The Northern Ireland local group of the RSS held an online meeting using MS Teams on Wednesday, 10 November, 2021, at 1pm. The speaker was Dr Michelle Carey, UCD, Dublin, Ireland.
From the outset Dr Carey emphasised the need to 'follow the Physics' when modelling dynamical systems and argued that understanding the system was paramount. She outlined the dynamical model proposed by Náraigh and Byrne for modelling Covid-19. This was an extended SEIR compartment model with 10 compartments (and differential equations) and 11 parameters denoted by θ. The equations also depended on c, the average number of contacts between a susceptible person and an infectious person per day. Michelle's focus was on estimating the θ parameters and x, the ten components of the system.
The usual estimation approach was via non-linear Least Squares (NLS); given an initial value θ0 and a set of p initial values, { Drx(0) } for r = 0, ..., p-1, a numerical solution is found as x̂(t, θ0;D0x(0), ..., Dp-1x(0)) at the N time points t. Next, θ̂ is found by minimising the discrepancy between the observations yi and x̂(ti, θ0,...), i = 1, ..., N by NLS. This is a two-stage procedure. However, the resulting surfaces may have multiple minima and optimising by NLS, or other methods, such as: Simulated Annealing (SA), Parameter Cascading (PC) and Smooth Functional Tempering (SFT), is non-trivial and computationally expensive.
Michelle introduced a new procedure called Data2LD which she claimed had better properties and was computationally efficient. Using a spline to model the data, y, she estimated the coefficients in the spline by minimising a penalised discrepancy function, J(c|θ, ρ), composed of two terms. The first term measures the fidelity to the data and the second term, a penalty, the fidelity to the dynamical system, J(c|θ, ρ), involves a new parameter ρ which controls the balance. A second stage is required to estimate by minimising H(θ|ρ)). (See Carey & Ramsay, CSDA, 2021, for details). When ρ → 0 the estimate is the Least Squares solution, free of the dynamical system, and when ρ → 1, the estimate complies with the dynamical system with parameter (desired). In addition, as ρ → 1 the surface becomes less convex.
Next Michelle showed a low order example in which solutions with ρ near 0 were only slightly biased. However, this finding could be exploited to move from θp≈0 (convex surfaces) → θp≈1 (non-convex) thereby speeding up the optimisation algorithm which did not now waste time being trapped in local minima. In the example she showed that Data2LD outperformed all other estimation methods considered: NLS, SA, PC and SFT. For example, the closest alternative method was SA, but the computation time was ≈500 times longer.
She went on to apply the Data2LD method to analyse the Irish COVID-19 data and discuss the findings.
This was an excellent talk on an innovative computational method in a challenging area. It was received with acclaim by an online audience of c25 participants. Michelle was asked several questions. One participant asked whether the regularisation of the surface in the toy model readily extended to higher dimensions? She was reassuring on this point. Another participant asked about robustness and if understanding the model could explain why one of the parameters in the analysis of the Irish data was unusually low? Michelle said that she was re-examining the structure of the relevant compartments in that part of the model. Finally, a third participant asked about the uncertainty in the estimates? Michelle said that work on this issue was in hand.
The chair congratulated the speaker on a very stimulating presentation and the participants thanked the speaker in the usual way.