From isolated patch to coupled structures: Nonlinear dynamic analysis and practical application of patch models in the context of infectious diseases

After the recovery of the COVID-19 epidemic, society faces challenges such as lax epidemic prevention behaviors and the continuous evolution of pathogens. To capture the spread patterns of infectious diseases and take timely action, we constructed a patch model based on population distribution, considering secondary transmission mechanisms. Initially, we explored bifurcation theory and analyzed the high-codimension Bogdanov–Takens and Hopf bifurcations on a single patch, without considering patch coupling. In the numerical simulation, Turing bifurcation dynamics were used to simulate the “outbreak” state of infectious diseases through the Turing patterns. Spatial expansion verification showed stronger connectivity in three-dimensional propagation. A sensitivity analysis was conducted on the basic reproduction number R0 using partial rank correlation coefficients and Latin square sampling to assess the influence of each parameter. Next, a migration matrix was constructed to represent coupling between patches, and an asymptotic stability analysis based on R0 in the disease-free equilibrium state revealed the critical relationship between R0 and the epidemiological threshold of 1. By establishing the local reproduction number under interpatch migration effects, we explored the relationship between the global and local reproduction numbers and verified the existence of a positive equilibrium point. In numerical studies, complex network topologies were used to quantify the effects of parameters on R0. We found that the Erdős–Renyi network’s uniform distribution characteristic naturally inhibits disease spread, as its basic reproduction number satisfies R0≡0. Finally, we analyzed the “resurgence” of H1N1 influenza using data from 38 countries across the Asia-Pacific, Europe, Africa and the Americas. The constrained nonlinear optimization algorithm and Monte Carlo simulation were used for model fitting, achieving a high determination coefficient of R2=0.9464 (using China as an example), confirming the model’s credibility. Future predictions of the global basic reproduction number R0 and trends were made through migration networks and a naive Bayes algorithm, showing a promising effect.