### Abstract

A model for the spread of an infectious disease in a population with constant recruitment of new susceptibles, developed in previous work, is further analyzed in the case that disease survivors are permanently immune and that the disease dynamics are much faster than the demographic dynamics. Though the model allows for arbitrarily many stages of infection, all of which have general length distributions and disease survival functions, the different time scales make it possible to find explicit formulas for the interepidemic period (distance between peaks or valleys of disease incidence) and the local stability or instability of the endemic equilibrium. It turns out that the familiar formula for the length of the interepidemic period of childhood diseases has to be reinterpreted when the exponential length distribution of the infectious period is replaced by a general distribution. Using scarlet fever in England and Wales (1897-1978) as an example, we illustrate how different assumptions for the length distributions of the exposed and infectious periods (under identical average lengths) lead to quite different values for the minimum length of the quarantine period to destabilize the epidemic equilibrium.

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

Pages (from-to) | 983-1012 |

Number of pages | 30 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 61 |

Issue number | 3 |

State | Published - 2000 |

### Fingerprint

### Keywords

- Arbitrary stage length distributions
- Average duration
- Average expectation of remaining duration
- Endemic equilibrium
- Hopf bifurcation
- Interepidemic period
- Isolation (quarantine)
- Many infection stages
- Scarlet fever
- Stage (or class) age

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Endemic models with arbitrarily distributed periods of infection II : Fast disease dynamics and permanent recovery.** / Feng, Zhilan; Thieme, Horst.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 61, no. 3, pp. 983-1012.

}

TY - JOUR

T1 - Endemic models with arbitrarily distributed periods of infection II

T2 - Fast disease dynamics and permanent recovery

AU - Feng, Zhilan

AU - Thieme, Horst

PY - 2000

Y1 - 2000

N2 - A model for the spread of an infectious disease in a population with constant recruitment of new susceptibles, developed in previous work, is further analyzed in the case that disease survivors are permanently immune and that the disease dynamics are much faster than the demographic dynamics. Though the model allows for arbitrarily many stages of infection, all of which have general length distributions and disease survival functions, the different time scales make it possible to find explicit formulas for the interepidemic period (distance between peaks or valleys of disease incidence) and the local stability or instability of the endemic equilibrium. It turns out that the familiar formula for the length of the interepidemic period of childhood diseases has to be reinterpreted when the exponential length distribution of the infectious period is replaced by a general distribution. Using scarlet fever in England and Wales (1897-1978) as an example, we illustrate how different assumptions for the length distributions of the exposed and infectious periods (under identical average lengths) lead to quite different values for the minimum length of the quarantine period to destabilize the epidemic equilibrium.

AB - A model for the spread of an infectious disease in a population with constant recruitment of new susceptibles, developed in previous work, is further analyzed in the case that disease survivors are permanently immune and that the disease dynamics are much faster than the demographic dynamics. Though the model allows for arbitrarily many stages of infection, all of which have general length distributions and disease survival functions, the different time scales make it possible to find explicit formulas for the interepidemic period (distance between peaks or valleys of disease incidence) and the local stability or instability of the endemic equilibrium. It turns out that the familiar formula for the length of the interepidemic period of childhood diseases has to be reinterpreted when the exponential length distribution of the infectious period is replaced by a general distribution. Using scarlet fever in England and Wales (1897-1978) as an example, we illustrate how different assumptions for the length distributions of the exposed and infectious periods (under identical average lengths) lead to quite different values for the minimum length of the quarantine period to destabilize the epidemic equilibrium.

KW - Arbitrary stage length distributions

KW - Average duration

KW - Average expectation of remaining duration

KW - Endemic equilibrium

KW - Hopf bifurcation

KW - Interepidemic period

KW - Isolation (quarantine)

KW - Many infection stages

KW - Scarlet fever

KW - Stage (or class) age

UR - http://www.scopus.com/inward/record.url?scp=0034927101&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034927101&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034927101

VL - 61

SP - 983

EP - 1012

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 3

ER -