Math. Biol. Group Meeting 25 July 2006 Joanne Turner

1. 4-group Network Model: Progress and Challenges Math. Biol. Group Meeting 25 July 2006 Joanne Turner

2. Outline • Background and motivation • previous 4-group dairy herd model • Modelling using a contact (adjacency) matrix • heterogeneous mixing • Output from 1-group model • theory and simulation • Output from first 4-group model: • heterogeneous direct (animal-animal) transmission • homogeneous indirect (environment-animal) transmission • movement and replacement (i.e. dynamic heterogeneity)

3. Background and Motivation • A model of a typical UK dairy herd was developed for a previous DEFRA project: • 4 management groups (unweaned, weaned, dry and lactating) • 4 group-specific environments (containing free-living infectious units) • 1 general environment (could represent contaminated personnel; allows between-group transmission) • Movement between groups (maturation, dry/lactating cycle) • Replacement with unweaned animals (herd is closed) • Direct transmission (animal to animal) • Indirect transmission (via the environment) • Pseudovertical transmission (some replacements are infected when they enter the herd; represents transmission from dam to calf within the first 24 to 48 hours) • Direct transmission was modelled by assuming homogeneous mixing of animals in the same management group: • assume all animals within a group are essentially the same and behave in the same way • use average contact rate for that group

4. Modelling using a Contact Matrix • Homogeneous mixing: • assume all animals within a group are essentially the same • use average contact rate for that group • Heterogeneous mixing: • assume all animals are different • use specific contact rate for each pair of individuals (contact matrix) • Each element mij of the contact (adjacency) matrix can be thought of as the probability of a suitable contact between animals i and j per unit time. • The matrix does not have to be symmetric. Here mij mji . i.e. animal 1 contacts animal 2 with probability 0.5, but animal 2 contacts animal 1 with probability 0.7 e.g. “contact” could be grooming

5. 1-Group Model • 1 group with 15 nodes representing 15 weaned calves. • Real (static) contact matrix (entries either 0 or 1) representing contact in the form of perineal allo-grooming. • Direct transmission rate is β x SIpairs • β is probability of transmission given an effective contact • ‘SIpairs’ is the number of effective contacts per unit time • Stochastic model advances by first calculating the next event time and then choosing the next event. • If transmission is chosen, one SI pair is selected at random and the susceptible is infected. • It is important to chose from those pairs where contact is from I to S. • For simplicity, indirect transmission via the environment was modelled homogeneously. However, it is possible to use the contact matrix approach given sufficient information.

6. increasing direct transmission Output from 1-Group Model • Distribution of contact rate • degree of heterogeneity in a real network • Prevalence over time and across simulations • effect of heterogeneity on disease transmission • Distribution of outbreak size and how this varies with different ‘interventions’ • Comparison with theoretical estimates for R0 and outbreak size • sensitivity of R0 and outbreak size can be used to suggest effective interventions

7. R0 for 1-Group Model • R0 is the average number of secondary infections produced by one primary infection during its infectious period. • When R0 > 1, major outbreaks are possible. • Plot shows R0 versus β (direct transmission parameter): • 3 theoretical estimates shown in green, magenta, black • mean from simulations shown in red • ‘ghost’ is the mean we would get if the population remained totally susceptible (i.e. if the supply of susceptibles was inexhaustible). • Results suggest that, in terms of finding a better theoretical estimate, incorporating the finite size of the resource is more important than trying to capture the heterogeneity within the contact matrix.

8. Results for 1-Group Model • A heterogeneous version of Ball’s method1 for calculating average outbreak size has been developed. • Mean (and standard error) are given below for 1-group SIS models with direct transmission only. • Currently, program only works for small N. 1Ball, F. (1999). Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math. Biosci. 156, 41-67.

9. First 4-group Model (Dairy Herd) • 4 groups representing the 4 main management groups (unweaned, weaned, dry and lactating). 112 animals in total. • Indirect transmission is modelled homogeneously for simplicity. • Direct transmission is modelled using a random contact matrix (which will be replaced later with a real matrix from Maz). • Only animals within the same group can be in direct contact. • Currently, contacts are assigned at random and individuals remain friends for life. • ‘Threshold’ mechanism controls average contact rate. e.g. • threshold = 0.72, all animals in D: actual contacts are animals 1 and 2. • threshold = 0.49, animal 2 in L: actual contacts are animals 1, 4 and 5. • threshold = 0.55, all animals in D: actual contacts are animals 1, 2 and 4.

10. First 4-group Model (Dairy Herd) • Entries in the contact matrix represent an individual’s rank in terms of attractiveness to the licker. • Threshold varies with group size and controls average contact rate only. • Effective contact matrix changes as a result of: • movement between groups (maturation, dry/lactating cycle) • death/culling and replacement • threshold mechanism (i.e. dynamic heterogeneity) • Questions: • What should the average contact rates for each group be? • How are ‘contacts’ acquired? • Should group structure be maintained, rather than average contact rate? • Should we try to control clustering too? • It might be possible to modify the threshold mechanism to encourage the formation of triangles.

11. Change in group size over time Change in average contact rate in U group Change in average contact rate in W, D and L Output from First 4-group Model

12. Change in prevalence over time Network Model vs Homogeneous Model • Transmission parameters from the previous study are the product of • probability of transmission given an effective contact • average number of effective contacts per unit time • Currently, there is not enough information to separate these components. • So, βi (direct transmission parameter for group i) is scaled to fit ni (average number of effective contacts per unit time in group i). • ni is chosen arbitrarily.

13. Future Work: Final 4-group Model • Heterogeneous indirect transmission (have done this for a 1-group model). • Real weighted contact matrices (from Maz’s study) • entries (between 0 and 1) representing the probability of contact • requires information on frequency and duration of behaviour • perineal allo-grooming (direct transmission) • allo-grooming and enviromental licking (indirect transmission) • Possibly a mechanism that ensures that the contact structure of each group remains fairly constant as individuals move in and out. • perhaps by animals assuming a certain role within each group, rather than acting completely independently. • Analysis: • comparisons with the homogeneous-mixing model and the random heterogeneous model. • sensitivity analysis to identify key parameters. • effects of interventions on R0, prevalence and outbreak size.