Abstract
A formal description of typical compartmental epidemic models obtained is presented by splitting the state into an infective substate, or infective compartment, and a noninfective substate, or noninfective compartment. A general formal study to obtain the reproduction number and discuss the positivity and stability properties of equilibrium points is proposed and formally discussed. Such a study unifies previous related research and it is based on linear algebraic tools to investigate the positivity and the stability of the linearized dynamics around the disease-free and endemic equilibrium points. To this end, the complete state vector is split into the dynamically coupled infective and noninfective compartments each one containing the corresponding state components. The study is then extended to the case of commensurate internal delays when all the delays are integer multiples of a base delay. Two auxiliary delay-free systems are defined related to the linearization processes around the equilibrium points which correspond to the zero delay, i.e., delay-free, and infinity delay cases. Those auxiliary systems are used to formulate stability and positivity properties independently of the delay sizes. Some examples are discussed to the light of the developed formal study.
Highlights
Epidemic models have been widely studied in the last decades involving several inter-actuating subpopulations with mutual coupled dynamics
The reachability of the endemic equilibrium point, as well as the stability properties of the equilibrium points, is definitely linked with the value of the so-called reproduction number. If such a reproduction number is less than unity the disease-free equilibrium point is locally asymptotically stable and the endemic one is unattainable since it has some negative component for some infective subpopulation
It turns out that there are no general analysis tools available in the background literature to discuss those properties based on general reasonable and generic assumptions independent of the particular epidemic model. These kinds of results are known from the background literature in a variety of epidemic models, they are revisited and presented here in a general framework as a fruitful combination of algebraic results based on positivity and stability of the linearized systems around the equilibrium points
Summary
Epidemic models have been widely studied in the last decades involving several inter-actuating subpopulations with mutual coupled dynamics. The positivity and stability of the solutions in both cases are formally discussed in a general context, rather than for specific models, based on the linearization analysis on the infective compartment for the disease-free and endemic equilibrium points.
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