Quantum Epidemic Model

• Quantum epidemic models are operator-based frameworks that use Hilbert-space methods and quantum stochastic dynamics to model classical epidemic transitions.
• They employ creation/annihilation operators, master equations, and quantum simulation techniques to capture detailed outbreak dynamics and memory effects.
• These models enhance the analysis of epidemic thresholds, optimize network control strategies, and bridge classical epidemiology with quantum information science.

Quantum epidemic models provide a formalism for epidemic processes using operator-theoretic methods, Hilbert-space representations, and quantum stochastic dynamics. These frameworks are not restricted to genuinely quantum systems but leverage algebraic techniques from quantum field theory and open quantum systems to describe stochastic birth–death processes, epidemic compartment transitions, spatial propagation in networks, and optimal control via combinatorial optimization. Applications span classical epidemiology (e.g., COVID-19, hepatitis C, influenza), quantum communication networks, and emerging computational paradigms utilizing quantum computing. The following exposition covers the mathematical foundations, representative model constructions, solution methodologies, and connections to physical systems.

1. Hilbert-Space and Operator Representations

The operator-theoretic approach to epidemic dynamics begins by constructing a Hilbert (or Fock) space spanned by occupation-number states. In the SIR compartment model, the bosonic Fock space is generated by the basis $\lvert n_S, n_I, n_R, n_D\rangle$, where $n_S, n_I, n_R, n_D \in \{0,1,2,\ldots\}$ count individuals in each compartment. Annihilation ($a_j$) and creation ($a_j^\dagger$) operators for each compartment obey canonical commutation relations $[a_i, a_j^\dagger] = \delta_{ij}$, and their action retrieves occupation numbers via $n_i = a_i^\dagger a_i$, with $$n_i\,|...,n_i,...\rangle = n_i\,|...,n_i,...\rangle$$ (Mondaini et al., 2021).

For network-based or agent-based epidemic systems, the state space may generalize to fermionic or multi-level subspaces (e.g., $C^2$, $C^3$) with basis vectors encoding presence/absence or specific compartment status across $N$ sites (Bagarello et al., 18 Mar 2025, Williams et al., 2011).

2. Quantum Epidemic Hamiltonians and Master Equations

The transitions between epidemic compartments are naturally encoded in operator-valued Hamiltonians reminiscent of second quantization:

$$\mathcal{H} = \beta n_S + \gamma n_I - [\beta a_I^\dagger a_S + (1-\sigma)\gamma a_R^\dagger a_I + \sigma\gamma a_D^\dagger a_I]$$

for an SIR model with fatality fraction $\sigma$ (Mondaini et al., 2021), or analogous constructions for extended compartment models integrating births, deaths, and chronic infection (Mondaini, 2015). The system's time evolution follows an “imaginary-time Schrödinger equation”:

$$\frac{d}{dt}|\nu(t)\rangle = -\mathcal{H}\,|\nu(t)\rangle$$

which is equivalent to the classical forward Kolmogorov (master) equation for compartment probabilities.

In quantum-inspired generalizations, the master equation can acquire memory effects and non-Markovianity via convolutional Lindblad terms (GKSL structure with memory kernels), allowing modeling of incubation periods, recovery/distribution tails, and history-dependent transitions (Bagarello et al., 18 Mar 2025). The density matrix evolves according to:

$$\frac{d\rho(t)}{dt} = \mathcal{L}_{nc}(t)[\rho(t)] + \int_0^t d\tau\,\mathcal{L}_{mem}(t-\tau)[\rho(\tau)]$$

with explicit jump operators for infection, recovery, and death.

3. Solution Methods: Mean-Field, Ensemble, and Exact Analytical Results

Quantum epidemic models yield multiple layers of solution techniques:

• Mean-field ODEs: Differentiating the moment-generating function or the operator equations leads to dynamical equations for the expected compartment populations. For instance,

$$\dot S = -\beta S\,,\quad \dot I = \beta S - \gamma I\,,\quad \dot R = (1-\sigma)\gamma I\,,\quad \dot D = \sigma\gamma I$$

• Exact recursions: In quantum time discretizations, the time grid $t_n = q^n t_0$ (with $q>1$) allows the formulation of recurrence relations for $S_n, I_n, R_n$, with the quantum derivative:

$D_q f(t) = \frac{f(qt) - f(t)}{(q-1)t}$

yielding closed-form product solutions for all compartment populations (Lemos-Silva et al., 9 Dec 2025).

• Operator-based ensemble solutions: Solving the master equation via matrix exponentials provides the full distribution of trajectories, enabling quantification of time-dependent uncertainty (variance, Shannon entropy) and limiting behavior at early/late times and outbreak peaks (Rojas-Venegas et al., 2024).
• Cluster-function method and n-point functions: In spatially structured models, e.g., 1D ring SIR, translationally-invariant cluster functions $K(n,t)$ and $G(n,t)$ enable exact computation of multi-point correlation functions and time-ordered probability densities for infected clusters, blocks, or rings (Williams et al., 2011).

4. Quantum Effects: Memory, Ballistic Spreading, and Dynamical Phenomena

Quantum epidemic frameworks allow the integration of effects that cannot be captured by classical Markovian models:

• Memory and semi-Markovian kernels: Gaussian or distributed recovery/death kernels introduce realistic waiting-time statistics beyond exponential, with wider tails generating protracted epidemic waves, echo outbreaks, and variability in compartment transitions (Bagarello et al., 18 Mar 2025).
• Ballistic propagation and interference: When epidemic spread is mapped onto quantum random walks (QRW) on spatial graphs, the dynamics interpolate between diffusive and super-diffusive (ballistic) scaling, with quantum coherence and interference dramatically increasing outbreak speed, cluster sizes, and the basic reproduction number $R_0$ (Manna et al., 6 Sep 2025).
• Quantum phase transitions and outbreak sequences: Open quantum spin system models, e.g., laser-driven Rydberg arrays, realize epidemic dynamics with coherent facilitation, long-range interactions, and dissipative processes, leading to discontinuous outbreak staircase transitions not seen in classical percolation (Pérez-Espigares et al., 2017). Survival probabilities and dead-density order parameters reveal phase boundaries and universality classes consistent with classical general epidemic processes (GEP) but with richer transient and recurrent outbreak patterns (Sturges et al., 28 Dec 2025).

5. Quantum Epidemics on Networks: Quantum Communication and Control

Quantum networks present unique constraints and opportunities for epidemic processes:

• Probabilistic link formation and epidemic resilience: In quantum networks, the weighted probability adjacency matrix $A_{ij}$ determines the spectral epidemic threshold $\tau_c = 1/\lambda_1(A)$, with quantum networks displaying higher thresholds due to sparser connectivity compared to classical counterparts. When matched for mean degree, thresholds converge, revealing the key importance of network topology over link physics (Hou et al., 20 Oct 2025).
• Quantum computation for epidemic control: Control strategies for network-based SIS/SIR processes (e.g., via mobility bans) can be posed as Quadratic Unconstrained Binary Optimization (QUBO) problems. Quantum annealing and gate-model algorithms enable near-real-time optimization of intervention policies, outperforming classical metaheuristics for large networks and short rolling horizons (Zino et al., 30 Aug 2025, Volpe et al., 17 Jul 2025).
• Quantum simulation protocols: Mapping classical SI/SIR models to spin-lattice (qubit) networks with parameterized Ising-type Hamiltonians enables simulation of epidemic trajectories on quantum devices. Bath coupling and reset protocols reproduce stochastic SI Markovian dynamics exactly, with bath parameters calibrated from empirical epidemiology (SAR, $R_0$, contact networks) (Wang et al., 2022).

6. Comparison to Classical Models and Interpretive Implications

There is deep structural equivalence between quantum-compliant epidemic models and classical stochastic processes:

• Second-quantization formalism as algebraic unification: The field-theoretic language of creation/annihilation operators compactly encodes all transition processes, permitting systematic derivation of moment equations, closure schemes, and path-integral techniques (Mondaini et al., 2021, Mondaini, 2015).
• Classical models embedding quantum phenomena: Tight-binding reformulations of SIS or network epidemics admit emergent “quantum” features such as interference, entanglement entropy, superposition, Rabi oscillations, and even Aharonov–Bohm analogs, blurring conceptual boundaries between quantum and classical stochastic dynamics (Pomorski, 2020).
• Intrinsic limits of predictability: Quantum-inspired operator models make explicit the stochastic ensemble of trajectories and reveal irreducible uncertainties in epidemic forecasting, set by the birth–death nature of contagion and recovery processes, independent of model complexity (Rojas-Venegas et al., 2024).

7. Extensions and Outlook

Quantum epidemic frameworks are extensible to multi-compartment systems (SEIR, age structure), more general stochastic processes, arbitrary network geometries, and advanced quantum computational implementations. The approach provides a unified methodological toolkit for exact, approximate, or simulation-based studies in population dynamics, epidemiological control, and quantum information science. Theoretical advances in memory kernels, quantum time discretizations, and operator decompositions offer new ways to analyze epidemic phase transitions, transient outbreaks, and the optimal allocation of interventions in large-scale systems with complex connectivity (Bagarello et al., 18 Mar 2025, Lemos-Silva et al., 9 Dec 2025, Volpe et al., 17 Jul 2025).