J. Appl. Math. Comput.
Global stability of an age-structured model for
· Toru Sasaki
· Yoji Otani
Received: 1 August 2017
© Korean Society for Computational and Applied Mathematics 2018
Abstract In this paper, we present an age-structured mathematical model for infec-
tious disease in vivo with infection age of cells. The model contains an immune variable
and the effect of absorption of pathogens into uninfected cells. We construct Lyapunov
functionals for the model and prove that the time derivative of the functionals are non-
positive. Using this, we prove the global stability results for the model. Especially, we
present the full mathematical detail of the proof of the global stability.
Keywords Lyapunov functionals · Age-structured equations · Immunity · Persistence
Mathematics Subject Classiﬁcation 35B15 · 45D05 · 92B05
Infectious diseases in vivo have been investigated extensively using ordinary differ-
ential equations, and differential equations with delay. Anderson et al.  and Nowak
and Bangham  proposed models containing immune variables explicitly. Local
stability analyses of the equilibria of such models are often difﬁcult because the sizes
of Jacobi matrices are large and a lot of symbolic calculations are needed or
transcendental equations appear as characteristic equations.
On the other hand, global stability analysis using Lyapunov functions were
extremely extended by Korobeinikov for ordinary differential equation models and
McCluskey for ordinary differential equation models with delay [13,15]. Pang et al.
 constructed Lyapunov functions for models of infectious disease in vivo with an
Graduate School of Environmental and Life Science, Okayama University, Okayama, Japan