Geometry and solutions of an epidemic SIS model permitting fluctuations and quantization

Abstract: Some recent works reveal that there are models of differential equations for the mean and variance of infected individuals that reproduce the SIS epidemic model at some point. This stochastic SIS epidemic model can be interpreted as a Hamiltonian system, therefore we wondered if it could be geometrically handled through the theory of Lie--Hamilton systems, and this happened to be the case. The primordial result is that we are able to obtain a general solution for the stochastic/ SIS-epidemic model (with fluctuations) in form of a nonlinear superposition rule that includes particular stochastic solutions and certain constants to be related to initial conditions of the contagion process. The choice of these initial conditions will be crucial to display the expected behavior of the curve of infections during the epidemic. We shall limit these constants to nonsingular regimes and display graphics of the behavior of the solutions. As one could expect, the increase of infected individuals follows a sigmoid-like curve. Lie--Hamiltonian systems admit a quantum deformation, so does the stochastic SIS-epidemic model. We present this generalization as well. If one wants to study the evolution of an SIS epidemic under the influence of a constant heat source (like centrally heated buildings), one can make use of quantum stochastic differential equations coming from the so-called quantum deformation.