Foot and Mouth Analysis

Mike Delorme Rachelle Miron

Outline

Foot and Mouth Information Discussion of Models

analysis Summary Questions

Disease History

Lots of information Disease that infects bovids (cows, pigs,

goats, sheep...) Humans are very rarely affected It can be spread through the air or through

contact There are vaccinations available

Canada does not allow vaccination

http://www.csiro.au/images/mediaReleases/FMDtrottersmall.jpg

http://www.mass.gov/agr/animalhealth/diseases/foot_mouth/pages/fmd_01.htm

2001 UK Outbreak The 2001 Pre Budget Report estimated the direct

economic impact of FMD at £2 billion - representing 0.2% of GDP. Tourism and related industries alone lost between £4.5 billion and £5.4 billion

The 2001 outbreak took 221 days to eradicate, 1 day less than the outbreak of 1967-1968

By the end of the outbreak on September 30 2001, 6,094,139 animals had been killed - around two million for welfare reasons, and around four million for disease control purposes

www.politics.co.uk

Statistic 1: (Source: Treasury, Pre Budget Report 2001 (CM53182)); Statistics 2 and 3: (Source: DEFRA, 2004)

Spread

Altered from http://news.bbc.co.uk/olmedia/1190000/images/_1193260_foot_mouth_25_03.gif

http://www.daffa.gov.au/animal-plant-health/animal/modelling/fmd

Building a Simple Model

Worked with a SIR model Continuous time

Susceptible

Born susceptible Can become infected Can die naturally

Infected

Enter from S Can die naturally Can die from disease Can recover

Recovered

Can die naturally Can lose their immunity

Become susceptible again

Model 1 – The Basics

Model 1 - Analysis

Set Infected=0 to get disease free equilibrium DFE

Set up Jacobian and evaluate at DFE Arino Method

Examines infected cows in the system Gives R0 value for model

Find I* Plot S,I,R vs. time

Model 1 - Equations

RISSdt

dS

ImIISIdt

dI

RRRdt

dR

Model 1

)0,0,(),,(

RIS

mR0

Model 1 - Analysis

Values for parameters

011.0

17.0

0064.0

1

89.0

00089.0

m

Model 1 – R0>1

Model 1 – R0>1

Model 1 - R0<1

Model 1 – R0 <1

Model 2 – More Infections

Model 2 – More Infections

Model 2 - Equations

RISrrDSSdt

dSi

SIrDDrSDdt

dDi

ImIIDdt

dI

RRIdt

dR

Model 2 - Analysis

)0,0,0,(),,,(

RIDS

m

rmrR i

)(

)(0

Model 2 – Parameter Values

0005.0

0005.0

011.0

17.0

0064.0

1

89.0

00089.0

ir

r

m

Model 2 – R0 >1

Model 2 – R0 <1

Model 3 - Vaccine

Can vaccinate cows so they never get the disease

Low numbers of susceptible cows at equilibrium can represent cows leaving the system due to vaccination

Model 3

Model 3 - Analysis

What will happen when different proportions of cattle are vaccinated?

Model 3 - Equations

RISrrDSSpdt

dSi )1(

SIrDDrSDdt

dDi

ImIIDdt

dI

pRRRIdt

dR

Model 3 - Equations

m

rmrpR i

)(

)()1(0

)0,0,0,)1(

(),,,(

pRIDS

Model 3 – Parameter values

10

0005.0

0005.0

011.0

17.0

0064.0

1

89.0

00089.0

p

r

r

m

i

Model 3 – R0 >1

Model 3 – R0 >1 - Zoomed

Model 3 – R0 < 1

Model 3 - Analysis

Very few Susceptible cows left Where have they gone?

Model 4 – Vaccine++

How many cows are vaccinated? Set up a “Perfect” class that keeps track of

immunized cows How many cows need to be vaccinated to

have no infected cows at equilibrium? Herd Immunity

Perfect Cows (P)

Created by vaccinating newborn cows vaccinating cows after

they recover from the disease Leave due to natural death

www.uoguelph.ca

Model 4

Model 4 - Equations

RISrrDSSpdt

dSi )1(

SIrDDrSDdt

dDi

ImIIDdt

dI

pRRRIdt

dR

PppRdt

dP

Model 4 - Equations

),0,0,0,)1(

(),,,,(

pp

PRIDS

m

rmrpR i

)(

)()1(0

Note: Same R0 as in Model 3

Same parameter values as in Model 3

Model 4 – R0 >1

Model 4 – R0 >1 Zoomed

Model 4 – R0 < 1

Model 5 – Keep track of dead

Improvement on Model 4 Now cows that die as a result of the disease are

tracked Z for Zombie

Model 5

Model 5 - Equations

RISrrDSSpdt

dSi )1(

SIrDDrSDdt

dDi

ImIIDdt

dI

pRRRIdt

dR

PppRdt

dP

mIdt

dZ

Model 5 - Equations

)0,,0,0,0,)1(

(),,,,,(

pp

ZPRIDS

m

rmrpR i

)(

)()1(0

Note: Same DFE and R0 as in Model 4

Same parameter values as in Model 3

Model 5 – R0 >1

Model 5 – R0 <1

Herd Immunity

What % of the population do you need to vaccinate in order to effectively stop the disease? Depends on vaccine and disease

Polio, small pox...

Assume there is 1 type of virus Assume the vaccine works 100% of the time

and as soon as administered

20

30 )1( ModelModel RpR

130 ModelR

Herd Immunity

We found this when going from Model 2 to Model 3 When we introduced vaccination

20

11

ModelRp

Herd Immunity

Plot I* vs. p see where the line crosses I*=0 For disease to die out

I* vs. p

What next?

Different ages Different virus strains Different vaccination strategies

Ring vaccination Economic strategy

Cost of killing a cow vs. Cost of vaccination

Does it make cents to vaccinate?

Cow costs ~$2190.90 Profit per cow ~$549.99 Vaccinate ~$47.74

Bates et al. Benefit:Cost ~10:1 Mike & Rachelle Benefit:Cost~11.5:1

Summary

The spread of Foot and Mouth can be modelled with a SIR type model

Analysis can determine the level of vaccination required to keep a cow immune

Equations can get complicated

Sources Pech, R. & Hone, J. (1988) A model of the dynamics and control of an

outbreak of foot and mouth disease in feral pigs in Australia. Journal of Applied Ecology, 25, 63-77. http://www.jstor.org/view/00218901/di996048/99p02395/0

Bates, T. & Carpenter, T. & Thurmond, M. (2003) Benefit-cost analysis of vaccination and preemptive slaughter as a means of eradicating foot-and-mouth disease. Vol. 64, No. 7, Pg 805-812. http://avmajournals.avma.org/doi/abs/10.2460/ajvr.2003.64.805?cookieSet=1&journalCode=ajvr

Kaneene, J. & Miller,R. (1995) Risk factors for metritis in Michigan dairy cattle using herd- and cow-based modelling approaches. Preventive Veterinary Medicine, Volume 23, Number 3, pp. 183-200. http://www.ingentaconnect.com/content/els/01675877/1995/00000023/00000003/art00438

How Much for that Cow? BBC News. David Brown, secretary of the Livestock Auctioneers Association. http://news.bbc.co.uk/1/hi/uk/1476141.stm