Neglected tropical diseases (NTDs) are a health crisis that is ignored by the developed world and only starts getting attention when there is an outbreak that kills a large population. AIDS associated Kaposi sarcoma (AIDS-KS) and Ebola Virus Disease (EVD) are viral infections which are just two of the many NTDs that require attention. This thesis presents novel within-host ordinary differential equation (ODE) mathematical models in conjunction with application of the optimal control theory of AIDS-KS dynamics. Additionally, we derive an epidemiological system of ODEs to model the transmission dynamics of EVD in the presence of public health education. AIDS-KS is caused by KS-associated herpesvirus (KSHV). Although the standard KS therapy has not changed in twenty years, not all cases of KS will respond to the same therapy. The goal of current AIDS-KS treatment modalities is to reconstitute the immune system and suppress HIV replication. We introduce novel within-host ODE mathematical models that consolidate the effect of both HIV and KSHV load on KS tumor progression by incorporating low or high viral loads (VLs) into the proliferation terms of the immune cell populations. Regulation of HIV/KSHV VL and viral reservoir cells is crucial for restoring a patient to an asymptomatic stage. Therefore, in order to device an optimal treatment plan, optimal control theory is applied by using Pontryagin’s maximum principle with control measures such as combined antiretroviral therapy (cART). The objective function is constituted to minimise the population of infected cells, VL and KS tumor and the harmful side effects of cART to the patient. The results indicate that the drug treatment strategies are capable of removing the viral reservoirs faster, consequently reducing the HIV and KS tumor burden. The predictions of the mathematical models have the potential to offer more effective therapeutic interventions based on VL and virus-infected cell load, hence support new studies addressing the superiority of VL over CD4 cell count when predicting life expectancy of HIV patients. iv Public participation in Ebola virus disease (EVD) prevention efforts is essential to reduce outbreaks. Targeted education through practical health information for specific populations and sub-populations is essential to combat this disease. We study the dynamics of EVD in he presence of public health education to assess the role of behavior change instigated by health education in the dynamics of an outbreak. The intensity of behaviour change is clear in two outbreaks of EVD that occurred in Sudan just three years apart. The first occurrence turned into the primary documented outbreak of EVD and produced a sizable number of infections. The subsequent outbreak delivered far less cases, apparently on the grounds that the population in the district gained from the primary episode. We derive a system of ODEs to model these two opposite behaviors, using data from these two EVD instances to estimate parameters significant to the two opposite behaviors. We then simulate a future EVD epidemic in Sudan using our model, which includes two susceptible populations, one of which is more knowledgeable about EVD than the other. Our results show how a better educated population leads to fewer cases of EVD and features the significance of ongoing education in public health.

Thesis (Msc Mathematics and Statistics Sciences) --Botswana International University of Science and Technology, 2021