Hamiltonian optimal control of quarantine against epidemic spreading on complex networks

Effective optimization of prevention and control measures can significantly organize the spread of infectious diseases. In this paper, we construct an SIQRSV (Susceptible-Infected-Quarantined-Recovered-Susceptible-Vaccinated) compartmental model for infectious diseases on complex networks to study the infection mechanism. Specifically, we analyze the impact mechanism of infection rates, consider network heterogeneity, and examine the influence of network topology on disease spread. Using a system of differential equations, we can elucidate the disease transmission process. Furthermore, we obtain the disease-free equilibrium point of the system in its steady state. By constructing an autonomous equation, we derive the basic reproduction number of the system, and further validate it using the next-generation matrix method. Additionally, through the Jacobian matrix, we demonstrate the stability of the disease-free equilibrium points. Subsequently, based on the compartmental model, we consider the costs of treatment, control measures, and vaccination to construct a Hamiltonian system to optimize the quarantine rate. Finally, we conduct simulation experiments based on our proposed model on various networks, including BA scale-free networks and four empirical networks. The results indicate that compared to random quarantine measures, our optimized measures can effectively suppress the spread of infectious diseases, thereby providing theoretical support for policymakers in formulating control measures.