Analytic solutions to stochastic epidemic models

Abstract

Even the simplest outbreaks might not be easily predictable. Fortunately, deterministic and stochastic models, systems differential equations and computational simulations have proved to be useful to a better understanding of the mechanics that leads to an epidemic outbreak. Whilst such systems are regularly studied from a modelling viewpoint using stochastic simulation algorithms, numerous potential analytical tools can be inherited from statistical and quantum physics, replacing randomness due to quantum fluctuations with low copy number stochasticity. Here, the Fock space representation, used in quantum mechanics, is combined with the symbolic algebra of creation and annihilation operators to consider explicit solutions for the master equations describing epidemics represented via the SIR model (Susceptible-Infected-Recovered), originally developed via Kermack and McKendrick’s theory. This is illustrated with an exact solution for a short size of population, including a consideration of very short time scales for the next infection, which emphasises when stiffness is present even for small copy numbers. Furthermore, we present a general matrix representation for the SIR model with an arbitrary number of individuals following diagonalization. This leads to the solution of this complex stochastic problem, including an explicit way to express the mean time of epidemic and basic reproduction number depending on the size of population and parameters of infection and recovery. Specifically, the project objective to apply use of the same tools in the approach of system governed by law of mass action, as previously developed for the Michaelis-Menten enzyme kinetics model [Santos et. al. Phys Rev. E 92, 062714 (2015)]. For this, a flexible symbolic Maple code is provided, demonstrating the prospective advantages of this framework compared to Gillespie stochastic simulation algorithms.