On the Jacobi Stability of Two SIR Epidemic Patterns with Demography

Abstract

In the present work, two SIR patterns with demography will be considered: the classical pattern and a modified pattern with a linear coefficient of the infection transmission. By reformulating of each first-order differential systems as a system with two second-order differential equations, we will examine the nonlinear dynamics of the system from the Jacobi stability perspective through the Kosambi–Cartan–Chern (KCC) geometric theory. The intrinsic geometric properties of the systems will be studied by determining the associated geometric objects, i.e., the zero-connection curvature tensor, the nonlinear connection, the Berwald connection, and the five KCC invariants: the external force εi—the first invariant; the deviation curvature tensor Pji—the second invariant; the torsion tensor Pjki—the third invariant; the Riemann–Christoffel curvature tensor Pjkli—the fourth invariant; the Douglas tensor Djkli—the fifth invariant. In order to obtain necessary and sufficient conditions for the Jacobi stability near each equilibrium point, the deviation curvature tensor will be determined at each equilibrium point. Furthermore, we will compare the Jacobi stability with the classical linear stability, inclusive by diagrams related to the values of parameters of the system.

Jacobi Stability for T-System

“…The main aim of this appendix is to show briefly the basic of the Kosambi-Cartan-Chern geometric theory, because all these notions and results are necessary to understand the obtained results about the Jacobi stability of the T-system [19,20,25,26,30,31,[35][36][37][38][39][40].…”

“…More precisely, the concept of Jacobi stability plays the role of proof of the resilience of a dynamical system defined by a system of second-order differential equations (semi-spray or SODE), where this resilience reflects the adaptability and the preservation of the system's basic behavior to changes in internal parameters and to influences from outside circumstances. Through the Kosambi-Cartan-Chern theory, i.e., from the perspective of the concept of Jacobi stability, the dynamics of various dynamical systems has recently been approached in [18][19][20][21]25,26,[30][31][32][33][34][35][36][37]. Therefore, the local behavior of the system is revealed through the use of geometric objects corresponding to the system of second-order differential equations (SODE), because this SODE is derived from the system of first-order differential equations [38][39][40].…”

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

“…With the help of the nonlinear and Berwald connections the five geometrical invariants can be constructed of which the second invariant plays an important role as it gives the Jacobi stability of dynamical system. Jacobi stability analysis for different systems like Lorenz system [12], Chua circuit system [13] and other systems [14][15][16][17][18][19][20][21] have been studied. According to the articles [22,23], one of the geometrical invariants that identifies the beginning of chaos is the deviation vector from the so-called Jacobi equation.…”