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

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 Species Contact Processes June 19, 2017 2 / 25

Bacteriophage Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 3 / 25

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Let’s use a Branching Process... Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 14 / 25

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How many connections? p con ∼ Pois ( c ) Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 15 / 25

How many connections? p con ∼ Pois ( c ) Probability gaining the infection from an infected parent � ∞ β 1 β 1 e − β 1 t e − ρ 1 t dt = p inf = β 1 + ρ 1 0 Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 15 / 25

How many connections? p con ∼ Pois ( c ) Probability gaining the infection from an infected parent � ∞ β 1 β 1 e − β 1 t e − ρ 1 t dt = p inf = β 1 + ρ 1 0 How many others will an individual infect? β 1 µ ∼ Pois ( c ) β 1 + ρ 1 Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 15 / 25

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µ ( t ′ | t ) = Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 18 / 25

� t µ ( t ′ | t ) = 0 ∨ ( t − t ′ ) Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 18 / 25

� t [ β 1 e − β 1 s ] µ ( t ′ | t ) = 0 ∨ ( t − t ′ ) Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 18 / 25

� t [ β 1 e − β 1 s ][ β 2 e − β 2 ( t ′ − t + s ) ] µ ( t ′ | t ) = 0 ∨ ( t − t ′ ) Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 18 / 25

� t [ β 1 e − β 1 s ][ β 2 e − β 2 ( t ′ − t + s ) ][ e − ρ 1 ( t − s ) ] µ ( t ′ | t ) = 0 ∨ ( t − t ′ ) Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 18 / 25

� t [ β 1 e − β 1 s ][ β 2 e − β 2 ( t ′ − t + s ) ][ e − ρ 1 ( t − s ) ][ e − (2 ρ 1 + ρ 2 )( t ′ − t + s ) ] ds µ ( t ′ | t ) = 0 ∨ ( t − t ′ ) Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 18 / 25

� t [ β 1 e − β 1 s ][ β 2 e − β 2 ( t ′ − t + s ) ][ e − ρ 1 ( t − s ) ][ e − (2 ρ 1 + ρ 2 )( t ′ − t + s ) ] ds µ ( t ′ | t ) = 0 ∨ ( t − t ′ ) � ∞ [ β 1 e − β 1 s ][ β 2 e − β 2 t ′ ][ e − ( ρ 1 + ρ 2 )( s − t ) ][ e − (2 ρ 1 + ρ 2 ) t ′ ] ds µ ( t ′ | t ) = t Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 18 / 25

Endemic? Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 19 / 25

Endemic? Distribution of types: ψ ( t ) Mean number of offspring: m Population: p Number of type t mp ψ ( t ) = Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 19 / 25

Endemic? Distribution of types: ψ ( t ) Mean number of offspring: m Population: p Number of type t � ∞ µ ( t | t ′ ) cp ψ ( t ′ ) dt ′ mp ψ ( t ) = 0 Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 19 / 25

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Questions? Sam Moore (University of Bath) Two Species Contact Processes June 19, 2017 25 / 25