RUMORS: How can they work ? Summer School Math Biology, 2007 Nuno & Sebastian

Outline What is a rumor ? Deterministic vs Stochastic Simple models Not so simple models Summary Discrete & Galton-Watson Process

“unverified proposition of belief that bears topical relevance for persons actively involved in its dissemination” “unauthenticated bits of information in that they are deprived of “secure standards of evidence”.” What is a rumor ? ?

Deterministic vs Stochastic How can we model a rumor ? Is a deterministic or stochastic approach better ? How do these approaches differ ?

Deterministic vs Stochastic Discrete Galton-Watson Process Markov Chain Esteban likes it !!!! WHY ?

Simple models Deterministic Natural recoveredForced recovered Mass action interaction between Infectious and the total population Natural recoveredForced recovered Infectious after 1 time step Infectious 1 time step before Recovered 1 time step before Recovered after 1 time step

Simple models Probability of infected someone Probability of doing nothing Number of infected after forced recovery Natural recoveredForced recovered Infectious after 1 time step Recovered 1 time step before Recovered after 1 time step Stochastic Probability of forgetting the rumor

Simple model assumptions 1. Total population size is extremely large; 2. The number of susceptibles remains roughly constant; 3. The size of the epidemic remains quite small; 4. Mass action interaction (homogeneous population); 5. In the stochastic model, forced recovery precedes other events;

SIMPLE MODEL - Deterministic Results Infected 0 2 “types” fixed points (I*,R*) = (0,0) and (I*,R*)=(0,R) eigenvalue of 1 ? “epidemic” if (αN/  ) > 1.

SIMPLE MODEL - Stochastic Results Stochastic model extinction of the rumor!!!

Effect of I 0 on rumor life-time (both models) Rumor life-time is inversely proportional to I 0

“Strange” Results: Effect of α on rumor lifetime αN/  < 1αN/  > 1

Simple model Extinction of the rumor √ × √ × ?

Not so simple models models Deterministic Susceptible after 1 time step Recovered that become susceptible again

Not so simple models models Stochastic Probability that recovered that become susceptible again

Not so simple model assumptions 1. Total population size is constant; 2.Mass action interaction (homogeneous population); 3.In the stochastic model, forced recovery precedes other events;

NOT SO SIMPLE MODEL - Deterministic results ELVIS IS ALIVE!?!?! Model with model extinction and “endemic” rumors None of the fixed points are stable...

NOT SO SIMPLE MODEL - Stochastic results

Effect of population size – stochastic model

Effect of population size – deterministic model For the deterministic case population size only changes the scale of the epidemic In the stochastic model however, increasing the population size generates very different behaviour

Not so simple model – comparison of deterministic & stochastic results For large  (~ p4) coexistence is observed in both deterministic and stochastic For small  deterministic predicts repeated outbreaks of the rumor. This is not possible in the stochastic model (by varying p4) For the deterministic model the population size does not make any difference, but population size affects the predictions of the stochastic model

Summary 1.Rumors can be modelled similarly to infectious diseases; 2. Not so different models can give us very different predictions; 3. Under certain conditions, stochastic models predict very different results from deterministic ones

Acknowledgments Julien Jungmin Thank you very much !! Group