Abstract
The initial data on SARS-CoV-2 (COVID-19) in South Africa showed seasonal transmission patterns of
five waves, with peaks occurring in winter and summer since the outbreaks began. The transmission dynamics
have mainly been driven by variations in environmental factors and virus evolution, and the two
were at the center of driving the different waves of the disease. In this thesis, we use three mathematical
models that incorporate seasonality, control measures, and the role of the environmental reservoir in the
transmission dynamics. In the first model, we considered the role of seasonality in the transmission dynamics
of COVID-19. A compartmental model with a time-dependent transmission rate is formulated and
the steady-state stabilities are analyzed. The model is fitted to data on new cases in South Africa for the
first four waves. The model results indicate the need to consider seasonality in the transmission dynamics
of COVID-19 and its importance in modeling fluctuations in the data for new cases. The potential impact
of seasonality in the transmission patterns of COVID-19 and the public health implications is discussed.
In the second model, a deterministic model incorporating the influences of seasonality, vaccination, and
the role of the environment is formulated to determine how these factors impact the epidemic. We analyze
the model, demonstrating its stabilities. We also demonstrate its application using the data reported by
the National Institute for Communicable Diseases, South Africa. We fitted our mathematical model to
the data from the third to the fifth wave and used a damping effect due to mandatory vaccination in the
fifth wave. Our analytical and numerical results indicate that different efficacies for vaccination have a
different influence on epidemic transmission at different seasonal periods. Our findings also indicate that
as long as the coronavirus persists in the environment, the epidemic will continue to affect the human population
and disease control should be geared toward the environment. In the third model, we introduce a
COVID-19 model with non-pharmaceutical interventions known as social processes, vaccination, and the
environmental reservoir. This was done by incorporating fear of infection and social distancing parameters
into the disease dynamics. The model steady states are determined and their stabilities are analyzed. To
ascertain the range of parameters that affect social processes, vaccination, and the environmental reservoir
numerical simulations are conducted. We fit the model to the COVID-19 new cases data from South
Africa. Our analysis suggests that South Africans experienced a significant decline after the first four
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waves of infection due to the implementation of mandatory vaccination coupled with social processes.
The basic reproduction number is calculated from the parameter values obtained and it indicates that the
largest disease risk is from the symptomatic individuals. Our analytical and numerical results, among other
findings, also indicate that the COVID-19 infection could die out if social processes and vaccination are
constantly carried out. Otherwise, it will remain endemic, necessitating long-term disease prevention and
intervention programs. The results obtained through mathematical analyses and numerical simulations of
this thesis have a pivotal role in controlling and managing the disease in the event of any reoccurrence of
COVID-19.