Bayesian Inference from Discretely Observed Epidemic Model Using Multidimensional Diffusion Approximation Approach
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Date
2018-08
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UNIVERSITY OF ILORIN
Abstract
Most epidemic data discretely observed are subjected to environmental influence. The dynamic of such data can well be described by discretised-version of Stochastic Differential Equations (SDEs). However, estimating the parameter for such a process proves to be challenging in practice. The difficulty that underlies this was the general intractability of the transition density, resulting into the likelihood functions that are not in closed forms. A direct implementation of Bayesian method of estimation results into convergence problems. Indeed, the literature has not adequately addressed these challenges for multi-dimensional cases. The aim of the study was to find the estimation procedure which would not suffer from these challenges. The objectives of these study were to: (i) derive epidemic models for explaining the behaviour of SDEs models, (ii) propose an improved diffusion bridge sampler capable of overcoming convergence problems, (iii) propose Bayesian data-augmentation method of parameters estimation which would not suffer from convergence, and (iv) compare the performance of the proposed methods with existing methods using simulated and real-life datasets.
The exact likelihood functions of SDEs are very rare in practice. A numerical approximation approach through discretization of the process using Euler-Maruyama scheme was adopted. Data were simulated by introducing augmented values between every two consecutive time points to allow Euler scheme converge to the true continuous-time SDEs. Then, Bayesian data-augmentation method of estimation was performed for such problems via Markov Chain Monte Carlo (MCMC) method of sampling. Posterior means (PM), posterior credible interval (PCI), acceptance probability, trace plot, Autocorrelation function (ACF) plot and the density plot were use to assess the performance of the estimators under different number of intermediate sub-interval data point.
The findings of the study were that:
(i) the evolution of SDEs (diffusion process) was found to be the most suitable trajectory that mirrored the exact dynamic epidemic models’ influence;
(ii) amongst the diffusion sampler considered, the proposed diffusion bridge sampler was found to be better when the number of augmented values tend to infinity;
(iii) the proposed method of parameter estimate was found not to suffer from convergence problem; and
(iv) results of assessment from epidemic outbreak in the simulation study and in two different real-life datasets revealed that the proposed method of parameter estimate was better than existing methods.
The study concluded that modeling from discretely-observed Stochastic Differential Equations (SDEs) using Bayesian data-augmentation approach provided alternative method of estimation to obtained parameter estimates of any natural dynamic phenomena that experience random variations. It is therefore recommended that the epidemiologists should be encouraged to apply diffusion approximation approaches whenever epidemic models are subjected to environmental influence.
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Keywords
Bayesian Inference, Discretely Observed Epidemic Model, Multidimensional Diffusion Approximation Approach, Stochastic Differential Equations, SDEs models