Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations

Rutwig Campoamor-Stursberg; Eduardo Fernández-Saiz; Francisco J. Herranz; Rutwig Campoamor-Stursberg; Eduardo Fernández-Saiz; Francisco J. Herranz

doi:10.3934/math.20231225

AIMS Mathematics

2023, Volume 8, Issue 10: 24025-24052. doi: 10.3934/math.20231225

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Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations

1.
Interdisciplinary Mathematical Institute (IMI), Complutense University, Madrid 28040, Spain
2.
Departament of Algebra, Geometry and Topology, Faculty of Mathematics, Complutense University, Madrid 28040, Spain
3.
Department of Mathematics and Data Science, San Pablo-CEU University, Alcorcón 28925, Spain
4.
Departament of Physics, University of Burgos, Burgos 09001, Spain

Received: 14 May 2023 Revised: 18 July 2023 Accepted: 24 July 2023 Published: 08 August 2023
MSC : 17B66, 34A26, 34C14, 92D25

Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that extend a recently proposed SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above-mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamiltonian models based on the book and oscillator algebras, denoted by $ \mathfrak{b}_2 $ and $ \mathfrak{h}_4 $, respectively. The last generalization corresponds to an SIS system possessing the so-called two-photon algebra symmetry $ \mathfrak{h}_6 $, according to the embedding chain $ \mathfrak{b}_2\subset \mathfrak{h}_4\subset \mathfrak{h}_6 $, for which an exact solution cannot generally be found but a nonlinear superposition rule is explicitly given.
- Lie systems,
- Lie-Hamilton systems,
- nonlinear differential equations,
- SIS models,
- exact solutions,
- nonlinear superposition rules
Citation: Rutwig Campoamor-Stursberg, Eduardo Fernández-Saiz, Francisco J. Herranz. Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations[J]. AIMS Mathematics, 2023, 8(10): 24025-24052. doi: 10.3934/math.20231225

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Abstract

Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that extend a recently proposed SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above-mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamiltonian models based on the book and oscillator algebras, denoted by $ \mathfrak{b}_2 $ and $ \mathfrak{h}_4 $, respectively. The last generalization corresponds to an SIS system possessing the so-called two-photon algebra symmetry $ \mathfrak{h}_6 $, according to the embedding chain $ \mathfrak{b}_2\subset \mathfrak{h}_4\subset \mathfrak{h}_6 $, for which an exact solution cannot generally be found but a nonlinear superposition rule is explicitly given.

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