In infectious disease modelling, the expected time from endemicity to extinction (of infection) may be analysed via WKB approximation, a method with origins in mathematical physics. The method is very general, but its uptake to date may have been limited by the practical difficulties of implementation. It is necessary to compute a trajectory of a (high dimensional) dynamical system, the `extinction path', and this trajectory is maximally sensitive to small perturbations, making numerical computation challenging. Our objective here is to make this methodology more accessible by presenting four computational algorithms, with associated Matlab code, together with discussion of various ways in which the algorithms may be tuned to achieve satisfactory convergence. We illustrate our methods using three standard infectious disease models. For each such model, we demonstrate that our algorithms are able to improve upon previously available results.