# Analysis and Control of an SEIR Epidemic System with Nonlinear Transmission Rate

Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 421)

## Introduction

Mathematical models describing the population dynamics of infectious diseases have been playing an important role in better understanding epidemiological patterns and disease control for a long time. In order to predict the spread of infectious disease among regions, many epidemic models have been proposed and analyzed in recent years (see [7, 4, 22, 25, 36, 19, 20]). However, most of the literature researched on epidemic systems (see [36, 24, 8, 13]) assumes that the disease incubation is negligible that, once infected, each susceptible individual (in class S) becomes infectious instantaneously (in class I) and later recovers (in class R) with a permanent or temporary acquired immunity. The model based on these assumptions is customarily called an SIR (susceptible-infectious-recovered) or SIRS (susceptible-infectious-recovered-susceptible) system (see [10, 9]). Many diseases such as measles, severe acute respiratory syndromes (SARS), and so on, however, incubate inside the hosts for a period of time before the hosts become infectious. So the systems that are more general than SIR or SIRS types need to be studied to investigate the role of incubation in disease transmission. We may assume that a susceptible individual first goes through a latent period (and is said to become exposed or in the class E) after infection before becoming infectious. Thus, the resulting models are of SEIR (susceptible-exposed-infectious-recovered) or SEIRS (susceptible-exposed-infectious-recovered-susceptible) types, respectively, depending on whether the acquired immunity is permanent or not.

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## Copyright information

© Springer-Verlag London Limited 2012

## Authors and Affiliations

1. 1.College of SciencesNortheastern UniversityShenyangChina, People’s Republic