On the Supervision of a Saturated SIR Epidemic Model with Four Joint Control Actions for a Drastic Reduction in the Infection and the Susceptibility through Time.

Asier Ibeas,Manuel De la Sen,Santiago Alonso-Quesada

doi:10.3390/ijerph19031512

Abstract

This paper presents and studies a new epidemic SIR (Susceptible–Infectious–Recovered) model with susceptible recruitment and eventual joint vaccination efforts for both newborn and susceptible individuals. Furthermore, saturation effects in the infection incidence terms are eventually assumed for both the infectious and the susceptible subpopulations. The vaccination action on newborn individuals is assumed to be applied to a fraction of them while that on the susceptible general population is of linear feedback type reinforced with impulsive vaccination actions (in practice, very strong and massive vaccination controls) at certain time points, based on information on the current levels of the susceptible subpopulation. Apart from the above vaccination controls, it is also assumed that the average of contagion contacts can be controlled via intervention measures, such as confinements or isolation measures, social distance rules, use of masks, mobility constraints, etc. The main objectives of the paper are the achievement of a strictly decreasing infection for all time periods and that of the susceptible individuals over the initial period if they exceed the disease-free equilibrium value. The monitoring mechanism is the combined activation of intervention measures to reduce the contagion contacts together with the impulsive vaccination to reduce susceptibility. The susceptibility and recovery levels of the disease-free equilibrium point are suitably prefixed by the design of the regular feedback vaccination on the susceptible subpopulation.

Highlights

Publisher’s Note: MDPI stays neutralEpidemiology is a scientific discipline whose objective is the study and distribution of frequency and determinant factors in the appearance and propagation of infectious diseases, mainly in humans but it is of interest in plants and animals
Note from Theorem 3 the important feature that, in the case that any eventual impulsive vaccination control action ends in finite time, either the values of the susceptible and recovered subpopulations, or the ratio of susceptible to recovered subpopulations at the disease-free equilibrium point can be pre-designed by the choices of the gain ld f and the fraction qd f which regulate the susceptible vaccination effort and the proportion of vaccinated newborn individuals, respectively
In addition to the previous actions (the value of β is fixed through (30) with λ = 0.9, the susceptible individuals are vaccinated at a rate of 20%, and the newborns are vaccinated at a rate of 70%), an impulsive vaccination is applied with different periodicities

Summary

Introduction

Epidemiology is a scientific discipline whose objective is the study and distribution of frequency and determinant factors in the appearance and propagation of infectious diseases, mainly in humans but it is of interest in plants and animals (both in the wildlife environment and in the farming or agriculture contexts). This paper develops a combined study of the influence of potentially joint vaccination efforts on both newborns and the susceptible subpopulation within the general population, taking as a basis a SIR (Susceptible–Infectious–Recovered) epidemic model This is claimed to be the first design of a proposal for deciding intervention measures through time combined with vaccination laws towards the achievement of a fast extinction of the disease. The impulsive control on the susceptible subpopulation is designed to achieve an evolution curve that is initially strictly decreasing through time to reach values under the disease-free equilibrium while the intervention measures are designed to achieve a similar effect on the infection curve through time by regulation of the maximum allowed average contagion rates. The proofs of the mathematical results are given in Appendix A in order to maintain easy and direct readability of the main body of the manuscript

The Epidemic Model

Impulsive Vaccination Control for a Drastic Reduction in the Susceptible

Simulation Examples

Findings

Conclusions