Epidemic Simulations and Interventions CSC 570 Christopher Siu 1 - - PowerPoint PPT Presentation

Epidemic Simulations and Interventions

CSC 570 Christopher Siu

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Mathematical Models of Epidemics

1 Compartmentalize the population based on state of infection:

SI SIS SIR SIRS SEIS SEIR MSIR Carriers Cross-immunity

2 Analyze the rate of movement between compartments.

Susceptible Exposed Infected Recovered β σ γ

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Mathematical Models of Epidemics

Full mixing assumes all individuals have an equal chance of meeting any other individuals. Using a contact network captures individual contact events. Susceptible Exposed Infected Recovered

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Mobility Patterns

Contact networks must often be generated randomly, and basic models assume that contacts never change. Humans make contact consistently, but not constantly.

Example (Granell and Mucha, 2018)

Epidemic threshold βc as a function of mobility probability p, for varying confjgurations of individuals’ residences

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Temporal Networks

Defjnition

A temporal network is one whose structure changes over time. Assuming discrete time steps, a temporal network is a sequence

• f “snapshots”, samples of the network at each step.

Example (Stopczynski et al., 2014)

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Epidemics and Interventions

Graph-based models enable a more granular simulation. Graph-based models provide more actionable information.

Example

1796 Edward Jenner creates the smallpox vaccine. 1853 Parliament bans variolation, requires vaccination. 1958 USSR calls for a global efgort to eradicate smallpox. 1977 Last natural case of smallpox occurs in Somalia.

Defjnition

Ring vaccination targets the contacts of infected individuals. 2017 Scientists recreate extinct horsepox for $100,000.

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Percolation

Once an individual is immunized, they generally can neither contract nor spread the disease. Once an individual is immunized, their corresponding vertex has no efgect on the network.

Defjnition

Percolation is the process of removing some fraction of vertices and their incident edges. This terminology is sometimes clarifjed as “site percolation”. Specifjcally removing edges is called “bond percolation”.

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Percolation

Defjnition

The occupation probability, denoted φ, is the probability that a vertex is active in the network. As φ → 0, there is often some threshold φc at which the “giant cluster” tends to break apart.

Example

φ = 1 φ = 0.6 φ = 0

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Uniform Vertex Removal

Suppose vertices are removed with uniform probability 1 − φ. Let P = (p1, p2, . . . , pk, . . . , p∆) be the degree probability distribution of the network. Let u denote the average probability that a vertex is not connected to the “giant cluster” via one of its neighbors.

Theorem

The expected fraction of vertices in the “giant cluster” is: S = φ

• 1 −

k pkuk

Where: u = (1 − φ) + φ

• k

(k + 1) pk+1 k uk

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Uniform Vertex Removal

There is often no closed-form solution to these equations. These equations depend on the degree distribution.

Example

If P follows… …a Poisson distribution, then φc = 1

k.

…a geometric distribution pk = (1 − a) ak, then φc = 1 − a 2a . …a power law distribution, then φc can be very small. When dealing with real-world models, we often have no control

• ver the degree distribution or the occupation probability.

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H1N1: Targeted Class Closure

Gemmetto, Barrat, and Cattuto (2014) simulated the spread of H1N1 in primary schools. Children (232) and teachers (10) at a school in Lyon, France Contacts sampled every 20 seconds, October 1st and 2nd, 2009 Measured by RFID badges communicating within ≈1.5 meters

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H1N1: Targeted Class Closure

Infmuenza’s incubation period is 1–4 days; duration, 3–7 days. The Lyon dataset must be periodically repeated.

Collected data represents Monday, Tuesday, Thursday, Friday. Traditional full mixing in a larger community used otherwise.

Applied an SE(I/A)R model.

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H1N1: Community Movement Restrictions

Frías-Martínez, Williamson, and Frías-Martínez (2011) retrospectively analyzed the spread of swine fmu in 2009. Call Data Records provide rough times and coordinates of ≈2.4 million individuals in an afgected Mexican city A billion CDRs from January 1st to May 31st, 2009

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H1N1: Community Movement Restrictions

April 17th Medical alert — noticeable decrease in mobility April 27th Schools and universities closed May 1st All non-essential activities closed May 6th Restrictions lifted

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H1N1: Community Movement Restrictions

April 17th Medical alert — noticeable decrease in mobility April 27th Schools and universities closed May 1st All non-essential activities closed May 6th Restrictions lifted

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