Rumour Dynamics Ines Hotopp University of Osnabrück 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

Definition: Rumour A piece of information of questionable accuracy, from no known reliable source, usually spread by word of mouth.

Model SusceptiblesInfectivesRecovered α 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

Discrete, deterministic system

Discrete, deterministic system with scaling

Stochastic System

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)=2I 0 before I(t)=0.

R 0 versus doubling time

R 0 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 R 0 =1 the probability of success bigger for a smaller I 0 ? Different parameter sets Collecting experimental data for parameter estimation SIR α β 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, “A Course in Mathematical Biology”, G. de Vries, T. Hillen, M. Lewis, J. Müller, B. Schönfisch. SIAM, Philadelphia, Acknowledgements and References