rumour dynamics
DESCRIPTION
Rumour Dynamics. Ines Hotopp University of Osnabr ü ck Jeanette Wheeler Memorial University of Newfoundland. Outline. Introduction Model formulations Numerical experiments Basic reproduction number Comparison of stochastic and deterministic results Further areas for research. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/1.jpg)
Rumour Dynamics
Ines Hotopp University of Osnabrück
Jeanette WheelerMemorial University of Newfoundland
![Page 2: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/2.jpg)
![Page 3: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/3.jpg)
Outline
Introduction Model formulations Numerical experiments Basic reproduction number Comparison of stochastic and
deterministic results Further areas for research
![Page 4: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/4.jpg)
Definition: RumourA piece of information of
questionable accuracy, from no known reliable source, usually spread by word of
mouth.
![Page 5: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/5.jpg)
Model
Susceptibles Infectives Recoveredα
β
λ
δ
![Page 6: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/6.jpg)
Model Assumptions
Assume constant, homogeneous population, so that
N=S+I+R. Assume constant rates of transmission
(α), recovery (β, λ), and relapse to susceptibility (δ).
Assume movements from I to R by βRI and by λI are independent.
![Page 7: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/7.jpg)
Continuous, deterministic system
RIIRdt
dR
IIRSIdt
dI
SIRdt
dS
![Page 8: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/8.jpg)
Discrete, deterministic system
)()()()()()(
)()()()()()()(
)()()()()(
ttRttItRttItRttR
ttItRttItIttStIttI
tIttSttRtSttS
![Page 9: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/9.jpg)
Discrete, deterministic system with scaling
N
ttR
N
ttI
N
tRttItRttR
N
ttI
N
tRttI
N
tIttStIttI
N
tIttS
N
ttRtSttS
)()()()()()(
)()()()()()()(
)()()()()(
2
22
2
![Page 10: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/10.jpg)
Stochastic System
])()(1[)(
)1()(
))1()1)(1(()(
)1())1(()()(
])(,)(|1)(,)([
)(])(,)(|)(,1)([
)(])(,)(|1)(,1)([
,
1,
1,1
,1,
trtiirtiriNtp
trtp
tiritp
tiriNtpttp
trrtRitIrttRittIP
tiriNrtRitIrttRittIP
tiirrtRitIrttRittIP
ri
ri
ri
riri
![Page 11: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/11.jpg)
S,I,R trajectories
![Page 12: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/12.jpg)
3D Trajectory Plot
![Page 13: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/13.jpg)
Fixed point analysis
Trivial fixed point (S*,I*,R*)=(N,0,0) Jacobian matrix of (S *,I*,R*)
![Page 14: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/14.jpg)
Eigenvalues of J(S*,I*,R*)
![Page 15: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/15.jpg)
Basic Reproduction Number
Definition: Rumour spread
One can say a rumour spreads if I(t)=2I0 before I(t)=0.
tN
R
10
![Page 16: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/16.jpg)
R0 versus doubling time
![Page 17: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/17.jpg)
R0 versus probability of spread
![Page 18: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/18.jpg)
R0 versus probability of spread
![Page 19: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/19.jpg)
R0 versus probability of spread
![Page 20: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/20.jpg)
Further Research
Different model (Why is there a relapse from recovered to susceptible? Does this make sense?)
Variable population size Why is for R0=1 the probability of success bigger for a
smaller I0? Different parameter sets Collecting experimental data for parameter estimation
S I Rαβ
λ
δ
![Page 21: Rumour Dynamics](https://reader035.vdocuments.site/reader035/viewer/2022062222/56814f97550346895dbd5402/html5/thumbnails/21.jpg)
We would like to thank the following people: Jim Keener and William Nelson for assistance with model
formulation and technical help. Mark Lewis, Thomas Hillen, Gerda de Vries, Julien Arino for
their time and interest.We would like to reference the following works: “Comparison of deterministic and stochastic SIS and SIR
models in discrete time”, Linda J.S. Allen, Amy M. Burgin. In Mathematical Biosciences, no. 163, pp.1-33, 2000.
“A Course in Mathematical Biology”, G. de Vries, T. Hillen, M. Lewis, J. Müller, B. Schönfisch. SIAM, Philadelphia, 2006.
Acknowledgements and References