Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

SEIR epidemiological model with varying infectivity and infinite delay

1. Analysis and Stochastics Research Group, Hungarian Academy of Sciences, Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1.
2. Center for Disease Modeling & Dept. of Mathematics and Statistics, York University, Toronto 4700 Keele str., M3J 1P3, ON

A new SEIR model with distributed infinite delay is derived when the infectivity depends on the age of infection. The basic reproduction number R0, which is a threshold quantity for the stability of equilibria, is calculated. If $R_0$ < 1, then the disease-free equilibrium is globally asymptotically stable and this is the only equilibrium. On the contrary, if $R_0$ > 1, then an endemic equilibrium appears which is locally asymptotically stable. Applying a perma- nence theorem for infinite dimensional systems, we obtain that the disease is always present when $R_0$ > 1.
  Figure/Table
  Supplementary
  Article Metrics

Keywords mathematical epidemiology; permanence.; stability; SEIR model; infinite delay

Citation: Gergely Röst, Jianhong Wu. SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences and Engineering, 2008, 5(2): 389-402. doi: 10.3934/mbe.2008.5.389

 

This article has been cited by

  • 1. Jinliang Wang, Xiaoqing Yu, Heidi L. Tessmer, Toshikazu Kuniya, Ryosuke Omori, Modelling infectious diseases with relapse: a case study of HSV-2, Theoretical Biology and Medical Modelling, 2017, 14, 1, 10.1186/s12976-017-0059-4
  • 2. Pierre Magal, Connell McCluskey, Two-Group Infection Age Model Including an Application to Nosocomial Infection, SIAM Journal on Applied Mathematics, 2013, 73, 2, 1058, 10.1137/120882056
  • 3. Dávid Lehotzky, Tamás Insperger, A least-square spectral element method for stability analysis of time delay systems∗∗This work was supported by the Hungarian National Science Foundation under grant OTKA-K105433., IFAC-PapersOnLine, 2015, 48, 12, 382, 10.1016/j.ifacol.2015.09.408
  • 4. Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki, Lyapunov functionals for virus-immune models with infinite delay, Discrete and Continuous Dynamical Systems - Series B, 2015, 20, 9, 3093, 10.3934/dcdsb.2015.20.3093
  • 5. Paul Georgescu, Hong Zhang, An impulsively controlled predator–pest model with disease in the pest, Nonlinear Analysis: Real World Applications, 2010, 11, 1, 270, 10.1016/j.nonrwa.2008.10.060
  • 6. P. Magal, C.C. McCluskey, G.F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 2010, 89, 7, 1109, 10.1080/00036810903208122
  • 7. Maoxing Liu, Gergely Röst, Gabriella Vas, SIS model on homogeneous networks with threshold type delayed contact reduction, Computers & Mathematics with Applications, 2013, 66, 9, 1534, 10.1016/j.camwa.2013.02.009
  • 8. David Lehotzky, Tamas Insperger, Gabor Stepan, Numerical methods for the stability of time-periodic hybrid time-delay systems with applications, Applied Mathematical Modelling, 2018, 57, 142, 10.1016/j.apm.2017.12.029
  • 9. Andrey Melnik, Andrei Korobeinikov, Global asymptotic properties of staged models with multiple progression pathways for infectious diseases, Mathematical Biosciences and Engineering, 2011, 8, 4, 1019, 10.3934/mbe.2011.8.1019
  • 10. Michael Y. Li, Zhisheng Shuai, Chuncheng Wang, Global stability of multi-group epidemic models with distributed delays, Journal of Mathematical Analysis and Applications, 2010, 361, 1, 38, 10.1016/j.jmaa.2009.09.017
  • 11. Wei Zhong, Yushim Kim, Megan Jehn, Modeling dynamics of an influenza pandemic with heterogeneous coping behaviors: case study of a 2009 H1N1 outbreak in Arizona, Computational and Mathematical Organization Theory, 2013, 19, 4, 622, 10.1007/s10588-012-9146-6
  • 12. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Mathematical Biosciences and Engineering, 2012, 9, 4, 819, 10.3934/mbe.2012.9.819
  • 13. Xinxin Wang, Shengqiang Liu, An epidemic model with different distributed latencies and nonlinear incidence rate, Applied Mathematics and Computation, 2014, 241, 259, 10.1016/j.amc.2014.05.032
  • 14. Dessalegn Y. Melesse, Abba B. Gumel, Global asymptotic properties of an SEIRS model with multiple infectious stages, Journal of Mathematical Analysis and Applications, 2010, 366, 1, 202, 10.1016/j.jmaa.2009.12.041
  • 15. Kasia A. Pawelek, Shengqiang Liu, Faranak Pahlevani, Libin Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Mathematical Biosciences, 2012, 235, 1, 98, 10.1016/j.mbs.2011.11.002
  • 16. Jinhu Xu, Yan Geng, Yicang Zhou, Global stability of a multi-group model with distributed delay and vaccination, Mathematical Methods in the Applied Sciences, 2017, 40, 5, 1475, 10.1002/mma.4068
  • 17. Hongying Shu, Yuming Chen, Lin Wang, Impacts of the Cell-Free and Cell-to-Cell Infection Modes on Viral Dynamics, Journal of Dynamics and Differential Equations, 2017, 10.1007/s10884-017-9622-2
  • 18. Xiulan Lai, Xingfu Zou, Modeling HIV-1 Virus Dynamics with Both Virus-to-Cell Infection and Cell-to-Cell Transmission, SIAM Journal on Applied Mathematics, 2014, 74, 3, 898, 10.1137/130930145
  • 19. Mingwang Shen, Yanni Xiao, Global Stability of a Multi-group SVEIR Epidemiological Model with the Vaccination Age and Infection Age, Acta Applicandae Mathematicae, 2016, 144, 1, 137, 10.1007/s10440-016-0044-7
  • 20. Jinliang Wang, Jiying Lang, Yuming Chen, Global threshold dynamics of an SVIR model with age-dependent infection and relapse, Journal of Biological Dynamics, 2017, 11, sup2, 427, 10.1080/17513758.2016.1226436
  • 21. Gang Huang, Yasuhiro Takeuchi, Wanbiao Ma, Daijun Wei, Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate, Bulletin of Mathematical Biology, 2010, 72, 5, 1192, 10.1007/s11538-009-9487-6
  • 22. Jinliang Wang, Min Guo, Xianning Liu, Zhitao Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay, Applied Mathematics and Computation, 2016, 291, 149, 10.1016/j.amc.2016.06.032
  • 23. Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki, Lyapunov functionals for multistrain models with infinite delay, Discrete and Continuous Dynamical Systems - Series B, 2016, 22, 2, 507, 10.3934/dcdsb.2017025
  • 24. O. Sharomi, A.B. Gumel, Dynamical analysis of a sex-structured Chlamydia trachomatis transmission model with time delay, Nonlinear Analysis: Real World Applications, 2011, 12, 2, 837, 10.1016/j.nonrwa.2010.08.010
  • 25. C. McCluskey, Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds, Mathematical Biosciences and Engineering, 2015, 13, 2, 10.3934/mbe.2015008
  • 26. Yongzhen Pei, Li Changguo, Qianyong Wu, Yunfei Lv, Successive Vaccination and Difference in Immunity of a Delay SIR Model with a General Incidence Rate, Abstract and Applied Analysis, 2014, 2014, 1, 10.1155/2014/678723
  • 27. Xinzhi Liu, Peter Stechlinski, Hybrid stabilization and synchronization of nonlinear systems with unbounded delays, Applied Mathematics and Computation, 2016, 280, 140, 10.1016/j.amc.2016.01.023
  • 28. Yu Yang, Lan Zou, Shigui Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Mathematical Biosciences, 2015, 270, 183, 10.1016/j.mbs.2015.05.001
  • 29. Xichao Duan, Sanling Yuan, Zhipeng Qiu, Junling Ma, Global stability of an SVEIR epidemic model with ages of vaccination and latency, Computers & Mathematics with Applications, 2014, 68, 3, 288, 10.1016/j.camwa.2014.06.002
  • 30. Lianwen Wang, Zhijun Liu, Xingan Zhang, Global dynamics of an SVEIR epidemic model with distributed delay and nonlinear incidence, Applied Mathematics and Computation, 2016, 284, 47, 10.1016/j.amc.2016.02.058
  • 31. J. Martín-Vaquero, A. Queiruga-Dios, A. Martín del Rey, A.H. Encinas, J.D. Hernández Guillén, G. Rodríguez Sánchez, Variable step length algorithms with high-order extrapolated non-standard finite difference schemes for a SEIR model, Journal of Computational and Applied Mathematics, 2018, 330, 848, 10.1016/j.cam.2017.03.031
  • 32. Zhigui Lin, Yinan Zhao, Peng Zhou, The infected frontier in an SEIR epidemic model with infinite delay, Discrete and Continuous Dynamical Systems - Series B, 2013, 18, 9, 2355, 10.3934/dcdsb.2013.18.2355
  • 33. Shengqiang Liu, Shaokai Wang, Lin Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear Analysis: Real World Applications, 2011, 12, 1, 119, 10.1016/j.nonrwa.2010.06.001
  • 34. Jinliang Wang, Ran Zhang, Toshikazu Kuniya, The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes, Journal of Biological Dynamics, 2015, 9, 1, 73, 10.1080/17513758.2015.1006696
  • 35. David Lehotzky, Tamas Insperger, Gabor Stepan, Extension of the spectral element method for stability analysis of time-periodic delay-differential equations with multiple and distributed delays, Communications in Nonlinear Science and Numerical Simulation, 2016, 35, 177, 10.1016/j.cnsns.2015.11.007
  • 36. Jinliang Wang, Ran Zhang, Toshikazu Kuniya, The dynamics of an SVIR epidemiological model with infection age: Table 1., IMA Journal of Applied Mathematics, 2016, 81, 2, 321, 10.1093/imamat/hxv039
  • 37. Xinzhi Liu, Peter Stechlinski, , Infectious Disease Modeling, 2017, Chapter 4, 83, 10.1007/978-3-319-53208-0_4
  • 38. Hongying Shu, Dejun Fan, Junjie Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Analysis: Real World Applications, 2012, 13, 4, 1581, 10.1016/j.nonrwa.2011.11.016
  • 39. Stephen A. Gourley, Gergely Röst, Horst R. Thieme, Uniform Persistence in a Model for Bluetongue Dynamics, SIAM Journal on Mathematical Analysis, 2014, 46, 2, 1160, 10.1137/120878197
  • 40. Toshikazu Kuniya, Stability Analysis of an Age-Structured SIR Epidemic Model with a Reduction Method to ODEs, Mathematics, 2018, 6, 9, 147, 10.3390/math6090147
  • 41. Aekabut Sirijampa, Settapat Chinviriyasit, Wirawan Chinviriyasit, Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period, Advances in Difference Equations, 2018, 2018, 1, 10.1186/s13662-018-1805-6
  • 42. Ryosuke Omori, Hiroshi Nishiura, Theoretical basis to measure the impact of short-lasting control of an infectious disease on the epidemic peak, Theoretical Biology and Medical Modelling, 2011, 8, 1, 10.1186/1742-4682-8-2
  • 43. Wei Zhong, Tim Lant, Megan Jehn, Yushim Kim, , Simulation for Policy Inquiry, 2012, Chapter 10, 181, 10.1007/978-1-4614-1665-4_10
  • 44. Shengqiang Liu, Yasuhiro Takeuchi, Gang Huang, Jinliang Wang, Sveir epidemiological model with varying infectivity and distributed delays, Mathematical Biosciences and Engineering, 2011, 8, 3, 875, 10.3934/mbe.2011.8.875

Reader Comments

your name: *   your email: *  

Copyright Info: 2008, , licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved