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Intrinsic disease maps using persistent cohomology
1.  CUNY Graduate Center, 365 5th Avenue, New York, NY 10016 
2.  Department of Mathematics, CUNY College of Staten Island, 2800 Victory Boulevard, Staten Island, NY 10314 
We use persistent cohomology and circular coordinates to investigate three datasets related to infectious diseases. We show that all three datasets exhibit circular coordinates that carry information about the data itself. For one of the datasets we are able to recover time post infection from the circular coordinate itself – for the other datasets, this information was not available, but in one we were able to relate the circular coordinate to red blood cell counts and weight changes in the subjects.
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V. de Silva, D. Morozov and M. VejdemoJohansson, Persistent cohomology and circular coordinates, Discrete Comput. Geom., 45 (2011), 737759. doi: 10.1007/s004540119344x. 
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M. VejdemoJohansson and A. Leshchenko, Certified mapper: Repeated testing for acyclicity and obstructions to the nerve lemma, in Topological Data Analysis, Abel Symposia, 15, Springer, Cham, 2020, 491–515. doi: 10.1007/9783030434083_19. 
show all references
References:
[1] 
K. Cumnock, A. S. Gupta, M. Lissner, V. Chevee, N. M. Davis and D. S. Schneider, Host energy source is important for disease tolerance to malaria, Current Biology, 28 (2018), 16351642. doi: 10.1016/j.cub.2018.04.009. 
[2] 
V. de Silva, D. Morozov and M. VejdemoJohansson, Persistent cohomology and circular coordinates, Discrete Comput. Geom., 45 (2011), 737759. doi: 10.1007/s004540119344x. 
[3] 
F. Pedregosa, G. Varoquaux, A. Gramfort and al. et, Scikitlearn: Machine learning in {P}ython, J. Mach. Learn. Res., 12 (2011), 28252830. 
[4] 
B. R. Rosenberg, M. Depla, C. A. Freije, D. Gaucher and S. Mazouz, et al., Longitudinal transcriptomic characterization of the immune response to acute hepatitis c virus infection in patients with spontaneous viral clearance, PLoS Pathogens, 14 (2018). doi: 10.1371/journal.ppat.1007290. 
[5] 
B. Y. Torres, J. H. M. Oliveira, A. T. Tate, P. Rath, K. Cumnock and D. S. Schneider, Tracking resilience to infections by mapping disease space, PLoS biology, 14 (2016). doi: 10.1371/journal.pbio.1002436. 
[6] 
M. VejdemoJohansson and A. Leshchenko, Certified mapper: Repeated testing for acyclicity and obstructions to the nerve lemma, in Topological Data Analysis, Abel Symposia, 15, Springer, Cham, 2020, 491–515. doi: 10.1007/9783030434083_19. 
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