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