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 $W=0$, then $L={\displaystyle \frac{{\mu}_{3}I}{\delta}}$, $V={\displaystyle \frac{k(1-\u03f5)I}{{\mu}_{4}}}$ and$$I\left(\frac{k(1-\u03f5){\beta}_{1}S}{{\mu}_{4}}+{\beta}_{2}S-\frac{({\mu}_{2}+\delta +\rho ){\mu}_{3}}{\delta}\right)=0.$$
- (i)
- When $I=0$, we have $L=V=0$ and according to (3) we get $S={\displaystyle \frac{\sigma}{{\mu}_{1}}}$. Thus, model (1) admits an equilibrium point of the form ${E}_{0}=({\displaystyle \frac{\sigma}{{\mu}_{1}}},0,0,0,0)$. This point is called the infection-free equilibrium which corresponding to the healthy state of the patient.
- (ii)
- When $I\ne 0$, we have $S={\displaystyle \frac{{\mu}_{3}{\mu}_{4}({\mu}_{2}+\delta +\rho )}{\delta k(1-\u03f5){\beta}_{1}+\delta {\mu}_{4}{\beta}_{2}}}$. By summing (3) and (4), we obtain $L={\displaystyle \frac{\sigma -{\mu}_{1}S}{{\mu}_{2}+\delta}}={\displaystyle \frac{\delta \sigma [k(1-\u03f5){\beta}_{1}+{\mu}_{4}{\beta}_{2}]-{\mu}_{1}{\mu}_{3}{\mu}_{4}({\mu}_{2}+\delta +\rho )}{\delta ({\mu}_{2}+\delta )[k(1-\u03f5){\beta}_{1}+{\mu}_{4}{\beta}_{2}]}}$. Since $I>0$, we have $\delta \sigma [k(1-\u03f5){\beta}_{1}+{\mu}_{4}{\beta}_{2}]>{\mu}_{1}{\mu}_{3}{\mu}_{4}({\mu}_{2}+\delta +\rho )$. This leads to ${\mathcal{R}}_{0}>1$, where$${\mathcal{R}}_{0}={\displaystyle \frac{\sigma \delta [k(1-\u03f5){\beta}_{1}+{\mu}_{4}{\beta}_{2}]}{{\mu}_{1}{\mu}_{3}{\mu}_{4}({\mu}_{2}+\delta +\rho )}}.$$The threshold parameter ${\mathcal{R}}_{0}$ 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 ${\mathcal{R}}_{01}+{\mathcal{R}}_{02}$, where ${\mathcal{R}}_{01}={\displaystyle \frac{k\delta \sigma {\beta}_{1}(1-\u03f5)}{{\mu}_{1}{\mu}_{2}{\mu}_{4}({\mu}_{2}+\delta +\rho )}}$ is the basic reproduction number of the virus-to-cell transmission mode and ${\mathcal{R}}_{02}={\displaystyle \frac{\sigma \delta {\beta}_{2}}{{\mu}_{1}{\mu}_{3}({\mu}_{2}+\delta +\rho )}}$ is the basic reproduction number of the cell-to-cell transmission mode.When ${\mathcal{R}}_{0}>1$, model (1) has another biological equilibrium called the infection equilibrium without humoral immunity of the form ${E}_{1}=({S}_{1},{L}_{1},{I}_{1},{V}_{1},0)$, where ${S}_{1}={\displaystyle \frac{\sigma}{{\mu}_{1}{\mathcal{R}}_{0}}}$, ${L}_{1}={\displaystyle \frac{\sigma ({\mathcal{R}}_{0}-1)}{({\mu}_{2}+\delta ){\mathcal{R}}_{0}}}$, ${I}_{1}={\displaystyle \frac{\delta \sigma ({\mathcal{R}}_{0}-1)}{{\mu}_{3}({\mu}_{2}+\delta ){\mathcal{R}}_{0}}}$ and ${V}_{1}={\displaystyle \frac{k\delta \sigma (1-\u03f5)({\mathcal{R}}_{0}-1)}{{\mu}_{3}{\mu}_{4}({\mu}_{2}+\delta ){\mathcal{R}}_{0}}}$.
- If $W\ne 0$, then $V={\displaystyle \frac{{\mu}_{5}}{r}}$. It follows from (3) to (6) that $L={\displaystyle \frac{\sigma -{\mu}_{1}S}{{\mu}_{2}+\delta}}$, $I={\displaystyle \frac{\delta (\sigma -{\mu}_{1}S)}{{\mu}_{3}({\mu}_{2}+\delta )}}$, $W={\displaystyle \frac{rk\delta (1-\u03f5)(\sigma -{\mu}_{1}S)-{\mu}_{3}{\mu}_{4}{\mu}_{5}({\mu}_{2}+\delta )}{p{\mu}_{3}{\mu}_{5}({\mu}_{2}+\delta )}}$ and$$\frac{{\mu}_{5}({\mu}_{2}+\delta ){\beta}_{1}S}{r(1+{q}_{1}W)}}+{\displaystyle \frac{\delta {\beta}_{2}S(\sigma -{\mu}_{1}S)}{{\mu}_{3}(1+{q}_{2}W)}}=({\mu}_{2}+\delta +\rho )(\sigma -{\mu}_{1}S).$$Since $W\ge 0$, we have $S\le {\displaystyle \frac{\sigma}{{\mu}_{1}}}-{\displaystyle \frac{{\mu}_{3}{\mu}_{4}{\mu}_{5}({\mu}_{2}+\delta )}{rk\delta {\mu}_{1}(1-\u03f5)}}$. This implies that there is no biological equilibrium when $S>{\displaystyle \frac{\sigma}{{\mu}_{1}}}-{\displaystyle \frac{{\mu}_{3}{\mu}_{4}{\mu}_{5}({\mu}_{2}+\delta )}{rk\delta {\mu}_{1}(1-\u03f5)}}$ or $\frac{\sigma}{{\mu}_{1}}}-{\displaystyle \frac{{\mu}_{3}{\mu}_{4}{\mu}_{5}({\mu}_{2}+\delta )}{rk\delta {\mu}_{1}(1-\u03f5)}}\le 0$. Let ${s}^{*}={\displaystyle \frac{\sigma}{{\mu}_{1}}}-{\displaystyle \frac{{\mu}_{3}{\mu}_{4}{\mu}_{5}({\mu}_{2}+\delta )}{rk\delta {\mu}_{1}(1-\u03f5)}}$ and $\mathcal{F}$ be the function defined on the closed interval $[0,{s}^{*}]$ as follows$$\begin{array}{ccc}\hfill \mathcal{F}\left(S\right)& =& {\displaystyle \frac{{\mu}_{5}({\mu}_{2}+\delta ){\beta}_{1}S}{r(1+{q}_{1}g\left(S\right))}}+{\displaystyle \frac{\delta {\beta}_{2}S(\sigma -{\mu}_{1}S)}{{\mu}_{3}(1+{q}_{2}g\left(S\right))}}-({\mu}_{2}+\delta +\rho )(\sigma -{\mu}_{1}S),\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mathcal{F}}^{\prime}\left(S\right)& =& {\displaystyle \frac{{\mu}_{5}({\mu}_{2}+\delta ){\beta}_{1}\left(1+{q}_{1}g\left(S\right)-{q}_{1}S{g}^{\prime}\left(S\right)\right)}{r{\left(1+{q}_{1}g\left(S\right)\right)}^{2}}}\hfill \\ & & +{\displaystyle \frac{\delta {\beta}_{2}(\sigma -{\mu}_{1}S)\left(1+{q}_{2}g\left(S\right)-{q}_{2}S{g}^{\prime}\left(S\right)\right)}{{\mu}_{3}{\left(1+{q}_{2}g\left(S\right)\right)}^{2}}}\hfill \\ & & +{\mu}_{1}\left({\mu}_{2}+\delta +\rho -{\displaystyle \frac{\delta {\beta}_{2}S}{{\mu}_{3}(1+{q}_{2}g\left(S\right))}}\right).\hfill \end{array}$$When the humoral immune response has not been established, we have $r{V}_{1}-{\mu}_{5}\le 0$. Hence, we define another threshold parameter called the reproduction number for humoral immunity as follows$${\mathcal{R}}_{1}^{W}={\displaystyle \frac{r{V}_{1}}{{\mu}_{5}}},$$As $\mathcal{F}\left({s}^{*}\right)={\displaystyle \frac{{\mu}_{3}{\mu}_{4}{\mu}_{5}^{2}{({\mu}_{2}+\delta )}^{2}[k(1-\u03f5){\beta}_{1}+{\mu}_{4}{\beta}_{2}]}{\delta {\mu}_{1}{r}^{2}{k}^{2}{(1-\u03f5)}^{2}}}\left({\mathcal{R}}_{1}^{W}-1\right)>0$ if ${\mathcal{R}}_{1}^{W}>1$, we deduce that there exists a ${S}_{2}\in (0,{s}^{*})$ such that $\mathcal{F}\left({S}_{2}\right)=0$. Further, we have ${\mathcal{F}}^{\prime}\left({S}_{2}\right)>0$. This establishes the uniqueness of ${S}_{2}$ and therefore model (1) has an unique infection equilibrium point with humoral immunity ${E}_{2}=({S}_{2},{L}_{2},{I}_{2},{V}_{2},{W}_{2})$ when ${\mathcal{R}}_{1}^{W}>1$, where ${S}_{2}\in \left(0,{s}^{*}\right)$, ${L}_{2}={\displaystyle \frac{\sigma -{\mu}_{1}{S}_{2}}{{\mu}_{2}+\delta}}$ ${I}_{2}={\displaystyle \frac{\delta (\sigma -{\mu}_{1}{S}_{2})}{{\mu}_{3}({\mu}_{2}+\delta )}}$, ${V}_{2}={\displaystyle \frac{{\mu}_{5}}{r}}$ and ${W}_{2}={\displaystyle \frac{rk\delta (1-\u03f5)(\sigma -{\mu}_{1}{S}_{2})-{\mu}_{3}{\mu}_{4}{\mu}_{5}({\mu}_{2}+\delta )}{p{\mu}_{3}{\mu}_{5}({\mu}_{2}+\delta )}}$.
- (i)
- If ${\mathcal{R}}_{0}\le 1$, then model (1) has a unique infection-free equilibrium ${E}_{0}=({S}_{0},0,0,0,0)$, where ${S}_{0}={\displaystyle \frac{\sigma}{{\mu}_{1}}}$.
- (ii)
- If ${\mathcal{R}}_{0}>1$, then model (1) has a unique infection equilibrium without humoral immunity ${E}_{1}=({S}_{1},{L}_{1},{I}_{1},{V}_{1},0)$ besides ${E}_{0}$, where ${S}_{1}={\displaystyle \frac{\sigma}{{\mu}_{1}{\mathcal{R}}_{0}}}$, ${L}_{1}={\displaystyle \frac{\sigma ({\mathcal{R}}_{0}-1)}{({\mu}_{2}+\delta ){\mathcal{R}}_{0}}}$, ${I}_{1}={\displaystyle \frac{\delta \sigma ({\mathcal{R}}_{0}-1)}{{\mu}_{3}({\mu}_{2}+\delta ){\mathcal{R}}_{0}}}$ and ${V}_{1}={\displaystyle \frac{k\delta \sigma (1-\u03f5)({\mathcal{R}}_{0}-1)}{{\mu}_{3}{\mu}_{4}({\mu}_{2}+\delta ){\mathcal{R}}_{0}}}$.
- (iii)
- If ${\mathcal{R}}_{1}^{W}>1$, then model (1) has a unique infection equilibrium with humoral immunity ${E}_{2}=({S}_{2},{L}_{2},{I}_{2},{V}_{2},{W}_{2})$ besides ${E}_{0}$ and ${E}_{1}$, where${S}_{2}\in \left(0,{\displaystyle \frac{\sigma}{{\mu}_{1}}}-{\displaystyle \frac{{\mu}_{3}{\mu}_{4}{\mu}_{5}({\mu}_{2}+\delta )}{rk\delta {\mu}_{1}(1-\u03f5)}}\right)$, ${L}_{2}={\displaystyle \frac{\sigma -{\mu}_{1}{S}_{2}}{{\mu}_{2}+\delta}}$${I}_{2}={\displaystyle \frac{\delta (\sigma -{\mu}_{1}{S}_{2})}{{\mu}_{3}({\mu}_{2}+\delta )}}$, ${V}_{2}={\displaystyle \frac{{\mu}_{5}}{r}}$ and ${W}_{2}={\displaystyle \frac{rk\delta (1-\u03f5)(\sigma -{\mu}_{1}{S}_{2})-{\mu}_{3}{\mu}_{4}{\mu}_{5}({\mu}_{2}+\delta )}{p{\mu}_{3}{\mu}_{5}({\mu}_{2}+\delta )}}$.
3. Analytical Results
3.1. Well-Posedness
3.2. Stability Analysis
- (i)
- ${E}_{1}$ is globally asymptotically stable if ${\mathcal{R}}_{1}^{W}\le 1<{\mathcal{R}}_{0}$ and $\rho =0$.
- (ii)
- ${E}_{1}$ is globally asymptotically stable if ${\mathcal{R}}_{1}^{W}\le 1<{\mathcal{R}}_{0}$ 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 |
---|---|---|---|
$\sigma $ | Epithelial cells | $57.757$–$1.2\times {10}^{4}$ | [9] |
production rate | cells mL${}^{-1}$ day${}^{-1}$ | ||
${\mu}_{1}$ | Death rate of uninfected | ${10}^{-3}$ day${}^{-1}$ | [35] |
epithelial cells | |||
${\beta}_{1}$ | Virus-to-cell infection rate | $3.2\times {10}^{-8}$–$4.5\times {10}^{-5}$ | Estimated |
mL virion${}^{-1}$ day${}^{-1}$ | |||
${\beta}_{2}$ | Cell-to-cell infection rate | 0–1 mL cell${}^{-1}$ day${}^{-1}$ | Assumed |
${\mu}_{2}$ | Death rate of latently | $0.08$–$0.59$ day${}^{-1}$ | Assumed |
infected epithelial cells | |||
$\delta $ | Rate to become productively | 1–$7.88$ day${}^{-1}$ | Estimated |
infected cells | |||
${\mu}_{3}$ | Death rate of productively | $0.6$–$5.2$ day${}^{-1}$ | Estimated |
infected epithelial cells | |||
k | Virion production rate per | $22.71$–580 | Estimated |
infected epithelial cell | virions cell${}^{-1}$ day${}^{-1}$ | ||
${\mu}_{4}$ | Virus clearance rate | $2.44$–20 day${}^{-1}$ | Estimated |
r | Activation rate of | 0–1 mL virion${}^{-1}$ day${}^{-1}$ | Assumed |
antibodies | |||
${\mu}_{5}$ | Death rate of antibodies | 0–1 day${}^{-1}$ | Assumed |
p | Neutralization rate of | 0–1 | Assumed |
virus by antibodies | mL molecules${}^{-1}$ day${}^{-1}$ | ||
${q}_{1}$ | Non-lytic strength against | 0–1 mL molecules${}^{-1}$ | Assumed |
virus-to-cell infection | |||
${q}_{2}$ | Non-lytic strength against | 0–1 mL molecules${}^{-1}$ | Assumed |
cell-to-cell infection | |||
$\rho $ | Cure rate of latently | 0–1 day${}^{-1}$ | Assumed |
infected cells | |||
$\u03f5$ | Effectiveness of | 0–1 | Assumed |
antiviral treatment |
Parameter | Value | Sensitivity Index |
---|---|---|
$\sigma $ | 500 | 1 |
${\mu}_{1}$ | 0.001 | −1 |
${\beta}_{1}$ | 0.0000011 | 0.88 |
${\beta}_{2}$ | 0.00000012 | 0.0115 |
${\mu}_{2}$ | 0.088 | −0.687 |
$\delta $ | 4.5 | 0.844 |
${\mu}_{3}$ | 0.088 | −1 |
k | 88 | 0.88 |
${\mu}_{4}$ | 10 | −0.885 |
$\rho $ | 0.02 | −0.156 |
$\u03f5$ | 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