### Abstract

A method for estimating parameters in dynamic stochastic (Markov Chain) models based on Kurtz's limit theory coupled with inverse problem methods developed for deterministic dynamical systems is proposed and illustrated in the context of disease dynamics. This methodology relies on finding an approximate large-population behavior of an appropriate scaled stochastic system. The approach leads to a deterministic approximation obtained as solutions of rate equations (ordinary differential equations) in terms of the large sample size average over sample paths or trajectories (limits of pure jump Markov processes). Using the resulting deterministic model, we select parameter subset combinations that can be estimated using an ordinary-least-squares (OLS) or generalized-least-squares (GLS) inverse problem formulation with a given data set. The selection is based on two criteria of the sensitivity matrix: the degree of sensitivity measured in the form of its condition number and the degree of uncertainty measured in the form of its parameter selection score. We illustrate the ideas with a stochastic model for the transmission of vancomycin-resistant enterococcus (VRE) in hospitals and VRE surveillance data from an oncology unit.

Original language | English (US) |
---|---|

Pages (from-to) | 71-100 |

Number of pages | 30 |

Journal | Journal of Biological Systems |

Volume | 19 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2011 |

### Fingerprint

### Keywords

- Inverse Problems
- Large Population Sample Path Approximations
- Markov Chain Stochastic Models
- Parameter Estimation
- Parameter Selection

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Ecology
- Applied Mathematics

### Cite this

*Journal of Biological Systems*,

*19*(1), 71-100. https://doi.org/10.1142/S0218339011003798

**A deterministic methodology for estimation of parameters in dynamic markov chain models.** / Ortiz, A. R.; Banks, H. T.; Castillo-Chavez, Carlos; Chowell, G.; Wang, X.

Research output: Contribution to journal › Article

*Journal of Biological Systems*, vol. 19, no. 1, pp. 71-100. https://doi.org/10.1142/S0218339011003798

}

TY - JOUR

T1 - A deterministic methodology for estimation of parameters in dynamic markov chain models

AU - Ortiz, A. R.

AU - Banks, H. T.

AU - Castillo-Chavez, Carlos

AU - Chowell, G.

AU - Wang, X.

PY - 2011/3

Y1 - 2011/3

N2 - A method for estimating parameters in dynamic stochastic (Markov Chain) models based on Kurtz's limit theory coupled with inverse problem methods developed for deterministic dynamical systems is proposed and illustrated in the context of disease dynamics. This methodology relies on finding an approximate large-population behavior of an appropriate scaled stochastic system. The approach leads to a deterministic approximation obtained as solutions of rate equations (ordinary differential equations) in terms of the large sample size average over sample paths or trajectories (limits of pure jump Markov processes). Using the resulting deterministic model, we select parameter subset combinations that can be estimated using an ordinary-least-squares (OLS) or generalized-least-squares (GLS) inverse problem formulation with a given data set. The selection is based on two criteria of the sensitivity matrix: the degree of sensitivity measured in the form of its condition number and the degree of uncertainty measured in the form of its parameter selection score. We illustrate the ideas with a stochastic model for the transmission of vancomycin-resistant enterococcus (VRE) in hospitals and VRE surveillance data from an oncology unit.

AB - A method for estimating parameters in dynamic stochastic (Markov Chain) models based on Kurtz's limit theory coupled with inverse problem methods developed for deterministic dynamical systems is proposed and illustrated in the context of disease dynamics. This methodology relies on finding an approximate large-population behavior of an appropriate scaled stochastic system. The approach leads to a deterministic approximation obtained as solutions of rate equations (ordinary differential equations) in terms of the large sample size average over sample paths or trajectories (limits of pure jump Markov processes). Using the resulting deterministic model, we select parameter subset combinations that can be estimated using an ordinary-least-squares (OLS) or generalized-least-squares (GLS) inverse problem formulation with a given data set. The selection is based on two criteria of the sensitivity matrix: the degree of sensitivity measured in the form of its condition number and the degree of uncertainty measured in the form of its parameter selection score. We illustrate the ideas with a stochastic model for the transmission of vancomycin-resistant enterococcus (VRE) in hospitals and VRE surveillance data from an oncology unit.

KW - Inverse Problems

KW - Large Population Sample Path Approximations

KW - Markov Chain Stochastic Models

KW - Parameter Estimation

KW - Parameter Selection

UR - http://www.scopus.com/inward/record.url?scp=79953199801&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953199801&partnerID=8YFLogxK

U2 - 10.1142/S0218339011003798

DO - 10.1142/S0218339011003798

M3 - Article

VL - 19

SP - 71

EP - 100

JO - Journal of Biological Systems

JF - Journal of Biological Systems

SN - 0218-3390

IS - 1

ER -