Stability and bifurcation of an SIVS epidemic model with treatment and age of vaccination. Appl. Math. Model., 34: 437–450 (2010)
Liu, L., Wang, J., Liu, X. Global stability of an SEIR epidemic model with age-dependent latency and relapse. Nonlinear Anal. Real World Appl., 24: 18–35 (2015)
Liu, W.M., Hetchote, H.W., Levin, S.A. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol., 23: 187–204 (1986)
Liu, Y., Jiao, F., Hu, L. Modeling mosquito population control by a coupled system. J. Math. Anal. Appl., 506: 125671 (2022)
Magal, P., McCluskey, C.C., Webb, G.F. Lyapunov functional and global asymptotic stability for an infection-age model. Appl. Anal., 89: 1109–1140 (2010)
Magal, P., McCluskey, C.C. Two-group infection age model including an application to nosocomial infection. SIAM J. Appl. Math., 73: 1058–1095 (2013)
Magal, P., Zhao, X.-Q. Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal., 37: 251–275 (2005)
Ruan, S., Wang, W. Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Diff. Equ., 188: 135–163 (2003)
Shen, M., Xiao, Y. Global stability of a multi-group SVEIR epidemiological model with the vaccination age and infection Age. Acta Appl. Math., 144: 137–157 (2016)
Smith, H.L., Thieme, H.R. Dynamical Systems and Population Persistence. Amer., Math., Soc., (2010)
van den Driessche, P., Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180: 29–48 (2002)
Wang, J., Zhang, R., Kuniya, T. The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes. J. Biol. Dynam., 9: 73–101 (2015)
Webb, G. Theory of Nonlinear Age-Dependent Population Dynamics. Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, New York, 1985
Xiao, D., Ruan, S. Global analysis of an epidemic model with nonmonotone incidence rate. Math. Biosci., 208: 419–429 (2007)
Yang, Y., Ruan, S., Xiao, D. Global stability of an age-struncture virus dynamics model with Beddington-Deangelis infection function. Math. Biosci. Eng., 12: 859–577 (2015)
Zou, L., Zhang, W., Ruan, S. Modeling the transmission dynamics and control of hepatitis B virus in China. J. Theore. Biol., 262: 330–338 (2010)
National Bureau of Statistics of China. China Statistical Yearbook 2009, Birth Rate, Death Rate and Natural Growth Rate of Population. http://www.stats.gov.cn/tjsj/ndsj/2009/indexch.htm [Accessed on 10 April 2016].
http://www.who.int/immunization/topics/WHO45osition_paper_HepB.pdf
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This paper is supported by The National Natural Science Foundation of China [12026236, 12026222, 12061079, 11601293, 12071418], Science and Technology Activities Priority Program for Overseas Researchers in Shanxi Province [20210049], The Natural Science Foundation of Shanxi Province [201901D211160, 201901D211461, 201901D111295].
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Feng, Xm., Liu, Ll. & Zhang, Fq. Dynamical Behavior of SEIR-SVS Epidemic Models with Nonlinear Incidence and Vaccination. Acta Math. Appl. Sin. Engl. Ser. 38, 282–303 (2022). https://doi.org/10.1007/s10255-022-1075-7
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DOI: https://doi.org/10.1007/s10255-022-1075-7
Keywords
- vaccination
- age-structured epidemic model
- delay differential equation model
- stability
- Lyapunov functional