Abstract
The construction of mathematical models is one of the tools used today for the study of problems in Medicine, Biology, Physiology, Biochemistry, Epidemiology, and Pharmacokinetics, among other areas of knowledge; its primary objectives are to describe, explain and predict phenomena and processes in these areas. The simulation, through mathematical models, allows exploring the impact of the application of one or several control measures on the dynamics of the transmission of infectious diseases, providing valuable information for decision-making with the objective of controlling or eradicating them. The mathematical models in Epidemiology are not only descriptive but also predictive, helping to prevent pandemics (epidemics that spread through large areas and populations) or by intervening in vaccination and drug acquisition policies. In this article we study the existence of periodic orbits and the general stability of the equilibrium points for a susceptible-infected-susceptible model (SIS), with a non-linear incidence rate. This type of model has been studied in many articles with a very particular incidence rate, here the novelty of the problem is that the aforementioned incidence rate is very general, in this sense this research provides a solution to an open problem. The methodology used is the Dulac technique, proceeding by reduction to the absurdity of the statement to the main test. It shows that the only point of equilibrium is asymptotically stable global. It can be noted that this problem may be subject to discretion or for equations in timescales. This can generate other research.
Highlights
There are different factors of a disease that make us unable to study all in the same way, such as: the mode of transmission, infectious agents, the affected population and the states through which an individual can pass, the latter are: susceptible (S), healthy individuals who can get the disease; exposed (E), those infected but who cannot spread the disease; infected (I), infected individuals and who can infect others; resistant (R), those that are resistant to the disease, have usually overcome it or have been vaccinated and carriers (M), individuals who carry the disease, but may never suffer from it [1].For the study of this area of research, it is assumed that individuals are in one of several possible states
For F1(I,S) = 0 y F2(I,S) = 0, we find that the only point of trivial equilibrium is A1=(0,1) and nontrivial equilibrium points are obtained from the algebraic equation: h(I) = Ip-1. (1-I)q = C, where G 9 ; /, C >0, 0 < I < 1
Let us denote by 9 ∗, the periodic orbit corresponding to the solution [ . , ∗
Summary
There are different factors of a disease that make us unable to study all in the same way, such as: the mode of transmission, infectious agents, the affected population and the states through which an individual can pass, the latter are: susceptible (S), healthy individuals who can get the disease; exposed (E), those infected but who cannot spread the disease; infected (I), infected individuals and who can infect others; resistant (R), those that are resistant to the disease, have usually overcome it or have been vaccinated and carriers (M), individuals who carry the disease, but may never suffer from it [1]. The spread of infectious diseases and their control measures have been the subject of several studies; most of them have done so in models for the dynamics in a population. The objective of this work is to use susceptibleinfected-susceptible models (SIS) These models allow to know the influence of the migratory flows in the propagation of a disease and to understand the characteristics of the propagation in subpopulations, each with its own dynamics but connected by the movement of people with each other. In this sense, a study of the existence of periodic orbits and stability of equilibrium points for susceptible-infected-removed-susceptible (SIRS). We aim to present a technique that involves more geometric elements related to the theory of equations and precise with a technique different from that of the other authors, the asymptotic uniformity of the stability of the solutions, which predicts if the population will be controlled to avoid outbreaks of disease
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