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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

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

Pages: 615 - 622, Volume 4, Issue 3, August 2004      doi:10.3934/dcdsb.2004.4.615

 
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Tsuyoshi Kajiwara - Department of Environmental and Mathematical Science, Okayama University, 700-8530 Tsushima, Okayama, Japan (email)
Toru Sasaki - Department of Environmental and Mathematical Science, Okayama University, 700-8530 Tsushima, Okayama, Japan (email)

Abstract: 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:  Differential equation, stability, immune response, infectious diseases.
Mathematics Subject Classification:  34D05, 92C50.

Received: January 2003;      Revised: February 2004;      Available Online: May 2004.