The Immune Response Modeling to a Viral Load: A Minimal Model

Introduction

Mathematical immunology is an independent discipline, which was formed at the intersection of biology, medicine, mathematics, physics and chemistry and began to evolve only in the early 70s of the last century. The founder of modern immunology is considered to be Louis Pasteur, who was the first to formulate the basic principle of protection against the causative agent of any infectious disease and also weakened cultures of pathogens of infectious diseases he called vaccines back in the late 19th century (Aronova, 2006). A large contribution to the development of immunology was made by Elie Metchnikoff, who in 1883 discovered phagocytosis and introduced the concept of “cellular immunity”, and in 1898 Paul Ehrlich created the theory of humoral immunity (Perelson & Weisbuch, 1992). These models in immunology can be conceptual, including verbal models, graphs or charts; and experimental, such as a specific cell culture system or mouse strain; or may take the form of quantitative mathematical or computer models (Handel et al., 2020). The most common types of quantitative models are what we term phenomenological models (Lythe & Molina-París, 2018). Phenomenological models are applied to extract patterns or, more broadly, information from data. As such, whenever data are being analyzed in some mathematical manner, this type of model is in play. Computing a correlation coefficient between two quantities of interest is an example of a very simple model that tries to detect a pattern. Regression models, in which a mathematical function is specified and the distance between the data and the function is minimized, also fall into this category (Nickaeen et al., 2019).

In recent years, the increase of available data has led to greater use of more complex phenomenological models, which increasingly go by names such as machine learning or deep learning approaches (Shinde & Kurhekar, 2020). These models do not explicitly describe the mechanisms by which patterns arise, and this is both a strength and a weakness. On the plus side, one can determine correlations, find patterns, deduce potential causation and make predictions without having to understand the underlying mechanisms governing a given system. The drawback is that such models provide, at best, hints regarding potential mechanisms. To study mechanisms and processes, mechanistic models are ideal. Mechanistic simulation models explicitly specify processes describing the mechanisms of interaction between system components (Pélissier et al., 2020). Usually, these models are highly simplified—but if done well, still can be very powerful—for abstractions of the system under study. The advantage of this kind of model is that it can provide mechanistic insights, leading to a better and deeper understanding of the system, to a point where the model might allow for very precise predictions. The main disadvantage of this approach is that model construction requires considerable knowledge (or at least assumptions) about the system and how its components interact.

Both phenomenological and mechanistic models are useful tools with distinct advantages and disadvantages. Deciding which type of model to use depends on the question and study system. It is common to start with phenomenological models, to determine patterns and obtain clues regarding the underlying processes and mechanisms, and then move to a mechanistic model to analyze those processes, their interactions, and the resulting outcomes in more detail. For the purposes of this article, we use the term ‘model’ meaning a model that describes the dynamics of the components of a system in an explicit and mechanistic way through mathematical equations or computational algorithms. Those models are generally studied by simulating them on a computer. There are different ways that such models can be implemented. The most common types used in immunology are compartmental models, in which each compartment tracks the size of a given biological entity of interest — for example, pathogen load or cytokine concentrations (Mosa, 2020). The most common way to implement a compartmental model is with ordinary differential equations.