Study of Birth-Death Processes with Immigration

Abstract

Birth-death processes are applied in the modelling of many biological populations, such as tumour cells and viruses. Various studies have established that birth-death processes, which occur when the population size is zero, are not in-line with reality in many situations. Therefore, in this study, the birth-death processes with immigration were investigated. We considered two immigration policies. First, immigration is allowed if and only if the population size is zero. Second, immigration at a constant rate is allowed irrespective of the population size. Birth and death rates were chosen such that the mean population size is a Gompertz function when the immigration rate is zero. The transient population size probability was obtained for both cases. Several tumour growth datasets were fitted using the mean population size of the above models and standard birth-death model without immigration. The two models with immigration provided entirely different probabilities of the population size being zero at an arbitrary epoch when compared with the model without immigration. Moreover, all three models provided a similar fit to the data. For each of the datasets studied, the models that allowed immigration produced less variance than the non-immigration model.

Keywords

• birth-death process
• Gompertz function
• immigration
• tumour growth