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 γ β S I R µ
Basic SIR Model review γ β S I R µ dS dt = − β Sg ( I ) � �� � direct transmission dI dt = β Sg ( I ) µ I γ I − − ���� ���� � �� � recovery disease induced death direct transmission dR dt = γ I ���� recovery
Basic SIR Viral Model η V ω γ β S I R µ
Basic SIR Viral Model η V ρ f(V) ω γ β S I R µ
η V Basic SIR Viral Model ρ f(V) ω γ β S I R dS µ dt = − β Sg ( I ) ρ Sf ( V ) − � �� � � �� � direct trans environmental trans dI dt = β Sg ( I ) + ρ Sf ( V ) µ I γ I − − ���� ���� � �� � � �� � recovery death direct trans environmental trans dR dt = γ I ���� recovery dV dt = ω I η V − ���� ���� shed virions degradation
SI Viral Model without Recovery η V ρ f(V) ω β S I µ
SI Viral Model with Necrophagy η V ρ f(V) ω µ β I S I D δ
SI Viral Model with Necrophagy η V ρ f(V) ω µ β I S I D δ
SI Viral Model with Necrophagy η V ρ f(V) ω µ β I S I D α D δ
SI Viral Model with Necrophagy dS dt = − β Sg ( I ) ρ Sf ( V ) α Sg ( D ) − − � �� � � �� � � �� � direct environmental necrophagy transmission transmission transmission dI dt = β Sg ( I ) + ρ Sf ( V ) + α Sg ( D ) µ I − ���� � �� � � �� � � �� � viral induced direct environmental necrophagy transmission transmission transmission death dD dt = µ I δ D − ���� ���� necrophagy viral induced death dV dt = ω [ I + D ] η V − ���� � �� � degradation shed virions
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: V f ( 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 = φ c ω virion shedding rate µ diseased induced mortality κ ID50 1 /δ mean dead tadpole survival time α necrophagy transmission rate α = φ c 1 /η environmental virion persistence time
Parameterization φ probability of infection c contact rate ρ environmental contact rate β direct transmission rate β = φ c ω virion shedding rate µ diseased induced mortality κ ID50 1 /δ mean dead tadpole survival time α necrophagy transmission rate α = φ c 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 14" Average"#"of"contacts"" 12" 10" 8" 6" 4" 2" 0" Wood"Frogs"" Gray"Treefrogs"
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 Shedding!Rate!in!Water! 2! PFU/mL!(10^y)! 1.5! 1! 0.5! 0! 72! 96! 120! Hours!Past!3!Day!Exposure! ! ! !
Shedding Rate Parameter Shedding!Rate!in!Water! 2! PFU/mL!(10^y)! 1.5! 1! 0.5! 0! 72! 96! 120! Hours!Past!3!Day!Exposure! ! Consider slope between 72 and 96 hours = 10 0 . 8 − 10 0 . 2 PFU/mL ! ! = 5 . 11 24 hours Consider slope between 96 and 120 hours = 10 1 . 3 − 10 0 . 8 PFU/mL = 14 . 36 24 hours 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 η V ρ f(V) ω µ β I S D I α D δ µ = disease induced mortality
Disease Induced Mortality Parameter η V ρ f(V) ω µ β I S D I α D δ µ = disease induced mortality 1 µ = length of infection period
Disease Induced Mortality Parameter η V ρ f(V) ω µ β I S D I α D δ µ = disease induced mortality 1 µ = length of infection period Above model assumes this is exponentially distributed
Disease Induced Mortality Parameter η V ρ f(V) ω µ β I S D I α D δ 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 − µ t
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 η V ρ f(V) ω µ β I S D I α D δ
Full Model η V ρ f(V) ω ε ∑ nµ ( β I ) nµ nµ i i I I S D L n 1 α D δ
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 Wood ¡Frog ¡Survival ¡Curve ¡ 100 ¡ 90 ¡ 80 ¡ Survival ¡(%) ¡ 70 ¡ 60 ¡ 24hr ¡ ¡ 50 ¡ 48hr ¡ 40 ¡ 72hr ¡ ¡ 30 ¡ 96hr ¡ ¡ 20 ¡ 10 ¡ 0 ¡ 0 ¡ 1 ¡ 2 ¡ 3 ¡ 4 ¡ 5 ¡ 6 ¡ 7 ¡ 8 ¡ 9 ¡ 10 ¡ 11 ¡ 12 ¡ 13 ¡ 14 ¡ Days ¡ ¡ ¡ We assume 1 ǫ = 1 day
Base vs. Full Model Survival 1 # 10 4 Environmental Viral load Base model 14 Full model 0.9 Environmental Virus (PFU/mL) Base model Wood Frog % Survival Full model 0.8 12 0.7 10 0.6 0.5 8 0.4 6 0.3 0.2 4 0.1 2 0 0 5 10 15 20 time (days) 0 0 5 10 15 20 time (days)
Full Model Simulations: Vary Contact Rate (density) Survival # 10 4 Environmental Viral loads 1 3 Environmental Virus (PFU/mL) c=10 c=10 0.9 c=100 c=100 c=1000 c=1000 2.5 Wood Frog % Survival 0.8 0.7 2 0.6 0.5 1.5 0.4 1 0.3 0.2 0.5 0.1 0 0 0 5 10 15 20 0 5 10 15 20 time (days) time (days)
Full Model Simulations: Vary Population Size # 10 4 Environmental Viral loads 4 Environmental Virus (PFU/mL) N0=5000 N0=10,000 3.5 N0=15,000 3 2.5 2 1.5 1 0.5 0 0 5 10 15 20 time (days)
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