Mathematical Biosciences and Engineering, 2011, 8(4): 931-952. doi: 10.3934/mbe.2011.8.931.

Primary: 34K19, 34K20, 92D30.

Export file:

Format

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

Content

  • Citation Only
  • Citation and Abstract

An SEIR epidemic model with constant latency time and infectious period

1. CIMAB, University of Milano, via C. Saldini 50, I20133 Milano
2. Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, I33100 Udine

We present a two delays SEIR epidemic model with a saturation incidence rate. One delay is the time taken by the infected individuals to become infectious (i.e. capable to infect a susceptible individual), the second delay is the time taken by an infectious individual to be removed from the infection. By iterative schemes and the comparison principle, we provide global attractivity results for both the equilibria, i.e. the disease-free equilibrium $\mathbf{E}_{0}$ and the positive equilibrium $\mathbf{E}_{+}$, which exists iff the basic reproduction number $\mathcal{R}_{0}$ is larger than one. If $\mathcal{R}_{0}>1$ we also provide a permanence result for the model solutions. Finally we prove that the two delays are harmless in the sense that, by the analysis of the characteristic equations, which result to be polynomial trascendental equations with polynomial coefficients dependent upon both delays, we confirm all the standard properties of an epidemic model: $\mathbf{E}_{0}$ is locally asymptotically stable for $\mathcal{R}% _{0}<1$ and unstable for $\mathcal{R}_{0}>1$, while if $\mathcal{R}_{0}>1$ then $\mathbf{E}_{+}$ is always asymptotically stable.
  Figure/Table
  Supplementary
  Article Metrics

Keywords nonlinear incidence rate; time delays; local stability analysis.; permanence; global attractivity; delay differential equations; Epidemic model

Citation: Edoardo Beretta, Dimitri Breda. An SEIR epidemic model with constant latency time and infectious period. Mathematical Biosciences and Engineering, 2011, 8(4): 931-952. doi: 10.3934/mbe.2011.8.931

 

This article has been cited by

  • 1. Mohammad A. Safi, Salisu M. Garba, Global Stability Analysis of SEIR Model with Holling Type II Incidence Function, Computational and Mathematical Methods in Medicine, 2012, 2012, 1, 10.1155/2012/826052
  • 2. Isam Al-Darabsah, Yuan Yuan, A Stage-Structured Mathematical Model for Fish Stock with Harvesting, SIAM Journal on Applied Mathematics, 2018, 78, 1, 145, 10.1137/16M1097092
  • 3. Isaac Mwangi Wangari, Lewi Stone, Nakul Chitnis, Backward bifurcation and hysteresis in models of recurrent tuberculosis, PLOS ONE, 2018, 13, 3, e0194256, 10.1371/journal.pone.0194256
  • 4. Huaixing Li, Yoichi Enatsu, Yoshiaki Muroya, A note on the global stability of an SEIR epidemic model with constant latency time and infectious period, Discrete and Continuous Dynamical Systems - Series B, 2012, 18, 1, 173, 10.3934/dcdsb.2013.18.173
  • 5. D. Breda, O. Diekmann, W. F. de Graaf, A. Pugliese, R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), Journal of Biological Dynamics, 2012, 6, sup2, 103, 10.1080/17513758.2012.716454
  • 6. Anjana Das, M. Pal, Modeling and Analysis of an Imprecise Epidemic System with Optimal Treatment and Vaccination Control, Biophysical Reviews and Letters, 2018, 13, 02, 37, 10.1142/S1793048018500042
  • 7. Yukihiko Nakata, Yoichi Enatsu, Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate, Mathematical Biosciences and Engineering, 2014, 11, 4, 785, 10.3934/mbe.2014.11.785
  • 8. Bruno Buonomo, Marianna Cerasuolo, The effect of time delay in plant--pathogen interactions with host demography, Mathematical Biosciences and Engineering, 2015, 12, 3, 473, 10.3934/mbe.2015.12.473
  • 9. Anjana Das, M. Pal, A mathematical study of an imprecise SIR epidemic model with treatment control, Journal of Applied Mathematics and Computing, 2018, 56, 1-2, 477, 10.1007/s12190-017-1083-6
  • 10. Isam Al-Darabsah, Yuan Yuan, A periodic disease transmission model with asymptomatic carriage and latency periods, Journal of Mathematical Biology, 2017, 10.1007/s00285-017-1199-1
  • 11. Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi, Global stability for epidemic model with constant latency and infectious periods, Mathematical Biosciences and Engineering, 2012, 9, 2, 297, 10.3934/mbe.2012.9.297
  • 12. Dmitri 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, 10.3934/dcds.2012.32.2675
  • 13. Xavier Bardina, Marco Ferrante, Carles Rovira, Stochastic Epidemic SEIRS Models with a Constant Latency Period, Mediterranean Journal of Mathematics, 2017, 14, 4, 10.1007/s00009-017-0977-8
  • 14. A M Pasion, J A Collera, Delay-induced stability switches in an SIRS epidemic model with saturated incidence rate and temporary immunity, Journal of Physics: Conference Series, 2019, 1298, 012006, 10.1088/1742-6596/1298/1/012006

Reader Comments

your name: *   your email: *  

Copyright Info: 2011, , 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