Exact solution of a stochastic susceptible-infectious-recovered model
Gunter M Schütz1, Marian Brandaut, Steffen Trimper
1Forschungszentrum Jülich, IFF D-52425 Jülich, Germany. g.schuetz@fz-juelich.de
Summary
This study presents an exact analytical solution for the susceptible-infectious-recovered (SIR) model using a quantum formulation. The findings reveal a finite stationary distribution for susceptible individuals, differing from standard mean-field theory.
Area of Science:
- Epidemiology
- Statistical Physics
- Quantum Mechanics
Background:
- The susceptible-infectious-recovered (SIR) model is a fundamental epidemiological tool.
- Standard SIR models often rely on mean-field approximations, limiting their accuracy in certain scenarios.
- Understanding stochastic effects and individual-based interactions is crucial for realistic disease modeling.
Purpose of the Study:
- To develop an exact analytical solution for the SIR model using a novel quantum formulation.
- To investigate the impact of stochasticity and network structure on disease dynamics.
- To compare the exact results with standard mean-field theory and simulations.
Main Methods:
- Formulation of the SIR model as a quantum many-body problem using spin operators.
- Mapping the master equation to a solvable hierarchy of evolution equations.
- Exact analytical solution for the time evolution on a linear chain with uncorrelated initial conditions.
Main Results:
- An exact analytical solution for the time evolution of susceptible, infectious, and recovered individuals is derived.
- Unlike mean-field approaches, a finite stationary distribution for susceptible individuals is observed, even in large populations.
- The infectious population dynamics can exhibit an initial increase before decay, depending on parameters.
Conclusions:
- The quantum formulation provides an exact method to solve the stochastic SIR model.
- Stochastic fluctuations lead to distinct population dynamics compared to mean-field approximations.
- This approach offers a more accurate understanding of disease spread in structured populations.
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