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**A parameter sensitivity methodology in the context of HIV delay equation models.**
*(English)*
Zbl 1083.92025

From the introduction: Over the past several years, the use of mathematical models as an aid in understanding features of HIV and other virus infection dynamics has been substantial. Several papers published in the mid nineties provided strong evidence for the high rate of HIV-1 replication and clearance in infected individuals. By the end of the decade, the general consensus was that in vivo, on the order of \(10^{10}\) virions are assembled and cleared every day. In many of these papers, the viral clearance rate \(c\) was identified by modeling the disease pathogenesis with a system of deterministic differential equations, numerically calculating a solution, and then fitting the results with experimental data (using a nonlinear least squares (NLS) approach). The existence of such a high replication/clearance rate implies a high mutation rate, thus indicating that pharmacological mono-therapy will ultimately fail, since the virus can rapidly manifest a resistance to any one drug. More importantly, this knowledge directly contributed to the current practice of simultaneously administering multiple drugs to HIV positive individuals in an effort to counteract the high mutation rate of the virus.

Following its success in helping to identify this significant feature of the HIV pathogenesis, the use of mathematical modeling and parameter identification in the study of HIV experienced a dramatic increase. In particular, in the wake of the publication of A. S. Perelson et al. [Science 271, 1582–1586 (1996)], there were papers covering everything from additional and/or alternative compartment formulations to arguments for and against the use of delay differential equations in modeling the eclipse phase (inclduding those that addressed the solution stability). Moreover, in the context of delay equations, many of these papers focused heavily on the inter-relationship between the parameters describing the drag efficacy \(\eta\), the length of the eclipse phase \(\tau\), the infected \(T\)-cell death rate \(\delta\), and the virion clearance rate \(c\).

The purpose of this paper is to illustrate our approach, which allows one to develop new insights into HIV pathogenesis by utilizing a mathematical tool not typically associated with conventional NLS techniques. Indeed, there is a precedence for this approach, as is evidenced by previous papers within the HIV modeling literature that make use of stochastic analysis and inference, control theory, and nonlinear analysis.

Following its success in helping to identify this significant feature of the HIV pathogenesis, the use of mathematical modeling and parameter identification in the study of HIV experienced a dramatic increase. In particular, in the wake of the publication of A. S. Perelson et al. [Science 271, 1582–1586 (1996)], there were papers covering everything from additional and/or alternative compartment formulations to arguments for and against the use of delay differential equations in modeling the eclipse phase (inclduding those that addressed the solution stability). Moreover, in the context of delay equations, many of these papers focused heavily on the inter-relationship between the parameters describing the drag efficacy \(\eta\), the length of the eclipse phase \(\tau\), the infected \(T\)-cell death rate \(\delta\), and the virion clearance rate \(c\).

The purpose of this paper is to illustrate our approach, which allows one to develop new insights into HIV pathogenesis by utilizing a mathematical tool not typically associated with conventional NLS techniques. Indeed, there is a precedence for this approach, as is evidenced by previous papers within the HIV modeling literature that make use of stochastic analysis and inference, control theory, and nonlinear analysis.

### MSC:

92C60 | Medical epidemiology |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

93A30 | Mathematical modelling of systems (MSC2010) |

92C50 | Medical applications (general) |

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\textit{H. T. Banks} and \textit{D. M. Bortz}, J. Math. Biol. 50, No. 6, 607--625 (2005; Zbl 1083.92025)

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