Two Species Contact Processes Sam Moore 1 University of Bath June - - PowerPoint PPT Presentation

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Two Species Contact Processes Sam Moore 1 University of Bath June - - PowerPoint PPT Presentation

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Two Species Contact Processes Sam Moore 1 University of Bath June 19, 2017 1 with Tim Rogers, Peter M orters Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 1 / 25 Epidemics Sam Moore (University of Bath) Two

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Two Species Contact Processes

Sam Moore 1

University of Bath

June 19, 2017

1with Tim Rogers, Peter M¨

  • rters

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Epidemics

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Bacteriophage

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Let’s use a Branching Process...

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How many connections?

pcon ∼ Pois(c)

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How many connections?

pcon ∼ Pois(c)

Probability gaining the infection from an infected parent

pinf = ∞ β1e−β1te−ρ1tdt = β1 β1 + ρ1

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How many connections?

pcon ∼ Pois(c)

Probability gaining the infection from an infected parent

pinf = ∞ β1e−β1te−ρ1tdt = β1 β1 + ρ1

How many others will an individual infect?

µ ∼ Pois( β1 β1 + ρ1 c)

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µ(t′|t) =

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µ(t′|t) = t

0∨(t−t′)

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µ(t′|t) = t

0∨(t−t′)

[β1e−β1s]

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µ(t′|t) = t

0∨(t−t′)

[β1e−β1s][β2e−β2(t′−t+s)]

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µ(t′|t) = t

0∨(t−t′)

[β1e−β1s][β2e−β2(t′−t+s)][e−ρ1(t−s)]

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µ(t′|t) = t

0∨(t−t′)

[β1e−β1s][β2e−β2(t′−t+s)][e−ρ1(t−s)][e−(2ρ1+ρ2)(t′−t+s)]ds

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µ(t′|t) = t

0∨(t−t′)

[β1e−β1s][β2e−β2(t′−t+s)][e−ρ1(t−s)][e−(2ρ1+ρ2)(t′−t+s)]ds µ(t′|t) = ∞

t

[β1e−β1s][β2e−β2t′][e−(ρ1+ρ2)(s−t)][e−(2ρ1+ρ2)t′]ds

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Endemic?

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Endemic?

Distribution of types: ψ(t) Mean number of offspring: m Population: p

Number of type t

mpψ(t) =

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Endemic?

Distribution of types: ψ(t) Mean number of offspring: m Population: p

Number of type t

mpψ(t) = ∞ µ(t|t′)cpψ(t′) dt′

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Questions?

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