Ranavirus SIR Model Angela Peace Department of Mathematics and - - PowerPoint PPT Presentation

Ranavirus SIR Model

Angela Peace Department of Mathematics and Statistics Texas Tech University March 23, 2016 Global Ranavirus Consortium Course

Outline

Review basic SIR differential equations model Formulate model for Ranavirus

direct transmission environmental transmission necrophagy transmission

Parameterize model

Possible due to lots of work done by: Suzanne O’Regan, Jennifer A. Spatz, Patrick N. Reilly, Rachel D. Hill, E. Davis Carter, Rebecca P. Wilkes, Debra L. Miller, Matt Gray

Model simulations Update model to be more realistic

Basic SIR Model review β µ

S I R

γ

Basic SIR Model review β µ

S I R

γ

dS dt = −βSg(I)

• direct transmission

dI dt = βSg(I)

direct transmission

− µI

• disease induced death

− γI

• recovery

dR dt = γI

• recovery

Basic SIR Viral Model

β µ

S I R

γ ω η

V

Basic SIR Viral Model

β µ

S I R

γ ω η

V

ρf(V)

Basic SIR Viral Model

β µ

S I R

γ ω η

V

ρf(V)

dS dt = −βSg(I)

• direct trans

− ρSf (V )

environmental trans

dI dt = βSg(I)

direct trans

+ ρSf (V )

environmental trans

− µI

• death

− γI

• recovery

dR dt = γI

• recovery

dV dt = ωI

• shed virions

− ηV

• degradation

SI Viral Model without Recovery

β µ

S I

ω η

V

ρf(V)

SI Viral Model with Necrophagy

βI

S I

ρf(V) η

V

ω δ

D

µ

SI Viral Model with Necrophagy

βI

S I

ρf(V) η

V

ω δ

D

µ

SI Viral Model with Necrophagy

βI

S I

ρf(V) η

V

ω δ

D

µ αD

SI Viral Model with Necrophagy

dS dt = −βSg(I)

• direct

transmission

− ρSf (V )

environmental transmission

− αSg(D)

necrophagy transmission

dI dt = βSg(I)

direct transmission

+ ρSf (V )

environmental transmission

+ αSg(D)

necrophagy transmission

− µI

• viral induced

death

dD dt = µI

• viral induced

death

− δD

• necrophagy

dV dt = ω[I + D]

• shed virions

− ηV

• degradation

Frequency-dependent vs density-dependent transmission

frequency-dependent tranmission per-individual contact rate is independent of population density Total population: N(t) = S(t) + I(t) g(I) = I/N(t) density-dependent transmission transmission scales with population density g(I) = I

Environmental transmission

The environmental contact rate function takes the following form: f (V ) = V V + κ where κ is the ranavirus ID50

Parameterization

φ probability of infection c contact rate ρ environmental contact rate β direct transmission rate β = φc ω virion shedding rate µ diseased induced mortality κ ID50 1/δ mean dead tadpole survival time α necrophagy transmission rate α = φc 1/η environmental virion persistence time

Parameterization

φ probability of infection c contact rate ρ environmental contact rate β direct transmission rate β = φc ω virion shedding rate µ diseased induced mortality κ ID50 1/δ mean dead tadpole survival time α necrophagy transmission rate α = φc 1/η environmental virion persistence time We’ll talk about parameterizing these values today based on recent empirical data.

Contact Rate Experiment

1 infected frog in a 12-L tub with 20 susceptible frog. monitored the number of contacts between infected frog with susceptible frogs over 10 minutes monitored at 2, 4, and 6 hours

Contact Rate Experiment

1 infected frog in a 12-L tub with 20 susceptible frog. monitored the number of contacts between infected frog with susceptible frogs over 10 minutes monitored at 2, 4, and 6 hours

0" 2" 4" 6" 8" 10" 12" 14"

Wood"Frogs"" Gray"Treefrogs" Average"#"of"contacts""

Contact Rate Parameters

Parameter Description unit c contact rate 1/day ρ environmental contact rate 1/day

Contact Rate Parameters

Parameter Description unit c contact rate 1/day ρ environmental contact rate 1/day average 12 contacts in 10 minutes = ⇒ 1.2 contacts/min c = 1728 / day assume ρ = 1728 / day

Shedding Rate Parameter

Parameter Description unit ω virion shedding rate PFU/mL/day/individual

Shedding Rate Parameter

Parameter Description unit ω virion shedding rate PFU/mL/day/individual 1 infected individual in 1L fresh water took water samples at 3, 6, 12, 24, 48 and 72 hours measured viral load

Shedding Rate Parameter

Parameter Description unit ω virion shedding rate PFU/mL/day/individual 1 infected individual in 1L fresh water took water samples at 3, 6, 12, 24, 48 and 72 hours measured viral load

! ! !

0! 0.5! 1! 1.5! 2! 72! 96! 120!

PFU/mL!(10^y)! Hours!Past!3!Day!Exposure!

Shedding!Rate!in!Water!

Shedding Rate Parameter

! ! !

0! 0.5! 1! 1.5! 2! 72! 96! 120!

PFU/mL!(10^y)! Hours!Past!3!Day!Exposure!

Shedding!Rate!in!Water!

Consider slope between 72 and 96 hours = 100.8−100.2 PFU/mL

24 hours

= 5.11 Consider slope between 96 and 120 hours = 101.3−100.8 PFU/mL

24 hours

= 14.36 Average these 2 values to get ω = 9.97 PFU/mL/day/individual

Disease Induced Mortality Parameter

Experiment: 1 infected frog (exposed 96 hours ago) Contact with Susceptible frogs Monitored mortality over time

Disease Induced Mortality Parameter

Experiment: 1 infected frog (exposed 96 hours ago) Contact with Susceptible frogs Monitored mortality over time

Disease Induced Mortality Parameter

βI

S I

ρf(V) η

V

ω δ

D

µ αD

µ = disease induced mortality

Disease Induced Mortality Parameter

βI

S I

ρf(V) η

V

ω δ

D

µ αD

µ = disease induced mortality

1 µ = length of infection period

Disease Induced Mortality Parameter

βI

S I

ρf(V) η

V

ω δ

D

µ αD

µ = disease induced mortality

1 µ = length of infection period

Above model assumes this is exponentially distributed

Disease Induced Mortality Parameter

βI

S I

ρf(V) η

V

ω δ

D

µ αD

µ = disease induced mortality

1 µ = length of infection period

Above model assumes this is exponentially distributed This means µ is constant and does not depend on the time spent in the compartment

ie: A frog that has been infected for 1 day is just as likely to die as a frog that has been infected for 3 days. A unrealistic assumption of the model!

Disease Induced Mortality Parameter

Fit exponential function y = e−µt

Model Simulations

Update Model

Add in a Latent compartment

A frog exposed to the virus isn’t immediately infectious

Update Model

Add in a Latent compartment

A frog exposed to the virus isn’t immediately infectious

Consider a gamma distribution for mortality

probability of mortality increases the longer the individual resides in the infection class Can achieve this by adding in stages of infection (multiple I compartments) This works because the sum of a sequence of independent exponentially-distributed random variables is gamma- distributed

Base Model

βI

S I

ρf(V) η

V

ω δ

D

µ αD

Full Model

∑ ( β I ) nµ

S L V

ρf(V) ω η

D

δ αD

I I

ε n

1

nµ nµ

i i

Full Model: Disease Induced Mortality Parameter

µ diseased induced mortality

Full Model: Disease Induced Mortality Parameter

µ diseased induced mortality

Full Model: Disease Induced Mortality Parameter

µ diseased induced mortality

Full Model: Disease Induced Mortality Parameter

µ diseased induced mortality

Full Model: Disease Induced Mortality Parameter

µ diseased induced mortality Using 5 stages we get µ = 0.3329 /day

Full Model: Incubation Parameter

1/ǫ incubation period

¡

0 ¡ 10 ¡ 20 ¡ 30 ¡ 40 ¡ 50 ¡ 60 ¡ 70 ¡ 80 ¡ 90 ¡ 100 ¡ 0 ¡ 1 ¡ 2 ¡ 3 ¡ 4 ¡ 5 ¡ 6 ¡ 7 ¡ 8 ¡ 9 ¡ 10 ¡ 11 ¡ 12 ¡ 13 ¡ 14 ¡

Survival ¡(%) ¡

Days ¡ ¡

Wood ¡Frog ¡Survival ¡Curve ¡

24hr ¡ ¡ 48hr ¡ 72hr ¡ ¡ 96hr ¡ ¡

We assume 1

ǫ = 1 day

Base vs. Full Model

time (days)

5 10 15 20

Wood Frog % Survival

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Survival

Base model Full model

time (days)

5 10 15 20

Environmental Virus (PFU/mL)

#104 2 4 6 8 10 12 14

Environmental Viral load

Base model Full model

Full Model Simulations: Vary Contact Rate (density)

time (days)

5 10 15 20

Wood Frog % Survival

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Survival

c=10 c=100 c=1000

time (days)

5 10 15 20

Environmental Virus (PFU/mL)

#104 0.5 1 1.5 2 2.5 3

Environmental Viral loads

c=10 c=100 c=1000

Full Model Simulations: Vary Population Size

time (days)

5 10 15 20

Environmental Virus (PFU/mL)

#104 0.5 1 1.5 2 2.5 3 3.5 4

Environmental Viral loads

N0=5000 N0=10,000 N0=15,000