Material for “ On the role of CD 8 T cells in the control of persistent infections , ” by

The modeling of complex systems typically involves reducing the number of free parameters in a model of the system, to a tractable quantity, while keeping the essential features of the phenomena under study (1). Ecological models frequently start with an assumption of functional forms such as the logistic term in Eq. 4 or the saturating function in Eq. 1 as these functional forms can arise in many ways. Here we present our derivation of these functional forms using more detailed models of the T cell, pathogen dynamics, and innate immunity, and derive the equations presented in the main text from them. The T cell subsection considers the integration of a within cell model for PD-1 expression in T cells, and the population dynamics of T cells, dendritic cells, and pathogen. The pathogen subsection considers a common model of free pathogen, uninfected target cell, and infected target cell and reduces it to the logistic equation used in the main text. The innate immunity model considers an alternate derivation of the logistic equation starting from an explicit innate immunity model, underscoring the ability of the logistic term to describe either resource limitations or innate immunity. These reductions are performed using quasi-static approximations. The quasi-static approximations become exact in the behavior of the steady states of the model, which much of the main text is focused on. The quasi-static approximations also yield a model in qualitative agreement with experimental results (2–4).