A Constrained Optimisation Framework for Parameter Identification of the SIRD Model

TL;DR

The paper addresses parameter identification for the SIRD model using an optimise-then-discretise approach, formulating a tracking-type cost $J(\rho,\alpha_v)$ subject to $\dot{\rho}=f(\rho,(\alpha_v,\alpha_f),t)$. It derives the state–adjoint optimality system and a reduced gradient $\tfrac{dj}{d\alpha_v}$, then assesses four gradient-based and quasi-Newton algorithms (PGD, FISTA, nmAPG, LM-BFGS) with projection onto admissible parameter sets. Numerical experiments across known-solution, regularisation, and data-driven calibration scenarios show LM-BFGS often delivering the best objective values with reasonable iteration counts, while nmAPG remains competitive; regularisation stabilises solutions and data-driven tests demonstrate practical applicability and data-related challenges. The framework lays groundwork for calibrating more complex compartmental models, including time-dependent controls and potential spatial extensions, enabling robust data-driven epidemiological analysis without ad hoc tuning.

Abstract

We consider a numerical framework tailored to identifying optimal parameters in the context of modelling disease propagation. Our focus is on understanding the behaviour of optimisation algorithms for such problems, where the dynamics are described by a system of ordinary differential equations associated with the epidemiological SIRD model. Applying an optimise-then-discretise approach, we examine properties of the solution operator and determine existence of optimal parameters for the problem considered. Further, first-order optimality conditions are derived, the solution of which provides a certificate of goodness of fit, which is not always guaranteed with parameter tuning techniques. We then propose strategies for the numerical solution of such problems, based on projected gradient descent, Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), nonmonotone Accelerated Proximal Gradient (nmAPG), and limited memory BFGS trust region approaches. We carry out a thorough computational study for a range of problems of interest, determining the relative performance of these numerical methods. Our results provide insights into the effectiveness of these strategies, contributing to ongoing research into optimising parameters for accurate and reliable disease spread modelling. Moreover, our approach paves the way for calibration of more intricate compartmental models.

Figures (5)

• Figure 1: Contour plots for the reduced objective $j$ with zero target $\widehat{\rho} = 0$. The panel on the left is the result of evaluating the scaled reduced objective function $( j(\alpha_\mathrm{v})/\mathsf{n}^2 )$ over the set of feasible controls $\mathcal{A}_{\mathrm{v}} = [0,1]^2$. The horizontal axis is plotted in symmetrical logarithmic scale for better visualisation of the minimal curves. The panel on the right zooms into the subregion $[ 0,1/20] \times [0,1]$. The white x mark at $\alpha \approx \left(\mathtt{0.007}\mathtt{0.213}\right)^\top \space$ corresponds to the smallest objective found in this grid search. The + marks in both plots depict the non--convex slice $j (\lambda \alpha_{\mathrm{v},1} + (1-\lambda)\alpha_{\mathrm{v},2} ) / {\mathsf{n}^2}$ for $\alpha_{\mathrm{v},1} = \left(10\right)^\top \space$, $\alpha_{\mathrm{v},2} = \left(01\right)^\top \space$, and different values of $\lambda \in [0,1]$.
• Figure 2: Contour plots for the reduced objective $j$ with known target $\widehat{\rho}_a$. The panel on the left is the result of evaluating the scaled reduced objective function $( j(\alpha_\mathrm{v})/\mathsf{n}^2 )$ over the set of feasible controls $\mathcal{A}_{\mathrm{v}} = [0,1]^2$. The panel on the right zooms into the subregion $[ 0.028,0.032] \times [0.5, 0.7]$. In each panel, the optimisation path given by the sequence of iterates $\{\alpha_{\mathrm{v},k}\}$ generated by each algorithm is illustrated. The + marker at the left panel is the starting point, while the x marker in both plots points at the optimal solution.
• Figure 3: Evolution of the reduced objective against number of iterations for each algorithm.
• Figure 4: State curves for different values of the regularisation parameter $\theta$. The intensity of the colour of each curve is associated with the value of $\theta$. Visually, most of the curves overlap, meaning an effective fitting of the data has been achieved. The dotted curves in the three panels display the target $\widehat{\rho}_b$.
• Figure 5: Results of the fitting. In all plots, time has been scaled to daily information and $\alpha$ has been scaled to real population units. First panel: State curves (continuous lines) of best fit against data (dotted line). We report $1-\rho_{\mathsf{S}}$ for visualisation purposes. All curves were normalised by $\mathsf{n}$ and are displayed in logarithmic scale. Second panel: Obtained controls (in logarithmic scale). Third panel: Gradient curves associated with each time--dependent parameter.

Theorems & Definitions (15)

• Remark 1
• Theorem 1
• proof
• Theorem 2: Positively Invariant Regions, (Amann_1990, Ch. 4 §16)
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof