In this paper, we consider a modified SIR model, implementing a population of Pathogens interacting with a human population of Susceptibles, with which we will have 4 ordinary differential equations in our system. The objective of this work is to analyze the stability of the disease free point (local and global) and the endemic equilibrium point (local) of this mathematical model. In addition, numerical simulations to the model are presented to contrast the effects of nonlinear transmission rates and other parameters.

“Given a closed and convex subset C of a cone Banach space E with norm ∥x∥P = d (x, 0) and a map T : C → C that satisfies the condition for all x, y ∈ C 0 ≤ s + |a| − 2b < 2(a + b) ad (T x, T y) + b (d (x, T x) + d (y, T y)) ≤ sd (x, y) The general objective of this article is to demonstrate the existence of at least one fixed point for the map T , for which we will use a particular case of the Krasnoselskij iteration.