COVID-19 transmission dynamics and the impact of vaccination: modelling, analysis and simulations

Abstract

Despite the lifting of COVID-19 restrictions, the COVID-19 pandemic and its effects remain a global challenge including the sub-Saharan Africa (SSA) region. Knowledge of the COVID-19 dynamics and its potential trends amidst variations in COVID-19 vaccine coverage is therefore crucial for policy makers in the SSA region where vaccine uptake is generally lower than in high-income countries. Using a compartmental epidemiological model, this study aims to forecast the potential COVID-19 trends and determine how long a wave could be, taking into consideration the current vaccination rates. The model is calibrated using South African reported data for the first four waves of COVID-19, and the data for the fifth wave are used to test the validity of the model forecast. The model is qualitatively analysed by determining equilibria and their stability, calculating the basic reproduction number $R_0$ and investigating the local and global sensitivity analysis with respect to $R_0$. The impact of vaccination and control interventions are investigated via a series of numerical simulations. Based on the fitted data and simulations, we observed that massive vaccination would only be beneficial (deaths averting) if a highly effective vaccine is used, particularly in combination with non-pharmaceutical interventions. Furthermore, our forecasts demonstrate that increased vaccination coverage in SSA increases population immunity leading to low daily infection numbers in potential future waves. Our findings could be helpful in guiding policy makers and governments in designing vaccination strategies and the implementation of other COVID-19 mitigation strategies.

Keywords: COVID-19; bifurcation analysis; mathematical modelling; parameter estimation; sensitivity analysis; vaccinations.

Figure 1.

A schematic diagram of the model framework.

Figure 2.

Simulation output results using fitted parameter values in the first wave shown in table 2. (a) Simulation output plotted alongside cumulative recoveries. (b) Simulation output plotted alongside daily reported cases.

Figure 3.

Simulation output results using fitted parameter values in the second wave shown in table 2. (a) Simulation output plotted alongside cumulative recoveries. (b) Simulation output plotted alongside daily reported cases.

Figure 4.

Simulation output results using fitted parameter values in the third wave shown in table 2. (a) Simulation output plotted alongside cumulative recoveries. (b) Simulation output plotted alongside daily reported cases.

Figure 5.

Simulation output results using fitted parameter values in the fourth wave shown in table 2. (a) Simulation output plotted alongside cumulative recoveries. (b) Simulation output plotted alongside daily reported cases.

Figure 6.

Combined waves with parameter values listed in table 2 and λ from equation (2.1), where blue dots are wave joints. (a) Cumulative recovery; (b) daily infection numbers; (c) vaccinated population.

Figure 7.

Simulation results of model (2.2) showing the effect of NPIs (ϕ) on the infected population for the four waves of the South Africa COVID-19 epidemic: (a) first wave, (b) second wave, (c) third wave and (d) fourth wave.

Figure 8.

Effect of varying parameter values of θ and $\epsilon$ which, respectively, represent vaccination rate and vaccination efficacy, on the disease dynamics among infectious population. (a) First wave; (b) second wave; (c) third wave; (d) fourth wave. The figure depicts a reduction in disease incidence with increasing vaccination rates. Both parameters were more influential in the third wave than in the fourth.

Figure 9.

Model forecast of the COVID-19 dynamics to generate the fifth wave, varying the date at which the wave is initiated. The solid red curve shows model simulation to 24 February 2022, the end of region considered for the model fitting, while the dashed curves indicate forecasted dynamics.

Figure 11.

The combined effect of varying vaccination rate, θ, and vaccine efficacy, $\epsilon$, on the forecasted results of model (2.2). (a) Vaccination rate is fixed to θ = 0.01 while varying $\epsilon$; (b) varying ϕ.

Figure 10.

Investigating the effect of varying vaccination rate θ and vaccine efficacy $\epsilon$ on the forecasted COVID-19 dynamics corresponding to model (2.2). (a) Varying vaccination rate, θ; (b) varying vaccine efficacy, $\epsilon$.

Figure 12.

Local sensitivity analysis results of the basic reproduction number ${\mathcal{R}}_0$ (3.2) corresponding to (a) first wave, (b) second wave, (c) third wave and (d) fourth wave. For each wave, the parameters from the respective columns of table 2 were used as nominal parameter values.

Figure 13.

Partial rank correlation coefficient (PRCC) values showing the sensitivity of the basic reproduction number ${\mathcal{R}}_0$ (3.2) with respect to the input parameters.

Figure 14.

Heat map of ${\mathcal{R}}_0$ as a function of control parameters, where ${\mathcal{R}}_0<1$ in the colourless region. (a) ${\mathcal{R}}_0$ as a function of θ and $\epsilon$, (b) ${\mathcal{R}}_0$ as a function of θ and ϕ and (c) ${\mathcal{R}}_0$ as a function of ϕ and $\epsilon$. The remaining parameters are fixed as shown in table 2.