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The immune system recognizes a myriad of invading pathogens and their toxic products. It does so with a finite repertoire of antibodies and T cell receptors. We here describe theories that quantify the immune system dynamics. We describe…

We study a classical many-particle system with an external control represented by a time-dependent extensive parameter in a Lagrangian. We show that thermodynamic entropy of the system is uniquely characterized as the Noether invariant…

We model the immune surveillance of a pathogen which passes through $n$ immunologically distinct stages. The biological parameters of this system induce a partial order on the stages, and this, in turn, determines which stages will be…

T cells are central to the adaptive immune response, capable of detecting pathogenic antigens while ignoring healthy tissues with remarkable specificity and sensitivity. Quantitatively understanding how T cell receptors (TCRs) discriminate…

Innate immunity now occupies a central role in immunology. However, artificial immune system models have largely been inspired by adaptive not innate immunity. This paper reviews the biological principles and properties of innate immunity…

The physical interpretation of the functioning of the adaptive immune system, which has been thoroughly characterized on genetic and molecular levels, provides a unique opportunity to define an adaptive self-organizing biological system in…

A Lagrangian formulation with nonlocality is investigated in this paper. The nonlocality of the Lagrangian is introduced by a new nonlocal argument that is defined as a nonlocal residual satisfying the zero mean condition. The nonlocal…

Biological organisms have evolved a wide range of immune mechanisms to defend themselves against pathogens. Beyond molecular details, these mechanisms differ in how protection is acquired, processed and passed on to subsequent generations…

The immune system is a real-time example of an evolving system that navigates the essentially infinite complexity of protein sequence space. How this system responds to disease and vaccination is discussed. Of particular focus is the case…

Pathogens drive changes in host immune systems that in turn exert pressure for pathogens to evolve. Quantifying and understanding this constant coevolutionary process has clear practical global health implications. Yet its relatively easier…

The vertebrate adaptive immune system provides a flexible and diverse set of molecules to neutralize pathogens. Yet, viruses such as HIV can cause chronic infections by evolving as quickly as the adaptive immune system, forming an…

In the study of the basic properties observed in the immune system and, in a broader view, in biological systems, several concepts have already been mathematically formulated or treated in an analytical perspective, such as degeneracy,…

Organisms have evolved immune systems that can counter pathogenic threats. The adaptive immune system in vertebrates consists of a diverse repertoire of immune receptors that can dynamically reorganize to specifically target the…

Simple ideas, endowed from the mathematical theory of control, are used in order to analyze in general grounds the human immune system. The general principles are minimization of the pathogen load and economy of resources. They should…

The phenomenon of immunological memory has been known for a long time. But, the underlying mechanism is poorly understood. According to the theory of clonal selection the response to a specific invading antigen (e.g., bacteria) is offered…

In this paper we demonstrate how the Legendre transform connects the statements of Noether's theorem in Hamiltonian and Lagrangian mechanics. We give precise definitions of symmetries and conserved quantities in both the Hamiltonian and…

The adaptive immune system provides a diverse set of molecules that can mount specific responses against a multitude of pathogens. Memory is a key feature of adaptive immunity, which allows organisms to respond more readily upon…

An interesting inference drawn by some Covid-19 epidemiological models is that there exists a proportion of the population who are not susceptible to infection -- even at the start of the current pandemic. This paper introduces a model of…

The Noether theorem connecting symmetries and conservation laws can be applied directly in a Hamiltonian framework without using any intermediate Lagrangian formulation. This requires a careful discussion about the invariance of the…

The immune system is a cognitive system of complexity comparable to the brain and its computational algorithms suggest new solutions to engineering problems or new ways of looking at these problems. Using immunological principles, a two (or…