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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. Environmental Infection Paratuberculosis in cattle, or Johne’s disease, has properties which are difficult to observe in the field, since it exhibits a long and variable subclinical period during which animals may actively shed bacteria. Therefore it is useful to develop a mathematical model of the infection. However: • The long and variable time period typically seen between infection and clinical disease will be poorly modelled by an exponential transition distribution. The times for which calves and adults remain in the sub-clinically infected classes are modelled as Environmental bacterial contamination c(t) is Gamma random variables, fitted to data presented modelled by a deterministic ODE which is linear in Rankin (1961, 1962). with respect to the infected cattle populations and the removal rate: Distributions of Times from Infection dc i Zi (t ) c to onset of Clinical Disease dt i where Z i (t ) is the number of cattle infected with 0.03 paratuberculosis in infection class i, i is the 0.025 shedding level for these animals, and is the 0.02 decay rate of bacteria in the environment. Conditional on the state of the system at time , • Many infected animals in an untested dairy herd this equation can be solved analytically, giving will never develop clinical signs and be identified i Z i ( ) as infected, because they will have been removed from the herd for other reasons. c(t ) c( )e (t ) i (1 e (t ) ). Probability Distribution Function Introduction Adults Calves 0.015 0.01 0.005 0 0 50 100 150 200 • The volume of bacteria shed by infected 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 appropriate candidate distributions for the • Animal infection may arise from poorly •the nature of any dose-response relationship. parameter values. Latin Hypercube sampling quantified interactions with a farm environment. 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 A variety of control methods have been modelled: to the epidemiology of paratuberculosis. 0 c c min f (c ) c max c min 1 Dairy Herd Model Information about individual cattle is stored, defining age, calving status and infection status. Months if c c min if c min c c max if c max c Cattle Infection Model • Slaughter of clinically infected animals • Slaughter of the dams, siblings and offspring of clinically infected animals • Testing by faecal culture or ELISA of the 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 animals, followed by conditional slaughter equilibrium. Cattle are infected through one of 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 The outcomes for each control policy as applied to with a specified probability. different scenarios are highly variable. Direct contact with an infectious animal. Animal to animal infection is modelled using a standard Only policies combining husbandry measures with true-mass action transition probability. testing and culling or vaccination can guarantee with a contaminated environment. to reduce the prevalence to negligible levels. Indirect infection is modelled using the link References functions fa and fc. Contact Once infected, animals pass through three subclinical infection classes, corresponding to zero, moderate and high levels of bacterial shedding. Iman, R., Conover, W., 1980, Small Sample Sensitivity Analysis Techniques for Computer Models, with an Application to Risk Assessment, Commun. Statist.-Theor. Meth. A9(17), 1749-1874. 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 research was funded by the Scottish Executive Rural Affairs offspring of infected animals. Animal status is updated for each animal This Department (project BSS/827/98). The authors would like to thank 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.