  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. 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), 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 efficacy 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.
