### Abstract

A model is developed for the spread of an infectious disease in a population with constant recruitment of new susceptibles and the fundamental properties of its solutions are analyzed. The model allows for arbitrarily many stages of infection all of which have general length distributions and disease mortalities. Existence and uniqueness of solutions to the model equations are established. A basic reproduction ratio is derived and related to the existence of an endemic equilibrium, to the stability of the disease-free equilibrium, and to weak and strong endemicity (persistence) of the disease. A characteristic equation is found, the zeros of which determine the local stability of the endemic equilibrium, and sufficient stability conditions are given for the case that infected individuals do not return into the susceptible class. In a subsequent paper, explicit sufficient and necessary stability conditions will be derived for the case that the disease dynamics are much faster than the demographics.

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

Pages (from-to) | 803-833 |

Number of pages | 31 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 61 |

Issue number | 3 |

State | Published - 2000 |

### Fingerprint

### Keywords

- Abstract Cauchy problems
- Arbitrary stage durations
- Endemic equilibrium
- Integrated semigroups
- Many infection stages
- Persistence
- Semiflows
- Stage (or class) age
- Volterra integral equations

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Endemic models with arbitrarily distributed periods of infection I : Fundamental properties of the model.** / Feng, Zhilan; Thieme, Horst.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 61, no. 3, pp. 803-833.

}

TY - JOUR

T1 - Endemic models with arbitrarily distributed periods of infection I

T2 - Fundamental properties of the model

AU - Feng, Zhilan

AU - Thieme, Horst

PY - 2000

Y1 - 2000

N2 - A model is developed for the spread of an infectious disease in a population with constant recruitment of new susceptibles and the fundamental properties of its solutions are analyzed. The model allows for arbitrarily many stages of infection all of which have general length distributions and disease mortalities. Existence and uniqueness of solutions to the model equations are established. A basic reproduction ratio is derived and related to the existence of an endemic equilibrium, to the stability of the disease-free equilibrium, and to weak and strong endemicity (persistence) of the disease. A characteristic equation is found, the zeros of which determine the local stability of the endemic equilibrium, and sufficient stability conditions are given for the case that infected individuals do not return into the susceptible class. In a subsequent paper, explicit sufficient and necessary stability conditions will be derived for the case that the disease dynamics are much faster than the demographics.

AB - A model is developed for the spread of an infectious disease in a population with constant recruitment of new susceptibles and the fundamental properties of its solutions are analyzed. The model allows for arbitrarily many stages of infection all of which have general length distributions and disease mortalities. Existence and uniqueness of solutions to the model equations are established. A basic reproduction ratio is derived and related to the existence of an endemic equilibrium, to the stability of the disease-free equilibrium, and to weak and strong endemicity (persistence) of the disease. A characteristic equation is found, the zeros of which determine the local stability of the endemic equilibrium, and sufficient stability conditions are given for the case that infected individuals do not return into the susceptible class. In a subsequent paper, explicit sufficient and necessary stability conditions will be derived for the case that the disease dynamics are much faster than the demographics.

KW - Abstract Cauchy problems

KW - Arbitrary stage durations

KW - Endemic equilibrium

KW - Integrated semigroups

KW - Many infection stages

KW - Persistence

KW - Semiflows

KW - Stage (or class) age

KW - Volterra integral equations

UR - http://www.scopus.com/inward/record.url?scp=0034923771&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034923771&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034923771

VL - 61

SP - 803

EP - 833

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 3

ER -