# Assessing the Impact of Vaccination on the Dynamics of COVID-19 in Africa: A Mathematical Modeling Study

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Formulation

- (i)
- Vaccination is administered to unvaccinated individuals that are susceptible, exposed, pre-symptomatic, asymptomatic, and naturally recovered from the virus. The model does not consider the vaccination of symptomatic and confirmed infectious individuals.
- (ii)
- The COVID-19 vaccine administered is imperfect, i.e., it provides only partial protection against COVID-19 infections. Thus, infections for the vaccinated can occur but at a reduced rate compared to that of the unvaccinated susceptible individuals.
- (iii)
- (iv)
- We assume that there is homogeneous mixing among the population, which means that every individual in the community is equally likely to mix and acquire infections from each member when they make contact.
- (v)
- Since the COVID-19 pandemic has persisted for a long time, we include the vital dynamics (birth and natural death) in the model.

#### 2.2. Data

#### 2.3. Model Fitting and Parameter Estimation Procedure

## 3. Results

#### 3.1. Analytical Results

#### 3.1.1. Computation of Control Reproduction Number

- Let $\mathcal{F}$ be a column vector for all new infections and $\mathrm{F}=\Im \mathcal{F}$ be the jacobian of $\mathcal{F}$ at disease-free equilibrium, ${X}_{0}$$$\mathcal{F}=\left(\begin{array}{c}{S}_{u}{\lambda}_{u}\\ 0\\ 0\\ 0\\ 0\\ {S}_{v}{\lambda}_{v}\\ 0\\ 0\\ 0\\ 0\end{array}\right)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}\mathrm{F}=\left(\begin{array}{cccccccccc}0& {A}_{1}& {A}_{2}& {A}_{3}& {A}_{4}& 0& {A}_{5}& {A}_{6}& {A}_{7}& {B}_{8}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& {B}_{1}& {B}_{2}& {B}_{3}& {B}_{4}& 0& {B}_{5}& {B}_{6}& {B}_{7}& {B}_{8}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right)$$$$\begin{array}{ccc}{A}_{1}=\frac{{S}_{u}^{0}{b}_{uu}{\theta}_{{p}_{u}}(1-{\psi}_{u})}{N}\hfill & {A}_{2}=\frac{{S}_{u}^{0}{b}_{uu}{\theta}_{{A}_{u}}(1-{\psi}_{u})}{N}\hfill & {A}_{3}=\frac{{S}_{u}^{0}{b}_{uu}{\theta}_{{s}_{u}}(1-{\psi}_{u})}{N}\hfill \\ {A}_{4}=\frac{{S}_{u}^{0}{b}_{uu}{\theta}_{{c}_{u}}(1-{\psi}_{u})}{N}\hfill & {A}_{5}=\frac{{S}_{u}^{0}{b}_{vu}{\theta}_{pv}(1-{\psi}_{u})}{N}\hfill & {A}_{6}=\frac{{S}_{u}^{0}{b}_{vu}{\theta}_{{c}_{v}}(1-{\psi}_{u})}{N}\hfill \\ {A}_{7}=\frac{{S}_{u}^{0}{b}_{vu}{\theta}_{{s}_{v}}(1-{\psi}_{u})}{N}\hfill & {A}_{8}=\frac{{S}_{u}^{0}{b}_{vu}{\theta}_{{c}_{v}}(1-{\psi}_{u})}{N}\hfill \\ {B}_{1}=\frac{{S}_{v}^{0}{b}_{uv}{\theta}_{{P}_{v}}(1-{\psi}_{v})}{N}\hfill & {B}_{2}=\frac{{S}_{v}^{0}{b}_{uv}{\theta}_{{A}_{v}}(1-{\psi}_{v})}{N}\hfill & {B}_{3}=\frac{{S}_{v}^{0}{b}_{uv}{\theta}_{{S}_{u}}(1-{\psi}_{v})}{N}\hfill \\ {B}_{4}=\frac{{S}_{v}^{0}{b}_{uv}{\theta}_{{C}_{v}}(1-{\psi}_{v})}{N}\hfill & {B}_{5}=\frac{{S}_{v}^{0}{b}_{vv}{\theta}_{{P}_{v}}(1-{\psi}_{v})}{N}\hfill & {B}_{6}=\frac{{S}_{v}^{0}{b}_{vv}{\theta}_{{A}_{v}}(1-{\psi}_{v})}{N}\hfill \\ {B}_{7}=\frac{{S}_{v}^{0}{b}_{vv}{\theta}_{{S}_{v}}(1-{\psi}_{v})}{N}\hfill & {B}_{8}=\frac{{S}_{v}^{0}{b}_{vv}{\theta}_{{C}_{v}}(1-{\psi}_{v})}{N}\hfill \end{array}$$
- Let $\mathcal{V}$ be the matrix of net transitions$$\mathcal{V}=\left(\begin{array}{c}\left({\alpha}_{E}+\nu +\mu \right){E}_{u}\\ -{\alpha}_{E}{E}_{u}+\left({\alpha}_{P}+\mu +\nu \right){I}_{{P}_{u}}\\ -(1-{\rho}_{1}){\alpha}_{P}{I}_{{P}_{u}}+(\mu +\nu +{\gamma}_{{a}_{1}}+{q}_{{a}_{1}}){I}_{{A}_{u}}\\ -{\rho}_{1}{\alpha}_{P}{I}_{{P}_{u}}+(\mu +{\gamma}_{{s}_{1}}+{q}_{{s}_{1}}+{\delta}_{{s}_{1}}){I}_{{S}_{u}}\\ -{q}_{{a}_{1}}{I}_{{A}_{u}}-{q}_{{s}_{1}}{I}_{{S}_{u}}+({\delta}_{{c}_{1}}+{\gamma}_{{c}_{1}}+\mu ){C}_{u}\\ -\nu {E}_{u}+\left({\alpha}_{E}+\mu \right){E}_{v}\\ -\nu {I}_{{P}_{u}}-{\alpha}_{E}{E}_{v}+\left({\alpha}_{P}+\mu \right){I}_{{P}_{v}}\\ -\nu {I}_{{A}_{u}}-(1-{\rho}_{2}){\alpha}_{P}{I}_{{P}_{v}}+(\mu +{\gamma}_{{a}_{2}}+{q}_{{a}_{2}}){I}_{{a}_{v}}\\ -{\rho}_{2}{\alpha}_{P}{I}_{{P}_{v}}-(\mu +{\gamma}_{{s}_{2}}+{q}_{{s}_{2}}+{\delta}_{{s}_{2}}){I}_{{S}_{v}}\\ -{q}_{{a}_{2}}{I}_{{A}_{v}}-{q}_{{s}_{2}}{I}_{{S}_{v}}+({\delta}_{{c}_{2}}+{\gamma}_{{c}_{2}}+\mu ){C}_{v}\end{array}\right)$$
- The Jacobian matrix ($\mathrm{V}=\Im \mathcal{V}$) of matrix V, at disease-free equilibrium, ${X}_{0}$ is given as:$$\mathrm{V}=\left(\begin{array}{cccccccccc}{\mathrm{a}}_{1}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -{\alpha}_{E}& {\mathrm{a}}_{2}& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -{\alpha}_{p}(1-{\rho}_{1})& {\mathrm{a}}_{3}& 0& 0& 0& 0& 0& 0& 0\\ 0& -{\alpha}_{p}{\rho}_{1}& 0& {\mathrm{a}}_{4}& 0& 0& 0& 0& 0& 0\\ 0& -{q}_{{p}_{1}}& -{q}_{{a}_{1}}& -{q}_{{s}_{1}}& {\mathrm{a}}_{5}& 0& 0& 0& 0& 0\\ -\nu & 0& 0& 0& 0& {\mathrm{a}}_{6}& 0& 0& 0& 0\\ 0& -\nu & 0& 0& 0& -{\alpha}_{E}& {\mathrm{a}}_{7}& 0& 0& 0\\ 0& 0& 0& -\nu & 0& 0& -{\alpha}_{p}(1-{\rho}_{2})& {\mathrm{a}}_{8}& 0& 0\\ 0& 0& 0& 0& 0& 0& -{\alpha}_{p}{\rho}_{2}& 0& {\mathrm{a}}_{9}& 0\\ 0& 0& 0& 0& 0& 0& -{q}_{p2}& -{q}_{a2}& -{q}_{s2}& {\mathrm{a}}_{0}\end{array}\right)$$$$\begin{array}{ccc}{a}_{1}={\alpha}_{E}+\mu +\nu \hfill & {a}_{2}={\alpha}_{p}+\mu +\nu +{q}_{p1}\hfill & {a}_{3}={\alpha}_{p}+\mu +\nu +{q}_{p1}\hfill \\ {a}_{4}={\gamma}_{a1}+\mu +\nu +{q}_{a1}\hfill & {a}_{5}={\gamma}_{c1}+\mu +{\delta}_{c1}\hfill & {a}_{6}={\alpha}_{E}+\mu \hfill \\ {a}_{7}={\alpha}_{p}+\mu +{q}_{p2}\phantom{\rule{4pt}{0ex}}\hfill & {a}_{8}={\gamma}_{a2}+\mu +{q}_{a2}\hfill & {a}_{9}={\delta}_{s2}+{\gamma}_{s2}+\mu +{q}_{s2}\hfill \\ {a}_{0}={\gamma}_{c2}+\mu +{\delta}_{c2}.\hfill \end{array}$$

#### 3.1.2. Computation of Basic Reproduction Number

#### 3.2. Numerical Results

#### 3.2.1. Dynamics of COVID-19 Infectious Classes Over Time

#### 3.2.2. Impact of Vaccination on the Control Reproduction Number per Country

#### 3.2.3. Impact of Vaccination on the Transmission Dynamics

#### 3.2.4. Impact of Vaccination on COVID-19 Incidence among the Vaccinated and Unvaccinated Individuals

#### 3.2.5. Impact of Vaccination on COVID-19 Mortality among the Vaccinated and Unvaccinated Individuals

#### 3.2.6. Impact of Vaccine Coverage with Different Levels of Reduction in the Transmission Rate due to NPIs ($\psi $) among Unvaccinated and Vaccinated Individuals

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Selected Countries | Start Date | End Date |
---|---|---|

DR Congo | 25 April 2021 | 30 November 2021 |

Rwanda | 5 March 2021 | 13 December 2021 |

Kenya | 30 May 2021 | 19 October 2021 |

Algeria | 4 January 2021 | 29 October 2021 |

Libya | 13 January 2021 | 13 December 2021 |

Namibia | 25 March 2021 | 25 November 2021 |

South Africa | 18 February 2021 | 8 November 2021 |

Nigeria | 16 May 2021 | 28 November 2021 |

Parameter | Value | References |
---|---|---|

${d}_{u}$ | $1/270\phantom{\rule{4.pt}{0ex}}\mathrm{day}\phantom{\rule{4.pt}{0ex}}{}^{-1}$ | [50] |

${d}_{v}$ | $1/270\phantom{\rule{4.pt}{0ex}}\mathrm{day}\phantom{\rule{4.pt}{0ex}}{}^{-1}$ | [51] |

$\omega $ | $1/180\phantom{\rule{4.pt}{0ex}}\mathrm{day}\phantom{\rule{4.pt}{0ex}}{}^{-1}$ | [52] |

${\alpha}_{E}$ | $3.3\phantom{\rule{4.pt}{0ex}}\mathrm{days}\phantom{\rule{4.pt}{0ex}}$ | [53] |

${\alpha}_{p}$ | $3.2\phantom{\rule{4.pt}{0ex}}\mathrm{days}\phantom{\rule{4.pt}{0ex}}$ | [25] |

${\rho}_{1}$ | 0.6 | [54] |

${\rho}_{2}$ | 0.1 | [55] |

${\gamma}_{{a}_{1}}$ | $1/5\phantom{\rule{4.pt}{0ex}}\mathrm{day}\phantom{\rule{4.pt}{0ex}}{}^{-1}$ | [29,54] |

${\gamma}_{{a}_{2}}$ | $1/3.4\phantom{\rule{4.pt}{0ex}}\mathrm{day}\phantom{\rule{4.pt}{0ex}}{}^{-1}$ | [29,54] |

${\gamma}_{{s}_{1}}$ | $1/10\phantom{\rule{4.pt}{0ex}}\mathrm{day}\phantom{\rule{4.pt}{0ex}}{}^{-1}$ | [29,54] |

${\gamma}_{{s}_{2}}$ | $1/8\phantom{\rule{4.pt}{0ex}}\mathrm{day}\phantom{\rule{4.pt}{0ex}}{}^{-1}$ | [29,54] |

${\gamma}_{{c}_{1}}$ | $1/11\phantom{\rule{4.pt}{0ex}}\mathrm{day}\phantom{\rule{4.pt}{0ex}}{}^{-1}$ | [29,54] |

${\gamma}_{{c}_{2}}$ | $1/10\phantom{\rule{4.pt}{0ex}}\mathrm{day}\phantom{\rule{4.pt}{0ex}}{}^{-1}$ | [29,54] |

**Figure A1.**Plots depicting the fitting of the model to the cumulative confirmed COVID-19 cases for selected African countries.

**Figure A2.**Dynamic trend for the force of infection over time during COVID-19 vaccination. The blue and red curves represent the force of infection for the unvaccinated (${\lambda}_{u}$) and vaccinated (${\lambda}_{v}$) individuals, respectively.

**Figure A3.**Bar plot presenting the estimated infection probabilities for vaccinated and unvaccinated individuals for selected African countries.

**Figure A4.**Plots presenting the estimated relative infectiousness for vaccinated and unvaccinated individuals for selected African countries. (

**A**) Pre-symptomatic infectious, (

**B**) Asymptomatic infectious, (

**C**) Symptomatic infectious, and (

**D**) Confirmed infectious.

Parameters | Countries | |||||||
---|---|---|---|---|---|---|---|---|

Algeria | DR Congo | Kenya | Libya | Namibia | Nigeria | Rwanda | South Africa | |

L.E | 77.5 | 61.6 | 67.5 | 73.4 | 64.9 | 55.8 | 70.0 | 64.9 |

${N}_{0}$ | 44,177,969 | 95,894,118 | 53,005,614 | 6,735,277 | 2,530,151 | 213,401,323 | 13,461,888 | 59,392,255 |

$\mu $ | $3.5\times {10}^{-5}$ | $4.4\times {10}^{-5}$ | $4\times {10}^{-5}$ | $3.7\times {10}^{-5}$ | $4.2\times {10}^{-5}$ | $4.9\times {10}^{-5}$ | $3.9\times {10}^{-5}$ | $4.2\times {10}^{-5}$ |

$\Lambda $ | 6227.61 | 17,591.26 | 10,285.07 | 433.56 | 264.94 | 39,787.80 | 1424.71 | 4999.12 |

VC | 13.62% | 0.15% | 6.38% | 27.5% | 13.69% | 3.00% | 35.5% | 26.6% |

$VP$ | 160 | 220 | 143 | 258 | 246 | 197 | 284 | 264 |

$\nu $ | 0.00085 | 0.00001 | 0.00045 | 0.00107 | 0.00056 | 0.00015 | 0.00125 | 0.00101 |

${\psi}_{1}$ | 0.35 | 0.30 | 0.15 | 0.35 | 0.30 | 0.45 | 0.45 | 0.45 |

${\psi}_{2}$ | 0.25 | 0.25 | 0.18 | 0.25 | 0.25 | 0.25 | 0.25 | 0.45 |

Country | ${\mathit{S}}_{\mathit{u}\mathbf{0}}\left(\times {\mathbf{10}}^{\mathbf{6}}\right)$ | ${\mathit{E}}_{\mathit{u}\mathbf{0}}$ | ${\mathit{I}}_{\mathit{pu}\mathbf{0}}$ | ${\mathit{I}}_{\mathit{Au}\mathbf{0}}$ | ${\mathit{I}}_{\mathit{Su}\mathbf{0}}$ | ${\mathit{R}}_{\mathit{u}\mathbf{0}}$ |

Namibia | 1.66 [1.57, 1.74] | 1500.00 [1365.29, 1634.70] | 700.15 [641.35, 758.9] | 706.06 [675.15, 736.97] | 700.00 [672.02, 727.97] | 144.62 [105.10, 184.14] |

South Africa | 44.87 [43.13, 46.60] | 4763.61 [4179.73, 5347.49] | 5499.92 [5214.45, 5785.39] | 1322.84 [970.52, 1675.16] | 265.16 [223.47, 306.86] | 455.28 [416.41, 494.15] |

Nigeria | 49.94 [30.06, 69.72] | 76,207.26 [72,089.64, 80,324.87] | 39,377.52 [36,424.84, 242,330.20] | 12,989.71 [11,127.50, 14,851.92] | 571.77 [425.02, 718.52] | 346.74 [194.97, 498.51] |

Libya | 4.19 [3.80, 4.58] | 10043.62 [9799.05, 10,288.17] | 7097.94 [6808.63, 7387.25] | 5144.00 [4593.28, 5694.72] | 2500.26 [2252.15, 2748.37] | 1159.89 [1058.219, 1261.57] |

DR.Congo | 7.14 [2.09, 12.19] | 69,889.75 [66,287.98, 73,491.51] | 1002.16 [902.45, 1101.87] | 541.19 [442.15, 640.22] | 784.58 [734.78, 834.37] | 249.44 [230.28, 268.59] |

Rwanda | 13.35 [12.58, 14.13] | 3191.59 [2882.33, 3500.86] | 1120.60 [1041.19, 1200.01] | 1011.78 [858.52, 1165.04] | 741.58 [714.24, 768.93] | 251.03 [208.27, 293.79] |

Algeria | 9.58 [5.42, 13.72] | 93,208.19 [88,602.33, 97,814.04] | 4344.26 [3988.51, 4700.00] | 530.67 [115.97, 945.38] | 581.77 [ 542.56, 620.99] | 100.51 [63.13, 137.89] |

Kenya | 4.32 [4.06, 4.55] | 77,571.99 [68,155.25, 86,988.73] | 44,589.27 [41,245.57, 47,932.96] | 34,186.64 [31,497.56, 36,875.72] | 13,641.08 [12,529.16, 14,753.01] | 962.79 [924.76, 1000.82] |

**Table A5.**Estimated (fitted) parameter values and their $95\%$ CI for the model for each selected country.

Parameter | Namibia | South Africa | Nigeria | Libya |
---|---|---|---|---|

${b}_{uu}$ | 0.69943 [0.66876, 0.73010] | 0.75138 [0.70816, 0.79461] | 0.68895 [0.64075, 0.73716] | 0.34566 [0.31123, 0.38009] |

${b}_{uv}$ | 0.49986 [0.46100, 0.53872] | 0.47487 [0.43008, 0.51967] | 0.29131 [0.25009, 0.33253] | 0.19777 [0.18971, 0.20583] |

${b}_{vu}$ | 0.49949 [0.44789, 0.55109] | 0.69237 [0.62429, 0.76045] | 0.20114 [0.14281, 0.25947] | 0.13348 [0.11327, 0.15354] |

${b}_{vv}$ | 0.09929 [0.09199, 0.10658] | 0.09903 [0.09701, 0.10105] | 0.09582 [0.09138, 0.10026] | 0.05061 [0.04610, 0.05513] |

${\theta}_{Pu}$ | 0.99999 [0.90132, 1.09866] | 0.94265 [0.85095, 1.03434] | 0.81294 [0.69935, 0.92650] | 0.97356 [0.88567, 1.06144] |

${\theta}_{Au}$ | 0.25624 [0.17032, 0.34215] | 0.32720 [0.24394, 0.41045] | 0.32603 [0.22519, 0.42687] | 0.54310 [0.44715, 0.63904] |

${\theta}_{Su}$ | 0.00012 [−0.0679, 0.06822] | 0.11758 [0.02434, 0.21083] | 0.64576 [0.83210, 0.96535] | 0.51002 [0.47033, 0.54970] |

${\theta}_{Cu}$ | 0.01583 [−0.0229, 0.05461] | 0.28121 [0.20859, 0.35382] | 0.88343 [0.79233, 0.97452] | 0.87620 [0.78317, 0.96922] |

${\theta}_{Pv}$ | 0.68982 [0.61379, 0.76585] | 0.69391 [0.61708, 0.77070] | 0.89522 [0.77477, 1.01567] | 0.49071 [0.44117, 0.52506] |

${\theta}_{Av}$ | 0.27906 [0.20984, 0.34828] | 0.59369 [0.52832, 0.65905] | 0.91138 [0.79100, 1.03177] | 0.47209 [0.42120, 0.52289] |

${\theta}_{Sv}$ | 0.19411 [0.12131, 0.26691] | 0.00502 [-0.0760, 0.08605] | 0.85105 [0.75183, 0.95027] | 0.59134 [0.52606, 0.65661] |

${\theta}_{Cv}$ | 0.64782 [0.57741, 0.71824] | 0.21637 [0.13351, 0.29921] | 0.63620 [0.56903, 0.70337] | 0.37854 [0.32847, 0.42861] |

${q}_{p1}$ | 0.00499 [0.00492, 0.00506] | 0.00499 [0.00496, 0.00502] | 0.00006 [−0.0002, 0.00019] | 0.00486 [0.00440, 0.00532] |

${q}_{p2}$ | 0.00067 [0.00063, 0.00072] | 0.00131 [0.00126, 0.00135] | 0.00026 [0.00018, 0.00034] | 0.00139 [0.00129, 0.00149] |

${q}_{a1}$ | 0.00498 [0.00465, 0.00532] | 0.00497 [0.00489, 0.00505] | 0.00027 [−0.00004, 0.0006] | 0.00351 [0.00309, 0.00392] |

${q}_{a2}$ | 0.00395 [0.00371, 0.00419] | 0.00256 [0.00253, 0.00265] | 0.00004 [−0.0003, 0.00040] | 0.00256 [0.00237, 0.00276] |

${q}_{s1}$ | 0.00868 [0.00824, 0.00913] | 0.00839 [0.00829, 0.00851] | 0.00031 [−0.0005, 0.00114] | 0.00862 [0.00792, 0.00933] |

${q}_{s2}$ | 0.00294 [0.00280, 0.00309] | 0.00356 [0.00341, 0.00372] | 0.000004 [−0.0004, 0.0004] | 0.00439 [0.00407, 0.00472] |

${\delta}_{s1}$ | 0.00029 [0.00028, 0.00031] | 0.19703 [0.17974, 0.00211] | 0.00567 [0.00458, 0.00677] | 0.00197 [0.00183, 0.02114] |

${\delta}_{s2}$ | 0.00019 [0.00018, 0.00020] | 0.09581 [0.08846, 0.10316] | 0.00015 [−0.0007, 0.00098] | 0.00099 [0.00182, 0.00217] |

${\delta}_{c1}$ | 0.00040 [0.00038, 0.00042] | 0.19959 [0.17916, 0.22004] | 0.00298 [0.00221, 0.00375] | 0.00199 [0.01899, 0.02094] |

${\delta}_{c2}$ | 0.00019 [0.00018, 0.00020] | 0.02278 [0.01553, 0.03003] | 0.00382 [0.00308, 0.00457] | 0.00099 [0.00090, 0.00108] |

Parameter | Rwanda | Algeria | Kenya | DR Congo |
---|---|---|---|---|

${b}_{uu}$ | 0.56684 [0.54223, 0.59140] | 0.79907 [0.74159, 0.85655] | 0.63721 [0.58228, 0.69215] | 0.59252 [0.55302, 0.63202] |

${b}_{uv}$ | 0.32961 [0.28843, 0.37078] | 0.44401 [0.39858, 0.48943] | 0.69956 [0.66150, 0.73763] | 0.25625 [0.18964, 0.32285] |

${b}_{vu}$ | 0.34471 [0.30639, 0.38305] | 0.54705 [0.48178, 0.61232] | 0.49691 [0.46349, 0.53033] | 0.29970 [0.26933, 0.33008] |

${b}_{vv}$ | 0.18714 [0.17146, 0.20282] | 0.05621 [0.05062, 0.06181] | 0.04087 [0.02406, 0.05769] | 0.01336 [0.00516, 0.02154] |

${\theta}_{Pu}$ | 0.49657 [0.45476, 0.53838] | 0.95831 [0.90050, 1.01612] | 0.46168 [0.42345, 0.49992] | 0.99997 [0.94617, 1.05378] |

${\theta}_{Au}$ | 0.38102 [0.32949, 0.43255] | 0.36306 [0.26965, 0.45647] | 0.24275 [0.21254, 0.27297] | 0.42697 [0.33455, 0.51938] |

${\theta}_{Su}$ | 0.43582 [0.41439, 0.45727] | 0.22221 [0.11050, 0.33392] | 0.31683 [0.26989, 0.36376] | 0.00335 [−0.0765, 0.08316] |

${\theta}_{Cu}$ | 0.41064 [0.35984, 0.46145] | 0.17918 [0.09015, 0.26821] | 0.08219 [0.03981, 0.12458] | 0.10705 [0.05563, 0.15847] |

${\theta}_{Pv}$ | 0.48203 [0.43416, 0.52992] | 0.23469 [0.15878, 0.31059] | 0.45516 [0.41124, 0.49907] | 0.22848 [0.17245, 0.28450] |

${\theta}_{Av}$ | 0.49791 [0.44987, 0.54594] | 0.48487 [0.42802, 0.54172] | 0.25320 [0.23014, 0.27626] | 0.01724 [−0.0390, 0.07350] |

${\theta}_{Sv}$ | 0.35278 [0.31528, 0.39029] | 0.62145 [0.55725, 0.68564] | 0.39507 [0.35426, 0.43587] | 0.48985 [ 0.44160, 0.5380] |

${\theta}_{Cv}$ | 0.03734 [0.00201, 0.07265] | 0.73060 [0.65886, 0.80235] | 0.00433 [−0.0367, 0.04540] | 0.40973 [0.35467, 0.4648] |

${q}_{p1}$ | 0.00019 [0.00018, 0.00022] | 0.00163 [0.00109, 0.00215] | 0.00019 [0.00018, 0.00021] | 0.00073 [0.00053, 0.00092] |

${q}_{p2}$ | 0.00010 [0.00009, 0.00011] | 0.00008 [0.00007, 0.00009] | 0.00006 [0.00005, 0.00010] | 0.00028 [0.00020, 0.00035] |

${q}_{a1}$ | 0.00254 [0.00227, 0.00281] | 0.00025 [−0.0003, 0.00079] | 0.00288 [0.00269, 0.00306] | 0.00095 [0.00077, 0.00113] |

${q}_{a2}$ | 0.00189 [0.00170, 0.00208] | 0.00003 [−0.0002, 0.00026] | 0.00199 [0.00185, 0.00213] | 0.00097 [0.00089, 0.00106] |

${q}_{s1}$ | 0.00018 [−0.0003, 0.00064] | 0.00130 [0.00042, 0.00218] | 0.00192 [0.00171, 0.00213] | 0.000009 [−0.0004, 0.0005] |

${q}_{s2}$ | 0.00091 [0.00086, 0.00096] | 0.003461 [0.0031, 0.00387] | 0.00198 [0.00180, 0.00216] | 0.00001 [0.00000, 0.00002] |

${\delta}_{s1}$ | 0.00023 [0.00020, 0.00025] | 0.00025 [0.00022, 0.00027] | 0.00029 [0.00027, 0.00032] | 0.00026 [0.00023, 0.00028] |

${\delta}_{s2}$ | 0.00019 [0.00018, 0.00020] | 0.00017 [0.00012, 0.00021] | 0.00012 [0.00010, 0.00013] | 0.000001 [0.0000, 0.00002] |

${\delta}_{c1}$ | 0.00021 [0.00019, 0.00023] | 0.00052 [0.00048, 0.00056] | 0.00014 [0.00013, 0.00015] | 0.00029 [0.00028, 0.00031] |

${\delta}_{c2}$ | 0.00019 [0.00019, 0.00021] | 0.00002 [−0.00001, 0.00005] | 0.00006 [0.00005, 0.00010] | 0.00018 [0.00016, 0.00020] |

**Table A7.**Estimated values for the basic and control reproduction numbers and its components for each country with 95% confidence interval.

Parameter | ${\mathit{R}}_{\mathit{c}1}$ | ${\mathit{R}}_{\mathit{c}2}$ | ${\mathit{R}}_{\mathit{c}3}$ | ${\mathit{R}}_{\mathit{c}}$ | ${\mathit{R}}_{0}$ |
---|---|---|---|---|---|

Namibia | 0.813 [0.775, 0.851] | 0.011 [0.009, 0.013] | 0.889 [0.851, 0.927] | 1.713 [1.639, 1.788] | 2.569 [2.449, 2.688] |

South Africa | 0.670 [0.658, 0.683] | 0.002 [0.015, 0.017] | 0.800 [0.786, 0.820] | 1.500 [1.464, 1.515] | 3.131 [3.099, 3.163] |

Nigeria | 0.777 [0.749, 0.829] | 0.021 [0.019, 0.024] | 0.801 [0.772, 0.829] | 1.599 [1.554, 1.643] | 3.157 [3.049, 3.265] |

Libya | 0.699 [0.682, 0.716] | 0.010 [0.009, 0.012] | 0.723 [0.707, 0.739] | 1.432 [1.399, 1.464] | 2.586 [2.525, 2.647] |

DR Congo | 0.841 [0.778, 0.903] | 0.00 [0.000, 0.001] | 0.841 [0.778, 0.903] | 1.682 [1.557, 1.808] | 2.407 [2.228, 2.587] |

Rwanda | 0.853 [0.841, 0.865] | 0.040 [0.036, 0.044] | 0.912 [0.896, 0.928] | 1.806 [1.781, 1.831] | 2.817 [2.777, 2.856] |

Algeria | 0.908 [0.805, 1.011] | 0.007 [0.005, 0.009] | 0.987 [0.8797, 1.094] | 1.902 [1.026, 2.237] | 3.640 [2.893, 3.634] |

Kenya | 0.225 [0.222, 0.227] | 0.836 [0.801, 0.872] | 0.007 [0.004, 0.011] | 1.911 [1.864, 1.986] | 2.438 [2.335, 2.543] |

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**Figure 2.**Evolution trend of each infectious compartment for vaccinated and unvaccinated individuals over time in each country. The first line (purple) in the legend depicts the ratio of the number of confirmed infectious vaccinated individuals to confirmed infectious unvaccinated individuals, the second line (green) depicts the ratio of the number of symptomatic infectious vaccinated individuals to the symptomatic infectious unvaccinated, the third line (red) depicts the ratio of the number of asymptomatic infectious vaccinated individuals to asymptomatic infectious unvaccinated, and the last line (blue) depicts the ratio of the number of pre-symptomatic infectious vaccinated individuals to pre-symptomatic infectious unvaccinated.

**Figure 3.**Vaccine effectiveness against COVID-19 infections and deaths. Panels (

**a–c**) depict vaccine effectiveness against pre-symptomatic, asymptomatic, and symptomatic new infections Panels (

**d,e**) present vaccine effectiveness against symptomatic and confirmed deaths.

**Figure 4.**Contour plots of the control reproduction number (${R}_{c}$) as functions of the vaccine coverage for different levels of reduction in the SARS-CoV-2 transmission rate due to control measures ($\psi $) among the unvaccinated and vaccinated individuals.

State Variable | Description |
---|---|

${S}_{u}\left({S}_{v}\right)$ | Susceptible unvaccinated (vaccinated) population |

${E}_{u}\left({E}_{v}\right)$ | Exposed unvaccinated (vaccinated) population |

${I}_{{p}_{u}}\left({I}_{{p}_{v}}\right)$ | Pre-symptomatic infectious unvaccinated (vaccinated) population |

${I}_{{A}_{u}}\left({I}_{{A}_{v}}\right)$ | Asymptomatic infectious unvaccinated (vaccinated) population |

${I}_{{S}_{u}}\left({I}_{{S}_{v}}\right)$ | Symptomatic infectious unvaccinated (vaccinated) population |

${C}_{u}\left({C}_{v}\right)$ | Confirmed infectious unvaccinated (vaccinated) population |

${R}_{u}\left({R}_{v}\right)$ | Recovered unvaccinated (vaccinated) population |

${D}_{u}\left({D}_{v}\right)$ | COVID-deceased unvaccinated (vaccinated) population |

Parameter | Description | Unit |
---|---|---|

$\Lambda $ | Recruitment rate | Individual ${\mathrm{day}}^{-1}$ |

$\mu $ | Natural death rate | ${\mathrm{day}}^{-1}$ |

$\nu $ | Vaccination rate | ${\mathrm{day}}^{-1}$ |

$\omega $ | Vaccine-derived immunity rate | ${\mathrm{day}}^{-1}$ |

$1/{\alpha}_{E}$ | Latent period | days |

$1/{\alpha}_{p}$ | Pre-symptomatic period | days |

${d}_{u}$ (${d}_{v}$) | Rate at which recovered unvaccinated (vaccinated) individuals from COVID-19 lose acquired immunity | ${\mathrm{day}}^{-1}$ |

${\rho}_{1}\left({\rho}_{2}\right)$ | Proportion of pre-symptomatic infectious unvaccinated (vaccinated), who develop COVID-19 symptoms | dimensionless |

${b}_{ij}$ | Infection probability of a susceptible individual in class i by an infectious individual in class j, for $(i,j\in u,v)$ | dimensionless |

${\delta}_{{s}_{1}}\left({\delta}_{{s}_{2}}\right)$ | COVID-19 death rate of symptomatic infectious unvaccinated (vaccinated) individuals | ${\mathrm{day}}^{-1}$ |

${\delta}_{{c}_{1}}\left({\delta}_{{c}_{2}}\right)$ | COVID-19 death rate of confirmed infectious unvaccinated (vaccinated) individuals | ${\mathrm{day}}^{-1}$ |

${\gamma}_{{a}_{1}}\left({\gamma}_{{a}_{2}}\right)$ | Recovery rate of asymptomatic unvaccinated (vaccinated) individuals | ${\mathrm{day}}^{-1}$ |

${\gamma}_{{s}_{1}}\left({\gamma}_{{s}_{2}}\right)$ | Recovery rate of symptomatic unvaccinated (vaccinated) individuals | ${\mathrm{day}}^{-1}$ |

${\gamma}_{{c}_{1}}\left({\gamma}_{{c}_{2}}\right)$ | Recovery rate of symptomatic unvaccinated (vaccinated) individuals | ${\mathrm{day}}^{-1}$ |

${\theta}_{{p}_{u}}({\theta}_{{A}_{u}},{\theta}_{{S}_{u}},{\theta}_{{C}_{u}})$ | Relative infectiousness of unvaccinated pre-symptomatic (asymptomatic, symptomatic, confirmed) individuals | dimensionless |

${\theta}_{{p}_{v}}({\theta}_{{A}_{v}},{\theta}_{{S}_{v}},{\theta}_{{C}_{v}})$ | Relative infectiousness of unvaccinated pre-symptomatic (asymptomatic, symptomatic, confirmed) individuals | dimensionless |

${q}_{{p}_{1}}({q}_{{a}_{1}},{q}_{{s}_{1}})$ | Per capita rate at which unvaccinated individuals from the pre-symptomatic (asymptomatic, symptomatic) infectious class test positive | ${\mathrm{day}}^{-1}$ |

${q}_{{p}_{2}}({q}_{{a}_{2}},{q}_{{s}_{2}})$ | Per capita rate at which vaccinated individuals from the pre-symptomatic (asymptomatic, symptomatic) infectious class test positive | ${\mathrm{day}}^{-1}$ |

${\delta}_{{s}_{1}}\left({\delta}_{{s}_{2}}\right)$ | COVID-19 induced death rate of unvaccinated (vaccinated) symptomatic infectious individuals | ${\mathrm{day}}^{-1}$ |

${\delta}_{{c}_{1}}\left({\delta}_{{c}_{2}}\right)$ | COVID-19 induced death rate of unvaccinated (vaccinated) confirmed infectious individuals | ${\mathrm{day}}^{-1}$ |

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**MDPI and ACS Style**

Montcho, Y.; Nalwanga, R.; Azokpota, P.; Doumatè, J.T.; Lokonon, B.E.; Salako, V.K.; Wolkewitz, M.; Glèlè Kakaï, R.
Assessing the Impact of Vaccination on the Dynamics of COVID-19 in Africa: A Mathematical Modeling Study. *Vaccines* **2023**, *11*, 857.
https://doi.org/10.3390/vaccines11040857

**AMA Style**

Montcho Y, Nalwanga R, Azokpota P, Doumatè JT, Lokonon BE, Salako VK, Wolkewitz M, Glèlè Kakaï R.
Assessing the Impact of Vaccination on the Dynamics of COVID-19 in Africa: A Mathematical Modeling Study. *Vaccines*. 2023; 11(4):857.
https://doi.org/10.3390/vaccines11040857

**Chicago/Turabian Style**

Montcho, Yvette, Robinah Nalwanga, Paustella Azokpota, Jonas Têlé Doumatè, Bruno Enagnon Lokonon, Valère Kolawole Salako, Martin Wolkewitz, and Romain Glèlè Kakaï.
2023. "Assessing the Impact of Vaccination on the Dynamics of COVID-19 in Africa: A Mathematical Modeling Study" *Vaccines* 11, no. 4: 857.
https://doi.org/10.3390/vaccines11040857