American Institute of Mathematical Sciences

Advanced
January  2006, 6(1): 69-96. doi: 10.3934/dcdsb.2006.6.69

Mathematical analysis of an age-structured SIR epidemic model with vertical transmission

Hisashi Inaba 1,

1. 

Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan

Received  December 2004 Revised  August 2005 Published  October 2005

In this paper, we consider a mathematical model for the spread of a directly transmitted infectious disease in an age-structured population. We assume that infected population is recovered with permanent immunity or quarantined by an age-specific schedule, and the infective agent can be transmitted not only horizontally but also vertically from adult individuals to their newborns. For simplicity we assume that the demographic process of the host population is not affected by the spread of the disease, hence the host population is a demographic stable population. First we establish the mathematical well-posedness of the time evolution problem by using the semigroup approach. Next we prove that the basic reproduction ratio is given as the spectral radius of a positive operator, and an endemic steady state exists if and only if the basic reproduction ratio $R_0$ is greater than unity, while the disease-free steady state is globally asymptotically stable if $R_0 < 1$. We also show that the endemic steady states are forwardly bifurcated from the disease-free steady state when $R_0$ crosses the unity. Finally we examine the conditions for the local stability of the endemic steady states.
Keywords: endemic steady state., age structure, basic reproduction ratio, SIR epidemic model, vertical transmission, threshold.
Mathematics Subject Classification: Primary: 92D30 Secondary: 92D2.
Citation: Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 69-96. doi: 10.3934/dcdsb.2006.6.69
[1]

Shujing Gao, Dehui Xie, Lansun Chen. Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 77-86. doi: 10.3934/dcdsb.2007.7.77
[2]

Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170
[3]

Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109
[4]

Andrea Franceschetti, Andrea Pugliese, Dimitri Breda. Multiple endemic states in age-structured $SIR$ epidemic models. Mathematical Biosciences & Engineering, 2012, 9 (3) : 577-599. doi: 10.3934/mbe.2012.9.577
[5]

Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929
[6]

Hassan Tahir, Asaf Khan, Anwarud Din, Amir Khan, Gul Zaman. Optimal control strategy for an age-structured SIR endemic model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2535-2555. doi: 10.3934/dcdss.2021054
[7]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[8]

Toshikazu Kuniya, Yoshiaki Muroya, Yoichi Enatsu. Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1375-1393. doi: 10.3934/mbe.2014.11.1375
[9]

Yutaro Chiyo, Yuya Tanaka, Ayako Uchida, Tomomi Yokota. Global asymptotic stability of endemic equilibria for a diffusive SIR epidemic model with saturated incidence and logistic growth. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022163
[10]

Dandan Sun, Yingke Li, Zhidong Teng, Tailei Zhang. Stability and Hopf bifurcation in an age-structured SIR epidemic model with relapse. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022141
[11]

Dimitri Breda, Stefano Maset, Rossana Vermiglio. Numerical recipes for investigating endemic equilibria of age-structured SIR epidemics. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2675-2699. doi: 10.3934/dcds.2012.32.2675
[12]

Gabriela Marinoschi. Identification of transmission rates and reproduction number in a SARS-CoV-2 epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022128
[13]

Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191
[14]

Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291
[15]

Tongtong Chen, Jixun Chu. Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2023, 28 (1) : 408-425. doi: 10.3934/dcdsb.2022082
[16]

Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3069-3096. doi: 10.3934/dcdsb.2020220
[17]

Xueying Sun, Renhao Cui. Existence and asymptotic profiles of the steady state for a diffusive epidemic model with saturated incidence and spontaneous infection mechanism. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4503-4520. doi: 10.3934/dcdss.2021120
[18]

Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure and Applied Analysis, 2012, 11 (1) : 97-113. doi: 10.3934/cpaa.2012.11.97
[19]

Liming Cai, Maia Martcheva, Xue-Zhi Li. Epidemic models with age of infection, indirect transmission and incomplete treatment. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2239-2265. doi: 10.3934/dcdsb.2013.18.2239
[20]

Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (277)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy