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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[1]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837

[2]

Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate. Mathematical Biosciences & Engineering, 2016, 13 (4) : 723-739. doi: 10.3934/mbe.2016016

[3]

Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 93-112. doi: 10.3934/dcdsb.2011.15.93

[4]

Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 61-74. doi: 10.3934/dcdsb.2011.15.61

[5]

Yoichi Enatsu, Yukihiko Nakata. Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate. Mathematical Biosciences & Engineering, 2014, 11 (4) : 785-805. doi: 10.3934/mbe.2014.11.785

[6]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445

[7]

Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133

[8]

Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057

[9]

Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119

[10]

Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101

[11]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727

[12]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082

[13]

Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525

[14]

Ting Guo, Haihong Liu, Chenglin Xu, Fang Yan. Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-20. doi: 10.3934/dcdsb.2018134

[15]

Jianquan Li, Yicang Zhou, Jianhong Wu, Zhien Ma. Complex dynamics of a simple epidemic model with a nonlinear incidence. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 161-173. doi: 10.3934/dcdsb.2007.8.161

[16]

David Schley, S.A. Gourley. Linear and nonlinear stability in a diffusional ecotoxicological model with time delays. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 575-590. doi: 10.3934/dcdsb.2002.2.575

[17]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121

[18]

Hong Yang, Junjie Wei. Global behaviour of a delayed viral kinetic model with general incidence rate. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1573-1582. doi: 10.3934/dcdsb.2015.20.1573

[19]

Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483

[20]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977

2016 Impact Factor: 1.035

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]