# Mc Kendrick Poster

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```							              A Stochastic Model of
Paratuberculosis Infection
In Scottish Dairy Cattle
I.J.McKendrick1, J.C.Wood1, M.R.Hutchings2, A.Greig2
1.    Biomathematics & Statistics Scotland, King’s Buildings, Edinburgh, EH9 3JZ.
2. Scottish Agricultural College, Animal Health Group, West Mains Road, Edinburgh, EH9 3JG.

Introduction                                                 Environmental Infection                               The times for which calves and adults remain in
the sub-clinically infected classes are modelled as
Paratuberculosis in cattle, or Johne’s disease, has          Environmental bacterial contamination c(t) is Gamma random variables, fitted to data presented
properties which are difficult to observe in the             modelled by a deterministic ODE which is linear in Rankin (1961, 1962).
field, since it exhibits a long and variable sub-            with respect to the infected cattle populations
clinical period during which animals may actively            and the removal rate:        dc                                   Distributions of Times from Infection
shed bacteria. Therefore it is useful to develop a                                              i Z i (t )  c                  to onset of Clinical Disease
dt
mathematical model of the infection. However:                                                  i

where Z i (t ) is the number of cattle infected with         0.03
• The long and variable time period typically seen           paratuberculosis in infection class i,  i is the

Probability Distribution
between infection and clinical disease will be
0.025
shedding level for these animals, and  is the
poorly modelled by an exponential transition               decay rate of bacteria in the environment.                   0.02

Function
distribution.                                                                                                                                                 0.015

Conditional on the state of the system at time  ,                                                                                              Calves

• Many infected animals in an untested dairy herd this equation can be solved analytically, giving                                                               0.01

will never develop clinical signs and be identified

0.005
as infected, because they will have been                                              i Z i ( )
0
removed from the herd for other reasons.               c(t )  c( )e  (t  )  i              (1  e  (t  ) ).                                                0   50    100     150   200

• The volume of bacteria shed by infected
Months

animals, and hence the associated force of The infective impact of c(t) on individual animals
infection, will increase with time from infection. is difficult to model, since it depends on         Latin Hypercube Sampling
• Several routes of infection exist, defining a non- •the distribution of infection on the farm
Expert opinion, experimental or survey data and
homogeneous population of susceptibles.            •the feeding and mixing patterns of the animals    published estimates are used to define
• Animal infection may arise from poorly •the nature of any dose-response relationship.                 appropriate candidate distributions for the
quantified interactions with a farm environment.                                                      parameter values.      Latin Hypercube sampling
We assume that a given level of contamination c(t) (Iman and Conover, 1980) is used to generate
• There is high uncertainty and large between- will have a specific impact on the force of
parameter combinations (scenarios).
farm variability in parameter estimates.           infection for each calf and each adult,
These issues indicate a need for a stochastic, summarised via arbitrary functions fc(c) and fa(c). Control Methods
animal-oriented model with properties specific We use piecewise linear functions of the form
to the epidemiology of paratuberculosis.                                                                A variety of control methods have been modelled:
        0                   if c  c m in
 c  c m in
                                                • Slaughter of clinically infected animals
Dairy Herd Model                                                        f (c )                         if c m in  c  c m ax
 c m ax  c m in                                • Slaughter of the dams, siblings and offspring of
Information about individual cattle is stored,                                                                if c m ax  c

        1                                         clinically infected animals
defining age, calving status and infection status.
• Testing by faecal culture or ELISA of the
Cattle Infection Model                                                 dams, siblings and offspring of infected animals,
followed by conditional slaughter
An infected animal is introduced into the farm,
and the epidemic is allowed to progress to • The annual faecal or ELISA testing of all
equilibrium. Cattle are infected through one of   animals, followed by conditional slaughter
three routes:                                   • Husbandry measures to reduce animal exposure
Infection from an infected dam. Where a calf is • Vaccination.
born to an infected dam, the calf will be infected
with a specified probability.                      The outcomes for each control policy as applied to
different scenarios are highly variable.
Direct contact with an infectious animal. Animal
to animal infection is modelled using a standard Only policies combining testing, culling and
true-mass action transition probability.           husbandry measures can guarantee to reduce
Contact    with a contaminated environment. the prevalence to negligible levels.
Indirect infection is modelled using the link
functions fa and fc.                          References
Iman, R., Conover, W., 1980, Small Sample Sensitivity Analysis
Once infected, animals pass through three sub-                       Techniques for Computer Models, with an Application to Risk
clinical infection classes, corresponding to zero,                   Assessment, Commun. Statist.-Theor. Meth. A9(17), 1749-1874.
moderate and high levels of bacterial shedding.                      Rankin, J. D., 1961, The experimental infection of cattle with
Mycobacterium Johnei, III: Calves maintained in an infectious
environment, J. Comp. Path., 71, 10-15.
Transitions between different management states are modelled by Rankin, J. D., 1962, The experimental infection of cattle with
random variables or fixed time-lags, depending on the nature of the Mycobacterium Johnei, IV: Adult cattle maintained in an infectious
transition. This detailed treatment allows records to be kept of dam- environment, J. Comp. Path., 72, 113-117.
offspring relationships and hence the modelling of vertical transmission
Acknowledgements
of infection and of plausible control methods such as slaughtering the
Affairs
offspring of infected animals. Animal status is updated for each animal This research was funded by the Scottish Executive Ruralto thank
Department (project BSS/827/98). The authors would like
on a discrete-time basis, with a time step of one month.                 Basil Lowman, George Gunn and Michael Pearce of SAC for advice
detailing the typical management of Scottish dairy herds.

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