Abstract
A general model for a disease without immunity against reinfection having arbitrary distributions of exposed and infective periods was formulated by Hethcote, Stech and van den Driessche [5]. They showed that for contact numbers exceeding 1, the endemic equilibrium is asymptotically stable if either the exposed period of the infective period is exponentially distributed or if both exposed and infective period have fixed length, and they conjectured that the endemic equilibrium is always asymptotically stable. We show that the endemic equilibrium is asymptotically stable if the mean exposed period is less than the mean infective period, or if the contact number is sufficiently large, or if the exposed period distribution function is convex. However, we also show that for a more general type of model in which the infective period distribution can depend on the length of the exposed period it is possible to have instability of the endemic equilibrium and a Hopf bifurcation.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 57-66 |
| Number of pages | 10 |
| Journal | Rocky Mountain Journal of Mathematics |
| Volume | 25 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 1 1995 |
| Externally published | Yes |
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Keywords
- Exposed period
- Hpidemic models
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Models for diseases with exposed periods. / Brauer, Fred.
In: Rocky Mountain Journal of Mathematics, Vol. 25, No. 1, 01.01.1995, p. 57-66.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Models for diseases with exposed periods
AU - Brauer, Fred
PY - 1995/1/1
Y1 - 1995/1/1
N2 - A general model for a disease without immunity against reinfection having arbitrary distributions of exposed and infective periods was formulated by Hethcote, Stech and van den Driessche [5]. They showed that for contact numbers exceeding 1, the endemic equilibrium is asymptotically stable if either the exposed period of the infective period is exponentially distributed or if both exposed and infective period have fixed length, and they conjectured that the endemic equilibrium is always asymptotically stable. We show that the endemic equilibrium is asymptotically stable if the mean exposed period is less than the mean infective period, or if the contact number is sufficiently large, or if the exposed period distribution function is convex. However, we also show that for a more general type of model in which the infective period distribution can depend on the length of the exposed period it is possible to have instability of the endemic equilibrium and a Hopf bifurcation.
AB - A general model for a disease without immunity against reinfection having arbitrary distributions of exposed and infective periods was formulated by Hethcote, Stech and van den Driessche [5]. They showed that for contact numbers exceeding 1, the endemic equilibrium is asymptotically stable if either the exposed period of the infective period is exponentially distributed or if both exposed and infective period have fixed length, and they conjectured that the endemic equilibrium is always asymptotically stable. We show that the endemic equilibrium is asymptotically stable if the mean exposed period is less than the mean infective period, or if the contact number is sufficiently large, or if the exposed period distribution function is convex. However, we also show that for a more general type of model in which the infective period distribution can depend on the length of the exposed period it is possible to have instability of the endemic equilibrium and a Hopf bifurcation.
KW - Exposed period
KW - Hpidemic models
UR - http://www.scopus.com/inward/record.url?scp=84881296575&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84881296575&partnerID=8YFLogxK
U2 - 10.1216/rmjm/1181072268
DO - 10.1216/rmjm/1181072268
M3 - Article
AN - SCOPUS:84881296575
VL - 25
SP - 57
EP - 66
JO - Rocky Mountain Journal of Mathematics
JF - Rocky Mountain Journal of Mathematics
SN - 0035-7596
IS - 1
ER -