Mathematical Biosciences and Engineering (MBE)

An SEIR epidemic model with constant latency time and infectious period

Pages: 931 - 952, Volume 8, Issue 4, October 2011      doi:10.3934/mbe.2011.8.931

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Edoardo Beretta - CIMAB, University of Milano, via C. Saldini 50, I20133 Milano, Italy (email)
Dimitri Breda - Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, I33100 Udine, Italy (email)

Abstract: 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.

Keywords:  Epidemic model, time delays, nonlinear incidence rate, delay differential equations, permanence, global attractivity, local stability analysis.
Mathematics Subject Classification:  Primary: 34K19, 34K20, 92D30.

Received: September 2010;      Accepted: March 2011;      Available Online: August 2011.