*Article* **On the Use of Entropy Issues to Evaluate and Control the Transients in Some Epidemic Models**

**Manuel De la Sen 1,\*, Raul Nistal 1, Asier Ibeas <sup>2</sup> and Aitor J. Garrido 1,3**


Received: 13 March 2020; Accepted: 4 May 2020; Published: 9 May 2020

**Abstract:** This paper studies the representation of a general epidemic model by means of a first-order differential equation with a time-varying log-normal type coefficient. Then the generalization of the first-order differential system to epidemic models with more subpopulations is focused on by introducing the inter-subpopulations dynamics couplings and the control interventions information through the mentioned time-varying coefficient which drives the basic differential equation model. It is considered a relevant tool the control intervention of the infection along its transient to fight more efficiently against a potential initial exploding transmission. The study is based on the fact that the disease-free and endemic equilibrium points and their stability properties depend on the concrete parameterization while they admit a certain design monitoring by the choice of the control and treatment gains and the use of feedback information in the corresponding control interventions. Therefore, special attention is paid to the evolution transients of the infection curve, rather than to the equilibrium points, in terms of the time instants of its first relative maximum towards its previous inflection time instant. Such relevant time instants are evaluated via the calculation of an "ad hoc" Shannon's entropy. Analytical and numerical examples are included in the study in order to evaluate the study and its conclusions.

**Keywords:** Shannon entropy; epidemic model; transient behavior; vaccination and treatment intervention controls
