Dynamics of Epidemiological Models
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Dynamics of Epidemiological Models Alberto Pinto • Maı´ra Aguiar • Jose´ Martins Nico Stollenwerk
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Received: 14 June 2010 / Accepted: 5 July 2010 / Published online: 27 July 2010 Springer Science+Business Media B.V. 2010
Abstract We study the SIS and SIRI epidemic models discussing different approaches to compute the thresholds that determine the appearance of an epidemic disease. The stochastic SIS model is a well known mathematical model, studied in several contexts. Here, we present recursively derivations of the dynamic equations A. Pinto J. Martins LIAAD-INESC, Porto LA, Portugal A. Pinto Department of Mathematics, Faculty of Sciences, University of Porto, Rua Campo Alegre, 687, 4169-007 Porto, Portugal e-mail: [email protected] A. Pinto Department of Mathematics and Research Center of Mathematics of the University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal M. Aguiar N. Stollenwerk Centro de Matema´tica e Aplicac¸o˜es Fundamentais da Universidade de Lisboa, Avenida Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal M. Aguiar Fundac¸a˜o Ezequiel Dias, Laborato´rio de Dengue e Febre Amarela, Rua Conde Pereira Carneiro 80, 30510-010 Belo Horizonte, MG, Brazil e-mail: [email protected] J. Martins (&) Department of Mathematics, School of Technology and Management, Polytechnic Institute of Leiria, Campus 2, Morro do Lena, Alto do Vieiro, 2411-901 Leiria, Portugal e-mail: [email protected] N. Stollenwerk Research Center Ju¨lich, 52425 Ju¨lich, Germany e-mail: [email protected]
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for all the moments and we derive the stationary states of the state variables using the moment closure method. We observe that the steady states give a good approximation of the quasi-stationary states of the SIS model. We present the relation between the SIS stochastic model and the contact process introducing creation and annihilation operators. For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I, then recover and remain only partial immune against reinfection R, we present the phase transition lines using the mean field and the pair approximation for the moments. We use a scaling argument that allow us to determine analytically an explicit formula for the phase transition lines in pair approximation. Keywords
Epidemic models Quasi-stationary states Pair approximation
1 Introduction One of the simplest and best studied epidemiological models is the stochastic SIS model. Many authors worked on the SIS model considering, only, the dynamical evolution of the mean value and the variance of the infected individuals. In this survey, we present recursively derivations of the dynamic equations for all the moments, and we derive the stationary states of the state variables using the moment closure method. The stationary states of the state variables give, surprisingly, good approximated values not of the stationary states but of the quasi-stationary states of the SIS master equation (see Martins et al. 2010b; Pinto et al. 2009). We prese
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