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

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28 Citations (Scopus)

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, "Non-linear 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.

Original languageEnglish
Pages (from-to)615-622
Number of pages8
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume4
Issue number3
Publication statusPublished - Aug 2004

Fingerprint

Pathogens
Stability Analysis
Interaction
Software
Interior
Nonlinear Oscillations
Immune Response
Stability Theorem
Infectious Diseases
Nonlinear Ordinary Differential Equations
Viruses
Ordinary differential equations
Virus
Manipulation
Determinant
Model
Derivatives
Derivative

Keywords

  • Differential equation
  • Immune response
  • Infectious diseases
  • Stability

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

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title = "A note on the stability analysis of pathogen-immune interaction dynamics",
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, {"}Non-linear 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.",
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AB - 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, "Non-linear 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.

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