An equilibrium solution to the elective waiting list problem

In many countries, demand exceeds supply for elective (non-emergency) hospital treatment, such as hip replacements and cataract removals. The consequence of this is the formation of a waiting list, to which patients join on referral from the family doctor and leave with treatment or renege for other reasons (deconditioning, seeking private healthcare, etc). Adequate performance is commonly incentivised through the imposition of targets on waiting times. In the first study to do so, we develop an equilibrium solution to this problem. The specific problem statement is defined based on consultation with senior healthcare managers and policy advisers. After specifying the parameters and dynamical properties, we manipulate the problem into a linear programming formulation. Generally, we find that, based on the amount of reneging deemed tolerable, a continuum of solutions is possible. On application to Englands National Health Service, we quantify the range of solutions in terms of two key metrics: by how much does the waiting list need to reduce and by how much does treatment capacity need to increase. All data and code are publicly available at an online repository. In addition to the results presented here, the solution has been incorporated within a freely available online tool which is in use across several public hospitals as well as local, regional and national planning boards.