Immortal branching processes are branching processes where at least one new individual replaces each individual before death.
The researchers have put forth immortality by positing that either
Immortal Branching Process (IBP's) has been discussed in the research paper by P. L. Krapivsky & S. Redner titled “Immortal Branching Processes” that forms the basis of the following text.
Importance of this research
The dynamical aspect of immortal branching is discussed in the research paper. Understanding Immortal Branching Processes (IBPs) could have critical applications such as control epidemics. A mathematical analysis would also make our friends from the medical fraternity better equipped to fight dengue, cancer, etc.
Let us look at some common use-cases of building a mathematical model to study branching:
Understanding the current patient condition (platelets count) and the above two factors can help us model the platelet requirement of the patient that could be critical to saving the patient's life.
Percolation to understand branching
The researchers have dedicated this research paper to the memory of their dear friend, Dietrich Stauffer, whose major contribution was in the field of Percolation. Percolation can help us study population growths, the spread of epidemics, etc.
P → P + P ….. rate r
The above equation illustrates that P gives rise to P + P at rate r. Kindly note that P could refer to cells, infected individuals, etc.
The following 3 cases can occur, based on the value of r
The researchers have explained possibilities & derived mathematical equations for critical branching, branching without extinction, and branching with input in detail in the research paper.
In the words of the researchers,
We analyzed two immortal branching processes (IBPs). The first is a simple extension of critical branching in which extinction cannot occur. The emergent behaviors are remarkably subtle, that are replete with logarithmic corrections. The second is a branching process with a steady input of cells, or equivalently, immigration. Such a steady input arises in a variety of non-equilibrium manybody processes, such as aggregation with a steady input of monomers, fragmentation with a steady input of large clusters, and turbulence with a steady energy input at large scales. In these examples, many new features were uncovered by incorporating steady input. Rich behaviors also occur in spatially extended systems with a spatially localized input and the role of spatial degrees of freedom in IBPs poses interesting challenges.
Source: P. L. Krapivsky & S. Redner's “Immortal Branching Processes”