Global Dynamics of a Periodic SEIRS Model with General Incidence Rate

Received Jul 25, 2017; Revised Oct 3, 2017; Accepted Oct 10, 2017

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Epidemiological models in mathematics have been recognized as valuable tools in analyzing the dynamics of an infectious disease nowadays. They are used to describe the spread of disease and also to make control measures known to avoid its persistence, for example, via vaccination terms or treatment terms. These models consider the total population divided into compartments, given by the biological assumptions on the model and represented by functions depending on time t. The most common categories used are susceptible (S), infected (I), recovered (R), exposed (E), quarantined (Q), and vaccinated (V), and the dynamics of model is given by transmission rates from a compartment to another. We have then indicated that the models could be of type SIR, SIRS, SEIR, SEIRS, SEIVR, SEIQV, and so forth.

To ensure that the model can give a justified qualitative description of the disease, the choice of the incidence rate plays an important role. An incidence rate is defined as the number of new health related events or cases of a disease in a population exposed to the risk in a given time period. Some examples are the bilinear incidence rate, the saturated incidence rate, or a general incidence rate. The bilinear incidence rate has been repeatedly used by several authors. It is given by βSI, where β is the transmission rate and the product SI represents the contact between infected and susceptible individuals (based on the law of mass action). It was introduced by Kermack and McKendrick [1] in 1927, and even when it is mathematically simple...