JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (4): 50-59.doi: 10.6040/j.issn.1671-9352.0.2024.356

Previous Articles    

Threshold dynamics analysis of an age-structured epidemic model with periodic infection rate

DONG Ying, LYU Yunfei*   

  1. School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
  • Published:2025-04-08

PDF (PC)

45

Abstract: This paper studies an age-structured SEIR epidemic model with periodic infection rate and population flows. Firstly, the existence and uniqueness of non-negative solution of the model are proved. Subsequently, by using the operator fixed-point theorem and the periodic renewal theorem, the existence of endemic periodic solution and the global asymptotic stability of the disease-free periodic solution of the model are demonstrated. By introducing the spectral radius r(F )of the Fréchet derivative F of the periodic solution operator at point 0, it is shown that when r(F )>1, the model has an endemic periodic solution(disease outbreak); when r(F )<1, the disease-free periodic solution is globally asymptotically stable(disease extinction).

Key words: age structure, population mobility, vaccination, periodic infection rate, periodic solution

CLC Number: 

  • O175
[1] 马知恩,周义仓,王稳地,等. 传染病动力学的数学建模与研究[M]. 北京:科学出版社,2004. MA Zhien, ZHOU Yicang, WANG Wendi, et al. Mathematical modelling and research of epidemic dynamical systems[M]. Beijing: Science Press, 2004.
[2] WEBB G B. Theory of nonlinear age-dependent population dynamics[M]. New York and Basel: Marcel Dekker, 1985.
[3] HUANG Jicai, KANG Hao, LU Min, et al. Stability analysis of an age-structured epidemic model with vaccination and standard incidence rate[J]. Nonlinear Analysis: Real World App lications, 2022, 66:103525.
[4] HETHCOTE H, LEVIN S. Periodicity in epidemiological models[J]. Applied Mathematical Ecology, 1989, 18:193-211.
[5] HETHCOTE H. Asymptotic behavior in a deterministic epidemic model[J]. Bulletin of Mathematical Biology, 1973, 35:607-614.
[6] BACÄER N. Approximation of the basic reproduction number R0 for vector-borne diseases with a periodic vector population[J]. Bulletin of Mathematical Biology, 2007, 69(3):1067-1091.
[7] KUNIYA T, INABA H. Endemic threshold results for an age-structured SIS epidemic model with periodic parameters[J]. Mathematical Analysis and Applications, 2013, 402(2):477-492.
[8] KUNIYA T. Existence of a nontrivial periodic solution in an age-structured SIR epidemic model with time periodic coefficients[J]. Applied Mathematics Letters, 2014, 27:15-20.
[9] KANG Hao, RUAN Shigui, HUANG Qimin. Periodic solutions of an age-structured epidemic model with periodic infection rate[J]. Communications on Pure and Applied Analysis, 2020, 19(10):4955-4972.
[10] OKUWA K, INABA H, KUNIYA T. Mathematical analysis for an age-structured SIRS epidemic model[J]. Mathematical Biosciences Engineering, 2019, 16(5):6071-6102.
[11] KHAN A, ZAMAN G. Global analysis of an age-structured SEIR endemic model[J]. Chaos Soliton Fract, 2018, 108:154-165.
[12] LI Xuezhi, FANG Bin. Stability of an age-structured SEIR epidemic model with infectivity in latent period[J]. Applied Applied Mathematics, 2009, 4(1):218-236.
[13] LI X Z, GUPUR G, ZHU G T. Threshold and stability results for an age-structured SEIR epidemic model[J]. Computers and Mathematics with Applications, 2001, 42(6/7):883-907.
[14] BUSENBERG S, IANNELLI M, THIEME H. Global behavior of an age-structured epidemic model[J]. SIAM Journal on Mathematical Analysis, 1991, 22(4):1065-1080.
[15] MAREK I. Frobenius theory of positive operators: comparison theorems and applications[J]. SIAM Journal on Applied Mathematics, 1970, 19(3):607-628.
[16] SAWASHIMA I. On spectral properties of some positive operators[J]. Natural Science Reports of Ochanomizu University, 1964, 15(2):53-64.
[17] KRASNOSELSKII M A. Positive solutions of operator equations[M]. Groningen: Noordhoff, 1964.
[18] INABA H. Threshold and stability results for an age-structured epidemic model[J]. Mathematical Biology,1990, 28(4):411-434.
[19] LANGLAIS M, BUSENBERG S N. Global behaviour in age structured S.I.S. models with seasonal periodicities and vertical transmission[J]. Journal of Mathematical Analysis and Applications,1997, 213(2):511-533.
[20] BACÄER N, GUERNAUOI S. The epidemic threshold of vector-borne diseases with seasonality[J]. Journal of Mathematical Biology, 2006, 53(3):421-436.
[21] NAKATA Y, KUNIYA T. Global dynamics of a class of SEIRS epidemic models in a periodic environment[J]. Journal of Mathematical Analysis and Applications, 2010, 363(1):230-237.
[1] FAN Jianhang, WU Kuilin. Periodic solutions of a non-autonomous piecewise bilinear system [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(12): 114-121.
[2] Yufeng ZHAO,Guirong LIU. Stationary distribution and probability density function of a stochastic predation system [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(10): 74-88.
[3] WANG Qi, JIA Jian-wen. Nontrivial periodic solution of a stochastic non-autonomous SIS model with public health education [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(9): 28-34.
[4] ZHAO Jiao. Existence and multiplicity of positive periodic solutions for a class of nonlinear third-order difference equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(9): 50-58.
[5] WANG Ai-li. Transcritical bifurcation for a pest management model with impulse and virus infection [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(7): 1-8.
[6] YANG Hu-jun, HAN Xiao-ling. Existence of positive periodic solutions for a class of non-autonomous fourth-order ordinary differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 109-114.
[7] CHEN Rui-peng, LI Xiao-ya. Positive periodic solutions for second-order singular differential equations with damping terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(8): 33-41.
[8] ZHANG Huan, LI Yong-xiang. Positive periodic solutions of higher-order ordinary differential equations with delayed derivative terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(4): 29-36.
[9] ZHANG Shen-gui. Applications of variational method to impulsive differential systems with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(4): 22-28.
[10] ZHANG Shen-gui. Periodic solutions for a class of Kirchhoff-type differential systems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(10): 1-6.
[11] LI Le-le, JIA Jian-wen. Hopf bifurcation of a SIRC epidemic model with delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(1): 116-126.
[12] CHEN Yu-jia, YANG He. Existence of periodic solutions of a class of third order delay differential equations in Banach spaces [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(8): 84-94.
[13] . Periodic solutions for second order singular damped differential equations with a weak singularity [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 84-88.
[14] WANG Shuang-ming. Dynamical analysis of a class of periodic epidemic model with delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(1): 81-87.
[15] CHEN Bin. Third-order periodic boundary value problems with sign-changing Greens function [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(8): 79-83.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd