Rumour Dynamics Ines Hotopp University of Osnabrück Jeanette Wheeler Memorial University of...
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![Page 1: Rumour Dynamics Ines Hotopp University of Osnabrück Jeanette Wheeler Memorial University of Newfoundland.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649d2b5503460f94a01558/html5/thumbnails/1.jpg)
Rumour Dynamics
Ines Hotopp University of Osnabrück
Jeanette WheelerMemorial University of Newfoundland
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![Page 3: Rumour Dynamics Ines Hotopp University of Osnabrück Jeanette Wheeler Memorial University of Newfoundland.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649d2b5503460f94a01558/html5/thumbnails/3.jpg)
Outline
Introduction Model formulations Numerical experiments Basic reproduction number Comparison of stochastic and
deterministic results Further areas for research
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Definition: RumourA piece of information of
questionable accuracy, from no known reliable source, usually spread by word of
mouth.
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Model
Susceptibles Infectives Recoveredα
β
λ
δ
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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.
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Continuous, deterministic system
RIIRdt
dR
IIRSIdt
dI
SIRdt
dS
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Discrete, deterministic system
)()()()()()(
)()()()()()()(
)()()()()(
ttRttItRttItRttR
ttItRttItIttStIttI
tIttSttRtSttS
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Discrete, deterministic system with scaling
N
ttR
N
ttI
N
tRttItRttR
N
ttI
N
tRttI
N
tIttStIttI
N
tIttS
N
ttRtSttS
)()()()()()(
)()()()()()()(
)()()()()(
2
22
2
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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
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S,I,R trajectories
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3D Trajectory Plot
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Fixed point analysis
Trivial fixed point (S*,I*,R*)=(N,0,0) Jacobian matrix of (S *,I*,R*)
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Eigenvalues of J(S*,I*,R*)
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Basic Reproduction Number
Definition: Rumour spread
One can say a rumour spreads if I(t)=2I0 before I(t)=0.
tN
R
10
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R0 versus doubling time
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R0 versus probability of spread
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R0 versus probability of spread
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R0 versus probability of spread
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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
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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