Abstract
In this note we introduce the study of the global behaviour of the network-based SIS epidemic model recently proposed by Pastor-Satorras and Vespignani [Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86 (2001) 3200], characterized in case of homogeneous scale-free networks by a very small epidemic threshold, and extended by Olinky and Stone [Unexpected epidemic threshold in heterogeneous networks: the role of disease transmission, Phys. Rev. E 70 (2004) 03902(r)]. We show that the above model may be read as a particular case of the classical multi-group SIS model proposed by Lajmainovitch and Yorke [A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976) 221] and extended by Aronsson and Mellander [A deterministic model in biomathematics. Asymptotic behaviour and threshold conditions, Math. Biosci. 49 (1980) 207]. Thus, by applying the methods used for SIS multi-group models, we straightforwardly show, for the first time, that the local conditions identified in the physics literature also determine the global behaviour of a disease spreading on a network. Finally, we briefly study the case in which the force of infection is non-linear, by showing that multiple coexisting equilibria are possible, and by giving a global threshold condition for the extinction.
| Original language | English |
|---|---|
| Pages (from-to) | 1567-1572 |
| Number of pages | 6 |
| Journal | Nonlinear Analysis: Real World Applications |
| Volume | 9 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - Sep 2008 |
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Keywords
- Epidemics
- Floquet's theory
- Global stability
- Monotone dynamical systems
- Networks
- Scale free
ASJC Scopus subject areas
- Engineering (miscellaneous)
- Mathematics(all)
- Analysis
- Applied Mathematics
- Modelling and Simulation
Cite this
A note on the global behaviour of the network-based SIS epidemic model. / d'Onofrio, Alberto.
In: Nonlinear Analysis: Real World Applications, Vol. 9, No. 4, 09.2008, p. 1567-1572.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - A note on the global behaviour of the network-based SIS epidemic model
AU - d'Onofrio, Alberto
PY - 2008/9
Y1 - 2008/9
N2 - In this note we introduce the study of the global behaviour of the network-based SIS epidemic model recently proposed by Pastor-Satorras and Vespignani [Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86 (2001) 3200], characterized in case of homogeneous scale-free networks by a very small epidemic threshold, and extended by Olinky and Stone [Unexpected epidemic threshold in heterogeneous networks: the role of disease transmission, Phys. Rev. E 70 (2004) 03902(r)]. We show that the above model may be read as a particular case of the classical multi-group SIS model proposed by Lajmainovitch and Yorke [A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976) 221] and extended by Aronsson and Mellander [A deterministic model in biomathematics. Asymptotic behaviour and threshold conditions, Math. Biosci. 49 (1980) 207]. Thus, by applying the methods used for SIS multi-group models, we straightforwardly show, for the first time, that the local conditions identified in the physics literature also determine the global behaviour of a disease spreading on a network. Finally, we briefly study the case in which the force of infection is non-linear, by showing that multiple coexisting equilibria are possible, and by giving a global threshold condition for the extinction.
AB - In this note we introduce the study of the global behaviour of the network-based SIS epidemic model recently proposed by Pastor-Satorras and Vespignani [Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86 (2001) 3200], characterized in case of homogeneous scale-free networks by a very small epidemic threshold, and extended by Olinky and Stone [Unexpected epidemic threshold in heterogeneous networks: the role of disease transmission, Phys. Rev. E 70 (2004) 03902(r)]. We show that the above model may be read as a particular case of the classical multi-group SIS model proposed by Lajmainovitch and Yorke [A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976) 221] and extended by Aronsson and Mellander [A deterministic model in biomathematics. Asymptotic behaviour and threshold conditions, Math. Biosci. 49 (1980) 207]. Thus, by applying the methods used for SIS multi-group models, we straightforwardly show, for the first time, that the local conditions identified in the physics literature also determine the global behaviour of a disease spreading on a network. Finally, we briefly study the case in which the force of infection is non-linear, by showing that multiple coexisting equilibria are possible, and by giving a global threshold condition for the extinction.
KW - Epidemics
KW - Floquet's theory
KW - Global stability
KW - Monotone dynamical systems
KW - Networks
KW - Scale free
UR - http://www.scopus.com/inward/record.url?scp=43649094607&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=43649094607&partnerID=8YFLogxK
U2 - 10.1016/j.nonrwa.2007.04.001
DO - 10.1016/j.nonrwa.2007.04.001
M3 - Article
AN - SCOPUS:43649094607
VL - 9
SP - 1567
EP - 1572
JO - Nonlinear Analysis: Real World Applications
JF - Nonlinear Analysis: Real World Applications
SN - 1468-1218
IS - 4
ER -