Author: Ing. Tadeáš Chrapovič
The main concern of this work is the process of calculating the reproductive number in non-linear models consisting of several infectious classes. The aim of the work is to demonstrate the calculation of the reproduction number in such models using a new generation matrix. In the introduction, we are going to show a way how we can modify these models while considering additional influence. We will explain the importance of the reproductive numbers in epidemiology. We will look at the generation of the reproductive numbers, general description of the new generation matrix and possibilities for its simplification. This part will also consist of a motivational model, which we analyze in terms of asymptotic stability, by which we are going to show the importance of the reproductive number. The work's core structure is divided into two parts. In the first part we are dealing with three epi- demiological models, by which we can show the use of the new generation matrix to calculate the reproductive number and the principle of its simplification. We are also describing the consequence of individual modifications and observing how the system gets simplified in terms of calculation of the reproduction number due to reductions. In the second part we are dealing with modification of the epidemiologi- cal models dedicated for the simulation, or more precisely the analysis of influenza. We analyze the models by examining the main features of generations composing a particular model and subsequently scrutinising their behaviour using a graphical representation of the infection's progress.
epidemic models, differential equations, reproduction number, new generation mat- rix, asymptotic stability, influenza models
This work deals with the issue of spreading the disease among the human popu- lation, considering the changes in its characteristics such as (reproductive number, stable number of contaminated ones) with the change of certain factors affecting the disease. The aim of this work is to show its possible use as well as to point at the possibilities of adaptation and the consequences of various factors through mathematical models dealing with the prediction of disease behavior. The core of this work is divided into three parts. In the first part of this work we deal with the compilation and more detailed analysis of basic epidemiological models. We are also describing a way in which strongly nonlinear models can be implemented using a Matlab environment. In the second part of the work we deal with the modification of basic epidemiological models for the most accurate predictions of certain situations, while we monitor the changes in the development of the disease caused by changing certain factors that affect the continuance of the disease. In the last part of the work we deal with unrealistic situations, so-called zombies, and the ideas by which we understand this word. In this part of the work we have assembled three own models, where the main idea is to change from a dead member of the population to a zombie.
epidemic models, differential equations, reproduction number, model SIR