Many large hospitals include an intensive care unit (ICU) for the care and treatment of patients who are critically ill or in urgent need of a surgical procedure. ICUs in different hospitals rarely act in isolation to one another, and in some situations, patients can be transferred from one ICU to another. This blog post describes the strategic behaviours of ICUs that emerge when patient transfer is possible. The importance of accurately modelling the system of interacting ICUs and the exciting journey I had simulating the system is also discussed.

Before commencing my AMSI Vacation Research Scholarship (VRS), I had never fully appreciated how valuable mathematics can be in assessing and evaluating the efficiency of a healthcare system. As a society, we set very high standards for the care and treatment of patients who are critically ill or in urgent need of a surgical procedure. Many large hospitals include an intensive care unit (ICU) to accommodate such patients. However, ICUs in different hospitals rarely act in isolation to one another.

When one ICU declares it is ‘busy’, it becomes possible for a patient to be transferred to an ICU that is ‘quiet’, if there is one. Without a mathematical model, it is difficult to appreciate the complex interactions that take place in a system of multiple ICUs. In the model for this project, each ICU chooses a bed occupancy threshold. On or above its threshold, an ICU has a status of high occupancy (‘busy’), but otherwise has a status of low occupancy (‘quiet’).

These decisions give each ICU the capability to behave strategically and look after their own interests. When all ICUs are ‘busy’, uncoordinated behaviour typically emerges in the system. There arises a misalignment of interests between society and the individual ICUs. It therefore is crucial to design a model that is fair to all ICUs, but also does not compromise social welfare.

In my research, a bed utilisation target was investigated to help align these interests and keep the system of ICUs operating efficiently. Developing a realistic model and finding an optimal target is of great value for managers and administrators that oversee and set policies for a given system of ICUs.

To achieve this task, I began simulating the interaction between ICUs using MATLAB and conducted multiple experiments with different parameter sets. I quickly discovered that whilst some aspects of research take less time than expected, others can take much longer! The simulations themselves also vary significantly in execution time. Some tests took only a few seconds to run, whereas others took over a day. It was extremely satisfying to run a simulation one day and wake up the next to see the appearance of plots on MATLAB, which indicated a test had run successfully.

As I analysed more and more plots, it became immensely tempting to continue to simulate additional plots using a wider range of parameter sets. Many of the tests revealed unexpected results, which shows just how complicated the interaction between multiple ICUs can be, and why it is so beneficial to study the system closely.

Overall, my VRS experience has demonstrated that an initial idea for research can rapidly lead into the exploration of many interesting extensions. Whilst I originally thought my six-week project would answer all there is to know about interacting ICUs, I finished the project with an extensive list of future directions! It has been highly rewarding to apply all the knowledge I have developed in my undergraduate studies to a useful and practical application of mathematics.

Ashley Hanson
The University of Melbourne

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