# Dinâmica, modelagem e controle de epidemias (Modelling and Control of Epidemics)

Nepomuceno, Erivelton (2005) Dinâmica, modelagem e controle de epidemias (Modelling and Control of Epidemics). Masters thesis, University of Minas Gerais (UMG).

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## Abstract

Epidemic is an alteration of one or more characteristics in a significant number of individuals of a population. Normally these characteristics are related to health. Individuals are considered as unique entities, for example, the human beings, ani- mals and even though machines or computers. The interaction between indivi- duals and environment consists in an epidemiological system. The classification of individuals in states is the most used approach to study an epidemiological system. Kermack and McKendrick developed the SIR model, which classifies the individuals in three states: susceptible, infectious and recovered. These three states are related by means of nonlinear differential equations. In this work the following aspects are investigated: i) influence of the vaccination and isolation on the dynamics of the SIR model; ii) models based on individuals (MBI); iii) use of optimal control and pulsed vaccination. The main contributions of this thesis are the following. First, it was verified that the vaccination and isolation consist in actions of control that modify the localization of transcritical bifurcation point. This change occurs proportionally to number of isolated individuals and inversely proportional to number of non-vaccinated individuals. The simultaneous use of the vaccination and isolation seems to be useful in certain circumstances. Secon- dly, a mathematical expression and an algorithm for the MBI was developed. It was evaluated that the MBI tends to present same results that the SIR model for very large populations and infinitesimal time intervals. An expression to calculate probability of eradication of an illness in a population was proposed. This proba- bility tends to increase with a reduction of population size. Finally, Pontryagin’s maximum principle was used to calculate an optimal control of vaccination using the SIR model. An analytical formula for the control law was obtained, which indicates a proportionality between population size and vaccination rate. After that, the intensity and the interval of pulsed vaccination using the SIR model were optimized by means of the Nelder-Mead’s algorithm. It was observed that an in- crease in the time interval of pulses can cause peaks in the number of infected individuals equivalent to the situation without vaccination. The proposed controls were applied in the MBI showing coherence with the results achieved in in the SIR model.

Item Type: Thesis (Masters) Modelling; Control; Epidemics; Faculty of Science and Engineering > Computer Science 16869 Erivelton Nepomuceno 19 Jan 2023 12:38 Yes This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available here