Age Structured Epidemic Modeling by Xue-Zhi Li & Junyuan Yang & Maia Martcheva

Author:Xue-Zhi Li & Junyuan Yang & Maia Martcheva
Language: eng
Format: epub
ISBN: 9783030424961
Publisher: Springer International Publishing

(5.44)

Define a Lyapunov function as where f ∗(s)ds.

Lemma 5.5

If is bounded, then

(5.45)

where

Proof

Note that

(5.46)

where we denote

It follows from the assumption of Lemma 5.5 that V 1(t) is well defined for all t ≥ 0. Deviating it along the solution of (5.43), we obtain

(5.47)

Observing (5.46), we have

(5.48)

With the help of integrating by parts, we obtain

(5.49)

here we used the fact and M′(a) = −η(a)f ∗(a).

Theorem 5.8

If γ(a) = γ and R 0 > 1, then the endemic equilibrium E ∗ is the global attractor of system (5.6).

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