Modeling the transmission dynamics of the COVID-19 Pandemic in South Africa

Abstract

Since its emergence late in 2019, the COVID-19 pandemic continues to exude major public health and socio-economic burden globally. South Africa is currently the epicenter for the pandemic in Africa. This study is based on the use of a compartmental model to analyze the transmission dynamics of the disease in South Africa. A notable feature of the model is the incorporation of the role of environmental contamination by COVID-infected individuals. The model, which is fitted and parametrized using cumulative mortality data from South Africa, is used to assess the impact of various control and mitigation strategies. Rigorous analysis of the model reveals that its associated continuum of disease-free equilibria is globally-asymptotically stable whenever the control reproduction number is less than unity. The epidemiological implication of this result is that the disease will eventually die out, particularly if control measures are implemented early and for a sustainable period of time. For instance, numerical simulations suggest that if the lockdown measures in South Africa were implemented a week later than the 26 March, 2020 date it was implemented, this will result in the extension of the predicted peak time of the pandemic, and causing about 10% more cumulative deaths. In addition to illustrating the effectiveness of self-isolation in reducing the number of cases, our study emphasizes the importance of surveillance testing and contact tracing of the contacts and confirmed cases in curtailing the pandemic in South Africa.

Keywords: COVID-19; Control reproduction number; Environmental contamination; Isolation; Social-distancing.

Fig. 1

Profile of the time-varying effective contact rate ($\beta \left(t\right)$), as a function of time, for various values of the compliance parameter, $\omega$.

Fig. 2

Flow diagram of the model (2).

Fig. 3

Time series plot showing a least square fit of Eq. (4) coupled with system (2), using South Africa COVID-19 reported cases for cumulative number of deaths. Parameter values used are as given in Table 3.

Fig. 4

Simulations of the model (2), showing the decrease in numbers of COVID-19 infected individuals as the social-distancing parameter $\omega$ increases: (A) Exposed, (B) Asymptomatic, (C) Symptomatic, and (D) Isolated individuals, respectively. Parameter values used are as given in Table 3 with various values of $\omega$.

Fig. 5

While the cumulative total number of deaths significantly increases in the absence of interventions i.e.$\omega =0$ (D), there is a decrease in cumulative numbers of carriers of infections (A), recoveries (B), and deaths (D) as $\omega$ increases. Parameter values used are as given in Table 3 with various values of the compliance parameter $\omega$.

Fig. 6

Simulations of the model (2), showing changes in the numbers of COVID-19 infected individuals, as the isolation rates (${\gamma }_1$ and ${\gamma }_2$) vary: (A) Exposed, (B) Asymptomatic, (C) Symptomatic, and (D) Isolated individuals, respectively. Parameter values used are as given in Table 3.

Fig. 7

Simulations of the model (2), showing changes in the cumulative number of COVID-19 affected individuals: (A) Cases and (B) Deaths. Parameter values used are as given in Table 3.

Fig. 8

Simulations of the model (2), showing changes in the number of COVID-19 infected individuals, as the parameter ${\tau }_0$ for the starting time of the lockdown varies:(A) Exposed, (B) Asymptomatic, (C) Symptomatic, and (D) Isolated individuals, respectively. Parameter values used are as given in Table 3.

Fig. 9

Simulations of the model (2) for the computation of the final size relations of the COVID-19 pandemic, namely the number $S_{\infty }$ of susceptible individuals who escaped the epidemic and the attack rate $\alpha$ of the epidemic. (A) In the absence of any control measures ($\beta ={\beta }_0$). (B) Under strict lockdown ($\beta ={\beta }_1$). Parameter values used are as given in Table 3, while initial conditions are $S\left(0\right)=59\times 10^7$, $I\left(0\right)=65$, $E\left(0\right)=A\left(0\right)=J\left(0\right)=P\left(0\right)=R\left(0\right)=0$.

Fig. 10

Simulations of the model (2), showing changes in the cumulative numbers of COVID-19 related cases (A) and deaths (B) as the environmental transmission factor (${\eta }_3$) changes. Parameter values used are as given in Table 3.

Fig. 11

Simulations of the model (2), showing changes in the cumulative numbers of COVID-19 related cases (A) and deaths (B) as the cleaning rate of virus ($\upsilon$) in the environment changes. Parameter values used are as given in Table 3.