ANZIAM  J.  48 (2006), 119-134
SEIR epidemic model with delay

Ping Yan
  Department of Mathematics
  University of Helsinki
Shengqiang Liu
  Department of Mathematics
  Xiamen University
  Xiamen 361005
  P. R. China

A disease transmission model of SEIR type with exponential demographic structure is formulated, with a natural death rate constant and an excess death rate constant for infective individuals. The latent period is assumed to be constant, and the force of the infection is assumed to be of the standard form, namely, proportional to $I(t)/N(t)$ where $N(t)$ is the total (variable) population size and $I(t)$ is the size of the infective population. The infected individuals are assumed not to be able to give birth and when an individual is removed from the $I$-class, it recovers, acquiring permanent immunity with probability $f$ $(0\leq f\leq 1)$ and dies from the disease with probability $1-f$. The global attractiveness of the disease-free equilibrium, existence of the endemic equilibrium as well as the permanence criteria are investigated. Further, it is shown that for the special case of the model with zero latent period, $R_0>1$ leads to the global stability of the endemic equilibrium, which completely answers the conjecture proposed by Diekmann and Heesterbeek.
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