Rumour Dynamics

Ines Hotopp University of Osnabrück

Jeanette WheelerMemorial University of Newfoundland

Outline

Introduction Model formulations Numerical experiments Basic reproduction number Comparison of stochastic and

deterministic results Further areas for research

Definition: RumourA piece of information of

questionable accuracy, from no known reliable source, usually spread by word of

mouth.

Model

Susceptibles Infectives Recoveredα

β

λ

δ

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.

Continuous, deterministic system

RIIRdt

dR

IIRSIdt

dI

SIRdt

dS

Discrete, deterministic system

)()()()()()(

)()()()()()()(

)()()()()(

ttRttItRttItRttR

ttItRttItIttStIttI

tIttSttRtSttS

Discrete, deterministic system with scaling

N

ttR

N

ttI

N

tRttItRttR

N

ttI

N

tRttI

N

tIttStIttI

N

tIttS

N

ttRtSttS

)()()()()()(

)()()()()()()(

)()()()()(

2

22

2

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

S,I,R trajectories

3D Trajectory Plot

Fixed point analysis

Trivial fixed point (S*,I*,R*)=(N,0,0) Jacobian matrix of (S *,I*,R*)

Eigenvalues of J(S*,I*,R*)

Basic Reproduction Number

Definition: Rumour spread

One can say a rumour spreads if I(t)=2I0 before I(t)=0.

tN

R

10

R0 versus doubling time

R0 versus probability of spread

R0 versus probability of spread

R0 versus probability of spread

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

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