A note on fractional linear pure birth and pure death processes in epidemic models |
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| Authors: | Roberto Garra Federico Polito |
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| Institution: | 1. Dipartimento di Fisica, “Sapienza” Università di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy;2. Dipartimento di Matematica, Università degli studi di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy;1. School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, PR China;2. School of Science, Changchun University, Changchun 130022, Jilin, PR China;3. Department of Mathematics, National University of Ireland, Galway, Ireland;1. Department of Computer Science, Rochester Institute of Technology, Rochester, NY 14623, USA;2. Department of Computer & Information Science, University of Konstanz, Box 67, D-78457 Konstanz, Germany;1. Dipartimento di Ingegneria dell’informazione, Ingegneria elettrica e Matematica Applicata (DIEM), Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy;2. Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli, Gruppo Collegato di Salerno, Italy;3. Dipartimento di Ingegneria Industriale (DIIN), Università degli Studi di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy;4. Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia (CNISM), Unitá di Salerno, Italy;5. Dipartimento di Studi e Ricerche Aziendali (Managment & Information Technology) (DISTRA), Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy;1. School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, PR China;2. School of Mathematics and Information Science, Guangxi Colleges and Universities Key Lab of Complex System Optimization and Large Data Processing, Yulin Normal University, Yulin, Guangxi 537000, PR China;3. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 121589, Saudi Arabia;4. College of Science, China University of Petroleum (East China), Qingdao 266580, PR China;5. Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan |
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| Abstract: | In this note we highlight the role of fractional linear birth and linear death processes, recently studied in Orsingher et al. (2010) 5] and Orsingher and Polito (2010) 6], in relation to epidemic models with empirical power law distribution of the events. Taking inspiration from a formal analogy between the equation for self-consistency of the epidemic type aftershock sequences (ETAS) model and the fractional differential equation describing the mean value of fractional linear growth processes, we show some interesting applications of fractional modelling in studying ab initio epidemic processes without the assumption of any empirical distribution. We also show that, in the framework of fractional modelling, subcritical regimes can be linked to linear fractional death processes and supercritical regimes to linear fractional birth processes.Moreover we discuss a simple toy model in order to underline the possible application of these stochastic growth models to more general epidemic phenomena such as tumoral growth. |
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