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

An SEIR epidemic model with constant latency time and infectious period

1. 

CIMAB, University of Milano, via C. Saldini 50, I20133 Milano, Italy

2. 

Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, I33100 Udine, Italy

Received  September 2010 Revised  March 2011 Published  August 2011

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.
Citation: Edoardo Beretta, Dimitri Breda. An SEIR epidemic model with constant latency time and infectious period. Mathematical Biosciences & Engineering, 2011, 8 (4) : 931-952. doi: 10.3934/mbe.2011.8.931
References:
[1]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters,, SIAM J. Math. Anal., 33 (2002), 1144.   Google Scholar

[2]

V. Capasso and G. Serio, A generalization of the Kermack-McKendric deterministic epidemic model,, Math. Biosci., 42 (1978), 43.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[3]

M. Giaquinta and G. Modica, "Mathematical Analysis. An Introduction to Functions of Several Variables,", Birkhauser Boston, (2009).   Google Scholar

[4]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125.   Google Scholar

[5]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infection,, SIAM J. Appl. Math., 70 (2010), 2693.  doi: 10.1137/090780821.  Google Scholar

[6]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,, Bull. Math. Biol., 72 (2010), 1192.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[7]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

[8]

Y. Kuang, "Delay Differential Equations with Application in Population Dynamics,", Dynamics in Science and Engineering, (1993).   Google Scholar

[9]

M. A. Safi and A. B. Gumel, Global asymptotic dynamics of a model of quarantine and isolation,, Discrete Contin. Dyn. S., 14 (2010), 209.  doi: 10.3934/dcdsb.2010.14.209.  Google Scholar

[10]

H. L. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,", Texts in Applied Mathematics, (2011).  doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[11]

R. Xu and Y. Du, \ A delayed SIR epidemic model with saturation incidence and constant infectious period,, J. Appl. Math. Comp., 35 (2010), 229.  doi: 10.1007/s12190-009-0353-3.  Google Scholar

[12]

R. Xu and Z. Ma, Global stability of a delayed SEIRS epidemic model with saturation incidence rate,, Nonlinear Dynam., 61 (2010), 229.  doi: 10.1007/s11071-009-9644-3.  Google Scholar

[13]

F. Zhang, Z. Li and F. Zhang, Global stability of an SIR epidemic model with constant infectious period,, Appl. Math. Comput, 199 (2008), 285.  doi: 10.1016/j.amc.2007.09.053.  Google Scholar

show all references

References:
[1]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters,, SIAM J. Math. Anal., 33 (2002), 1144.   Google Scholar

[2]

V. Capasso and G. Serio, A generalization of the Kermack-McKendric deterministic epidemic model,, Math. Biosci., 42 (1978), 43.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[3]

M. Giaquinta and G. Modica, "Mathematical Analysis. An Introduction to Functions of Several Variables,", Birkhauser Boston, (2009).   Google Scholar

[4]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125.   Google Scholar

[5]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infection,, SIAM J. Appl. Math., 70 (2010), 2693.  doi: 10.1137/090780821.  Google Scholar

[6]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,, Bull. Math. Biol., 72 (2010), 1192.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[7]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

[8]

Y. Kuang, "Delay Differential Equations with Application in Population Dynamics,", Dynamics in Science and Engineering, (1993).   Google Scholar

[9]

M. A. Safi and A. B. Gumel, Global asymptotic dynamics of a model of quarantine and isolation,, Discrete Contin. Dyn. S., 14 (2010), 209.  doi: 10.3934/dcdsb.2010.14.209.  Google Scholar

[10]

H. L. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,", Texts in Applied Mathematics, (2011).  doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[11]

R. Xu and Y. Du, \ A delayed SIR epidemic model with saturation incidence and constant infectious period,, J. Appl. Math. Comp., 35 (2010), 229.  doi: 10.1007/s12190-009-0353-3.  Google Scholar

[12]

R. Xu and Z. Ma, Global stability of a delayed SEIRS epidemic model with saturation incidence rate,, Nonlinear Dynam., 61 (2010), 229.  doi: 10.1007/s11071-009-9644-3.  Google Scholar

[13]

F. Zhang, Z. Li and F. Zhang, Global stability of an SIR epidemic model with constant infectious period,, Appl. Math. Comput, 199 (2008), 285.  doi: 10.1016/j.amc.2007.09.053.  Google Scholar

[1]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[2]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[3]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[4]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[5]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[6]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[7]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[8]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[9]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[10]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[11]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[12]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[13]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[14]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[15]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275

[16]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[17]

Telmo Peixe. Permanence in polymatrix replicators. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020032

[18]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[19]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[20]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (102)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]