The polysymplectic formalism in local field theory, developed by Günther [J. Diff. Geom. 25, 23 (1987)], is revised. A new approach and new results on momentum maps and reduction are given.

In the present study we introduce a deterministic mathematical model in order to study the interaction between the human immune system and a virus, like COVID 19. The mathematical analysis is based on the tools of dynamical systems theory, by modeling the interactions between the immune system and the virus, using a predator-prey method and following the ideas of G. Moza, from \cite{TGV}. It will be obtained some conclusions with medical relevance and the main three of them are the followings: 1) A deficiency of a single type of immunity in the early stages of virus proliferation, may lead to a large virus multiplication and the human body can loses the fight against this virus; 2) If the level of all components of the immunity system are at the normal threshold from the first moment of the infection and the immune system kill the virus at higher rate than the rate of virus reproduction, then the human body has the ability to stop the multiplication of the virus and liquidate it, that means the virus will be destroyed; 3) If the level of at least one type of immunity can be increased beyond the normal threshold, by medical interventions (like vaccination) in the early stages of virus infection, then the immune system has a better chance to win the fight with the virus.

Using the feedback linearization method, a state feedback control for the two-dimensional Kermack-McKendrick model for the spread of epidemics is obtained. This form of the dynamical system is suitable to carry out a qualitative analysis of the model. An optimal control problem for a stochastic version of the model is also set up and solved explicitly in a particular instance.

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By reformulating the circuit system of Lü as a set of two second order differential equations, we investigate the nonlinear dynamics of Lü’s circuit system from the Jacobi stability point of view, using Kosambi–Cartan–Chern geometric theory. We will determine the five KCC invariants, which express the intrinsic geometric properties of the system, including the deviation curvature tensor. Finally, we will obtain necessary and sufficient conditions on the parameters of the system to have the Jacobi stability near the equilibrium points.