In this work, we consider two SIR patterns with demography: the classical pattern and a modified pattern with a linear transmission coefficient of the infection. By reformulating of each first order differential systems as a system with two second-order differential equations, we investigate the nonlinear dynamics of the system from the Jacobi stability point of view by using the KCC geometric theory. We will study the intrinsic geometric properties of the systems by determining the geometric associated objects: the zero-connection curvature tensor, the nonlinear connection, the Berwald connection, and the five KCC invariants: the first invariant - the external force εi, the second invariant - the deviation curvature tensor Pji, the third invariant - the torsion tensor Pjki, the fourth invariant - the Riemann-Christoffel curvature tensor Pjkli, and the fifth invariant - the Douglas tensor Djkli. In order to obtain necessary and sufficient conditions for the Jacobi stability near the equilibrium points, the deviation curvature tensor will be determined at each equilibrium points. Furthermore, we will compare the Jacobi stability with the classical linear stability, inclusive by diagrams related to the values of parameters of the system.

In this paper, we will consider an autonomous two-dimensional ODE Kolmogorov type 1 system with three parameters, which is a particular system of the general predator–prey systems with 2 a Holling type II. By reformulating this system as a set of two second order differential equations, we 3 will investigate the nonlinear dynamics of the system from the Jacobi stability point of view, using 4 the Kosambi-Cartan-Chern (KCC) geometric theory. We will determine the nonlinear connection, the 5 Berwald connection and the five KCC-invariants which express the intrinsic geometric properties 6 of the system, including the deviation curvature tensor. Furthermore, we will obtain necessary and 7 sufficient conditions on the parameters of the system in order to have the Jacobi stability near the 8 equilibrium points and we will point out these on a few examples.

In this paper it will be considered three deterministic models for the study of the interaction between the human immune system and a virus: the logistic model, the Gompertz model, and the generalized logistic model (or Richards model). A qualitative analysis of this models based on dynamical systems theory will be performed, by studying the local behavior of the equilibria and obtaining local dynamics properties from the linear stability point of view. Also, we will compare this models in order to understanding which is more appropriate to models the interaction between the human immune system and a virus. It will be obtained some natural medical interpretations which are available for all three models and can be useful to the medical community.

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.