Two-Type Reducible Age-Dependent Branching Processes With Non-Homogeneous Poisson Immigration

Abstract

Two type reducible age-dependent branching stochastic processes with non-homogeneous Poisson immigration are considered as models of renewal cell population dynamics. The asymptotic behaviour of the first moments of the process with or without immigration is investigated. Several classes of asymptotic behavior are identified for the population dynamics. Our results are also useful for developing associated methods of statistical inference.

1. Introduction

This paper deals with a generalization of a class of branching stochastic processes considered by Hyrien and Yanev [¹] and motivated by analyses of the generation of terminally differentiated oligodendrocytes, which are the myelin-forming cells of the central nervous system, and by the proliferation of leukemia cells. In both biological systems, the cell population is maintained or expands by using self-renewal division and/or through an influx of stem/progenitor cells in upstream compartments that become further committed (or differentiated). The consideration of these cellular systems motivated the formulation of an age-dependent branching process with non-homogeneous Poisson immigration to be defined below. Yakovlev and Yanev [²] and Hyrien and Yanev [¹] investigated the one-type version of this process. Because biological experiments are able to distinguish between cell types, it is equally important to study the multitype case. In the present paper we deal with two-type reducible age-dependent branching stochastic processes allowing an immigration component with a possibly time inhomogeneous rate.

Multitype Markov branching processes were first introduced in the literature in the well known papers of Kolmogorov and Dmitriev and Kolmogorov and Sevastyanov (for references and details see [³]). The asymptotic behavior of multitype processes has been well studied in the non-reducible case (e.g., Sevastyanov [³], Mode [⁴] or Athreya and Ney [⁵]). In contrast, no general theory exists for reducible processes so that every particular case is of interest, even from a mathematical standpoint. Biological applications of branching processes to biology can be found in the well-known books by Jagers [¹⁴], Yakovlev and Yanev [¹⁵], Kimmel and Axelrod [¹⁶] and Haccou et al. [¹⁷]. For applications specific to cell biology, we refer also to [^6–13].

This paper is organized as follows. Section 2 presents the modeling assumptions and gives equations for the associated probability generating functions (p.g.f.) and for the first-order moments of the processes (with or without immigration). In particular, we show that the moments of the process without immigration satisfy renewal-type equations. Section 3 summarizes well-known asymptotic results of renewal theory (Theorem A), and extends these results to a general class of renewal-type equations (Theorem 1). Using these results, we investigate the asymptotic behavior of the expectations of the processes with and without immigration (see Theorems 2 and 3, and Corollaries 1 and 2 of Section 3). We deduce that the “critical” parameter governing the asymptotic behavior of the process is α = max{α₁, α₂}, where α_i denotes the Malthusian parameter associated with type-i cells, i = 1, 2. The asymptotic results can also be used to develop an asymptotic estimation theory.

2. Models and equations

We consider modeling a cell population that consists of two cell-types using a process that begins at t = 0 with a single type-1 cell of zero age. In applications to oligodendrocytes, type-1 and type-2 cells could correspond to oligodendrocytes type-2 astrocytes precursor cells and to oligodendrocytes, for instance. The lifespan of every type-1 cell is described by a random variable (r.v.) τ₁ with cumulative distribution function (c.d.f.) G₁(t) = P(τ₁ ≤ t) satisfying G₁(0+) = 0. Upon completion of its lifespan, every type-1 cell produces a random number of offsprings ξ = (ξ₁, ξ₂), where ξ₁ and ξ₂ denote the number of type-1 and the number of type-2 offsprings, respectively. Put $h_1(s_1,s_2)=E\{s_1^{{\xi }_1}{s}_2^{{\xi }_2}\}$, |s_i| ≤ 1, i = 1, 2, for the offspring p.g.f., and define the associated expectations $a_1=E{\xi }_1=\frac{\partial h_1(s_1,s_2)}{\partial s_1}{|}_{s_1=s_2=1},a_2=E{\xi }_2=\frac{\partial h_1(s_1,s_2)}{\partial s_2}{|}_{s_1=s_2=1}$. Likewise, the lifespan of every type-2 cell is described by a r.v. τ₂ with c.d.f. G₂(t) = P(τ₂ ≥ t) satisfying G₂(0+) = 0. Upon completion of its lifespan, every type-2 cell produces η₂ type-2 cells, where η₂ denotes a r.v. with p.g.f. $h_2(s_2)=E\{s_2^{{\eta }_2}\}$. Write $b_2=E{\eta }_2=\frac{{\mathit{\text{dh}}}_2(s_2)}{{\mathit{\text{ds}}}_2}{|}_{s_2=1}$ for the mean number of offsprings of type-2 cells. Lastly, every cell is of age zero at birth and evolves independently of every other cell. These assumptions therefore define a reducible two-type age-dependent branching process without immigration.

Let ${\mu }_1={\displaystyle {\int }_0^{\mathrm{\infty }}}{\mathit{\text{xdG}}}_1(x)<\mathrm{\infty }\text{ and }{\mu }_2={\displaystyle {\int }_0^{\mathrm{\infty }}}{\mathit{\text{xdG}}}_2(x)<\mathrm{\infty }$ denote the mean lifespans of type-1 and type-2 cells. Introduce the process without immigration Z(t) = (Z₁(t), Z₂(t)), where Z₁(t) and Z₂(t) denotes the number of type-1 and of type-2 cells at time t. Write s = (s₁, s₂), and define the p.g.f.

$$\begin{array}{c}\hfill F_1(t;s)=E\{s_1^{Z_1(t)}{s}_2^{Z_2(t)}|Z_1(0)=1\},\hfill \\ \hfill F_2(t;s_2)=E\{s_2^{Z_2(t)}|Z_2(0)=1\}.\hfill \end{array}$$

It can be shown that F₁ and F₂ satisfy the system of nonlinear integral equations

$$F_1(t;\mathbf{s})={\displaystyle {\int }_0^t}{h}_1(F_1(t-u;\mathbf{s}),F_2(t-u;s_2)){\mathit{\text{dG}}}_1(u)+s_1(1-G_1(t))$$ (1)

$$F_2(t;s_2)={\displaystyle {\int }_0^t}{h}_2(F_2(t-u;s_2)){\mathit{\text{dG}}}_2(u)+s_2(1-G_2(t)),$$ (2)

with initial conditions F₁(0; s₁, s₂) = s₁ and F₂(0; s₂) = s₂. Introduce the associated means:

$$\begin{array}{c}{A}_1(t)=E\{Z_1(t)|Z_1(0)=1\}=\frac{\partial }{\partial s_1}{F}_1(t;s_1,s_2){|}_{s_1=s_2=1},\hfill \\ A_2(t)=E\{Z_2(t)|Z_1(0)=1\}=\frac{\partial }{\partial s_2}{F}_1(t;s_1,s_2){|}_{s_1=s_2=1},\hfill \\ B_2(t)=E\{Z_2(t)|Z_2(0)=1\}=\frac{\partial }{\partial s_2}{F}_2(t;s_2){|}_{s_2=1}.\hfill \end{array}$$

It follows from equations (1) and (2) that A₁, A₂ and B₂ satisfy the system of renewal-type integral equations:

$$A_1(t)=a_1{\displaystyle {\int }_0^t}{A}_1(t-u){\mathit{\text{dG}}}_1(u)+1-G_1(t),A_1(0)=1,$$ (3)

$$A_2(t)=a_1{\displaystyle {\int }_0^t}{A}_2(t-u){\mathit{\text{dG}}}_1(u)+a_2{\displaystyle {\int }_0^t}{B}_2(t-u){\mathit{\text{dG}}}_1(u),A_2(0)=0,$$ (4)

$$B_2(t)=b_2{\displaystyle {\int }_0^t}{B}_2(t-u){\mathit{\text{dG}}}_2(u)+1-G_2(t),B_2(0)=1.$$ (5)

In order to formulate the process with immigration, let 0 = S₀ < S₁ < S₂ < S₃ < ⋯ denote time points generated by a non-homogeneous Poisson process Π(t) ∈ P_o(R(t)) with instantaneous and cumulative rates r(t) and $R(t)={\displaystyle {\int }_0^t}r(u)\mathit{\text{du}}$, with r(t) ≥ 0. The r.v.s S_k represent the times at which new immigrants (differentiated stem/progenitor cells) enter into the population of type-1 cells. Let U_k = S_k − S_k−1 denote inter-arrival times. Clearly, we have that $S_k={\displaystyle {\sum }_{i=1}^k}{U}_i,k=1,2\cdots$. Let I_k denote the number of immigrants entering the population at time S_k. We assume that these cells are all of zero age at time S_k, and that {I_k}_k=1,2⋯ are i.i.d. r.v.s with p.g.f. $g(s)={\mathit{\text{Es}}}^{I_k}={\displaystyle {\sum }_{i=0}^{\mathrm{\infty }}}{g}_i{s}^i,|s|\le 1$.

When {U_k}_k=1,2⋯ are r.v.s with c.d.f. G₀(x) = P(U_i ≤ x) = 1 − e^−rx, x ≥ 0, Π(t) reduces to an ordinary Poisson process with cumulative rate R(t) = rt.

Let Y₁(t) (resp., Y₂(t)) denote the number of type-1 (resp., type-2) cells at time t in the process with immigration. Write Y(t) = (Y₁(t), Y₂(t)), and assume Y(0) = (0, 0). This process admits the following representation

$$\begin{array}{cc}\mathbf{Y}(t)={\displaystyle \sum _{k=1}^{\mathrm{\Pi }(t)}}{\mathbf{Z}}^{I_k}(t-S_k)\hfill & \text{if}\mathrm{\Pi }(t)>0,\hfill \end{array}$$ (6)

and Y(t) = 0 if Π(t) = 0, where {Z^I_k (t)}_k=1,2⋯ denotes a collection of i.i.d. copies of the branching process without immigration Z(t) started with a random number of ancestors I_k. In fact, the process Y(t) begins with the first non-empty set of immigrants.

Introduce the p.g.f. $\mathrm{\Psi }(t;s_1,s_2)=E\{s_1^{Y_1(t)}{s}_2^{Y_2(t)}|\mathbf{Y}(0)=(0,0)\}$. It follows from equation (6) that

$$\mathrm{\Psi }(t;s_1,s_2)=\text{exp}\{-{\displaystyle {\int }_0^t}r(t-u)[1-g(F_1(u;s_1,s_2))]\mathit{\text{du}}\}$$ (7)

where Ψ(0; s₁, s₂) = 1 and where the p.g.f. F₁(u; s₁, s₂) satisfies equation (1). The proof is similar to that of the one-dimensional case considered in Yakovlev and Yanev ([⁷], Theorem 1), and requires applications of the Order Statistics Property and random time change methods. The process Y(t), t ≥ 0, is clearly non-homogeneous in time and non-Markov.

Let $\gamma =E\{I_k\}=\frac{\mathit{\text{dg}}(s)}{\mathit{\text{ds}}}{|}_{s=1}$ be the immigration number mean. Introduce the expectations of the process with immigration

$$\begin{array}{c}{M}_1(t)=E\{Y_1(t)|\mathbf{Y}(0)=(0,0)\}=\frac{\partial }{\partial s_1}\mathrm{\Psi }(t;s_1,s_2){|}_{s_1=s_2=1},\hfill \\ M_2(t)=E\{Y_2(t)|\mathbf{Y}(0)=(0,0)\}=\frac{\partial }{\partial s_2}\mathrm{\Psi }(t;s_1,s_2){|}_{s_1=s_2=1}.\hfill \end{array}$$

Using equation (7), we obtain that

$$\begin{array}{cc}{M}_1(t)=\gamma {\displaystyle {\int }_0^t}r(t-u)A_1(u)\mathit{\text{du}},\hfill & M_1(0)=0\hfill \end{array},$$ (8)

$$\begin{array}{cc}{M}_2(t)=\gamma {\displaystyle {\int }_0^t}r(t-u)A_2(u)\mathit{\text{du}},\hfill & M_2(0)=0\hfill \end{array},$$ (9)

where A₁(t) and A₂(t) are determined by equations (3) – (5).

Remark 1. In biological applications, the most interesting parameterizations of the offspring p.g.f., where as usual h₁(1, 1) = 1, are

• Model 1: $h_1(s_1,s_2)=p_0+p_1{s}_1^2+p_2{s}_2$, which means that, upon completing its lifespan, every type-1 cell either dies with probability p₀, or it divides in two type-1 cells with probability p₁, or it differentiates into a single type-2 cell with probability p₂;

• Model 2: $h_1(s_1,s_2)=p_0+p_1{s}_1^2+p_2{s}_2^2$;

• Model 3: $h_1(s_1,s_2)=p_0+p_1{s}_1^2+p_2{s}_1{s}_2$;

• Model 4: $h_1(s_1,s_2)=p_0+p_1{s}_1^2+p_2{s}_2^2+p_3{s}_1{s}_2$;

In all 4 cases, we have $h_2(s_2)=1-q+{\mathit{\text{qs}}}_2^2,0\le q\le 1$, meaning that every type-2 cell either die with probability 1 − q or it divides into two type-2 cells with probability q.

3. Renewal type equations and asymptotic behaviour of the means

Note that the moments of the process without immigration (see (3)–(5)) satisfy renewal equations given in the following general form

$$U(t)=\varkappa {\displaystyle {\int }_0^t}U(t-x)\mathit{\text{dG}}(x)+f(t),\varkappa >0,$$ (10)

where G(x) is a c.d.f. with a Laplace transform $\widehat{G}(\lambda )={\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-\lambda x}\mathit{\text{dG}}(x)$. If ϰG(0+) < 1, (10) has a unique solution bounded on all bounded intervals. The Malthus parameter α determined by the equation ϰĜ(α) = 1 plays an important role in the asymptotic behavior of U(t). Specifically, ϰĜ(α) = 1 has a unique real solution α ≥ 0 when ϰ ≥ 1. A solution may not exist when ϰ < 1, but if it does it has to be negative. In what follows we assume that the Malthus parameter exists. Introduce the c.d.f. $\tilde{G}(t)=\varkappa {\displaystyle {\int }_0^t}{e}^{-\alpha x}\mathit{\text{dG}}(x)$, and define the means $\mu ={\displaystyle {\int }_0^{\mathrm{\infty }}}\mathit{\text{xdG}}(x)\text{ and }\tilde{\mu }={\displaystyle {\int }_0^{\mathrm{\infty }}}\mathit{\text{xd}}\tilde{G}(x)$, assumed finite when required. Theorem A recall classical asymptotic results for the renewal processes (e.g., see Feller [¹⁸]).

Theorem A.

1. Let ϰ = 1 and f(t) be a directly Riemann integrable (d.R.i) function. Then lim_t→∞ $U(t)={\displaystyle {\int }_0^{\mathrm{\infty }}}f(x)\mathit{\text{dx}}/\mu$.

2. Let ϰ = 1 and lim_t→∞ f(t) = C, 0 < C < ∞. Then U(t) ~ Ct/μ, t → ∞.

3. Let ϰ < 1 and lim_t→∞ f(t) = C, 0 < C < ∞. Then lim_t→∞ U(t) = C/(1 − ϰ).

Note that case (i) of Theorem A is well-known as the Key Renewal Theorem. Mitov and Yanev [¹⁹] obtained further developments (including when μ = ∞). Theorem 1 offers some extensions to Theorem A, which will prove useful for investigating the asymptotic behaviour of the moments of our process.

Theorem 1. Let μ < ∞, μ̃ < ∞ and f(t) ~ Ct^ρe^βt, 0 < C < ∞, ρ ≥ 0, t → ∞.

1. If α < β, then U(t) ~ Ct^ρe^βt/[1 − ϰĜ(β)];

2. If α > β, then $U(t)\sim e^{\alpha t}{\displaystyle {\int }_0^{\mathrm{\infty }}}f(x)e^{-\alpha x}\mathit{\text{dx}}/\tilde{\mu }$;

3. If α = β, then U(t) ~ Ct^ρ+1e^αt/μ̃(ρ + 1).

Remark 2. When ρ = β = 0, case (i) of Theorem 1 reduces to case (iii) of Theorem A, and case (iii) of Theorem 1 leads to case (ii) of Theorem A.

Setting s₂ = 1 in (1) and (2), it is not difficult to see that Z₁(t) and Z₂(t) can be considered as classical age-dependent Bellman-Harris branching processes with initial conditions Z₁(0) = Z₂(0) = 1. Introduce the associated Malthus parameters α₁ and α₂ solving the equations

$$\begin{array}{ccc}{a}_1{\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-{\alpha }_{1}x}{\mathit{\text{dG}}}_1(x)=1\hfill & \text{and}\hfill & b_2{\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-{\alpha }_{2}x}{\mathit{\text{dG}}}_2(x)=1\hfill \end{array}.$$ (11)

The one-dimensional process Z_i(t), i = 1, 2, is classified as subcritical if α_i < 0 (a₁ < 1 or b₂ < 1), critical if α_i = 0 (a₁ = 1 or b₂ = 1) and supercritical if α_i > 0 (a₁ > 1 or b₂ > 1) (e.g., Harris, 1963; Athreya and Ney, 1972). Moreover, we have A₁(t) ≡ 1 and B₂(t) ≡ 1 in the critical case, and, in the non-critical cases,

$$\begin{array}{ccc}{A}_1(t)\sim C_1{e}^{{\alpha }_{1}t}\hfill & \text{and}\hfill & B_2(t)\sim C_2{e}^{{\alpha }_{2}t}\hfill \end{array},t\to \mathrm{\infty },$$ (12)

where

$$C_1=(a_1-1)/{\alpha }_1{a}_1{\tilde{\mu }}_1<\mathrm{\infty },C_2=(b_2-1)/{\alpha }_2{b}_2{\tilde{\mu }}_2<\mathrm{\infty },$$ (13)

and where

$$\begin{array}{c}{\tilde{\mu }}_1=a_1{\displaystyle {\int }_0^{\mathrm{\infty }}}{\mathit{\text{xe}}}^{-{\alpha }_{1}x}{\mathit{\text{dG}}}_1(x)={\displaystyle {\int }_0^{\mathrm{\infty }}}\mathit{\text{xd}}{\tilde{G}}_1(x),\hfill \\ {\tilde{\mu }}_2=b_2{\displaystyle {\int }_0^{\mathrm{\infty }}}{\mathit{\text{xe}}}^{-{\alpha }_{2}x}{\mathit{\text{dG}}}_2(x)={\displaystyle {\int }_0^{\mathrm{\infty }}}\mathit{\text{xd}}{\tilde{G}}_2(x),\hfill \end{array}$$ (14)

assumed finite. Notice that (11) implies G̃₁(∞) = G̃₂(∞) = 1.

The two-type process Z(t) is reducible, and the asymptotic behaviour of A₂(t) depends of both Malthus parameters, as stated in Theorem 2.

Theorem 2. We have, as t → ∞, that

	∼	K1teα1t	δ = 0	K1 = a2C2/a1μ̃1,
A2(t)	∼	K2eα1t	δ < 0	$K_2=\frac{a_2}{a_1{\tilde{\mu }}_1}{\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{\delta x}{\overline{B}}_2(x)\mathit{\text{dx}},$
	∼	K3eα2t	δ > 0	K3 = a2C2Ĝ1(α2)/[1 − a1 Ĝ1(α2)],

where δ = α₂ − α₁, B̅₂(t) = e^−α₂t B₂(t) and ${\widehat{G}}_1({\alpha }_2)={\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-{\alpha }_{2}x}{\mathit{\text{dG}}}_1(x)<\mathrm{\infty }$.

Corollary 1. (Markov case) When G₁(t) = 1 − e^−β₁t and G₂(t) = 1 − e^−β₂t, the Malthus parameters are α₁ = β₁(a₁ − 1) and α₂ = β₂(b₂ − 1). Therefore, solving (3)–(5) directly yields A₁(t) = e^α₁t and B₂(t) = e^α₂t. Likewise, A₂(t) = a₂β₁te^α₁t when α₁ = α₂, and $A_2(t)=\frac{{\alpha }_2{\beta }_1}{{\alpha }_2-{\alpha }_1}(e^{{\alpha }_{2}t}-e^{{\alpha }_{1}t})$ when α₁ ≠ α₂. Therefore, in accordance with δ = α₂ − α₁, we have

	∼	K1teα1t	δ = 0	K1 = a2β1,
A2(t)	∼	K2eα1t	δ < 0	K2 = a2β1/(−δ),
	∼	K3eα2t	δ > 0	K3 = a2β1/δ.

These results follows also directly from Theorem 2.

The asymptotic behaviour of the means for the process with immigration can now be deduced from equations (8)–(9). Let α = max{α₁, α₂},

$$\begin{array}{c}{K}_{\alpha }=\{K_1,\text{ if }\alpha ={\alpha }_1={\alpha }_2\}\vee \{K_2,\text{ if }\alpha ={\alpha }_1>{\alpha }_2\}\vee \{K_3,\text{ if }\alpha ={\alpha }_2>{\alpha }_1\},\hfill \\ \overline{R}(t)={\displaystyle {\int }_0^t}R(u)\mathit{\text{du}},{\widehat{R}}_z(t)={\displaystyle {\int }_0^t}{e}^{\mathit{\text{zx}}}\mathit{\text{dR}}(x),{\tilde{R}}_{{\alpha }_1}(t)={\displaystyle {\int }_0^t}{\widehat{R}}_{{\alpha }_1}(u)\mathit{\text{du}},\hfill \\ R_{\alpha }(t)=\{{\tilde{R}}_{{\alpha }_1}(t),\alpha ={\alpha }_1={\alpha }_2\}\vee \{{\widehat{R}}_{{\alpha }_1}(t),\alpha ={\alpha }_1\}\vee \{{\widehat{R}}_{{\alpha }_2}(t),\alpha ={\alpha }_2\}.\hfill \end{array}$$

Theorem 3. We have, as t → ∞, that

• (3.1)M₁(t) = γR(t), α₁ = 0; M₁(t) ~ γC₁e^α₁t R̂_α1 (t), α₁ ≠ 0;
• (3.2)M₂(t) ~ γK_αe^αt R_α(t), t → ∞.

Corollary 2. (time-homogeneous immigration). When R(t) = rt, the results of Theorem 3 simplify as follows:

• M₁(t) ~ rγC₁e^α₁t/α₁, α₁ > 0; M₁(t) → rγC₁/(−α₁), α₁ < 0,; and M₁(t) = rγt, when α₁ = 0.

• If α₁ ≠ α₂: M₂(t) ~ rγK_αe^αt/α, α > 0; ~ rγK_αt, α = 0; → rγK_α/(−α), α < 0.

• If α₁ = α₂ = α: M₂(t) ~ rγK₁te^αt/α, α > 0; $\sim \frac{r\gamma }{2{\mu }_1}{t}^2,\alpha =0;\to r\gamma K_1/(-\alpha ),\alpha <0$.

Comment

Based on the asymptotic behavior of the means, we can classify the two-type process as subcritical if α < 0, critical if α = 0, and supercritical if α > 0. Moreover, in the time-homogeneous case R(t) = rt, the means grow exponentially in the supercritical case, converge to some constants in the subcritical case, and increase linearly or even quadratically in the critical case. When the immigration is time-inhomogeneous, the asymptotic behavior of the means depends also on the cumulative rate R(t).

Let Q₁₂(t) = P{Z₁(t) = 0, Z₂(t) = 0 | Z₁(0) = 1, Z₂(0) = 0} = F₁(t; 0, 0) denote the probability of extinction (degeneration) at t, which can be characterized by (1) and (2). The role of the critical parameter α = max{α₁, α₂} is further confirmed by the following Theorem.

Theorem 4. We have q₁₂ = lim_t→∞ Q₁₂(t) = 1 (extinction or degeneration) if α ≤ 0, and q₁₂ < 1 (non-extinction or non-degeneration) if α > 0.

Note that the asymptotic behavior of Q₁₂(t) can be deduced from the results of Vatutin [²⁰].

Finally, it is worth mentioning that equations for the second factorial moments can be deduced from (1), (2) and (7). These equations are also renewal-type equations of the form given in equation (10). Using the results presented in this paper, it is possible to derive the asymptotic behavior of these second-order moments, which appears to be much more complicated and diverse. These results will be presented in another paper.