Mathematical models have long been used to describe complex biological interactions with the aim of predicting mechanistic interactions hard to distinguish from data. This dissertation uses modeling, mathematical analyses, and data fitting techniques to provide hypotheses on the mechanisms of immune response formation and function. The immune system, comprised of the innate and adaptive immune responses, is responsible for protecting the body against invading pathogens, with disease or vaccine induced immune memory leading to fast responses to subsequent infections. While there is some agreement about the underlying mechanisms of adaptive immune memory, innate immune memory is poorly understood. Stimulation with lipopolysaccharide induces differential phenotypes in innate immune cells depending on the strength of the stimulus, such that a secondary lipopolysaccharide encounter of a constant dose results in either strong or weak inflammatory cytokine expression. We model the biochemical kinetics of three molecules involved in macrophages responses to lipopolysaccharide and find that once a macrophage is programed to show a weak inflammatory response this cannot be reverted. Contrarily, a secondary lipopolysaccharide stimulus of a very high dose or applied prior to waning of the effects of the primary stimulus can induce a phenotype switch in macrophages initially programed to show strong inflammatory responses. Some pathogens, such as the hepatitis B virus, have developed strategies that hinder an efficient innate immune response. Hepatitis B virus infection is a worldwide pandemic with approximately 257 million chronically infected people. One beneficial event in disease progression is the seroclearance of hepatitis B e antigen often in combination with hepatitis B antibody formation. We propose mathematical models of within-host interactions and use them to predict that hepatitis B e antibody formation causes hepatitis B e antigen seroclearance and the subsequent reactivation of cytotoxic T cell immune responses. We use the model to quantify the time between antibody formation and antigen clearance and the average monthly hepatocyte turnover during that time. We further expand the study of hepatitis B infection, by investigating the kinetics of the virus under an experimental drug administered during a clinical trial. Available drugs usually fail to induce hepatitis B s antigen clearance, defined as the functional cure point of chronic hepatitis B infections. Drug therapy clinical trials that combined RNA interference drug ARC-520 with entecavir have shown promising results in reducing hepatitis B s antigen titers. We develop pharmacokinetic-pharmacodynamic models describing the mechanistic interactions of the drugs, hepatitis B virus DNA, and virus proteins. We fit the model to clinical trial data and predict that ARC-520 alone is responsible for the reduction of hepatitis B s and e antigens, while entecavir is the driving force behind viral reduction. This work was supported by Simons Foundation, Grant No. 427115, and National Science Foundation, Grant No. 1813011. / Doctor of Philosophy / Mathematical models have long been used to describe complex biological interactions with the aim of predicting interactions that explain observed data and informing new experiments. This dissertation uses modeling, mathematical analyses, and data fitting techniques to provide hypotheses on the mechanisms of immune response formation and function. The immune system, comprised of the innate and adaptive immune responses, is responsible for protecting the body against invading pathogens, such as viruses, bacteria, or fungi. If an immune response to a secondary pathogen encounter differs from the response when the body first encounters the specific pathogen, this is called immune memory. The mechanisms underlying the memory of immune responses are well understood in the context of adaptive immune responses, but less so for innate immune responses. Stimulation with lipopolysaccharide, a cell wall component of many bacteria, programs innate immune cells, such as macrophages, to be in one of two states, called phenotypes, depending on the strength of the stimulus. Based on their phenotype the macrophages show either a weak or strong inflammatory response upon a secondary lipopolysaccharide encounter of a constant dose. We model the biochemical kinetics of three molecules involved in macrophages responses to lipopolysaccharide. We find that once a macrophage is programed to show a weak inflammatory response this cannot be reverted. Contrarily, a secondary lipopolysaccharide stimulus that is either of a very high dose or applied before the effects of the primary stimulus have waned, can induce a phenotype switch in macrophages initially programed to show strong inflammatory responses. Some pathogens, such as the hepatitis B virus, have developed strategies that hinder an efficient innate immune response. Hepatitis B virus infection is a worldwide pandemic with approximately 257 million chronically infected people. Hepatitis B e antigen is a protein that infected liver cells release into blood and that impairs adaptive immune responses. It is considered a beneficial event in disease progression, and called hepatitis B e antigen clearance, when hepatitis B e antigen becomes indetectable in a patient's blood. We propose mathematical models of interactions between liver cells, the virus, hepatitis B e antigens and hepatitis B e antibodies, which neutralize the antigens. We predict that antibody formation causes antigen clearance and a reactivation of immune responses. We furthermore use the model to quantify the time between antibody formation and antigen clearance and the average number of liver cells killed during that time. We further expand the study of hepatitis B infection, by investigating the kinetics of the virus under an experimental drug administered during a clinical trial. Available drugs rarely induce hepatitis B s antigen clearance, but clinical trials that combined a novel drug, called ARC-520, with the commonly used drug entecavir have shown promising results in reducing hepatitis B s antigen titers in the blood of infected patients. Following the clearance of hepatitis B s antigen, a protein that is released by infected cells and impairs adaptive immunity, the body usually has the capability to control the infection without medication. We develop mathematical models describing the interactions of the drugs, hepatitis B virus, and virus proteins. We fit the model to clinical trial data and predict that ARC-520 alone is responsible for the reduction of hepatitis B s and e antigens, while entecavir is the driving force behind viral reduction.