  The aim of this paper is to study the steady states of the mathematical
models with delay kernels which describe pathogen-immune dynamics of many kinds
of infectious diseases. In the study of mathematical models of infectious
diseases it is important to predict whether the infection disappears or the
pathogens persist. The delay kernel is described by the memory function that
reflects the influence of the past density of pathogen in the blood.
  By using the coefficient of kernel k, as a bifurcation parameter, the models
are found to undergo a sequence of Hopf bifurcation. The direction and the
stability criteria of bifurcation periodic solutions are obtained by applying
the normal form theory and the center manifold theorems. Some numerical
simulation examples for justifying the theoretical results are also given.
