Improved small-world network --model definition and generative function derivation in infectious disease scenarios

In recent years, there has been growing interest in predicting and simulating the spread and prevention of infectious diseases. The most commonly used model is the SEIR model, which characterizes the spread process by introducing parameters such as the infection rate, contact numbers, incubation period, and recovery rate. Despite its widespread application, this model has some limitations. Firstly, it is a quantity-based model that only reflects changes in quantity and cannot capture the spatial spread of the disease. Secondly, in the transmission of infectious diseases, there is a noticeable threshold phenomenon in terms of quantity, which the SEIR model cannot depict. The use of SEIR in specific scenarios can partially address the aforementioned issues. Common approaches include considering individuals as organized within a small-world network, where the proximity of individuals on this network influences the spread of infectious diseases. The model constructed in this paper attempts to introduce geographical space on this basis, defining a small-world network in geographical space and providing a generating function for the percolation phenomenon of disease transmission within it.