Mathematical Modeling and Analysis of the Dynamics of RNA Viruses in Presence of Immunity and Treatment: A Case Study of SARS-CoV-2
Abstract
:1. Introduction
2. Materials and Methods
2.1. Mathematical Formulation
2.2. Equilibria and Threshold Parameters
- If , then , and
- (i)
- (ii)
- The threshold parameter is called the basic reproduction number. Biologically, this threshold parameter represents the average number of secondary infections produced by one productively infected cell at the beginning of infection. It can be rewritten as , where is the basic reproduction number of the virus-to-cell transmission mode and is the basic reproduction number of the cell-to-cell transmission mode.When , model (1) has another biological equilibrium called the infection equilibrium without humoral immunity of the form , where , , and .
- Since , we have . This implies that there is no biological equilibrium when or . Let and be the function defined on the closed interval as followsWhen the humoral immune response has not been established, we have . Hence, we define another threshold parameter called the reproduction number for humoral immunity as followsAs if , we deduce that there exists a such that . Further, we have . This establishes the uniqueness of and therefore model (1) has an unique infection equilibrium point with humoral immunity when , where , , and .
3. Analytical Results
3.1. Well-Posedness
3.2. Stability Analysis
- (i)
- is globally asymptotically stable if and .
- (ii)
- is globally asymptotically stable if and δ is sufficiently large.
4. Numerical Results
4.1. Parameters Estimation
4.2. Sensitivity Analysis
4.3. Numerical Simulation
5. Conclusions and Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameter | Definition | Value | Source |
---|---|---|---|
Epithelial cells | – | [9] | |
production rate | cells mL day | ||
Death rate of uninfected | day | [35] | |
epithelial cells | |||
Virus-to-cell infection rate | – | Estimated | |
mL virion day | |||
Cell-to-cell infection rate | 0–1 mL cell day | Assumed | |
Death rate of latently | – day | Assumed | |
infected epithelial cells | |||
Rate to become productively | 1– day | Estimated | |
infected cells | |||
Death rate of productively | – day | Estimated | |
infected epithelial cells | |||
k | Virion production rate per | –580 | Estimated |
infected epithelial cell | virions cell day | ||
Virus clearance rate | –20 day | Estimated | |
r | Activation rate of | 0–1 mL virion day | Assumed |
antibodies | |||
Death rate of antibodies | 0–1 day | Assumed | |
p | Neutralization rate of | 0–1 | Assumed |
virus by antibodies | mL molecules day | ||
Non-lytic strength against | 0–1 mL molecules | Assumed | |
virus-to-cell infection | |||
Non-lytic strength against | 0–1 mL molecules | Assumed | |
cell-to-cell infection | |||
Cure rate of latently | 0–1 day | Assumed | |
infected cells | |||
Effectiveness of | 0–1 | Assumed | |
antiviral treatment |
Parameter | Value | Sensitivity Index |
---|---|---|
500 | 1 | |
0.001 | −1 | |
0.0000011 | 0.88 | |
0.00000012 | 0.0115 | |
0.088 | −0.687 | |
4.5 | 0.844 | |
0.088 | −1 | |
k | 88 | 0.88 |
10 | −0.885 | |
0.02 | −0.156 | |
0.2 | −0.2212 |
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Hattaf, K.; El Karimi, M.I.; Mohsen, A.A.; Hajhouji, Z.; El Younoussi, M.; Yousfi, N. Mathematical Modeling and Analysis of the Dynamics of RNA Viruses in Presence of Immunity and Treatment: A Case Study of SARS-CoV-2. Vaccines 2023, 11, 201. https://doi.org/10.3390/vaccines11020201
Hattaf K, El Karimi MI, Mohsen AA, Hajhouji Z, El Younoussi M, Yousfi N. Mathematical Modeling and Analysis of the Dynamics of RNA Viruses in Presence of Immunity and Treatment: A Case Study of SARS-CoV-2. Vaccines. 2023; 11(2):201. https://doi.org/10.3390/vaccines11020201
Chicago/Turabian StyleHattaf, Khalid, Mly Ismail El Karimi, Ahmed A. Mohsen, Zakaria Hajhouji, Majda El Younoussi, and Noura Yousfi. 2023. "Mathematical Modeling and Analysis of the Dynamics of RNA Viruses in Presence of Immunity and Treatment: A Case Study of SARS-CoV-2" Vaccines 11, no. 2: 201. https://doi.org/10.3390/vaccines11020201