During an immune response, it is understood that there are key differences between the cells and cytokines that are present in a primary response versus those present in subsequent responses. Specifically, after a primary response, memory cells are present and drive the clearance of antigen in these later immune responses. When comparing acute infections to chronic infections, there are also differences in the dynamics of the immune system. In this dissertation, we develop three mathematical models to explore these differences in the immune response to acute and chronic infections through the creation, activation, regulation, and long term maintenance of T cells. We mimic this biological behavior through the use of delayed differential equation (DDE) models.

The first model explores the dynamics of adaptive immunity in primary and secondary responses to acute infections. It is shown that while we observe similar amounts of antigen stimulation from both immune responses, with the incorporation of memory T cells, we see an increase in both the amount of effector T cells present and the speed of activation of the immune system in the secondary response. We conclude that our model is robust and can be applied to study different types of antigen from viral to bacterial.

Extending our work to chronic infections, we develop our second and third models to explore breast cancer dormancy and T cell exhaustion. For our breast cancer dormancy model, we find that our model behaves similar to acute infections, but with constant antigen stimulation. Moreover, we observe the importance of immune protection on the long term survival of breast cancer cells.

In our third model we find that while memory T cells play a major role in the effectiveness of the immune system in acute infection, in chronic infections, over long periods of time, T cell exhaustion prevents proper immune function and clearance of antigen. We also observe how the lack of long term maintenance of memory T cells plays an important role in the final outcome of the system. Finally, we propose two potential extensions to the three models developed: creating a simplified acute infection model and creating a combined breast cancer dormancy model with T cell exhaustion.