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Stability and bifurcation analysis of a nested multi-scale model for COVID-19 viral infection Cover

Stability and bifurcation analysis of a nested multi-scale model for COVID-19 viral infection

Computational and Mathematical Biophysics
Volume 12 (2024): Issue 1 (January 2024)
By: Bishal Chhetri and  Krishna Kiran Vamsi Dasu  
Open Access
|Jun 2024

Figures & Tables

Figure 1

Local asymptotic stability of E 0 {E}_{0} of Systems (2.18)–(2.20).
Local asymptotic stability of E 0 {E}_{0} of Systems (2.18)–(2.20).

Figure 2

Global asymptotic stability of E 0 {E}_{0} of Systems (2.18)–(2.20).
Global asymptotic stability of E 0 {E}_{0} of Systems (2.18)–(2.20).

Figure 3

Local asymptotic stability of E 1 {E}_{1} of Systems (2.18)–(2.20) starting from the initial state ( S 0 , E 0 , I 0 ) = ( 1,000 , 100 , 50 ) . \left({S}_{0},{E}_{0},{I}_{0})=\left(\hspace{0.1em}\text{1,000}\hspace{0.1em},\hspace{0.33em}100,50).
Local asymptotic stability of E 1 {E}_{1} of Systems (2.18)–(2.20) starting from the initial state ( S 0 , E 0 , I 0 ) = ( 1,000 , 100 , 50 ) . \left({S}_{0},{E}_{0},{I}_{0})=\left(\hspace{0.1em}\text{1,000}\hspace{0.1em},\hspace{0.33em}100,50).

Figure 4

Trans-critical bifurcation exhibited by Systems (2.18)–(2.20) at R 0 = 1 . {R}_{0}=1. The change in the stability of the equilibria with variation in R 0 {R}_{0} can be observed.
Trans-critical bifurcation exhibited by Systems (2.18)–(2.20) at R 0 = 1 . {R}_{0}=1. The change in the stability of the equilibria with variation in R 0 {R}_{0} can be observed.

Figure 5

Heat plots for parameters μ \mu and d d .
Heat plots for parameters μ \mu and d d .

Figure 6

Heat plots for parameters π \pi and d d .
Heat plots for parameters π \pi and d d .

Figure 7

Heat plots for parameters μ \mu and π \pi .
Heat plots for parameters μ \mu and π \pi .

Figure 8

Heat plots for parameters β \beta and d d .
Heat plots for parameters β \beta and d d .

Figure 9

Effect of variation of burst rate of virus on infected population.
Effect of variation of burst rate of virus on infected population.

Figure 10

Effect of variation of rate of clearance of infected cells by the immune system on infected population.
Effect of variation of rate of clearance of infected cells by the immune system on infected population.

Figure 11

Effect of variation of rate of clearance of virus by the immune system on infected population.
Effect of variation of rate of clearance of virus by the immune system on infected population.

Parameter values

SymbolsValuesSource
ω \omega 2[10]
k k 0.05[10]
μ c {\mu }_{c} 0.1 0.1 [10]
μ v {\mu }_{v} 0.1[10]
α \alpha 0.24 0.24 [28]
x x 0.795[10]
y y 0.56[10]
Λ \Lambda μ N ( 0 ) = 71.3 \mu N\left(0)=71.3 [39]
β \beta 0.0115[39]
μ \mu 0.062[34]
π \pi 0.09[39]
d d 0.0018[39]
γ 1 , γ 2 {\gamma }_{1},{\gamma }_{2} 0.05, 0.0714[39]

Meanings of the parameters

ParametersBiological meaning
ω \omega Natural birth rate of cells
Λ \Lambda Birth rate of human population
α \alpha Burst rate of the virus
μ \mu Natural death rate of human population
μ c {\mu }_{c} Natural death rate of cells
μ v {\mu }_{v} Natural death rate of virus
π \pi Infection rate of exposed population
k k Infection rate of susceptible cell
γ 1 , γ 2 {\gamma }_{1},{\gamma }_{2} Recovery rate of the exposed and infected human population
β \beta Transmission rate of the susceptible human population
x x Rate at which the infected cells are cleared by the individual immune response
y y Rate at which the virus particles are cleared by the individual immune response
d d Death rate of the infected human population

Elasticity indices of R 0 {R}_{0}

ParametersElastic indexValue
β \beta ϕ β R 0 {\phi }_{\beta }^{{R}_{0}} 1
π \pi ϕ π R 0 {\phi }_{\pi }^{{R}_{0}} 1
Λ \Lambda ϕ Λ R 0 {\phi }_{\Lambda }^{{R}_{0}} 1
μ \mu ϕ μ R 0 {\phi }_{\mu }^{{R}_{0}} ‒ 0.2785 ‒0.2785
γ 1 {\gamma }_{1} ϕ γ 1 R 0 {\phi }_{{\gamma }_{1}}^{{R}_{0}} ‒ 0.3196 ‒0.3196
γ 2 {\gamma }_{2} ϕ γ 2 R 0 {\phi }_{{\gamma }_{2}}^{{R}_{0}} ‒ 0.00082 ‒0.00082
N h {N}_{h} ϕ N h R 0 {\phi }_{{N}_{h}}^{{R}_{0}} 0.0018
d d ϕ d R 0 {\phi }_{d}^{{R}_{0}} − 0.99 -0.99

Comparative effectiveness for ℛ 0 {{\mathcal{ {\mathcal R} }}}_{0}

No.Indicator% AgeCEL% AgeCEM% AgeCEH
1 ℛ 0 {{\mathcal{ {\mathcal R} }}}_{0} 010101
2 ℛ E ρ {{\mathcal{ {\mathcal R} }}}_{{E}_{\rho }} 305605905
3 ℛ E δ {{\mathcal{ {\mathcal R} }}}_{{E}_{\delta }} 0.10630.2420.454
4 ℛ E ε {{\mathcal{ {\mathcal R} }}}_{{E}_{\varepsilon }} 0.07520.2630.383
5 ℛ E ρ δ {{\mathcal{ {\mathcal R} }}}_{{E}_{\rho \delta }} 30.07760.12790.457
6 ℛ E ρ ε {{\mathcal{ {\mathcal R} }}}_{{E}_{\rho \varepsilon }} 30.05660.1690.236
7 ℛ E ε δ {{\mathcal{ {\mathcal R} }}}_{{E}_{\varepsilon \delta }} 0.2240.3540.672
8 ℛ E ρ δ ε {{\mathcal{ {\mathcal R} }}}_{{E}_{\rho \delta \varepsilon }} 30.16860.34890.58

Initial values of the variables

VariableInitial valuesSource
U ( s ) U\left(s) 3.2 × 1 0 5 3.2\times 1{0}^{5} [10]
U * ( s ) {U}^{* }\left(s) 0[10]
V ( s ) V\left(s) 5.2[10]
S ( t ) S\left(t) 1,000Assumed
E ( t ) E\left(t) 100Assumed
I ( t ) I\left(t) 50Assumed
DOI: https://doi.org/10.1515/cmb-2024-0006 | Journal eISSN: 2544-7297
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Language: English
Submitted on: Dec 21, 2023
Accepted on: May 6, 2024
Published on: Jun 21, 2024
Published by: Sciendo
In partnership with: Paradigm Publishing Services
Keywords:
multi-scale model,
well-posedness,
elasticity,
comparative effectiveness,
heat plot
Related subjects:
Physics,
Biophysics,
Mathematics,
Applied mathematics,
Life sciences,
Molecular biology,
Chemistry,
Computational chemistry and molecular modeling

© 2024 Bishal Chhetri, Krishna Kiran Vamsi Dasu, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 License.

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