Competition of SARS-CoV-2 Variants in Cell Culture and Tissue: Wins the Fastest Viral Autowave
- Can these experiments be described by the mathematical model (the simplest possible one)?
- Could the situations of one strain overtaking another strain after some prolonged time interval be observed qualitatively if the spatial effects are taken into account?
- Could spatial effects be very important: for example, could they result in any new information that cannot be obtained in the homogeneous system?
2.1. Model Description and Governing Equations
2.2. Steady Travelling Solution
2.3. Two-Strain Model with Competition for Cells
2.4. Model Parameters and Comparison with Experimental Data
- Arbitrary non-dimensional values (typically, of the order of 1) for plotting the explanatory graphs when deriving analytical formulas in the Appendices. For this set, in numerical calculations we take .
- Parameters which correspond to the SARS-CoV-2 variants Delta and Omicron replicating in human nasal epithelial cultures (hNECs) in the in vitro experiment  (see Table 1). We estimated them from physical reasons (e.g., D), derived/estimated from experimental results and kept fixed (e.g., , ), or varied under the strict limitations as described below:
- The diffusion coefficient D was estimated using the Stockes–Einstein formula at 300 K assuming virus diameter 100 nm  and water viscosity.
- The experimental in vitro system in  is considered as homogeneous, because (a) only average concentrations are presented in that article, (b) the full size of the experimental system is 6.5 mm , which is comparable to the width of the front in our numerical calculations (about 1–3 mm, see Figure 5 below), and (c) substantial convection should be expected during inoculation and everyday sampling.
- Since the virus concentration determined by RT-qPCR for the E gene in  (see Figure 2C,D) was approximately 1000 times greater than the PFU concentration for all available time points and cell lineages (crossed vs. filled markers), that is only 1 out of 1000 viral particles was able to effectively infect a cell and reproduce with its help, we compared the model variable with the united data for (filled markers) and (open markers). In particular, we estimated as .
- Initial cell concentration was limited in the range .
- The characteristic time of infection () was assumed to be 1 h for both strains. Characteristic times if transition () and cell death () were limited as 2 h and 0.1 h from below, respectively. Virus death times () were limited in the range [0.1, 100] h.
- To avoid large differences between the corresponding parameters of Delta and Omicron variants, the rations , , and were limited to the range of from to 10.
- After the fitting, all parameters were rounded up to 1 decimal digit, and those that were close to each other were set as equal.
- For the obtained set of kinetic parameters, in spatial numerical calculations (with diffusion) we set L = 2 cm or more to be able to track the transition of autowave to the steady propagation regime.
2.5. Numerical Methods
3.1. Virus Replication Number Provides the Condition for the Infection Progression,
3.2. Estimates of the Steady Wave Speed
3.3. Equations for the Final Concentration of Intact Cells and the Total Spatial Viral Load
3.4. Homogeneous Case without Competition: Omicron Is “Quick” and Wins the Start but Delta Can Overtake It after 1–2 Days’ Lag
3.5. Spatially Distributed Case without Competition: Omicron Can Win the Race despite Low Concentration and
3.6. Spatially Distributed Case with Competition: Omicron Can Win and Completely Suppress Delta
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
|URT||upper respiratory tract|
|LRT||lower respiratory tract|
|hNECs||culture of human nasal epithelial cells|
|KPP or KPPF equation||Kolmogorov–Petrovskii–Piskunov–Fisher equation|
Appendix A.1. Condition for Existence of a Positive Solution to Equation (5)
Appendix A.2. Estimates of the Steady Wave Speed
Appendix A.3. Derivation of Equations for the Final Concentration of Intact Cells and the Total Spatial Viral Load
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|Equal for both strains:|
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Tokarev, A.; Mozokhina, A.; Volpert, V. Competition of SARS-CoV-2 Variants in Cell Culture and Tissue: Wins the Fastest Viral Autowave. Vaccines 2022, 10, 995. https://doi.org/10.3390/vaccines10070995
Tokarev A, Mozokhina A, Volpert V. Competition of SARS-CoV-2 Variants in Cell Culture and Tissue: Wins the Fastest Viral Autowave. Vaccines. 2022; 10(7):995. https://doi.org/10.3390/vaccines10070995Chicago/Turabian Style
Tokarev, Alexey, Anastasia Mozokhina, and Vitaly Volpert. 2022. "Competition of SARS-CoV-2 Variants in Cell Culture and Tissue: Wins the Fastest Viral Autowave" Vaccines 10, no. 7: 995. https://doi.org/10.3390/vaccines10070995