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 ?

tRtItItRttR

tRtItINtItIttI

Simple models

Deterministic

Natural recovered Forced recovered

Mass action interaction between Infectious and the total population

Natural recovered Forced recovered

Infectious after 1 time step

Infectious 1 time step before

Recovered 1 time step before

Recovered after 1 time step

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ppptRtIptIttI

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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 I0 on rumor life-time (both models)

Rumor life-time is inversely proportional to I0

“Strange” Results: Effect of α on rumor lifetime

αN/ < 1 αN/ > 1

Simple model Extinction of the rumor

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× ?

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tRtItItItStIttI

tRtItStSttS

Not so simple models models

DeterministicSusceptible after 1 time step

Recovered that become susceptible again

Not so simple models models

Stochastic

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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