Interaction of maturation delay and nonlinear birth in population and epidemic models |
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Authors: | K Cooke P van den Driessche X Zou |
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Institution: | (1) Department of Mathematics, Pomona College, Claremont, CA, 91711-6348, USA, US;(2) Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3P4, CA |
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Abstract: | A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e−
d
1
T−dN(t) governs the adult population, with the death rate in previous life stages d
1≧0. Standard assumptions are made on B(N) so that a unique equilibrium N
e
exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d
1>0, as T increases the equilibrium N
e
can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease
that does not cause death is introduced into the population, a threshold parameter R
0 is identified. When R
0<1, the disease dies out; when R
0>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations
indicate that oscillations can also be induced by disease related death in a model with maturation delay.
Received: 2 November 1998 / Revised version: 26 February 1999 |
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Keywords: | : Maturation delay Epidemic model Global stability Periodic solutions |
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