# American Institute of Mathematical Sciences

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.

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##### 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.
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