Extinction Moment for a Branching Tree Evolution with Birth Rate and Death Rate Both Depending on Age

Abstract

In this paper, a branching tree evolution is established, in which the birth rate and the death rate are both dependent on node’s age. The extinction probability and the t-pre-extinction (extinct before time ) probability are studied, by which the distribution of the extinction moment can be given. The analytical formula and the approximation algorithm for the distribution of extinction moment are given; furthermore, the analytical formula and the approximation algorithm of extinction probability are given, and a necessary and sufficient condition of extinction with probability 1 is given. It is the first time to study the distribution of extinction time for the branching process with birth rate and the death rate both depending on node’s age, and the results will do great help in the theory of branching process. It is expected to be applied in the fields of biology, genetics, medicine, epidemiology, demography, nuclear physics, actuarial mathematics, algorithm, and data structures, etc.

1. Introduction

The classical biological reproduction model G-W branching process [1] has been extended to different biological reproduction models, such as branching processes in random environments [2–4] and branching population evolution models [5–11]. The age-dependent branching process was introduced by Bellman and Harris [6]. In branching models, the population extinction problem is one of the primary research contents. Many problems in branching models related to the population extinction are studied, but the distribution of extinction moment is hardly involved. In this paper, a branching tree evolution is established, in which the birth rate and death rate are both dependent on node’s age. The extinction probability and the t-pre-extinction (extinct before time ) probability are studied, by which the distribution of the extinction moment can be given.

The paper is organized as follows. The model is described and the existence theorem is presented in Section 2. In Section 3, the extinction probability is studied, and the analytical formula and the approximation algorithm of the extinction probability are given. A necessary and sufficient condition of extinction with probability 1 is also given. In Section 4, the t-pre-extinction probability is studied, the iterative integral equation with unique solution is established, which is satisfied by the t-pre-extinction probability, and the analytical formula and the approximation algorithm of t-pre-extinction probability are given. The stochastic order of extinction moment is studied in Section 5. The conclusions are presented in Section 6.

2. Description and Existence Theorem for the Model

In this paper, based on the mechanism of asexual reproduction of biological population, a continuous time random graph evolution is constructed, in which a node’s birth rate and death rate are both dependent on the node’s age.

Given a population is composed of biological individuals (nodes). The evolution of the population is based on the following basic assumptions:(1)All nodes in the population are homogeneous and mutually independent(2)The node’s death rate in the population is a nonnegative function dependent on the node’s age, such that (3)The node’s birth rate in the population is a nonnegative function dependent on the node’s age(4)Conditioned under a node being alive, the node’s reproduction behaviors in the future are conditional independent(5)Conditioned under a node being alive, the node’s death is conditional independent with the node’s reproduction(6)At initial time , there is only one initial node in the population (this condition is not essential, only for convenience of presentation)(7)In addition to the initial node, each of other nodes in the population has only one parent node

Based on the above assumptions, the branching tree evolution is described as follows.

Given a node in the population, its age is at time . For a sufficiently small period , conditioned under node being alive at time , the conditional probability for node being dead in the period is , the conditional probability for node producing one child node in the period is , and the conditional probability for node producing more than one child node in the period is .

In the population, if node is a child of node , then there is a directed link from node to node . When at least one of the parent and child dies, the link between them is a virtual (dotted) line, and the dead node is called a virtual node. Otherwise, it is called a real node, and so on. At time , all nodes (real and virtual) and directed links (real and virtual) construct a directed random tree, denoted by . And thus, the process of reproduction is an evolution of random trees, denoted by . As the evolution is characterized by the birth rate and the death rate , therefore, the model is referred to as “branching tree evolution with birth rate and death rate both depending on age,” denoted by .

According to the definition of the model, , the number of offspring born in period is finite, and no more than one offspring will be born at the same time. Therefore, the initial node and all its offspring nodes can be ordered as according to the order of birth time.

, denotewhere is the vector of the labeled nodes; is the vector of the adjacency relation (parent-child relation) between nodes: means that the initial node has no parent node. indicates that node is the parent of node , ; is the vector of node’s alive-death status: denotes that node is alive and denotes that node is dead, ; and is the birth time vector: represents there is an initial node at time , is the time when node is born, , and implies that no more than one node is born at the same time.

, denote

, denote . , denote

, and , define the function on . ,where is a indicative function, and

, and , define the function on :

Let be a Lebesgue measure on , for a given , and , then is a measure on . Let be a count measure on , denote , and then is a measure on . Define ,

Then, , , is a measure on , , , is a measurable function on .

Let

Let be the number of nodes in the random branching tree , then can be expressed by a matrix, i.e.,

The birth time and the alive-death status of the nodes in can be expressed by the matrix , and denote

We have the following theorem.

Theorem (existence) is the marginal process of the nonhomogeneous Markov process in the state space , where the transfer function of is here wherewhere . Substituting (12) and (13) into (11), the transfer function of is obtained.

It is not difficult to prove that is a Borel subset of the separable complete distance space , and the existence theorem of can be proved by the existence theorem of Markov process. is a marginal process of , and thus, the existence of is proved.

3. The Extinction Probability

Define

is called the extinction moment, at which the population extinct. The probability is called t-pre-extinction probability, which is the probability of the population extinct before time , and the probability is called extinction probability. The distribution of extinction moment is given by t-pre-extinction probability and . If , i.e., . Then, is a real-valued random variable, so the t-pre-extinction probability is the distribution function of .

In this section, the extinction probability for the branching tree evolution is studied, the analytical formula and the approximation algorithm of extinction probability are given, and a necessary and sufficient condition of extinction with probability 1 is also given.

Let be the number of nodes that are alive in the population at time , then . Obviously, when , and .

It is obvious that has the following properties:(1)(2) is continuous on

Lemma 1. Given a node in the population, its lifespan is , and then has the probability density function:where is the node’s death rate.
It is easy to prove Lemma 1.

Theorem 1. The following iterative integral equation is satisfied by the t-pre-extinction probability where is the node’s birth rate and is the node’s death rate.

Proof. , Let be the node’s lifespan, then we getIn the following, we first calculate . Equally divide the interval into intervals, denote , and let be the random event: the initial node produces a child node in the period and the offspring of this child node extinct before time or the initial node does not produce a child node in the period , .
When is sufficiently small, the probability that the initial node does not produce a child node in the period is ; the probability of producing more than one child node is ; the probability of producing one child node is , and this child node’s offspring extinct before time with probability , soNoting the independent assumptions of the model, we haveThen,That isSoThus, the theorem is proved.
Denote

Lemma 2. The extinction probability is a solution of the equation

Proof. ,, and let be large enough, such thati.e., , which imply .
Thus, Lemma 2 is proved.
It is easy to prove that the function has the following properties.

Lemma 3. (1)(2) is increasing on (3) is a strictly concave function on

Theorem 2. The extinction probability is the smallest solution of the equation

Proof. By Lemma 3, is a strictly concave function on , and thus, is also a strictly concave function on . It is easy to see that any strictly concave function has at most two different roots in its definition domain; hence, has at most two different solutions on , one of which is . Let be the smallest solution of the equation .(1)If , since is the smallest solution of the equation, then the equation has no solution in . But, by Lemma 2, is the solution of the equation, infer that , i.e., is the smallest solution of the equation .(2), let such that , since is the unique solution of the equation in , it is easy to see that .It is easy to prove . In fact, suppose contrarily , such that , thenNoting that is increasing, thenThis contradicts , so the above assumption is not true. By Lemma 2, is the solution of the equation, i.e., . By the fact that is the smallest solution of the equation, imply that . So is the smallest solution of the equation. Therefore, the theorem is proved.

Corollary 1. If , thenFurthermore, .

Proof. If , thenLet , that is,Obviously, ; the two roots of the above equation are as follows:Then,thus,The proof is completed.
As a consequence of Theorem 2, a sufficient condition for is given.

Corollary 2. , if , then .

Proof. According to the assumptions, there isIf is a solution of the equation , then satisfiesThus, , i.e., , so , deduce .
Corollary 2 shows that when the death rate is greater than the birth rate, the population is certainly extinct, which is intuitive.

Corollary 3. Let and be two branching tree evolutions with different birth rates and the same death rate. The corresponding extinction probabilities are denoted by and , respectively. If , then .

Proof. , .
If , then by the definition of , it is easy to see that , and , .
Since is a continuous function with at least one smallest root on (), and , therefore, the smallest root of on is less than or equal to the smallest root of on , and by Theorem 2, there isThe proof is completed.
Corollary 3 implies that, for two models with the same death rate, the higher the birth rate is, the lower the extinction probability is, which is intuitive.

Corollary 4. Let and be two branching tree evolutions with different death rates but with the same birth rate. The corresponding extinction probabilities are denoted by and , respectively, , if , then .

Proof. Denote , , if , then .
So for any decreasing function .
The corresponding functions to are denoted by and , respectively. Noting that is a decreasing function with , so by the definition ofimply ,, then , ,
Since is a continuous function with at least one smallest root on , , and , therefore, the smallest root of on is less than or equal to the smallest root of on , and by Theorem 2,Corollary 4 shows that, for two models with the same birth rate, the randomly longer the lifespan is, the smaller the extinction probability is, which is intuitive.

Theorem 3. .

Proof. (1)Sufficiency: assume , let , then ; i.e., is the decreasing function on . Noting that , thus is the smallest root of on , and by Theorem 2, we get .(2)Necessity: assume , because is a strictly concave function on , is also a strictly concave function on , and . In addition, by Theorem 2 and the assumptions, it is obvious that is the smallest root of on , so is decreasing on . Hence, , that is, .Thus, the theorem is proved.
For introduced above, noting that , , denotei.e., is the times iteration of , and then, there is the following conclusion.

Theorem 4. , there is .

Proof. (1)If , by and Theorem 2, we have , . For the increasing property of , so , in addition, is continuous, then i.e., , and then, we can get by Theorem 2, i.e., .(2)If , , then , and ; thus, By Theorem 2, , i.e., .(3)If , , then , and , and thus, Noting that the equation has no root on the interval , so , i.e.,Note: the significance of Theorem 4 is obvious. It gives a numerical method to calculate the asymptotic value of extinction probability. For any initial value , iteration value is the asymptotic value of the extinction probability .

4. The -Pre-Extinction Probability

In this section, the analytic formula and the approximation algorithm of t-pre-extinction probability are given, and the iterative integral equation with unique solution is established, which is satisfied by the t-pre-extinction probability.

Let , , denote . Divide the interval equally into intervals . Step function is defined as follows:where

We always assume that the birth rate function is bounded in any finite interval; denote

Theorem 5. (1), is nondecreasing on (2) is a monotonic increasing sequence of functions(3),

Proof. (1)To prove is a nondecreasing function on because  So, is nondecreasing on ; suppose inductively that is nondecreasing on , then  Thus, is nondecreasing on . It is deduced by mathematical induction that is nondecreasing on .(2), to prove on , .  Hence, on . Suppose inductively that on , by the definitions of and , we have  where  Thus, it can be proved by mathematical induction that , , i.e., is a monotonic increasing sequence of functions.(3)For simplicity, denote , . Because is uniform continuous on , so , when is sufficiently large, that is sufficiently small.The following conclusion can be deduced by mathematical induction:It is easy to prove that , .When ,When ,So,Suppose inductively that .
Then,When ,,SoIt is obvious that when , there isSoIt is proved deductively that , .
Especially, .
That is,The proof is complete.

Theorem 6. The t-pre-extinction probability is the unique solution of the iterative integral equation in Theorem 1.

Proof. According to Theorem 1, the t-pre-extinction probability is the solution of the iterative integral equation in Theorem 1, and it is not difficult to deduce by Theorem 5 that the solution of the iterative integral equation is unique. Therefore, the t-pre-extinction probability is the unique solution of the iterative integral equation. The theorem is proved.

5. The Stochastic Order of Extinction Moment

If , then the extinction moment is a real-valued random variable, and the t-pre-extinction probability is the distribution function of . In this section, we study the stochastic order of the extinction moment for different branching tree evolutions.

Theorem 7. (1)Let and be, respectively, the extinction probability and extinction moment for the branching tree evolutions ; let and be, respectively, the extinction probability and extinction moment for the branching tree evolutions . If , , and , then is stochastically smaller than , that is .(2)Let and be, respectively, the extinction probability and extinction moment for the branching tree evolutions ; let and be, respectively, the extinction probability and extinction moment for the branching tree evolutions . If , , and , , then is stochastically smaller than , that is .

Proof. (1)Corresponding to the branching tree evolution and , similarly to Theorem 5, define the step function series as and , respectively. By the hypothesis of and the definition of , applying the mathematical induction, it is easy to prove that , , , and thus . By Theorem 5,  Thus, , .  Because , so . Thus, (1) is proved.(2)Corresponding to the branching tree evolution and , similarly to Theorem 5, define the step function series as and , respectively, and denote ; then , . Denote ; then is a decreasing function of . Applying the mathematical induction, we haveThus, , and by Theorem 5,Then, , so . Thus, (2) is proved and theorem is proved.

6. Conclusions

This paper addresses an important problem in the field of branching process. The extinction probability and the t-pre-extinction probability are studied by constructing a branching tree evolution model in which the birth rate and the death rate are both dependent on node’s age. The analytical formula and the approximation algorithm for the distribution of extinction moment are given; furthermore, the analytical formula and the approximation algorithm of extinction probability are given, and a necessary and sufficient condition of extinction with probability 1 is given.

Due to publishing constraints, only the population extinction is studied, the graph-topological properties and the age structure of nodes will be studied in subsequent papers.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by Shanghai Natural Science Foundation (no. 16ZR1414000).