Abstract
The SARS-CoV-2 virus is the source of the contagious illness, COVID-19, which is still a problem in
South Africa. In this thesis, three mathematical models for the dynamics of COVID-19 incorporate the
application to optimal control, modelling the effects of stigma, and comparing the dynamics of COVID-19
in deterministic and stochastic models to understand the impact of stochasticity in COVID-19 disease dynamics.
The exploratory data analysis was obtained to indicate hidden features in the COVID-19 data of
South Africa. The mathematical models were analysed, and equilibrium points and their basic reproduction
numbers, R0, were determined. Using a Lyapunov function, we determined the global stability of the
equilibrium points. An optimal control problem was solved using the Pontryagain Maximum Principle.
The models were fitted to the data to obtain parameter values. The continuous-time Markov chain model
for the dynamics of COVID-19 was developed and analysed. The probability of disease persistence and extinction
using the Bienaym´e–Galton–Watson branching processes theory was calculated. The expectation
matrix of the Bienaym´e–Galton–Watson branching process was evaluated. The results of the numerical
simulations for the first model point to the use of masks and physical separation as the best strategies for
reducing COVID-19 infection, followed by control of testing and screening. These strategies will also
be more successful in reducing the rate at which humans who are not infected contract the disease. In
the second model, we see either low, moderate, or high; stigma significantly sustains COVID-19 in the
population. Thus, by informing people about the illness, COVID-19 eradication programs should lower
COVID-19-related stigma. Lastly, the third model reveals that stigmatized and self-isolated individuals
have an impact on spreading COVID-19 disease.
Keywords: COVID-19 · Optimal control · Stigma · Isolation · Mathematical model · Basic reproduction
number · Sensitivity analysis · Model fitting · Stochastic model · Branching process.