Modeling Epidemics with Memory Effects: an Open Quantum System Approach

TL;DR

The paper addresses modeling epidemics with memory by casting the disease dynamics into an open quantum system framework and employing a GKSL-based evolution with memory terms. The core method uses a reduced density operator $\rho(t)$ evolving as $\frac{d}{dt}\rho(t)=\mathcal{L}(t)[\rho(t)]+\int_0^t d\tau\,\mathcal{K}(\tau)[\rho(t-\tau)]$, with Gaussian memory kernels for infection, recovery, and death, and a finite Hilbert space that encodes compartments via Lindblad operators. The authors derive explicit generalized master equations for populations and coherences, propose positivity conditions through a dynamical map with survival probabilities $g_n(t)$, and demonstrate through numerical experiments that the model captures epidemic peaks and memory effects while maintaining physical positivity under suitable parameters. They also discuss limitations regarding complete positivity in non-Markovian GKSL dynamics and outline potential extensions to other domains where memory plays a crucial role, such as decision-making processes.

Abstract

In this work, we introduce a quantum-inspired epidemic model to study the dynamics of an infectious disease in a population divided into compartments. By treating the healthy population as a large reservoir, we construct a framework based on open quantum systems and a Hilbert space formalism to model the spread of the infection. This approach allows for a mathematical framework that captures both Markovian and semi-Markovian dynamics in the evolution equations. Through numerical experiments, we examine the impact of varying memory parameters on the epidemic evolution, focusing in particular on the conditions under which the model remains physically admissible.

Figures (2)

• Figure 1: (a) Time evolutions of the expectation values $\langle {\cal I} \rangle,\langle {\cal R} \rangle,\langle {\cal D} \rangle$. (b) Time evolutions of the survival probability functions $g_{|100\rangle}(t),g_{|101\rangle}(t),g_{|110\rangle}(t)$. (c) Region of positivity-negativity of the evolution for the survival probability function $g_{|100\rangle}$ in the $\kappa_R-\kappa_D$ plane: white region admissible region, black region non admissible region. (d) Time evolutions of the entropy and purity.
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