Chaotic Epidemic Outbreaks: Deterministic or Random? Lora Billings - - PowerPoint PPT Presentation

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Chaotic Epidemic Outbreaks: Deterministic or Random? Lora Billings - - PowerPoint PPT Presentation

Nov 26, 2022 580 likes •852 views

Chaotic Epidemic Outbreaks: Deterministic or Random? Lora Billings Montclair State University, Upper Montclair, NJ Ira Schwartz Naval Research Laboratory, Washington, DC Erik Bollt Clarkson University, Potsdam, NY Supported by ONR

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Chaotic Epidemic Outbreaks: Deterministic or Random?

Lora Billings Montclair State University, Upper Montclair, NJ Ira Schwartz Naval Research Laboratory, Washington, DC Erik Bollt Clarkson University, Potsdam, NY

Supported by ONR N00173-01-1-G911

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Measles

http://science-education.nih.gov

Total number of U.S. cases

Graph by Alun Lloyd (2002)

Vaccine developed in 1963.

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Why do so many people study measles?

The biological system is fairly simple

We can test and improve models (design vaccination strategies) These models can be used for many applications

(other diseases, computer viruses, etc.)

Excellent data is available

We can ask detailed questions about spatial and temporal dynamics The data exhibits periodic or more complex behavior

Some have conjectured that the dynamics could be chaotic

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  • D. Earn, et al. Science, 2000

Question: Is the pre-vaccine time series chaotic? Answer: Undetermined (Not enough data)

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Outline

SEIR model - a model for epidemics in childhood diseases

(Yorke and London (1973); May and Anderson (1979); Schwartz (1983); Grenfell et al. (2000); Hethcote (2000))

Add stochastic perturbations to represent noise in population size Bifurcation to stochastic chaos Possible vaccination strategies to control and prevent future

  • utbreaks
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Modeling Epidemics: Assumptions

The population:

Assume the population is large and well mixed. Variables and parameters: Normalize the population: S + E + I + R = 1 Infectives I: Recovered R: Exposed E: Susceptibles S: birth and death rate µ: µ: µ: µ: contact rate (for S & I) β: β: β: β: mean infectious period γ γ γ γ −

− − −1 1 1 1:

: : : mean latent exposed period α α α α−

− − −1 1 1 1:

: : :

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The standard SEIR model

R I dt dR I I E dt dI E E IS t dt dE S IS t dt dS µ

  • γ

µ

  • γ
  • α

µ

  • α
  • )

β( µ

  • )

β( µ = = = − = S E I R

µ µ µ µ µ µ µ µ

β β β βI α α α α

γ γ γ γ µ µ µ µ µ µ µ µ µ µ µ µ

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Our flavor

The contact rate: The infectives are roughly proportional to the exposed

[Schwartz, J. Math. Biol. 1985]

) π 2 cos 1 ( ) 1 ( ) ( t δ β t β t β + = + = ) ( γ µ α ) ( t E t I         + ≈

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The model we study

)) δcos(2π (1 β ) β( α) (µ

  • )

β( γ µ α µ

  • )

β( µ t t I IS t dt dI S IS t dt dS + = +         + = − = The modified SI model (MSI)

The paremeter we vary is δ

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Bifurcation diagram

The system is driven periodically, so consider the discrete map on the Poincare section.

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Adding noise

The system is driven periodically, so add noise as if it is a map. (Additive noise) Noise: normal distribution, mean=0, vary the standard deviation (σ)

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Noisy dynamics

Time series Phase space diagram Probability density function

log(S) log(S) log(S) log(S) time time log(S) log(I) log(I) log(I) log(I) log(I)

σ σ σ σ=0.01 σ σ σ σ=0.05

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Stochastic Chaos?

Deterministic definition (numerical)

Compact set Positive Lyapunov exponent Not asymptotically periodic

Stochastic version?

Compact set Positive Lyapunov exponent Homoclinic/heteroclinic topology

(makes chaotic orbits possible)

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Lyapunov exponents

Red: σ = 0.01 (noisy) Blue: σ = 0.05 (chaotic)

But Lyapunov exponents can yield false results

log(S) log(I)

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Unstable manifolds

Random trajectories follow the unstable manifolds of the period three saddle What is the role of the manifolds?

log(S) log(I)

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Smale Horseshoe Topology

Homoclinic Orbit Heteroclinic orbit

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Stochastic Chaotic Saddle

log(S) log(I)

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New tool to detect transport

Use a Galerkin approximation of the Stochastic Frobenius- Perron Operator to detect the flux across a basin boundaries and predict the most probable regions of transport created by noise.

A B

Phase space

Leakage from B to A

A

Leakage from A to B

Galerkin matrix

B

Ordered subsets Ordered subsets

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Area Flux

log(S) log(I)

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PDF Flux

log(S) log(I)

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Probability Density Function

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How do we use this information?

Predict the occurrence of chaos Control the dynamics/prevent outbreaks

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Predicting chaos

How much noise is needed to induce chaos?

  • = value of σ so that the largest

Lyapunov exponent is positive (20 trials)

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Controlling the dynamics

If we can identify points in the bull’s eye, then we can predict future outbreaks

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Controlling the dynamics

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Conclusions

Stochastic perturbations can induce new, emergent dynamics in models Chaotic behavior can be induced in models by additive noise The topology reveals the mechanism that facilitates these dynamics We can use the topology to our advantage and control the system