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04-04-2022 | Regular Paper

Eigenvalue analysis of SARS-CoV-2 viral load data: illustration for eight COVID-19 patients

Author: Till D. Frank

Published in: International Journal of Data Science and Analytics

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Abstract

Eigenvalue analysis is an important tool in economics and nonlinear physics to analyze industrial processes and instability phenomena, respectively. A model-based eigenvalue analysis of viral load data from eight symptomatic COVID-19 patients was conducted. The eigenvalues and eigenvectors of the instabilities were determined that give rise to COVID-19. For all eight patients, it was found that the virus dynamics followed the unstable eigenvectors until the viral load reached the respective peak values. At the peak virus values, the virus dynamics branched off from the directions specified by the eigenvectors. The temporal course of the unstable eigenvalues was determined as well. For all patients, it was found that the eigenvalues switched from positive to negative values just when the virus load reached peak values. These findings suggest that the fixed, instability-related eigenvalues and eigenvectors determine initial stages of SARS-CoV-2 infections during which virus load increases. In contrast, the time-dependent eigenvalues show a sign-switching phenomenon that indicates when the virus dynamics switches from the growth stage (increasing virus load) to the decay stage (decreasing virus load). The virus dynamics model was a standard three-variable virus dynamics model frequently used in the literature.
Literature
1.
go back to reference Abbad, A., Abdelmalek, S., Bendoukha, S., Gambino, G.: A generalized Degn-Harrison reaction-diffusion system: asymptotic stability and non-existence results. Nonlinear analysis: real world applications 57, article 103,191 (2021) Abbad, A., Abdelmalek, S., Bendoukha, S., Gambino, G.: A generalized Degn-Harrison reaction-diffusion system: asymptotic stability and non-existence results. Nonlinear analysis: real world applications 57, article 103,191 (2021)
2.
go back to reference Baccam, P., Beauchemin, C., Macken, C.A., Hayden, F.G., Perelson, A.S.: Kinetics of influenza A virus infection in humans. J. Virol. 80, 7590–7599 (2006) Baccam, P., Beauchemin, C., Macken, C.A., Hayden, F.G., Perelson, A.S.: Kinetics of influenza A virus infection in humans. J. Virol. 80, 7590–7599 (2006)
3.
go back to reference Best, B., Guedj, J., Madelain, V., de Lamballerie, X., Lim, S.Y., Osuna, C.E., Whitney, J.B., Perelson, A.S.: Zika plasma viral dynamics in nonhuman primates provides insights into early infection and antiviral strategies. PNAS 114, 8847–8852 (2017) Best, B., Guedj, J., Madelain, V., de Lamballerie, X., Lim, S.Y., Osuna, C.E., Whitney, J.B., Perelson, A.S.: Zika plasma viral dynamics in nonhuman primates provides insights into early infection and antiviral strategies. PNAS 114, 8847–8852 (2017)
4.
go back to reference Bhattacharya, M., Chatterjee, S., Sharam, A.R., Agoramoorthy, G., Chakraborty, C.: D614G mutation and SARS-CoV-2: impact on S-protein structure, function, infectivity, and immmunity. Appl. Microbiol. Biothechnol. 105, 9035–9045 (2021) Bhattacharya, M., Chatterjee, S., Sharam, A.R., Agoramoorthy, G., Chakraborty, C.: D614G mutation and SARS-CoV-2: impact on S-protein structure, function, infectivity, and immmunity. Appl. Microbiol. Biothechnol. 105, 9035–9045 (2021)
5.
go back to reference Böhmer, M.M., Buchholz, U., Corman, V.M., Hoch, M., Katz, K., et al.: Investigation of a COVID-19 outbreak in Germany resulting from a single travel-associated primary case: a case series. Lancet Infect. Dis. 20, 920–928 (2020) Böhmer, M.M., Buchholz, U., Corman, V.M., Hoch, M., Katz, K., et al.: Investigation of a COVID-19 outbreak in Germany resulting from a single travel-associated primary case: a case series. Lancet Infect. Dis. 20, 920–928 (2020)
6.
go back to reference Bressloff, P.C., Cowan, J.D., Golubitsky, M., Thomas, P.J., Wiener, M.C.: Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex. Phil. Trans. R. Soc. Lond. B 356, 299–330 (2001) Bressloff, P.C., Cowan, J.D., Golubitsky, M., Thomas, P.J., Wiener, M.C.: Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex. Phil. Trans. R. Soc. Lond. B 356, 299–330 (2001)
8.
go back to reference Cheung, O.Y., Graziano, P., Smith, M.W.: Acute lung injury. In: Leslie, K.O., Wick, M.R. (Eds.) Practical pulmonary pathology: a diagnostic approach, pp. 125-146.e3. Elsevier, New York (2018) Cheung, O.Y., Graziano, P., Smith, M.W.: Acute lung injury. In: Leslie, K.O., Wick, M.R. (Eds.) Practical pulmonary pathology: a diagnostic approach, pp. 125-146.e3. Elsevier, New York (2018)
9.
go back to reference Chua, T., Lukassen, S., Trum, S., Hennig, B.P., Wnedisch, D., Pott, F., Debnath, O., Thuermann, L., Kurth, F., Voelker, M.T., et al.: COVID-19 severity. Nat. Biotechnol. 38, 970–979 (2020) Chua, T., Lukassen, S., Trum, S., Hennig, B.P., Wnedisch, D., Pott, F., Debnath, O., Thuermann, L., Kurth, F., Voelker, M.T., et al.: COVID-19 severity. Nat. Biotechnol. 38, 970–979 (2020)
10.
go back to reference Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993) MATH Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993) MATH
11.
go back to reference Czuppon, P., Debarre, F., Goncalves, A., Tenaillon, O., Perelson, A.S., Guedj, J., Blanquart, F.: Success of prophylactic antiviral therapy for SARS-CoV-2: predicted critical efficiacies and impact of different drug-specific mechanisms of action. PLoS Comput. Biol. 17, article e1008752 (2021) Czuppon, P., Debarre, F., Goncalves, A., Tenaillon, O., Perelson, A.S., Guedj, J., Blanquart, F.: Success of prophylactic antiviral therapy for SARS-CoV-2: predicted critical efficiacies and impact of different drug-specific mechanisms of action. PLoS Comput. Biol. 17, article e1008752 (2021)
12.
go back to reference Daoxiang, Z., Sun, G., Zhao, L., Yan, P.: Pattern formation and selection in a diffusive predator–prey system with ratio-dependent functional response. Acta Ecol. Sin. 37, 290–297 (2017) Daoxiang, Z., Sun, G., Zhao, L., Yan, P.: Pattern formation and selection in a diffusive predator–prey system with ratio-dependent functional response. Acta Ecol. Sin. 37, 290–297 (2017)
13.
go back to reference Davies, N.G., Jarvis, C.I., van Zandvoort, K., Clifford, S., Sun, F.Y, Funk, S, et al.: Increased mortality in community-tested cases of SARS-CoV-2 lineage B.1.1.7. Nature 593, 270–274 (2021) Davies, N.G., Jarvis, C.I., van Zandvoort, K., Clifford, S., Sun, F.Y, Funk, S, et al.: Increased mortality in community-tested cases of SARS-CoV-2 lineage B.1.1.7. Nature 593, 270–274 (2021)
14.
go back to reference Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases. Wiley, Chichester (2000) MATH Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases. Wiley, Chichester (2000) MATH
15.
go back to reference Dufiet, V., Boissonade, J.: Dynamics of Turing pattern monolayers close to onset. Phys. Rev. E 53, 4883–4892 (1996) Dufiet, V., Boissonade, J.: Dynamics of Turing pattern monolayers close to onset. Phys. Rev. E 53, 4883–4892 (1996)
16.
go back to reference Dutt, A.K.: Turing pattern amplitude equations for a model glycolytic reaction-diffusion system. J. Math. Chem. 48, 841–855 (2010) MathSciNetMATH Dutt, A.K.: Turing pattern amplitude equations for a model glycolytic reaction-diffusion system. J. Math. Chem. 48, 841–855 (2010) MathSciNetMATH
17.
go back to reference Frank, T.: Determinism and Self-organization of Human Perception and Performance. Springer, Berlin (2019) Frank, T.: Determinism and Self-organization of Human Perception and Performance. Springer, Berlin (2019)
18.
go back to reference Frank, T.D.: Nonlinear Fokker–Planck Equations: Fundamentals and Applications. Springer, Berlin (2005) MATH Frank, T.D.: Nonlinear Fokker–Planck Equations: Fundamentals and Applications. Springer, Berlin (2005) MATH
19.
go back to reference Frank, T.D.: Multistable pattern formation systems: candidates for physical intelligence. Ecol. Psychol. 24, 220–240 (2012) Frank, T.D.: Multistable pattern formation systems: candidates for physical intelligence. Ecol. Psychol. 24, 220–240 (2012)
20.
go back to reference Frank, T.D.: COVID-19 interventions in some European countries induced bifurcations stabilizing low death states against high death states: an eigenvalue analysis based on the order parameter concept of synergetics. Chaos, Solitons Fractals 140, article 110194 (2020) Frank, T.D.: COVID-19 interventions in some European countries induced bifurcations stabilizing low death states against high death states: an eigenvalue analysis based on the order parameter concept of synergetics. Chaos, Solitons Fractals 140, article 110194 (2020)
21.
go back to reference Frank, T.D.: COVID-19 order parameters and order parameter time constants of Italy and China: a modeling approach based on synergetics. J. Biol. Syst. 28, 589–608 (2020) MathSciNetMATH Frank, T.D.: COVID-19 order parameters and order parameter time constants of Italy and China: a modeling approach based on synergetics. J. Biol. Syst. 28, 589–608 (2020) MathSciNetMATH
22.
go back to reference Frank, T.D.: Emergence and subsiding of the first-wave COVID-19 pandemic in Pakistan 2020: an eigenvalue analysis based on synergetics. Proc. Pak. Acad. Sci. B 57, 1–7 (2020) Frank, T.D.: Emergence and subsiding of the first-wave COVID-19 pandemic in Pakistan 2020: an eigenvalue analysis based on synergetics. Proc. Pak. Acad. Sci. B 57, 1–7 (2020)
23.
go back to reference Frank, T.D.: Rise and decay of the COVID-19 epidemics in the USA and the State of New York in the first half of 2020: A nonlinear physics perspective yielding novel insights. BioMed Res. Int. 2021, 6645688 (2021) Frank, T.D.: Rise and decay of the COVID-19 epidemics in the USA and the State of New York in the first half of 2020: A nonlinear physics perspective yielding novel insights. BioMed Res. Int. 2021, 6645688 (2021)
24.
go back to reference Frank, T.D.: SARS-Coronavirus-2 nonlinear dynamics in patients: three-dimensional state and amplitude state description. J. Phys. Soc. Jpn. 90, 073802 (2021) Frank, T.D.: SARS-Coronavirus-2 nonlinear dynamics in patients: three-dimensional state and amplitude state description. J. Phys. Soc. Jpn. 90, 073802 (2021)
25.
go back to reference Frank, T.D., Chiangga, S.: SEIR order parameters and eigenvectors of the three stages of completed COVID-19 epidemics: with an illustration for Thailand January to May 2020. Phys. Biol. 18, 046002 (2021) Frank, T.D., Chiangga, S.: SEIR order parameters and eigenvectors of the three stages of completed COVID-19 epidemics: with an illustration for Thailand January to May 2020. Phys. Biol. 18, 046002 (2021)
26.
go back to reference Frank, T.D., Daffertshofer, A., Peper, C.E., Beek, P.J., Haken, H.: Towards a comprehensive theory of brain activity: coupled oscillator systems under external forces. Physica D 144, 62–86 (2000) MathSciNetMATH Frank, T.D., Daffertshofer, A., Peper, C.E., Beek, P.J., Haken, H.: Towards a comprehensive theory of brain activity: coupled oscillator systems under external forces. Physica D 144, 62–86 (2000) MathSciNetMATH
27.
go back to reference Gambino, G., Lombardo, M.C., Rubino, G., Sammartino, M.: Pattern selection in the 2D FitzHug-Nagumo model. Ricerche mat. 68, 535–549 (2019) MathSciNetMATH Gambino, G., Lombardo, M.C., Rubino, G., Sammartino, M.: Pattern selection in the 2D FitzHug-Nagumo model. Ricerche mat. 68, 535–549 (2019) MathSciNetMATH
28.
go back to reference Gambino, G., Lombardo, M.C., Sammartino, M., Sciacca, V.: Turing pattern formation in the Brusselator system with nonlinear diffusion. Phys. Rev. E 88, article 042925 (2013) Gambino, G., Lombardo, M.C., Sammartino, M., Sciacca, V.: Turing pattern formation in the Brusselator system with nonlinear diffusion. Phys. Rev. E 88, article 042925 (2013)
29.
go back to reference Gambino, G., Lombardo, M.L., Sammartino, M.: Turing instability and traveling fronts for a nonlinear reaction-diffusion system with cross-diffusion. Math. Comput. Simul. 82, 1112–1132 (2012) MathSciNetMATH Gambino, G., Lombardo, M.L., Sammartino, M.: Turing instability and traveling fronts for a nonlinear reaction-diffusion system with cross-diffusion. Math. Comput. Simul. 82, 1112–1132 (2012) MathSciNetMATH
30.
go back to reference Goncalves, A., Bertrand, Y., Ke, R., Comets, E., de Lamballerie, X., Malvy, D., Pizzorno, D., Terrier, O., Calatrava, M.R., Mentre, F., Smith, P., Perelson, A.S., Guedj, J.: Timing of antiviral treatment initiation is critical to reduce SARS-CoV-2 viral load. CPT: Pharm. Syst. Pharmacol. 9, 509–514 (2020) Goncalves, A., Bertrand, Y., Ke, R., Comets, E., de Lamballerie, X., Malvy, D., Pizzorno, D., Terrier, O., Calatrava, M.R., Mentre, F., Smith, P., Perelson, A.S., Guedj, J.: Timing of antiviral treatment initiation is critical to reduce SARS-CoV-2 viral load. CPT: Pharm. Syst. Pharmacol. 9, 509–514 (2020)
31.
go back to reference Goncalves, P.: Behavior modes, pathways and overall trajectories: eigenvector and eigenvalue analysis of dynamic systems. Syst. Dyn. Rev. 25, 35–62 (2009) Goncalves, P.: Behavior modes, pathways and overall trajectories: eigenvector and eigenvalue analysis of dynamic systems. Syst. Dyn. Rev. 25, 35–62 (2009)
32.
go back to reference Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1983) MATH Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1983) MATH
33.
go back to reference Haagmans, B.L., Kuiken, T., Martina, B.E., Fouchier, R.A.M., Rimmelzwaan, G.F., van Amerongen, G., van Riel, D., de Jong, T., Itamura, S., Chan, K.H., Tashiro, M., Osterhaus, A.D.M.E.: Pegylated interferon-alpha protects type 1 pneumocytes against SARS coronavirus infection in macaques. Nat. Med. 10, 290–293 (2004) Haagmans, B.L., Kuiken, T., Martina, B.E., Fouchier, R.A.M., Rimmelzwaan, G.F., van Amerongen, G., van Riel, D., de Jong, T., Itamura, S., Chan, K.H., Tashiro, M., Osterhaus, A.D.M.E.: Pegylated interferon-alpha protects type 1 pneumocytes against SARS coronavirus infection in macaques. Nat. Med. 10, 290–293 (2004)
34.
go back to reference Hadjichrysanthou, C., Lawrence, E.C.E., Vegvari, C., de Wolf, F., Anderson, R.M.: Understanding the within-host dynamics of influenza a virus: from theory to clinical implications. J. R. Soc. Interface 13, article 20160289 (2016) Hadjichrysanthou, C., Lawrence, E.C.E., Vegvari, C., de Wolf, F., Anderson, R.M.: Understanding the within-host dynamics of influenza a virus: from theory to clinical implications. J. R. Soc. Interface 13, article 20160289 (2016)
35.
go back to reference Haken, H.: Synergetics. An Introduction. Springer, Berlin (1977) MATH Haken, H.: Synergetics. An Introduction. Springer, Berlin (1977) MATH
36.
go back to reference He, X., Lau, E.H.Y., Wu, P., Deng, X., Wang, J., et al.: Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat. Med. 26, 672–675 (2020) He, X., Lau, E.H.Y., Wu, P., Deng, X., Wang, J., et al.: Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat. Med. 26, 672–675 (2020)
37.
go back to reference Hernandez-Vargas, E.N., Velasco-Hernandez, J.X.: In-host mathematical modelling of COVID-19 in humans. Annu. Rev. Control. 50, 448–456 (2020) MathSciNet Hernandez-Vargas, E.N., Velasco-Hernandez, J.X.: In-host mathematical modelling of COVID-19 in humans. Annu. Rev. Control. 50, 448–456 (2020) MathSciNet
38.
go back to reference Kampmann, C.E., Oliva, R.: Loop eigenvalue elasticity analysis: three case studies. Syst. Dyn. Rev. 22, 141–162 (2006) Kampmann, C.E., Oliva, R.: Loop eigenvalue elasticity analysis: three case studies. Syst. Dyn. Rev. 22, 141–162 (2006)
39.
go back to reference Mackey, M.C., Glass, L.: Oscillations and chaos in physiological control systems. Science 197, 287–289 (1977) MATH Mackey, M.C., Glass, L.: Oscillations and chaos in physiological control systems. Science 197, 287–289 (1977) MATH
40.
go back to reference Martines, R.B., Ritter, J.M., Matkovic, E., Gary, J., Bollweg, B.C., Bullock, H., Goldsmith, C.S., et al.: Pathology and pathogenesis of SARS-CoV-2 associated with fatal coronavirus disease, United States. Emerg. Infectious Diseases 26, 2005–2015 (2020) Martines, R.B., Ritter, J.M., Matkovic, E., Gary, J., Bollweg, B.C., Bullock, H., Goldsmith, C.S., et al.: Pathology and pathogenesis of SARS-CoV-2 associated with fatal coronavirus disease, United States. Emerg. Infectious Diseases 26, 2005–2015 (2020)
41.
go back to reference Mochan, E., Sego, T.J., Gaona, L., Rial, E., Ermentrout, G.B.: Compartmental model suggests importance of innate immune response to COVID-19 Bull. Math. Biol. 83, 79 (2021) MATH Mochan, E., Sego, T.J., Gaona, L., Rial, E., Ermentrout, G.B.: Compartmental model suggests importance of innate immune response to COVID-19 Bull. Math. Biol. 83, 79 (2021) MATH
42.
go back to reference Murray, J.D.: Mathematical Biology. Springer, Berlin (1993) MATH Murray, J.D.: Mathematical Biology. Springer, Berlin (1993) MATH
43.
go back to reference Neant, N., Lingas, G., Le Hingrat, Q., Ghosn, J., Engelmann, I., et al.: Modeling SARS-CoV-2 viral kinetics and association with mortality in hospitalized patients from the French COVID cohort. PNAS 118, article e2017962118 (2021) Neant, N., Lingas, G., Le Hingrat, Q., Ghosn, J., Engelmann, I., et al.: Modeling SARS-CoV-2 viral kinetics and association with mortality in hospitalized patients from the French COVID cohort. PNAS 118, article e2017962118 (2021)
44.
go back to reference Novikov, F.N., Stroylov, V.S., v. Svitanko, I., Nebolsin, V.E.: Molecular basis of COVID-19 pathogenesis. Russ. Chem. Rev. 89, 858–878 (2020) Novikov, F.N., Stroylov, V.S., v. Svitanko, I., Nebolsin, V.E.: Molecular basis of COVID-19 pathogenesis. Russ. Chem. Rev. 89, 858–878 (2020)
45.
go back to reference Nowak, M.A., May, R.M.: Viral Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, New York (2000) MATH Nowak, M.A., May, R.M.: Viral Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, New York (2000) MATH
46.
go back to reference Oliva, R.: On structural dominance analysis. Syst. Dyn. Rev. 36, 8–28 (2020) Oliva, R.: On structural dominance analysis. Syst. Dyn. Rev. 36, 8–28 (2020)
47.
go back to reference Perelson, A.S., Ribeiro, R.M.: Modeling the within-host dynamics of HIV infection. BMC Biol. 11, article 96 (2013) Perelson, A.S., Ribeiro, R.M.: Modeling the within-host dynamics of HIV infection. BMC Biol. 11, article 96 (2013)
48.
go back to reference Saha, A., Saha, B.: Novel coronavirus SARS-CoV-2 (Covid-19) dynamics inside the human body. Rev. Med. Virol. 30, article e2140 (2020) Saha, A., Saha, B.: Novel coronavirus SARS-CoV-2 (Covid-19) dynamics inside the human body. Rev. Med. Virol. 30, article e2140 (2020)
49.
go back to reference Saleh, M., Oliva, R., Davidsen, P., Kampmann, C.E.: Eigenvalue analysis of system dynamics models: another perspective. In: Spencer, R.L. (ed.) Conference Proceedings: 24th International Conference of the System Dynamics Society, July 23–27, 2006. System Dynamics Society, New York (2006) Saleh, M., Oliva, R., Davidsen, P., Kampmann, C.E.: Eigenvalue analysis of system dynamics models: another perspective. In: Spencer, R.L. (ed.) Conference Proceedings: 24th International Conference of the System Dynamics Society, July 23–27, 2006. System Dynamics Society, New York (2006)
50.
go back to reference Sallenave, J.M., Guillot, L.: Innate immune signaling and preteolytic pathways in the resolution or exacerbation of SARS-CoV-2 in COVID-19: key therapeutic targets? Front. Immunol. 11, article 1229 (2020) Sallenave, J.M., Guillot, L.: Innate immune signaling and preteolytic pathways in the resolution or exacerbation of SARS-CoV-2 in COVID-19: key therapeutic targets? Front. Immunol. 11, article 1229 (2020)
51.
go back to reference Walsh, K.A., Jordan, K., Clyne, B., Rohde, D., Drummond, L., et al.: SARS-CoV-2 detection, viral load and infectivity over the course of an infection. J. Infect. 81, 357–371 (2020) Walsh, K.A., Jordan, K., Clyne, B., Rohde, D., Drummond, L., et al.: SARS-CoV-2 detection, viral load and infectivity over the course of an infection. J. Infect. 81, 357–371 (2020)
52.
go back to reference Wang, S., Pan, Y., Wang, Q., Miao, H., Brown, A.N., Rong, L.: Modeling the viral dynamics of SARS-CoV-2 infection. Math. Biosci. 328, article 108438 (2020) Wang, S., Pan, Y., Wang, Q., Miao, H., Brown, A.N., Rong, L.: Modeling the viral dynamics of SARS-CoV-2 infection. Math. Biosci. 328, article 108438 (2020)
53.
go back to reference Wölfel, R., Corman, V.M., Guggemos, W., Seilmaier, M., Zange, S., et al.: Virological assessment of hospitalized patients with COVID-2019. Nature 581, 465–469 (2020) Wölfel, R., Corman, V.M., Guggemos, W., Seilmaier, M., Zange, S., et al.: Virological assessment of hospitalized patients with COVID-2019. Nature 581, 465–469 (2020)
54.
go back to reference Wolter, N., Jassat, W., Walaza, S., Welch, R., Moultrie, H., Groome, M., et al.: Early assessment of the clinical severity of the SARS-CoV-2 Omicron variant in South Africa: a data link study. Lancet 399, 437–446 (2022) Wolter, N., Jassat, W., Walaza, S., Welch, R., Moultrie, H., Groome, M., et al.: Early assessment of the clinical severity of the SARS-CoV-2 Omicron variant in South Africa: a data link study. Lancet 399, 437–446 (2022)
58.
go back to reference Xu, Z., Shi, L., Wang, Y., Zhang, J., Huang, L., Zhang, C., et al.: Pathological findings of COVID-19 associated with acute respiratory distress syndrome. Lancet Respir. Med. 8, 420–422 (2020) Xu, Z., Shi, L., Wang, Y., Zhang, J., Huang, L., Zhang, C., et al.: Pathological findings of COVID-19 associated with acute respiratory distress syndrome. Lancet Respir. Med. 8, 420–422 (2020)
59.
go back to reference Zahedipour, F., Hosseini, S.A., Sathyapalan, T., Majeed, M., Jamialahmadi, T., Al-Rasadi, K., Banach, M., Sahebkar, A.: Potential effects of curcumin in the treatment of COVID-19 infection. Phytother. Res. 34, 2911–2920 (2020) Zahedipour, F., Hosseini, S.A., Sathyapalan, T., Majeed, M., Jamialahmadi, T., Al-Rasadi, K., Banach, M., Sahebkar, A.: Potential effects of curcumin in the treatment of COVID-19 infection. Phytother. Res. 34, 2911–2920 (2020)
60.
go back to reference Zhang, L., Jackson, C.B., Mou, H., Ojha, A., Peng, H., Quinlan, B.D., et al.: SARS-CoV-2 spike-protein D614G mutation increases viron spike density and infectivity. Nature Commun. 11, article 6013 (2020) Zhang, L., Jackson, C.B., Mou, H., Ojha, A., Peng, H., Quinlan, B.D., et al.: SARS-CoV-2 spike-protein D614G mutation increases viron spike density and infectivity. Nature Commun. 11, article 6013 (2020)
61.
go back to reference Zhou, L., Niu, Z., Jiang, X., Zhang, Z., Zheng, Y., et al.: SARS-CoV-2 Tarets by the pscRNA profiling of ACE2, TMPRSS2 and furin proteases. iScience 23, article 101744 (2020) Zhou, L., Niu, Z., Jiang, X., Zhang, Z., Zheng, Y., et al.: SARS-CoV-2 Tarets by the pscRNA profiling of ACE2, TMPRSS2 and furin proteases. iScience 23, article 101744 (2020)
Metadata
Title
Eigenvalue analysis of SARS-CoV-2 viral load data: illustration for eight COVID-19 patients
Author
Till D. Frank
Publication date
04-04-2022
Publisher
Springer International Publishing
Published in
International Journal of Data Science and Analytics
Print ISSN: 2364-415X
Electronic ISSN: 2364-4168
DOI
https://doi.org/10.1007/s41060-022-00319-y

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