Hackathon multi-site SEI2R over a hexagonal grid graph
The notebook is based on previously developed framework for simulations with multi-site epidemiological models. [AA2, AA3, AAp1, AAp2, AAp3].
For the derivation of the graph and traveling patterns matrix used in [AA3] in this notebook we utilize an ad hoc solution using GeoHistogram’s output. (Replicating the previous approach using Voronoi mesh over USA population density data was problematic.)
The bigger picture for this notebook can seen in the project management files [AAo1, AAo2].
For the model in this notebook we use the dataset [WRI2] that associates $\approx 12500$ German cities with their populations and geographics coordinates. (The data was retrieved using Mathematica’s function CityData.)
General assumptions (for COVID-19)
The general assumptions about COVID-19 and related mathematical modeling are listed in [AA1].
The most important features of COVID-19 used in the model here are following.
- We have two types of infected populations: severely symptomatic and normally symptomatic.
- The severely symptomatic population is 20% of the infected population.
- Please see the full list in [AA1].
Specific assumptions (for this model)
- People from each cell of the obtained hexagonal grid can travel to the neighboring cells.
- All trips finish within a day.
- We consider grids that have cells with radius 50-70 km to be adequate.
- After a quarantine is enforced the usual, "everything is normal" traffic patterns become much less representative of the quarantine-time traffic patterns.
- Hackathon-wise: We did not produce actual traffic data quickly enough, and when we did, other tasks become more important in view of the approaching deadlines.
- A quarantine decreases the contact rates with a certain constant factor smaller than 1.
- Only severely symptomatic people are hospitalized.
- The hospitals have limited number of beds.
Additional points & observations
- Note that if quarantine scenarios are enacted / simulated then comprehensive traffic traveling patters data is less important — people would be staying at home.
The single-site models
SEI2R (almost standard)
The model SEI2R [AA2] is a modification on the standard SIR model, [Wk1, HH1]. The fairly simple modification uses two infected populations: severely symptomatic and normally symptomatic. The main workflow using the SEI2R implementation in [AAp1] is given in [AA2].
SEI2HR (hospitalization population)
SEI2HR is implemented in [AAp1] and detailed explanations of its equations and usage are given in [AA4].
SEI2HR-Econ (economics extensions)
The model SEI2HR-Econ deals with all rectangles in the diagram above.
The diagram and following table with the stocks in SEI2HR-Econ should give a pretty good idea of model’s scope:
SEI2HR-Econ is implemented in [AAp1] and detailed explanations of its equations and usage are given in [AA5].
The presented work can be extended in multiple ways:
Sensitivity analysis based on systematic runs of the scaled model (in this notebook.)
- For example, the changes of Deceased Infected Population with respect to different available number of beds or quarantine period lengths.
Combination of the hexagonal cell traffic patterns graph with traffic patterns graph based on dedicated traffic patterns studies.
- We hope that a union merge of the two graphs would produce closer to reality simulation of the traffic patterns.
- Calibration of the model with concrete, closer to reality parameter values.
- Combination of this model with a dedicated Economy-impact-of-COVID-19 model.
- Further tasks are shown (hinted) in [AAo2].
[Wk1] Wikipedia entry, "Compartmental models in epidemiology".
[HH1] Herbert W. Hethcote (2000). "The Mathematics of Infectious Diseases". SIAM Review. 42 (4): 599–653. Bibcode:2000SIAMR..42..599H. doi:10.1137/s0036144500371907.
[AA2] Anton Antonov, "Basic experiments workflow for simple epidemiological models", (2020), SystemModeling at GitHub.
[AA3] Anton Antonov, "Scaling of Epidemiology Models with Multi-site Compartments", (2020), SystemModeling at GitHub.
[AA4] Anton Antonov, “Extension of SEI2R with Hospitalized Population”, (2020), SystemModeling at GitHub.
[AA5] Anton Antonov, “Simple Economics Extension of Compartmental Epidemiological Models”, (2020), SystemModeling at GitHub.
Repositories, packages, data
[AAp2] Anton Antonov, "Epidemiology model modifications Mathematica package", (2020), SystemsModeling at GitHub.
[AAp3] Anton Antonov, "Epidemiology modeling visualization functions Mathematica package", (2020), SystemsModeling at GitHub.
[AAp4] Anton Antonov, "System dynamics interactive interfaces functions Mathematica package", (2020), SystemsModeling at GitHub.
Project management files
[AAo2] Anton Antonov, WirVsVirus-hackathon-Geo-spatial-temporal-model-mind-map, (2020), SystemsModeling at GitHub.
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