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
- A single particle cannot die; OR
- There exists an immortal stem cell that gives birth to normal cells that can subsequently undergo critical branching.
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.
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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:
- Model & control epidemics: For example, understanding the mathematics behind Covid-19 can help us predict waves and save human lives at large.
- Condition Monitoring for Patients: When a patient is affected by the Dengue virus, the body's auto-immune system destroys platelets. The body is also creating platelets at the same time. Platelet count in the patient at any given time could be modeled by understanding the below two forces:
- Platelet Depletion: Auto-immune system destroys platelets
- Platelet Addition: The body produces platelets naturally.
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
- Critical Branching Process: In this branching, r=1 means that particles give birth to a single offspring before dying. In this branching, the average population is constant in time.
- Sub-critical Branching: Here, r<1 meaning more deaths happen than births. In this case, the population quickly goes extinct.
- Supercritical branching: Here, r>1, which means more births happen than deaths. It results in unlimited growth.
The researchers have explained possibilities & derived mathematical equations for critical branching, branching without extinction, and branching with input in detail in the research paper.
Conclusion
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”