Abstract view
Dynamical Analysis of a StageStructured Model for Lyme Disease with Two Delays


Published:20160111
Printed: Jun 2016
Dan Li,
Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, P.R. China
Wanbiao Ma,
Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, P.R. China
Abstract
In this paper, a
nonlinear stagestructured model for Lyme disease is considered.
The model is a system of differential equations with two time
delays. The basic reproductive rate, $R_0(\tau_1,\tau_2)$, is
derived. If $R_0(\tau_1,\tau_2)\lt 1$, then the boundary equilibrium
is globally asymptotically stable. If $R_0(\tau_1,\tau_2)\gt 1$,
then there exists
a unique positive equilibrium whose local asymptotical stability
and the existence of
Hopf bifurcations are established by analyzing the distribution
of the characteristic values.
An explicit algorithm for determining the direction of Hopf bifurcations
and the
stability of the bifurcating periodic solutions is derived by
using the normal form and
the center manifold theory. Some numerical simulations are performed
to confirm the correctness
of theoretical analysis. At last, some conclusions are given.