American Institute of Mathematical Sciences

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August  2004, 4(3): 615-622. doi: 10.3934/dcdsb.2004.4.615

A note on the stability analysis of pathogen-immune interaction dynamics

Tsuyoshi Kajiwara 1, and  Toru Sasaki 1,

1. 

Department of Environmental and Mathematical Science, Okayama University, 700-8530 Tsushima, Okayama, Japan, Japan

Received  January 2003 Revised  February 2004 Published  May 2004

The stability analysis of the interior equilibria, whose components are all positive, of non linear ordinary differential equation models describing in vivo dynamics of infectious diseases are complicated in general. Liu, "Nonlinear oscillation in models of immune responses to persistent viruses, Theor. Popul. Biol. 52(1997), 224-230" and Murase, Sasaki and Kajiwara, "Stability analysis of pathogen-immune interaction dynamics (submitted)" proved the stability of the interior equilibria of such models using symbolic calculation software on computers. In this paper, proofs without using symbolic calculation software of the stability theorems given by Liu and Murase et al. are presented. Simple algebraic manipulations, properties of determinants, and their derivatives are used. The details of the calculation given by symbolic calculation software can be seen clearly.
Keywords: infectious diseases., stability, Differential equation, immune response.
Mathematics Subject Classification: 34D05, 92C5.
Citation: Tsuyoshi Kajiwara, Toru Sasaki. A note on the stability analysis of pathogen-immune interaction dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 615-622. doi: 10.3934/dcdsb.2004.4.615
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