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Rumour Dynamics

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.

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Rumour Dynamics

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  1. Rumour Dynamics Ines Hotopp University of Osnabrück Jeanette Wheeler Memorial University of Newfoundland

  2. Outline • Introduction • Model formulations • Numerical experiments • Basic reproduction number • Comparison of stochastic and deterministic results • Further areas for research

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

  4. Model δ β α Susceptibles Infectives Recovered λ

  5. 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.

  6. Continuous, deterministic system

  7. Discrete, deterministic system

  8. Discrete, deterministic system with scaling

  9. Stochastic System

  10. S,I,R trajectories

  11. 3D Trajectory Plot

  12. Fixed point analysis • Trivial fixed point (S*,I*,R*)=(N,0,0) • Jacobian matrix of (S *,I*,R*)

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

  14. Basic Reproduction Number Definition: Rumour spread One can say a rumour spreads if I(t)=2I0 before I(t)=0.

  15. R0 versus doubling time

  16. R0 versus probability of spread

  17. R0 versus probability of spread

  18. R0 versus probability of spread

  19. Further Research β α S I R λ δ • 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

  20. Acknowledgements and References 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.

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