Structure and Hierarchy of SARS-CoV-2 Infection Dynamics Models Revealed by Reaction Network Analysis
Abstract
:1. Introduction
2. Materials and Methods: Procedure for the Organizational Analysis
- Step 1—Deriving the set of reactions: Each summand of each ODE (or PDE) is translated into a reaction as illustrated by the transition from Subfigures (a) to (b) in Figure 1. On the left-hand side of each reaction formula, there is a set of species, the so-called support of a reaction. The support of a reaction is the unique set of species that are needed to run the reaction. If only one of the species of the support of the reaction is missing then that reaction is not active. The term (of the ODE (or PDE)) that belongs to that reaction must be zero if and only if the concentration of at least one of the species in the support of that reaction is zero. The number of the appearance of each species of a reaction on the right-hand side of a reaction is bigger or less than the number on the left-hand side depending on whether the regarding term has a positive or negative sign in the ODE (or PDE)) of the regarding species. As an example we consider reaction . The corresponding summand is . It is zero if and only if the concentration of at least one of or is zero. Thus the support of contains exactly the species E and V. On the left-hand side of the reaction equation of the species E resp. V appear only to the power of one because of the power of E resp. V is one in . Since the summand appears only in the ODE of V, namely with a negative sign, the right-hand side of the reaction equation of contains one less of V than the left-hand side. The number of E is equal on both sides of the reaction equation since the amount of E is not affected by the reaction .
- Step 2—Calculating the organizations from the set of reactions: The second step is to compute the organizations (as defined in [21]) from the derived reactions. Each organization consists of a subset of species that is
- closed and
- self-maintaining.
- A subset of species is closed if and only if for each reaction with its support contained in that subset, also all species appearing on the right-hand side of the reaction equation are contained in that subset. In other words, no reaction that is active on the subset S produces a species that is not contained in that subset.
- A subset of species is self-maintaining if and only if there is a feasible flux for S for S such that
- consisting of ODEs but also of PDEs,
- describing in-host dynamics but also host-to-host and mixed (in-host and host-to-host) models.
- Analyzed infection dynamics of SARS-CoV-2 but also compared to Influenza models
3. Analysis of the Models
3.1. In-Host Models
3.2. Host-To-Host Models
3.3. A Linked In-Host/Host-To-Host Model
3.4. Hierarchy of Models
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. List of the Reactions of All Models with Reactions Constants in Brackets
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Peter, S.; Dittrich, P.; Ibrahim, B. Structure and Hierarchy of SARS-CoV-2 Infection Dynamics Models Revealed by Reaction Network Analysis. Viruses 2021, 13, 14. https://doi.org/10.3390/v13010014
Peter S, Dittrich P, Ibrahim B. Structure and Hierarchy of SARS-CoV-2 Infection Dynamics Models Revealed by Reaction Network Analysis. Viruses. 2021; 13(1):14. https://doi.org/10.3390/v13010014
Chicago/Turabian StylePeter, Stephan, Peter Dittrich, and Bashar Ibrahim. 2021. "Structure and Hierarchy of SARS-CoV-2 Infection Dynamics Models Revealed by Reaction Network Analysis" Viruses 13, no. 1: 14. https://doi.org/10.3390/v13010014
APA StylePeter, S., Dittrich, P., & Ibrahim, B. (2021). Structure and Hierarchy of SARS-CoV-2 Infection Dynamics Models Revealed by Reaction Network Analysis. Viruses, 13(1), 14. https://doi.org/10.3390/v13010014