In recent years, there has been an increasing interest in the mathematical and computational modeling of the human immune system (HIS). Computational models of HIS dynamics may contribute to a better understanding of the relationship between complex phenomena and immune response; in addition, computational models will support the development of new drugs and therapies for different diseases. However, modeling the HIS is an extremely difficult task that demands a huge amount of work to be performed by multidisciplinary teams. In this study, our objective is to model the spatio-temporal dynamics of representative cells and molecules of the HIS during an immune response after the injection of lipopolysaccharide (LPS) into a section of tissue. LPS constitutes the cellular wall of Gram-negative bacteria, and it is a highly immunogenic molecule, which means that it has a remarkable capacity to elicit strong immune responses. We present a descriptive, mechanistic and deterministic model that is based on partial differential equations (PDE). Therefore, this model enables the understanding of how the different complex phenomena interact with structures and elements during an immune response. In addition, the model's parameters reflect physiological features of the system, which makes the model appropriate for general use.

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By June 2021, a new contagious disease, the Coronavirus disease 2019 (COVID-19), has infected more than 172 million people worldwide, causing more than 3.7 million deaths. Many aspects related to the interactions of the disease’s causative agent, SAR2-CoV-2, and the immune response are not well understood: the multiscale interactions among the various components of the human immune system and the pathogen are very complex. Mathematical and computational tools can help researchers to answer these open questions about the disease. In this work, we present a system of fifteen ordinary differential equations that models the immune response to SARS-CoV-2. The model is used to investigate the hypothesis that the SARS-CoV-2 infects immune cells and, for this reason, induces high-level productions of inflammatory cytokines. Simulation results support this hypothesis and further explain why survivors have lower levels of cytokines levels than non-survivors.

Bacterial infections can be of two types: acute or chronic. The chronic bacterial infections are characterized by being a large bacterial infection and/or an infection where the bacteria grows rapidly. In these cases, the immune response is not capable of completely eliminating the infection which may lead to the formation of a pattern known as microabscess (or abscess). The microabscess is characterized by an area comprising fluids, bacteria, immune cells (mainly neutrophils), and many types of dead cells. This distinct pattern of formation can only be numerically reproduced and studied by models that capture the spatiotemporal dynamics of the human immune system (HIS). In this context, our work aims to develop and implement an initial computational model to study the process of microabscess formation during a bacterial infection.

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