Simulated-likelihood-based Inferencefor an outbreak of influenza

Marc Baguelin

Health Protection Agency

marc.baguelin@hpa.org.uk

A few info

Work done at the School of Veterinary Medicine of the University of Cambridge with the Animal Health Trust in Newmarket

Funded by the Horserace Betting Levy Board

Now I work at the HPA currently on the modelling of swine flu in order to inform Public Health policies

Equine influenza

Currently circulating strains : H3N8

(current human strains H3N2 and H1N1 and now H1N1pdm, H5N1 does not transmit human to human)

H7N7 also exists for horses but no longer circulates (but still in the vaccine)

Two separate sub-lineages co-circulating

From the modelling point of view, the main difference with human influenza is the way the population is distributed

Moulton‘98Nt‘1/93

Miami ‘63

Leics ‘00

Suffolk ‘89KY ‘91

KY ‘98

S.Americanfamily

Nt‘2/93

Am

eric

an s

ub-li

neag

eE

uropean sub-lineageEI phylogenic tree

Why modelling?

To help understand epidemics

Risk assesment

Test different scenarios for vaccine policies

Etc.

The 2003 Newmarket outbreak

21 training yards involved over ~60

more than 1300 horses at risk (~2500)

The dynamics of the epidemics cannot be understood with a simple one yard model as previously for EI: need of a new model.

The map of the outbreak

S E I R

Ex: I5 will be the number of infectious horses in yard 5

SEIR Model

Latent and infectious periods

j

i

tij = rate of transmission from a horse infected in yard i to susceptible horses in yard j

T= (tij)

tij

tjj

Within and between yard transmission rates,

Mixing matrix

How to find the mixing matrix?

Depends of the contacts between the horses in the different yards: very difficult to quantify (shared facilities, contacts when moving, routes taken for going to training, vets, air spread…) usually considered as one spatial and one stochastic part

Assumptions can be made to reduce the number of parameters to find: necessity of the expertise from epidemiologists from the fields

Some of the available data

Number of horses in training for most of the yards with age structure (from Raceform ‘Horses in training 2003’)

Serological data giving antibodies level for some yards which allow us to estimate the level of protection of the horses

Geographical location of the yards

Estimation of the proportion of infected horses in each yards at the end of the epidemic

Date of first detected cases in the infected yards

Trainer questionnaires

But…

Though a huge amount of work done to collect data, few input for the model:

A lot of quantities have to be averaged

Stochastic (as opposed to deterministic) model means that each run leads to different output: in that sense, one ‘run’ available

Lack of temporal data

A classical assumption

Model with two-level of mixing λG (global rate; between yards) and λL (local rate; inside yards).

Number of susceptible

As all the horses are vaccinated, the status of the initial population is uncertain. Vaccine coverage (though theoretically 100%) and efficacy in horses is difficult to predict. Less data than for human + circulation of cross protecting but heterologous strains.

A statistical model (log regression) has been proposed to connect the probability that an animal will be infected given the virus entered its yard (using different variables among which the AB level)

Combining threshold theorem + the statistical modelaverage of the risk for the yard from stat model

mean infectious period

Data

Inference method

The likelihood is analytically and numerically impossible to calculate for each of the pairs (λL, λG)

Very easy to simulate the model

-> use of simulated likelihood to estimate λL, λG

Two approaches

1 Estimate simultaneously the pairs (λL, λG)

2 Estimate first λL and then λG, since the transmission is mostly locally driven (see 1)

First method

1) Use a grid of (λL, λG)

2) For each values of the grid simulate N realisations of the epidemic

3) Count anytime the output is close from the real data (ideally exactly-discrete data)

4) For N sufficiently big, the frequency does approximate the likelihood

This is ABC with “flat” priors

Result (first method)

Grey: give the final size (+/- 0.5%)

Black: final size + exact number of yards

λL = 1.03; λG = 1.5e−2) for the exponential distributions andλL = 0.7; λG = 1.5e−2 for the empirical ones.

Non-regular likelihood

The outbreak is essentially locally driven

It is possible to have a more efficient estimation for the local transmission by using the ten yards for which we have the final sizesAs less than 2% (0.63% on average) of cases will come from re-introduction

Second method

Estimating for a grid of local transmission the simulated likelihood to have simultaneously the exact count for the ten final sizes as independent sub-epidemic (seeded from outside) with the number of susceptible as given by the predicted risk

Second methods: results

The estimated values of the intra-yard transmissionλL were 0.78 for the exponential (grey) distribution and 0.69 (black) for the empirical distribution.

Then estimate the global transmission knowing the local

λG =1.7e−2 for the exponential distribution and 1.6e−2 for the empirical distribution

Conclusion

When likelihood are difficult to derive analytically and models easy to simulate, simulated-likelihood-based methods are an efficient solution

It’s the case in many models of transmission of infectious diseases

More work has to be done on the methodological side of this, especially the limits/accuracy of these methods, the most efficient way of implementing them, model selection issues and deviations from standard theory due to the threshold/phase transition behaviour of epidemic models

Acknowledgments

Horserace Betting Levy Board

CIDC

Epidemiology group in AHT (esp. Richard Newton)

Vet School at Cambridge University (esp. James Wood)

Prof Bryan Grenfell from Penn State

Dr Nikolaos Demiris from MRC-BSU, Cambridge, now in Athens