1. A Stochastic Model of Paratuberculosis Infection In Scottish Dairy Cattle I.J.McKendrick 1, J.C.Wood 1, M.R.Hutchings 2, A.Greig 2 1. Biomathematics & Statistics Scotland, King’s Buildings, Edinburgh, EH9 3JZ. 2.Scottish Agricultural College, Animal Health Group, West Mains Road, Edinburgh, EH9 3JG. Transitions between different management states are modelled by random variables or fixed time-lags, depending on the nature of the transition. This detailed treatment allows records to be kept of dam- offspring relationships and hence the modelling of vertical transmission of infection and of plausible control methods such as slaughtering the offspring of infected animals. Animal status is updated for each animal on a discrete-time basis, with a time step of one month. Introduction Paratuberculosis in cattle, or Johne’s disease, has properties which are difficult to observe in the field, since it exhibits a long and variable sub- clinical 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. Many infected animals in an untested dairy herd will never develop clinical signs and be identified as infected, because they will have been removed from the herd for other reasons. The volume of bacteria shed by infected animals, and hence the associated force of infection, will increase with time from infection. Several routes of infection exist, defining a non- homogeneous population of susceptibles. Animal infection may arise from poorly quantified interactions with a farm environment. There is high uncertainty and large between- farm variability in parameter estimates. These issues indicate a need for a stochastic, animal-oriented model with properties specific to the epidemiology of paratuberculosis. Dairy Herd Model Information about individual cattle is stored, defining age, calving status and infection status. where is the number of cattle infected with paratuberculosis in infection class i, is the shedding level for these animals, and is the decay rate of bacteria in the environment. Conditional on the state of the system at time, this equation can be solved analytically, giving Environmental Infection Environmental bacterial contamination c(t) is modelled by a deterministic ODE which is linear with respect to the infected cattle populations and the removal rate: The infective impact of c(t) on individual animals is difficult to model, since it depends on the distribution of infection on the farm the feeding and mixing patterns of the animals the nature of any dose-response relationship. We assume that a given level of contamination c(t) will have a specific impact on the force of infection for each calf and each adult, summarised via arbitrary functions f c (c) and f a (c). We use piecewise linear functions of the form Cattle Infection Model An infected animal is introduced into the farm, and the epidemic is allowed to progress to equilibrium. Cattle are infected through one of three routes: Infection from an infected dam. Where a calf is born to an infected dam, the calf will be infected with a specified probability. Direct contact with an infectious animal. Animal to animal infection is modelled using a standard true-mass action transition probability. Contact with a contaminated environment. Indirect infection is modelled using the link functions f a and f c. Once infected, animals pass through three sub- clinical infection classes, corresponding to zero, moderate and high levels of bacterial shedding. Adults Calves Acknowledgements This research was funded by the Scottish Executive Rural Affairs Department (project BSS/827/98). The authors would like to thank Basil Lowman, George Gunn and Michael Pearce of SAC for advice detailing the typical management of Scottish dairy herds. The times for which calves and adults remain in the sub-clinically infected classes are modelled as Gamma random variables, fitted to data presented in Rankin (1961, 1962). Latin Hypercube Sampling Expert opinion, experimental or survey data and published estimates are used to define appropriate candidate distributions for the parameter values. Latin Hypercube sampling (Iman and Conover, 1980) is used to generate parameter combinations (scenarios). Control Methods A variety of control methods have been modelled: 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 The annual faecal or ELISA testing of all animals, followed by conditional slaughter Husbandry measures to reduce animal exposure Vaccination. References 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. Rankin, J. D., 1962, The experimental infection of cattle with Mycobacterium Johnei, IV: Adult cattle maintained in an infectious environment, J. Comp. Path., 72, 113-117. The outcomes for each control policy as applied to different scenarios are highly variable. Only policies combining husbandry measures with testing and culling or vaccination can guarantee to reduce the prevalence to negligible levels.