Theoretical analysis and practical application of multi-patch infectious disease model

Human movement plays a key role in the spread of infectious diseases. However, in real life, mobility behavior often exhibits significant heterogeneity. Therefore, secondary contact is considered, which increases the probability that susceptible populations turn exposed. Based on these considerations, a multi-patch system that incorporates the above factors is proposed. First, the dynamics of this system are analyzed. The system’s well-posedness and the impact of the global basic reproduction number on disease transmission are then established. Additionally, the uniqueness and stability of the endemic equilibrium point is proved when the global basic reproduction number exceeds 1. The relationship between the patch-specific basic reproduction number and the global reproduction number is also established. Next, a local analysis examines the internal dynamics of each patch, and the bifurcation phenomena in the system are demonstrated. Finally, in the numerical simulation section, the impact of the basic reproduction number on disease transmission is analyzed through the control variable method and the partial rank correlation coefficient $\left(PRCC\right)$ method. The model is fitted to actual Human Immunodeficiency Virus $\left(HIV\right)$ data from Africa, with five countries selected as examples based on the clustering principle. The global basic reproduction number is calculated, and the results indicate that the disease does not outbreak when the global basic reproduction number falls below 1. Furthermore, the seasonal autoregressive integrated moving average $\left(SARIMA\right)$ method is used for prediction to verify our hypothesis, which provides practical significance to the model.