A new approach to pair approximation method for spatial model in ecology

A mathematical model using either mean-field approximation or pair approximation has both advantage and limitation. Mean-field approximation is suitable for describing behaviors of a large system; however, it neglects the scale of individual interaction. On the other hand, pair approximation is suitable for studying local interaction especially singles and pairs; however, the mathematical formulas are still limited. Thus, choosing the right model for the right purpose is recommended. The objective of this paper is to develop the techniques of pair approximation by using probability averaged value based on probability distribution; for example, multinomial distribution and Poisson distribution. These extended formulas are suitable for a model composed of individuals whose event rates are additionally affected by their nearby neighbor. Finally, we apply these tools to a simple SIS epidemic model.