Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Recently, new strains of influenza such as bird flu and swine flu have emerged. These strains have the capacity to infect people on a quite large scale and are characterized by their resistance to existing influenza treatment and their high mortality rates. In this thesis, we consider two models for influenza transmission dynamics that include both sensitive and resistant strains and accounts for disease induced mortality. The first model allows for immigration/migration and does not include any control measure. The second one explores the effects of vaccination and treatment of the sensitive strain but ignores immigration/migration. We studied the two models mathematically and numerically. We started with the model without any control measures; we calculated the basic reproductive numbers, determined the equilibrium points and investigated their stability. Our analysis showed that when the basic reproduction numbers of both strains are less than one then the two strains will die out. When at least one of the basic reproduction numbers is greater than one, then the strain with the higher basic reproduction number is the one that will persist. Numerical simulations were carried out to confirm the stability results and a bifurcation diagram was given. We also studied numerically the impact of the mortality rate of influenza on the dynamics of the disease. Especially, we investigated the effect of the mortality rate on the time needed for the pandemic to reach its peak, the value at the peak for each strain and, when eradication is possible, the time it takes for the disease to be eradicated. For the model with control, we also calculated the control reproductive number and the equilibrium points. The stability analysis was carried out numerically and bifurcation diagrams with vaccination and treatment parameters were given to determine the regions where eradication of the disease is possible. Our results suggest that in the presence of a resistant strain, treating more infected individuals will not eradicate the disease as the resistant strain will always persist. In such a case vaccination and antiviral treatment should be implemented simultaneously. / AFRIKAANSE OPSOMMING: Geen opsomming