### Abstract

Two SIR models for the spread of infectious diseases which were originally suggested by Greenhalgh and Das (1995, Theor, Popul. Biol. 47, 129-179; 1995, Mathematical Population Dynamics: Analysis of Hetrerogeneity, pp. 79-101, Winnipeg: Wuerz Publishing) are considered but with a time delay in the vaccination tenn. This reflects the fact that real vaccines do not immediately confer permanent immunity. The population is divided into susceptible, infectious, and immune classes. The contact rate is constant in model I but it depends on the population size in model n. The death rate depends on the population size in both models. There is an additional mortality due to the disease, and susceptibles are vaccinated and may become permanently immune after a lapse of some time. Using the time delay as a bifurcation parameter, necessary and sufficient conditions for Hopf bifurcation to occur are derived. Numerical results indicate that that for diseases in human populations Hopf bifurcation is unlikely to occur at realistic parameter values if the death rate is a concave function of the population size.

Original language | English |
---|---|

Pages (from-to) | 113-142 |

Number of pages | 30 |

Journal | IMA Journal of Mathematics Applied in Medicine and Biology |

Volume | 16 |

Issue number | 2 |

Publication status | Published - 1999 |

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### Keywords

- Density dependence
- Differential equations
- Epidemic model
- Hopf bifurcation
- Immunization
- Time delay

### ASJC Scopus subject areas

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

### Cite this

*IMA Journal of Mathematics Applied in Medicine and Biology*,

*16*(2), 113-142.

**Hopf bifurcation in epidemic models with a time delay in vaccination.** / Khan, Q. J A; Greenhalgh, David.

Research output: Contribution to journal › Article

*IMA Journal of Mathematics Applied in Medicine and Biology*, vol. 16, no. 2, pp. 113-142.

}

TY - JOUR

T1 - Hopf bifurcation in epidemic models with a time delay in vaccination

AU - Khan, Q. J A

AU - Greenhalgh, David

PY - 1999

Y1 - 1999

N2 - Two SIR models for the spread of infectious diseases which were originally suggested by Greenhalgh and Das (1995, Theor, Popul. Biol. 47, 129-179; 1995, Mathematical Population Dynamics: Analysis of Hetrerogeneity, pp. 79-101, Winnipeg: Wuerz Publishing) are considered but with a time delay in the vaccination tenn. This reflects the fact that real vaccines do not immediately confer permanent immunity. The population is divided into susceptible, infectious, and immune classes. The contact rate is constant in model I but it depends on the population size in model n. The death rate depends on the population size in both models. There is an additional mortality due to the disease, and susceptibles are vaccinated and may become permanently immune after a lapse of some time. Using the time delay as a bifurcation parameter, necessary and sufficient conditions for Hopf bifurcation to occur are derived. Numerical results indicate that that for diseases in human populations Hopf bifurcation is unlikely to occur at realistic parameter values if the death rate is a concave function of the population size.

AB - Two SIR models for the spread of infectious diseases which were originally suggested by Greenhalgh and Das (1995, Theor, Popul. Biol. 47, 129-179; 1995, Mathematical Population Dynamics: Analysis of Hetrerogeneity, pp. 79-101, Winnipeg: Wuerz Publishing) are considered but with a time delay in the vaccination tenn. This reflects the fact that real vaccines do not immediately confer permanent immunity. The population is divided into susceptible, infectious, and immune classes. The contact rate is constant in model I but it depends on the population size in model n. The death rate depends on the population size in both models. There is an additional mortality due to the disease, and susceptibles are vaccinated and may become permanently immune after a lapse of some time. Using the time delay as a bifurcation parameter, necessary and sufficient conditions for Hopf bifurcation to occur are derived. Numerical results indicate that that for diseases in human populations Hopf bifurcation is unlikely to occur at realistic parameter values if the death rate is a concave function of the population size.

KW - Density dependence

KW - Differential equations

KW - Epidemic model

KW - Hopf bifurcation

KW - Immunization

KW - Time delay

UR - http://www.scopus.com/inward/record.url?scp=0033039236&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033039236&partnerID=8YFLogxK

M3 - Article

VL - 16

SP - 113

EP - 142

JO - Mathematical Medicine and Biology

JF - Mathematical Medicine and Biology

SN - 1477-8599

IS - 2

ER -