How to run Distributed Lag Non-linear Model on dataset?

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Hello everyone.
I am trying to generate a certain graph that will show the relationship between a discomfort index (Tappmax, oC) and daily number of deaths for the timeseries 2006-2016. Attached you can find the shape of graph I want to produce and the dataset I am assessing.
Is anyone familiar with Distributed Lag Models? I have read https://www.mathworks.com/help/econ/what-are-time-series-regression-models.html about DLMs but I didn't quite get it.
I would appreciate any help!
PS. I am using R2019a.

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