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arXiv:1304.0189v1 [math.PR] 31 Mar 2013
Fractional Non-Linear, Linear and Sublinear Death Processes Enzo Orsingher, Federico Polito, & Ludmila Sakhno Abstract This paper is devoted to the study of a fractional version of nonlinear M ν (t), t > 0, linear M ν (t), t > 0 and sublinear Mν (t), t > 0 death processes. Fractionality is introduced by replacing the usual integerorder derivative in the difference-differential equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan–Caputo. We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions of their classical counterparts with the random time process T2ν (t), t > 0. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.
1
Introduction
We assume that we have a population of n0 individuals or objects. The components of this population might be the set of healthy people during an epidemic or the set of items being sold in a store, or even, say, melting ice pack blocks. However even a coalescence of particles can be treated in this same manner, leading to a large ensemble of physical analogues suited to the method. The main interest is to model the fading process of these objects and, in particular, to analyse how the size of the population decreases. The classical death process is a model describing this type of phenomena and, its linear version is analysed in [1], page 90. The most interesting feature of the extinguishing population is the probability distribution pk (t) = Pr {M (t) = k | M (0) = n0 } ,
t > 0, 0 ≤ k ≤ n0 ,
(1.1)
where M (t), t > 0 is the point process representing the size of the population at time t. If the death rates are proportional to the population size, the process is called linear and the probabilities (1.1) are solutions to the initial-value problem d dt pk (t) = (µ(k + 1)pk+1 (t) − µkpk (t), 0 ≤ k ≤ n0 , (1.2) 1, k = n0 , pk (0) = 0, 0 ≤ k < n0 , 1
with pn0 +1 (t) = 0. The distribution satisfying (1.2) is n0 −k n0 −µkt pk (t) = , e 1 − e−µt k
0 ≤ k ≤ n0 .
(1.3)
The equations (1.2) are based on the fact that the death rate of each component of the population is proportional to the number of existing individuals. In the non-linear case, where the death rates are µk , 0 ≤ k ≤ n0 , equations (1.2) must be replaced by d dt pk (t) = (µk+1 pk+1 (t) − µk pk (t), 0 ≤ k ≤ n0 , (1.4) 1, k = n0 , pk (0) = 0, 0 ≤ k < n0 . In this paper we consider fractional versions of the processes described above, where fractionality is obtained by substitution of the integer-order derivatives appearing in (1.2) and (1.4), with the fractional derivative called Caputo or Dzhrbashyan–Caputo derivative, defined as follows Z t 0 f (s) dν f (t) = 1 ν ds, 0 < ν < 1, dtν Γ(1−ν) (1.5) 0 (t − s) 0 f (t) , ν = 1. The main advantage of the Dzhrbashyan–Caputo fractional derivative over the usual Riemann–Liouville fractional derivatives is that the former requires only integer-order derivatives in the initial conditions. The fractional derivative operator is vastly present in the physical and mathematical literature. It appears for example in generalisations of diffusion-type differential equations (see [2], [3], [4] and [5]), hyperbolic equations such as telegraph equation (see [6]), reaction-diffusion equations (see [7]), or in the study of continuous time random walks (CTRW) scaling limits (see [8], [9]). Fractional calculus has also been considered by some authors to describe cahotic Hamiltonian dynamics in low dimensional systems (see e.g. [10], [11], [12], [13] and [14]). For a complete review of fractional kinetics the reader can consult [15] or the book by Zaslavsky [16]. In the literature are also present fractional generalisations of point processes, such as the Poisson process (see [17], [18], [19], [20], [21] and [22]) and the birth and birth-death processes (see [23], [24], [25]). Fractional models are also used in other fields, for example finance ([26], [27]). The population size is governed by dν = µk+1 pk+1 (t) − µk pk (t), 0 ≤ k ≤ n0 , dtν pk (t) ( (1.6) 1, k = n0 , ν pk (0) = 0, 0 ≤ k < n0 , and is denoted by M ν (t), t > 0. 2
Let us assume that a crack has the form of a process T2ν (t), t > 0. For ν = 1/2, this coincides with a reflecting Brownian motion and has been described and derived in [28]. For ν 6= 1/2, the process T2ν (t), t > 0, can be identified with a stable process (see for details on this point [29]). The ensemble of n0 particles moves on the fracture and, at the same time, undergoes a decaying process which respects the same probabilistic rules of the usual death process. For the number of existing particles, we have therefore Z ∞ pkν (t) = pk (s)Pr {T2ν (t) ∈ ds} . (1.7) 0
We observe that Pr {T2ν (t) ∈ ds} = q(s, t)ds,
(1.8)
is a solution to ∂2 ∂ 2ν q(s, t) = q(s, t), ∂t2ν ∂s2
s > 0, t > 0,
with the necessary initial conditions. Furthermore we recall that Z ∞ ν e−zt q(s, t)dt = z ν−1 e−z s , z > 0, s > 0.
(1.9)
(1.10)
0
The distribution q(s, t) is also a solution to ∂ν ∂ q(s, t) = − q(s, t), ν ∂t ∂s
s > 0,
(1.11)
as can be ascertained directly. If we take the fractional derivative in (1.7) we get Z ∞ dν ν ∂ν p (1.12) (t) = pk (s) ν Pr {T2ν (t) ∈ ds} k ν dt ∂t 0 Z ∞ ∂ =− pk (s) q(s, t)ds ∂s 0 Z ∞ ∞ dpk (s) q(s, t)ds = −q(s, t)pk (s) 0 + ds 0 Z ∞ = −µk pk (s) + µk+1 pk+1 (s) q(s, t)ds 0 ν = −µk pkν (t) + µk+1 pk+1 (t).
This shows that replacing the time derivative with the fractional derivative corresponds to considering a death process (annihilating process) on particles displacing on a crack. We now give some details about (1.11). By taking the Laplace transform of both members of (1.11) we have that Z ∞ Z ∞ ν ∂ −zt ∂ −zt e q(s, t)dt = − e q(s, t)dt (1.13) ∂tν ∂s 0 0 3
=−
ν ∂ ν−1 −szν z e = z 2ν−1 e−sz . ∂s
Furthermore, Z ∞ Z ∞ ν −zt ∂ ν e e−zt q(s, t)dt − z ν−1 q(s, 0) q(s, t)dt = z ∂tν 0 0 ν = z ν z ν−1 e−sz − z ν−1 δ(s),
(1.14)
and therefore for s > 0 this establishes that q(s, t) solves equation (1.11). We note that a gas particle moving on a fracture has inspired to different authors the iterated Brownian motion (see [30]). The distribution pkν (t) = Pr {M ν (t) = k | M ν (0) = n0 } ,
0 ≤ k ≤ n0 ,
is obtained explicitly and reads Eν,1 (−µn0 tν ), k = n0 , n0 ν n X Q0 Eν,1 (−µm t ) µj , 0 < k < n0 , n j=k+1 m=k Q0 (µ − µ ) h m pkν (t) = h=k h6=m n n 0 0 X Y µh 1 − Eν,1 (−µm tν ), k = 0, n0 > 1. µ − µ h m m=1
(1.15)
(1.16)
h=1 h6=m
Obviously, for k = 0, n0 = 1, p0ν (t) = 1 − Eν,1 (−µ1 tν ).
(1.17)
The Mittag-Leffler functions appearing in (1.16) are defined as Eν,γ (x) =
∞ X h=0
xh , Γ (νh + γ)
x ∈ R,
ν, γ > 0.
(1.18)
For ν = γ = 1, E1,1 (x) = ex and formulae (1.16) provide the explicit distribution of the classical non-linear death process. For µk = kµ the distribution of the fractional linear death process can be obtained either directly by solving the Cauchy problem (1.6) with µk = k · µ and pn0 +1 (t) = 0, or by specialising (1.16) resulting in the following form pνk (t) =
nX −k n0 0 n0 − k (−1)r Eν,1 (−(k + r)µtν ). k r=0 r
(1.19)
A technical tool necessary for our manipulations is the Laplace transform of Mittag-Leffler functions which we write here for the sake of completeness: Z ∞ 1 z ν−γ , R(z) > |ϑ| ν . (1.20) e−zt tγ−1 Eν,γ (±ϑtν )dt = ν z ∓ ϑ 0 4
Another special case is the so-called fractional sublinear death process (for sublinear birth processes consult [31]) where the death rates have the form µk = µ(n0 + 1 − k). In the sublinear process, the annihilation of particles or individuals accelerates with decreasing population size. The distribution pνk (t), 0 ≤ k ≤ n0 of the fractional sublinear death process ν M (t), t > 0, is strictly related to that of the fractional linear birth process N ν (t), t > 0 (see, for details on this point, [24]): Pr {Mν (t) = 0 | Mν (0) = n0 } = Pr {N ν (t) > n0 | N ν (0) = 1} .
(1.21)
In general, the connection between the fractional sublinear death process and the fractional linear birth process is expressed by the relation Pr {Mν (t) = n0 − (k − 1) | Mν (0) = n0 } ν
ν
= Pr {N (t) = k | N (0) = 1} ,
(1.22)
1 ≤ k ≤ n0 .
This shows a sort of symmetry in the evolution of fractional linear birth and fractional sublinear death processes. For all fractional processes considered in this paper, a subordination relationship holds. In particular, for the fractional linear death process we can write that M ν (t) = M (T2ν (t)), 0 < ν < 1, t > 0, (1.23) where T2ν (t) is a process for which Pr {T2ν (t) ∈ ds} = q(s, t)ds, is a solution to the following Cauchy problem (see [32]) 2ν ∂ ∂2 = ∂s t > 0, s > 0, 2 q(s, t), ∂t2ν q(s, t) ∂ ∂t q(s, t) s=0 = 0, q(s, 0) = δ(s), 0 < ν ≤ 1,
(1.24)
(1.25)
with the additional initial condition 1/2 < ν ≤ 1.
qt (s, 0) = 0,
(1.26)
In equation (1.23), M (t), t > 0, represents the classical linear death process. Subordination relations of this type are extensively treated in [29] and [33]. We also show that all the fractional death processes considered below can be viewed as classical death processes with rate µ · Ξ, where Ξ is a Wrightdistributed random variable.
2
The fractional linear death process and its properties
In this section we derive the distribution of the fractional linear death process as well as some interesting related properties and interpretations. 5
1 Figure 1: Plot of p0.7 n0 (t) (in black) and pn0 (t) (in grey), both with n0 = 10.
Theorem 1. The distribution of the fractional linear death process M ν (t), t > 0 with n0 initial individuals and death rates µk = µ · k, is given by pνk (t) = Pr {M ν (t) = k | M ν (0) = n0 } nX −k n0 − k n0 0 (−1)r Eν,1 (−(k + r)µtν ), = r k r=0
(2.1)
where 0 ≤ k ≤ n0 , t > 0 and ν ∈ (0, 1]. The function Eν,1 (x) is the MittagLeffler function previously defined in (1.18). Proof. The state probability pνn0 (t), t > 0 is readily obtained by applying the Laplace transform to equation (1.6), with µk = µ · k, and then transforming back the result, thus yielding pνn0 (t) = Eν,1 (−n0 µtν ),
t > 0, ν ∈ (0, 1].
(2.2)
When k = n0 − 1, in order to solve the related differential equation, we can write z ν−1 − µ(n0 − 1)L {pn0 −1 } (z) z ν + n0 µ 1 1 L {pn0 −1 } (z) = µn0 z ν−1 ν · z + n0 µ z ν + (n0 − 1)µ 1 1 L {pn0 −1 } (z) = n0 z ν−1 − . z ν + (n0 + 1)µ z ν + n0 µ
z ν L {pn0 −1 } (z) = µn0 ⇔ ⇔
6
(2.3)
1 Figure 2: Plot of p0.7 n0 −1 (t) (in black) and pn0 −1 (t) (in grey). Here n0 = 10.
By inverting equation (2.3), we readily obtain that pνn0 −1 (t) = n0 (Eν,1 (−(n0 − 1)µtν ) − Eν,1 (−n0 µtν )) .
(2.4)
For general values of k, with 0 ≤ k < n0 , we must solve the following Cauchy problem: dν n0 pk (t) = µ(k + 1) (2.5) dtν k+1 n0X −k−1 n0 − k − 1 (−1)r Eν,1 (−(k + 1 + r)µtν ) − µkpk (t), × r r=0 subject to the initial condition pk (0) = 0 and with ν ∈ (0, 1]. The solution can be found by resorting to the Laplace transform, as we see in the following. n0 z ν L {pk } (z) = µ(k + 1) (2.6) k+1 n0X −k−1 n0 − k − 1 z ν−1 − µkL {pk } (z). × (−1)r ν r z + (k + 1 + r)µ r=0 The Laplace transform L {pk } (z) can thus be written as L {pk } (z)
(2.7)
7
n0X −k−1 n0 − k − 1 n0 z ν−1 1 (−1)r ν · ν r k+1 z + (k + 1 + r)µ z + kµ r=0 n0X −k−1 n0 n0 − k 1 1 = (−1)r z ν−1 − k r+1 z ν + kµ z ν + (k + 1 + r)µ r=0 nX −k 1 n0 0 n0 − k 1 j−1 ν−1 − = (−1) z z ν + kµ z ν + (k + j)µ k j=1 j
= µ(k + 1)
=
n0 k
nX 0 −k j=1
n0 − k z ν−1 (−1)j ν j z + (k + j)µ
ν−1 nX 0 −k n0 z n0 − k − (−1)j k z ν + kµ j=1 j = =
n0 k
nX 0 −k
n0 k
nX 0 −k
j=1
j=0
ν−1 n0 z n0 − k z ν−1 + (−1)j ν ν z + (k + j)µ k z + kµ j n0 − k z ν−1 . (−1)j ν j z + (k + j)µ
By taking now the inverse Laplace transform of (2.7), we obtain the claimed result (1.19). Remark 1. When ν = 1, equation (1.19) easily reduces to the distribution of the classical linear death process, i.e. n0 −k n0 −kµt pk (t) = e 1 − e−µt , t > 0, 0 ≤ k ≤ n0 . (2.8) k In the following theorem we give a proof of an interesting subordination relation. Theorem 2. The fractional linear death process M ν (t), t > 0 can be represented as i.d. M ν (t) = M (T2ν (t)), t > 0, ν ∈ (0, 1], (2.9) where M (t), t > 0 is the classical linear death process (see e.g. [1], page 90) and T2ν (t), t > 0, is a random time process whose one-dimensional distribution coincides with the folded solution to the following fractional diffusion equation ( 2ν ∂ ∂2 ∂t2ν q(s, t) = ∂s2 q(s, t), t > 0, ν ∈ (0, 1], (2.10) q(s, 0) = δ(s), with the additional condition qt (s, 0) = 0 if ν ∈ (1/2, 1] (see [32]). 8
Proof. By evaluating the Laplace transform of the generating function of the fractional linear death process M ν (t), t > 0, we obtain that Z ∞ e−zt Gν (u, t)dt (2.11) 0
n0 −k n0 X n0 − k u = e (−1)r Eν,1 (−(k + r)µtν )dt k r=0 r 0 k=0 nX n0 0 −k X n0 − k z ν−1 k n0 = u (−1)r ν k r=0 r z + (k + r)µ 0 nX Z ∞X n0 0 −k ν n0 − k k n0 = u (−1)r z ν−1 e−s(z +(k+r)µ) ds k r 0 r=0 k=0 (n ) nX Z ∞ 0 0 −k X n0 − k −sz ν ν−1 k n0 r −s(k+r)µ = e z u (−1) e ds k r=0 r 0 k=0 ) (n Z ∞ nX 0 −k 0 X n0 − k k n0 −µsk r −srµ −sz ν ν−1 u e (−1) e ds = e z k r 0 r=0 k=0 (n ) Z ∞ 0 X k n0 −sz ν ν−1 −µsk −µs n0 −k u = e z e (1 − e ) ds k 0 k=0 Z ∞ ν = e−sz z ν−1 G(u, s)ds Z
∞
−zt
n0 X
k
0
Z
∞
e−zt
= 0
Z =
n0 ∞X
Z 0
(
∞
e−zt
0
uk Pr {M (s) = k} fT2ν (s, t)ds dt
k=0
∞ X
) uk Pr {M (T2ν (t)) = k} dt,
k=0
and this is sufficient to prove that (2.9) holds. Note that we used two facts. The first one is that Z ∞ ν e−zt fT2ν (s, t)dt = z ν−1 e−sz , s > 0, z > 0, (2.12) 0
is the Laplace transform of the solution to (2.10). The second fact is that the Laplace transform of the Mittag-Leffler function is Z ∞ z ν−1 . (2.13) e−zt Eν,1 (−ϑtν )dt = ν z +ϑ 0
In figures 1 and 2, we compare the behaviour of the fractional probabilities 0.7 1 1 p0.7 n0 (t) and pn0 −1 (t) with their classical counterparts pn0 (t) and pn0 −1 (t), t > 0. 9
What emerges from the inspection of both figures is that, for large values of t, the probabilities, in the fractional case, decrease more slowly than p1n0 (t) and p1n0 −1 (t). The probability pn0.7 (t), increases initially faster than p1n0 −1 (t), but 0 −1 1 after a certain time lapse, pn0 −1 (t) dominates p0.7 n0 −1 (t). Remark 2. For ν = 1/2, in view of the integral representation Z ∞ 2 2 e−w +2xw dw, x ∈ R, E 12 ,1 (x) = √ π 0
(2.14)
we extract from (1.19) that 2 pk (t) = √ π 1 2
Z = 0
∞
Z
∞
e
−w2
0
n0 k
nX 0 −k r=0
1 n0 − k (−1)r e−2w(k+r)µt 2 r
(2.15)
y2
e− 4t 1 √ pk (y)dy = Pr {M (|B(t)|) = k} , πt
where B(t), t > 0 is a Brownian motion with volatility equal to 2. Remark 3. We can interpret formula (1.19) in an alternative way, as follows. For each integer k ∈ [0, n0 ] we have that pνk (t) = Pr {M ν (t) = k | M ν (0) = n0 } Z ∞ = pk (s)Pr {T2ν (t) ∈ ds}
(2.16)
0
nX Z ∞ 0 −k n0 − k (−1)r e−µ(k+r)s Pr {T2ν (t) ∈ ds} r 0 r=0 nX Z ∞ 0 −k n0 − k n0 (−1)r e−µ(k+r)s t−ν W−ν,1−ν (−st−ν )ds = r k r=0 0 nX Z ∞ −k ν n0 0 n0 − k = (−1)r e−ξµ(k+r)t W−ν,1−ν (−ξ)dξ k r=0 r 0 Z ∞ = W−ν,1−ν (−ξ)Pr {Mξ (tν ) = k | Mξ (0) = n0 } dξ,
=
n0 k
0
where W−ν,1−ν (−ξ) is the Wright function defined as W−ν,1−ν (−ξ) =
∞ X r=0
(−ξ)r , r!Γ (1 − ν(r + 1))
0 < ν ≤ 1.
(2.17)
We therefore obtain an interpretation in terms of a classical linear death process MΞ (t), t > 0 evaluated on a new time scale and with random rate µ · Ξ, where Ξ is a random variable, ξ ∈ R+ , with Wright density fΞ (ξ) = W−ν,1−ν (−ξ), 10
ξ ∈ R+ .
(2.18)
From equation (1.6) with µk = k · µ, the related fractional differential equation governing the probability generating function, can be easily obtained, leading to ( ν ∂ ∂ ν ν ∂tν G (u, t) = −µu(u − 1) ∂u G (u, t), ν ∈ (0, 1], (2.19) ν n0 G (u, 0) = u . ∂ Gν (u, t) u=1 , we obtain that From this, and by considering that EM ν (t) = ∂u (
dν ν dtν EM (t) ν
= −µEM ν (t), EM (t) = n0 .
ν ∈ (0, 1],
(2.20)
Equation (2.20) is easily solved by means of the Laplace transforms, yielding EM ν (t) = n0 Eν,1 (−µtν ),
t > 0, ν ∈ (0, 1].
(2.21)
Remark 4. The mean value EM ν (t) can also be directly calculated. ν
EM (t) =
n0 X
kpνk (t)
(2.22)
k=0 n0 X
X n0 n0 n0 − k = k (−1)r−k Eν,1 (−rµtν ) k r−k k=0 r=k n0 r X X n0 n0 − k = Eν,1 (−rµtν )(−1)r k (−1)k r − k k r=0 k=1 r n0 X n0 − 1 X r − 1 ν r Eν,1 (−rµt )(−1) n0 (−1)k = k − 1 r − 1 r=1 k=1
= n0 Eν,1 (−µtν ). This last step in (2.22) holds because r−1 r X X r−1 r−1 (−1)k = (−1)k+1 = k−1 k
k=1
3
k=0
(
−1, 0,
r = 1, r > 1.
(2.23)
Related models
In this section we present two models which are related to the fractional linear death process. The first one is its natural generalisation to the non-linear case i.e. we consider death rates in the form µk > 0, 0 ≤ k ≤ n0 . The second one is a sublinear process (see [31]), namely with death rates in the form µk = µ(n0 + 1 − k); the death rates are thus an increasing sequence as the number of individuals in the population decreases towards zero.
11
3.1
Generalisation to the non-linear case
Let us denote by M ν (t), t > 0 the random number of components of a non-linear fractional death process with death rates µk > 0, 0 ≤ k ≤ n0 . The state probabilities pkν (t) = Pr {M ν (t) = k | M ν (0) = n0 }, t > 0, 0 ≤ k ≤ n0 , ν ∈ (0, 1] are governed by the following difference-differential equations ν d dtν pk (t) = µk+1 pk+1 (t) − µk pk (t), 0 < k < n0 , dν k = 0, dtνν p0 (t) = µ1 p1 (t), d k = n0 , (3.1) dtν pn0 (t) (= −µn0 pn0 (t), 0, 0 ≤ k < n0 , pk (0) = 1, k = n . 0 The fractional derivatives appearing in (3.1) provide the system with a global memory; i.e. the evolution of the state probabilities pkν (t), t > 0, is influenced by the past, as definition (1.5) shows. This is a major difference with the classical non-linear (and, of course, linear and sublinear) death processes, and reverberates in the slowly decaying structure of probabilities extracted from (3.1). In the non-linear process, the dependence of death rates from the size of the population is arbitrary, and this explains the complicated structure of the probabilities obtained. Further generalisation can be considered by assuming that the death rates depend on t (non-homogeneous, non-linear death process). We outline here the evaluation of the probabilities pkν (t), t > 0, 0 ≤ k ≤ n0 , which can be obtained, as in the linear case, by means of a recursive procedure (similar to that implemented in [24] for the fractional linear birth process). Let k = n0 . By means of the Laplace transform applied to equation (3.1) we immediately obtain that pnν 0 (t) = Eν,1 (−µn0 tν ).
(3.2)
When k = n0 − 1 we get z ν L pnν 0 −1 (z) = −µn0 −1 L pnν 0 −1 (z) + µn0
z ν−1 z ν + µn0
(3.3)
z ν−1 1 · z ν + µn0 z ν + µn0 −1 ν 1 1 1 ν−1 ⇔ L pn0 −1 (z) = µn0 z − ν z ν + µn0 z + µn0 −1 µn0 −1 − µn0 µn0 ν ν ν ⇔ pn0 −1 (t) = Eν,1 (−µn0 t ) − Eν,1 (−µn0 −1 t ) . µn0 −1 − µn0 ⇔ L pnν 0 −1 (z) = µn0
For k = n0 − 2 we obtain in the same way that z ν L pnν 0 −2 (z)
12
(3.4)
µn0 µn0 −1 = −µn0 −2 L pnν 0 −2 (z) + µn0 −1 − µn0
z ν−1 z ν−1 − , z ν + µn0 z ν + µn0 −1
so that L pnν 0 −2 (z)
(3.5)
µn0 µn0 −1 ν−1 1 1 1 z − ν µn0 −1 − µn0 z ν + µn0 z + µn0 −1 z ν + µn0 −2 µn0 µn0 −1 ν−1 1 1 1 = z − µn0 −1 − µn0 z ν + µn0 z ν + µn0 −2 µn0 −2 − µn0 1 1 1 − − . z ν + µn0 −1 z ν + µn0 −2 µn0 −2 − µn0 −1
=
By inverting the Laplace transform we readily arrive at the following result Eν,1 (−µn0 tν ) ν pn0 −2 (t) = µn0 µn0 −1 (3.6) (µn0 −1 − µn0 )(µn0 −2 − µn0 ) Eν,1 (−µn0 −1 tν ) Eν,1 (−µn0 −2 tν ) − − (µn0 −1 − µn0 )(µn0 −2 − µn0 ) (µn0 −1 − µn0 )(µn0 −2 − µn0 −1 ) Eν,1 (−µn0 −2 tν ) + (µn0 −1 − µn0 )(µn0 −2 − µn0 −1 ) Eν,1 (−µn0 tν ) = µn0 µn0 −1 (µn0 −1 − µn0 )(µn0 −2 − µn0 ) 1 Eν,1 (−µn0 −2 tν ) 1 + − (µn0 −1 − µn0 ) µn0 −2 − µn0 −1 µn0 −2 − µn0 ν Eν,1 (−µn0 −1 t ) − (µn0 −1 − µn0 )(µn0 −2 − µn0 −1 ) Eν,1 (−µn0 tν ) = µn0 µn0 −1 (µn0 −1 − µn0 )(µn0 −2 − µn0 ) Eν,1 (−µn0 −2 tν ) Eν,1 (−µn0 −1 tν ) + − . (µn0 −2 − µn0 −1 )(µn0 −2 − µn0 ) (µn0 −1 − µn0 )(µn0 −2 − µn0 −1 ) The structure of the state probabilities for arbitrary values of k = n0 − l, 0 ≤ l < n0 , can now be easily obtained. The proof follows the lines of the derivation of the state probabilities for the fractional non-linear pure birth process adopted in Theorem 2.1 in [24]. We have that l l−1 X Q Eν,1 (−µn0 −m tν ) , 1 ≤ l < n0 , µ n −j 0 l j=0 Q m=0 ν (µn0 −h − µn0 −m ) pn0 −l (t) = h=0 h6=m Eν,1 (−µn0 tν ), l = 0.
13
(3.7)
By means of some changes of indices, formula (3.7) can also be written as n0 n0 X Q Eν,1 (−µm tν ) , 0 < k < n0 , µ n0 j=k+1 j Q m=k ν (µ − µ ) h m pk (t) = (3.8) h=k h6=m Eν,1 (−µn0 tν ), k = n0 . For the extinction probability, we have to solve the following initial value problem: n0 n0 X Q Eν,1 (−µm tν ) dν , n0 > 1, p (t) = µ µ ν 0 1 j n0 dt Q j=2 m=1 (µh − µm ) h=1 (3.9) h6=m ν d ν n0 = 1, ν p0 (t) = µ1 Eν,1 (−µ1 t ), dt p0 (0) = 0, n0 ≥ 1. When n0 > 1, starting from (3.9) and by resorting to the Laplace transform once again, we have that n0 n0 X Q L p0ν (z) = µj j=1
1 n0 Q
m=1
·
(µh − µm )
z −1 . z ν + µm
(3.10)
h=1 h6=m
The inverse Laplace transform of (3.10) leads to p0ν (t)
=
n0 Q
µj
j=1
n0 X
1 n0 Q
m=1
tν Eν,ν+1 (−µm tν )
(3.11)
(µh − µm )
h=1 h6=m
=
n0 Q j=1
µj
n0 X
1 n0 Q
m=1
·
(µh − µm )
1 [1 − Eν,1 (−µm tν )] µm
h=1 h6=m
=
n0 Y n0 X
µh µh − µm
m=1 h=1 h6=m n0 Y n0 X
=1−
m=1 h=1 h6=m
−
µh µh − µm
n0 Y n0 X m=1 h=1 h6=m
µh µh − µm
Eν,1 (−µm tν )
Eν,1 (−µm tν ).
Note that, in the last step, we used the following fact: n0 Y n0 X µh = 1. µh − µm m=1 h=1 h6=m
14
(3.12)
This can be ascertained by observing that Y
(µh − µl ) = det A =
a1,j (−1)j+1 Min1,j
(3.13)
j=1
1≤h 1, j n0 j=k+1 Q m=k ν (µh − µm ) pk (t) = h=k h6 = m Eν,1 (−µn0 tν ), k = n0 , n0 ≥ 1,
(3.17)
and
p0ν (t)
=
n0 Y n0 X 1 − m=1 h=1 h6=m
µh µh − µm
Eν,1 (−µm tν ),
(3.18)
1 − Eν,1 (−µ1 tν ),
3.2
n0 > 1, n0 = 1.
A fractional sublinear death process
We consider in this section the process where the infinitesimal death probabilities have the form Pr {M(t, t + dt] = −1 | M(t) = k} = µ(n0 + 1 − k)dt + o(dt),
(3.19)
where n0 is the initial number of individuals in the population. The state probabilities pk (t) = Pr {M(t) = k | M(0) = n0 } , 15
0 ≤ k ≤ n0 ,
(3.20)
0.7 Figure 3: Plot of p0.7 n0 (t) (in black) and pn0 (t) (in grey), with n0 = 2.
satisfy the equations d dt pk (t) = −µ(n0 + 1 − k)pk (t) + µ(n0 − k)pk+1 (t), 1 ≤ k ≤ n0 , d p (t) = µn p (t), k = 0, dt 0 ( 0 1 1, k = n0 , pk (0) = 0, 0 ≤ k < n0 .
(3.21)
In this model the death rate increases with decreasing population size. The probabilities pνk (t) = Pr {Mν (t) = k | Mν (0) = n0 } of the fractional version of this process are governed by the equations dν dtν pk (t) = −µ(n0 + 1 − k)pk (t) + µ(n0 − k)pk+1 (t), 1 ≤ k ≤ n0 , dν p (t) = µn p (t), k = 0, 0 1 dtν 0 ( (3.22) 1, k = n0 , p (0) = k 0, 0 ≤ k < n0 . We first observe that the solution to the Cauchy problem ( ν d dtν pn0 (t) = −µpn0 (t), pn0 (0) = 1, is pνn0 (t) = Eν,1 (−µtν ), t > 0. In order to solve the equation ( ν d ν dtν pn0 −1 (t) = −2µpn0 −1 (t) + µEν,1 (−µt ), pn0 −1 (0) = 0, 16
(3.23)
(3.24)
we resort to the Laplace transform and obtain that 1 1 L pνn0 −1 (z) = µz ν−1 ν · z + µ z ν + 2µ 1 1 ν−1 − . =z z ν + µ z ν + 2µ
(3.25)
By inverting (3.25) we extract the following result pνn0 −1 (t) = Eν,1 (−µtν ) − Eν,1 (−2µtν ). By the same technique we solve ( ν d ν ν dtν pn0 −2 (t) = −3µpn0 −2 (t) + 2µ [Eν,1 (−µt ) − Eν,1 (−2µt )] , pn0 −2 (0) = 0,
(3.26)
(3.27)
thus obtaining 1 1 1 − ν (3.28) L pνn0 −2 (z) = 2µz ν−1 ν z + µ z + 2µ z ν + 3µ 1 1 1 1 1 1 = 2µz ν−1 − − − z ν + µ z ν + 3µ 2µ z ν + 2µ z ν + 3µ µ z ν−1 z ν−1 z ν−1 = ν −2 ν + ν . z +µ z + 2µ z + 3µ In light of (3.28), we infer that pνn0 −2 (t) = Eν,1 (−µtν ) − 2Eν,1 (−2µtν ) + Eν,1 (−3µtν ).
(3.29)
For all 1 ≤ n0 − m ≤ n0 , by similar calculations, we arrive at the general result m X m pνn0 −m (t) = (−1)l Eν,1 (− (l + 1) µtν ) , 1 ≤ n0 − m ≤ n0 . (3.30) l l=0
Introducing the notation n0 − m = k, we rewrite the state probabilities (3.30) in the following manner pνk (t)
=
nX 0 −k l=0
n0 − k (−1)l Eν,1 (− (l + 1) µtν ) , l
1 ≤ k ≤ n0 .
(3.31)
For the extinction probability we must solve the following Cauchy problem ( ν Pn0 −1 n0 −1 d (−1)l Eν,1 (− (l + 1) µtν ) , l=0 dtν p0 (t) = µn0 l (3.32) p0 (0) = 0. 17
0.7 Figure 4: Plot of p0.7 n0 −1 (t) (in black) and pn0 −1 (t) (in grey), with n0 = 2.
The Laplace transform of (3.32) yields z
ν
L {pν0 } (z)
= µn0
nX 0 −1 l=0
n0 − 1 z ν−1 . (−1)l ν z + µ(l + 1) l
(3.33)
The inverse Laplace transform can be written down as nX 0 −1
Z t n0 − 1 1 (−1)l Eν,1 (−(l + 1)µsν ) (t − s)ν−1 ds. Γ(ν) 0 l l=0 (3.34) The integral appearing in (3.34) can be suitably evaluated as follows pν0 (t) = µn0
Z
t
Eν,1 (−(l + 1)µsν ) (t − s)ν−1
0
=
Z ∞ X (−(l + 1)µ)m t νm s (t − s)ν−1 ds Γ(νm + 1) 0 m=0
=
∞ X (−(l + 1)µ)m tν(m+1) Γ(ν)Γ(νm + 1) Γ(νm + 1) Γ(νm + ν + 1) m=0
=
∞ X Γ(ν) (−(l + 1)µtν )m+1 (−µ(l + 1)) m=0 Γ(ν(m + 1) + 1)
=
Γ(ν) [Eν,1 (−(l + 1)µtν ) − 1] . (−µ(l + 1)) 18
(3.35)
By inserting result (3.35) into (3.34), we obtain pν0 (t)
nX 0 −1
n0 − 1 (−1)l+1 = n0 [Eν,1 (−(l + 1)µtν ) − 1] l+1 l l=0 nX 0 −1 n0 = (−1)l+1 [Eν,1 (−(l + 1)µtν ) − 1] l+1 l=0 n0 n0 X X n0 n0 l ν = (−1) Eν,1 (−lµt ) − (−1)l l l l=1 l=1 n0 X n0 =1+ (−1)l Eν,1 (−lµtν ) l l=1 n0 X n0 (−1)l Eν,1 (−lµtν ). = l
(3.36)
l=0
Remark 5. We check that the probabilities (3.31) and (3.36) sum up to unity. We start by analysing the following sum: n0 X
pνk (t) =
n0 nX 0 −k X n0 − k k=1 l=0
k=1
l
(−1)l Eν,1 (−(l + 1)µtν ).
(3.37)
In order to evaluate (3.37), we resort to the Laplace transform n0 X
L {pνk } (z) =
n0 nX 0 −k n0 − k 1 z ν−1 X (−1)l zν . µ l + 1+l µ
(3.38)
k=1 l=0
k=1
By using formula (6) of [34] (see also [35], formula (5.41), page 188), we obtain that n0 X k=1
n0 z ν−1 X Γ(n0 − k + 1) zν zν zν µ + 1 + 2 . . . + 1 + n − k 0 k=1 µ µ µ ν z n0 Γ µ + 1 Γ (n0 − k + 1) z ν−1 X = ν µ Γ zµ + 1 + n0 − k k=1 n0 Z 1 zν z ν−1 X = x µ (1 − x)n0 −k dx µ k=1 0 ν−1 Z 1 zν z = x µ −1 [1 − (1 − x)n0 ] dx µ 0 Z 1 z ν−1 1 zµν −1 = − x (1 − x)n0 dx z µ 0 Z n0 z ν−1 ∞ −y zµν (− ln x=y) 1 = − e 1 − e−y dy z µ 0
L {pνk } (z) =
19
(3.39)
Z n0 1 ν−1 ∞ −wzν e z 1 − e−µw dw z 0 Z ∞ n0 X ν n0 ν−1 k e−z w−µwk dw −z (−1) k 0 k=0 n 0 X n0 1 (−1)k ν − z ν−1 . k z + µk
(y/µ=w)
=
=
1 z
=
1 z
k=0
The inverse Laplace transform of (3.39) is therefore n0 X
pνk (t) = 1 −
k=1
=−
n0 X n0
k
k=0 n0 X k=1
(−1)k Eν,1 (−µktν )
(3.40)
n0 (−1)k Eν,1 (−µktν ). k
By putting (3.36) and (3.40) together, we conclude that n0 X
pνk (t) = 1,
(3.41)
k=0
as it should be. Remark 6. We observe that, in the linear and sublinear death processes, the extinction probabilities coincide. This implies that although the state probabilities pνk (t) and pνk (t) differ (see figures 3 and 4) for all 1 ≤ k ≤ n0 , we have that n0 n0 X X pνk (t). (3.42) pνk (t) = k=1
k=1
This can be checked by performing the following sum n0 X
L {pνk (t)} (z)
(3.43)
k=1
=
n0 nX −k X n0 0 n0 − k k=1
=
z ν−1 µ
=
z ν−1 µ
=
z ν−1 µ
k
(−1)r
z ν−1 z ν + µ(k + r)
r r=0 n0 n0 −k X n0 − k 1 n0 X (−1)r zν k r=0 r + k+r µ k=1 n0 X n0 (n0 − k)! ν ν z zν z k + k + k + 1 . . . + n 0 k=1 µ µ µ ν z n0 X n0 Γ (n0 − k + 1) Γ µ + k ν k Γ z +n +1 k=1
µ
20
0
z ν−1 = µ
Z
z ν−1 = µ
Z
=
1
(1 − x)
zν µ
−1
0
k
k=1 1
(1 − x) 0 ν−1
1 z − z µ
n0 X n0
zν µ
−1
(1 − xn0 ) dx
1
Z
xn0 −k (1 − x)k dx
xn0 (1 − x)
zν µ
−1
dx.
0
This coincides with the fourth-to-last step of (3.39) and therefore we can conclude that n0 n0 n0 X X X n0 pνk (t) = − (−1)k Eν,1 (−µktν ) = pνk (t). (3.44) k k=1
3.2.1
k=1
k=1
Mean value
ν Theorem 3. Consider the fractional sublinear death process t > 0 dePn0M (t), ν fined above. The probability generating function G (u, t) = k=0 uk pνk (t), t > 0, |u| ≤ 1, satisfies the following partial differential equation: ∂ 1 ∂ν ν − 1 [Gν (u, t) − pν0 (t)] + µ(u − 1) Gν (u, t). G (u, t) = µ(n0 + 1) ν ∂t u ∂u (3.45) subject to the initial condition Gν (u, 0) = un0 , for |u| ≤ 1, t > 0.
Proof. Starting from (3.22), we obtain that n0 dν X uk pνk (t) dtν
(3.46)
k=0
= −µ
n0 X
uk (n0 + 1 − k)pνk (t) + µ
k=1
nX 0 −1
uk (n0 − k)pνk+1 (t),
k=0
so that ∂ν ν ∂ G (u, t) = − µ(n0 + 1) [Gν (u, t) − pν0 (t)] + µu Gν (u, t) (3.47) ∂tν ∂u µ(n0 + 1) ν ∂ + [G (u, t) − pν0 (t)] − µ Gν (u, t) u ∂u 1 ∂ = µ(n0 + 1) − 1 [Gν (u, t) − pν0 (t)] + µ(u − 1) Gν (u, t). u ∂u
Theorem 4. The mean number of individuals EMν (t), t > 0 in the fractional sublinear death process, reads n0 X n0 + 1 EMν (t) = (−1)k+1 Eν,1 (−µktν ), t > 0, ν ∈ (0, 1]. (3.48) k+1 k=1
21
∂ Proof. From (3.45) and by considering that EMν (t) = ∂u Gν (u, t) u=1 , we directly arrive at the following initial value problem: ( ν d ν ν ν dtν EM (t) = −µ(n0 + 1) [1 − p0 (t)] + µEM (t), (3.49) EMν (0) = n0 , which can be solved by resorting to the Laplace transform, as follows: n0 X z ν−1 n0 1 z ν−1 L {EM (t)} (z) = n0 ν + µ(n0 + 1) (−1)k ν · z −µ k z + µk z ν − µ ν
k=1
= n0
z ν−1 + zν − µ
n0 X k=1
(3.50) ν−1 n0 + 1 z z ν−1 (−1)k ν − ν . k+1 z − µ z + µk
In (3.50), formula (3.36) must be considered. By inverting the Laplace transform we obtain that EMν (t)
(3.51)
n0 X n0 + 1 ν = n0 Eν,1 (µt ) + (−1)k [Eν,1 (µtν ) − Eν,1 (−µktν )] k+1 k=1 n0 X n0 + 1 ν ν (−1)k = n0 Eν,1 (µt ) + Eν,1 (µt ) k+1 k=1 n0 X n0 + 1 (−1)k Eν,1 (−µktν ) − k+1 k=1 n0 X n0 + 1 (−1)k+1 Eν,1 (−µktν ), = k+1 k=1
as desired. Remark 7. The mean value (3.48) can also be directly derived as follows. EMν (t) =
n0 X
kpνk (t)
(3.52)
k=0 n0 X
nX 0 −k
n0 − k = k (−1)l Eν,1 (−(l + 1)µtν ) l k=1 l=0 n0 n0X +1−k X n0 − k = k (−1)l−1 Eν,1 (−µltν ) l−1 k=1 l=1 n0 n0X +1−l X n0 − k = (−1)l−1 Eν,1 (−µltν ) k . l−1 l=1
k=1
22
It is now sufficient to show that n0X +1−l k=1
k
n0 − k n0 + 1 = . l−1 l+1
(3.53)
Indeed, n0X +1−l k=1
k
n0 − k l−1
=
nX 0 −1
(n0 − k)
k=l−1
=
nX 0 −1
k l−1
(3.54)
k (n0 + 1 − k − 1) l−1
k=l−1
= (n0 + 1)
nX 0 −1 k=l−1
nX 0 −1 k k+1 −l l−1 l
k=l−1
nX 0 +1 k−1 k−1 −l = (n0 + 1) l l−1 k=l+1 k=l n0 n0 + 1 = (n0 + 1) −l l l+1 n0 + 1 = . l+1 n0 X
The crucial step of (3.54) is justified by the following formula n0 X k−1 j n0 − 1 n0 =1+ + ··· + = . j−1 j−1 j−1 j
(3.55)
k=j
Figure 5 shows that in the sublinear case, the mean number of individuals in the population, decays more slowly than in the linear case, as expected. Note that (3.48) satisfies the initial condition EMν (0) = n0 . In order to check this, it is sufficient to show that n0 nX 0 +1 X n0 + 1 n0 + 1 (−1)k+1 = (−1)r (3.56) k+1 r r=2 k=1 "n +1 # 0 X n0 + 1 n0 + 1 r = (−1) − 1 + = n0 . r 1 r=0 The details in (3.56) explain also the last step of (3.51). 3.2.2
Comparison of Mν (t) with the fractional linear death process M ν (t) and the fractional linear birth process N ν (t)
The distributions of the fractional linear and sublinear processes examined above display a behaviour which is illustrated in Table 1 (see also Table 2 for the mean values). 23
Figure 5: Plot of EM 0.7 (t) (in black) and EM0.7 (t) (in grey), n0 = 2. The most striking fact about the models dealt with above, is that the linear probabilities decay faster than the corresponding sublinear ones, for small values of k; whereas, for large values of k, the sublinear probabilities take over and the extinction probabilities in both cases coincide. The reader should also compare the state probabilities of the death models examined here with those of the fractional linear pure birth process (with birth rate λ and one progenitor). These read k X k−1 ν (−1)j−1 Eν,1 (−λjtν ), k ≥ 1. (3.57) pˆk (t) = j − 1 j=1 Note that pˆν1 (t) = Eν,1 (−λtν ) is of the same form as pνn0 (t) = Eν,1 (−µtν ). We now show that ∞ X
pˆνk (t)
=1−
k=n0 +1
n0 X
pˆνk (t)
(3.58)
k=1
=1−
=1−
n0 X k X k−1 k=1 j=1 n0 X
j−1
(−1)j−1 Eν,1 (−λjtν )
(−1)j−1 Eν,1 (−λjtν )
j=1
=1−
n0 X
n0 X k−1 k=j
(−1)j−1
j=1
n0 Eν,1 (−λjtν ) j
= (3.36) with λ replacing µ. 24
j−1
Table 1: State probabilities pνk (t) for the fractional linear death process M ν (t), t > 0, and pνk (t) for the fractional sublinear death process Mν (t). State Probabilities pνn0 (t) = Eν,1 (−µn0 tν ) pνn0 (t) = Eν,1 (−µtν ) pνn0 −1 (t) = n0 [Eν,1 (−(n0 − 1)µtν ) − Eν,1 (−n0 µtν )] pνn0 −1 (t) = Eν,1 (−µtν ) − Eν,1 (−2µtν ) .. . Pn0 −k n0 −k (−1)l Eν,1 (−(k + l)µtν ) pνk (t) = nk0 l=0 l P 0 −k n0 −k pνk (t) = n (−1)l Eν,1 (− (l + 1) µtν ) l=0 l .. . Pn0 −1 n0 −1 pν1 (t) = n0 l=0 (−1)l Eν,1 (−(1 + l)µtν ) l P 0 −1 n0 −1 pν1 (t) = n (−1)l Eν,1 (− (l + 1) µtν ) l=0 l pν0 (t) = pν0 (t) =
P n0
n0 l ν l=0 l (−1) Eν,1 (−lµt ) Pn0 n0 l ν l=0 l (−1) Eν,1 (−lµt )
Table 2: Mean values for the fractional linear birth N ν (t), fractional linear death M ν (t) and fractional sublinear death Mν (t) processes. EN ν (t) = Eν,1 (λtν ) EM ν (t) = n0 Eν,1 (−µtν ) EMν (t) =
P n0
k=1
n0 +1 (−1)k+1 Eν,1 (−µktν ) k+1
Note that in the above step we used formula (3.55). By comparing formulae (3.4) of [24] and (3.31) above, we arrive at the conclusion that (for λ = µ) Pr {N ν (t) = k | N ν (0) = 1} k X k−1 = (−1)j−1 Eν,1 (−λj tν ) j − 1 j=1 = Pr {Mν (t) = n0 + 1 − k | M(0) = n0 } ,
(3.59)
1 ≤ k ≤ n0 .
pν0 (t)
corresponds to the probability of the event The probability of extinction {N ν (t) > n0 } for the fractional linear birth process. Acknowledgement: The authors wish to thank Francis Farrelly for having checked and corrected the manuscript. The authors are grateful to the referees 25
for drawing our attention to some relevant references and for useful remarks which improved the presentation of the paper.
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