### Abstract

We study a population model for an infectious disease that spreads in the host population through both horizontal and vertical transmission. The total host population is assumed to have constant density and the incidence term is of the bilinear mass-action form. We prove that the global dynamics are completely determined by the basic reproduction number R_{0}(p, q), where p and q are fractions of infected newborns from the exposed and infectious classes, respectively. If R_{0}(p, q) ≤ 1, the disease-free equilibrium is globally stable and the disease always dies out. If R_{0}(p, q) > 1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the vertical transmission to the basic reproduction number is also analyzed.

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

Pages (from-to) | 58-69 |

Number of pages | 12 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 62 |

Issue number | 1 |

State | Published - 2001 |

### Fingerprint

### Keywords

- Compound matrices
- Endemic equilibrium
- Epidemic models
- Global stability
- Latent period
- Vertical transmission

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*62*(1), 58-69.

**Global dynamics of an seir epidemic model with vertical transmission.** / Li, Michael Y.; Smith, Hal; Wang, Liancheng.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 62, no. 1, pp. 58-69.

}

TY - JOUR

T1 - Global dynamics of an seir epidemic model with vertical transmission

AU - Li, Michael Y.

AU - Smith, Hal

AU - Wang, Liancheng

PY - 2001

Y1 - 2001

N2 - We study a population model for an infectious disease that spreads in the host population through both horizontal and vertical transmission. The total host population is assumed to have constant density and the incidence term is of the bilinear mass-action form. We prove that the global dynamics are completely determined by the basic reproduction number R0(p, q), where p and q are fractions of infected newborns from the exposed and infectious classes, respectively. If R0(p, q) ≤ 1, the disease-free equilibrium is globally stable and the disease always dies out. If R0(p, q) > 1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the vertical transmission to the basic reproduction number is also analyzed.

AB - We study a population model for an infectious disease that spreads in the host population through both horizontal and vertical transmission. The total host population is assumed to have constant density and the incidence term is of the bilinear mass-action form. We prove that the global dynamics are completely determined by the basic reproduction number R0(p, q), where p and q are fractions of infected newborns from the exposed and infectious classes, respectively. If R0(p, q) ≤ 1, the disease-free equilibrium is globally stable and the disease always dies out. If R0(p, q) > 1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the vertical transmission to the basic reproduction number is also analyzed.

KW - Compound matrices

KW - Endemic equilibrium

KW - Epidemic models

KW - Global stability

KW - Latent period

KW - Vertical transmission

UR - http://www.scopus.com/inward/record.url?scp=0036222772&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036222772&partnerID=8YFLogxK

M3 - Article

VL - 62

SP - 58

EP - 69

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 1

ER -