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
Hopf bifurcation in epidemic models with a time delay in vaccination. / Khan, Q. J A; Greenhalgh, David.
In: IMA Journal of Mathematics Applied in Medicine and Biology, Vol. 16, No. 2, 1999, p. 113-142.Research output: Contribution to journal › Article
}
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
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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 -