Network-based SEIVQHR modelling with optimal mobility control and QUBO reformulation

Abstract Human mobility between interconnected geographical regions can substantially influence the spatial transmission of infectious diseases, motivating epidemic-control strategies that account for both disease dynamics and regional interactions. In this study, we develop a network-based discrete-time SEIVQHR model comprising susceptible, exposed, infectious, vaccinated, quarantined, hospitalized, and recovered compartments, with inter-regional transmission represented through a normalized mobility matrix. The formulation incorporates vaccination, quarantine, hospitalization, and mobility-dependent infection within a unified metapopulation framework. A key feature of the model is the explicit representation of breakthrough infection among vaccinated individuals. Vaccine efficacy $$\epsilon $$ determines the residual susceptibility $$1-\epsilon $$ , and breakthrough infections enter the exposed compartment before progressing through the infectious pathway. We establish positivity, boundedness, and positive invariance of the discrete-time system under admissibility conditions involving the time step $$\Delta t$$ . The basic reproduction number $$R_0$$ is derived using a next-generation matrix formulation that accounts for both vaccination coverage and network connectivity. The disease-free equilibrium is shown to be locally asymptotically stable for $$R_0<1$$ and unstable for $$R_0>1$$ . We further formulate a finite-horizon optimal-control problem with binary regional mobility restrictions and continuous vaccination allocation subject to vaccine-supply constraints. For a prescribed vaccination trajectory, the binary mobility-control component is approximated locally by a quadratic objective and reformulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. The resulting QUBO is mapped to an equivalent Ising Hamiltonian, enabling solution using simulated annealing, greedy heuristics, or quantum-annealing platforms. The framework is calibrated using COVID-19 surveillance data from India for 2021, with selected epidemiological parameters fixed at literature-based values to reduce overfitting and maintain biological plausibility. Numerical experiments examine parameter sensitivity, reproduction-number behaviour, admissible parameter regions, QUBO construction, and solver performance. The results identify the transmission rate as the dominant sensitivity factor and demonstrate the feasibility of the proposed QUBO-based formulation for binary mobility-control optimization. Overall, the proposed framework provides a mathematically consistent decision-support methodology that connects network-based epidemic modelling, continuous vaccination allocation, binary mobility intervention, and QUBO/Ising optimization while retaining the nonlinear epidemic dynamics for validation of the resulting intervention strategies.