On some mortality rate processes and mortality deceleration with age

Abstract

A specific mortality rate process governed by the non-homogeneous Poisson process of point events is considered and its properties are studied. This process can describe the damage accumulation in organisms experiencing external shocks and define its survival characteristics. It is shown that, although the sample paths of the unconditional mortality rate process are monotonically increasing, the population mortality rate can decrease with age and, under certain assumptions, even tend to zero. The corresponding analysis is the main objective of this paper and it is performed using the derived conditional distributions of relevant random parameters. Several meaningful examples are presented and discussed.

Appendix: Proof of Theorem 1

Proof

Let $0\le T_1 \le T_2 \le \cdots$ be the sequential arrival times of shocks in the NHPP $\{N(t), t\ge 0\}$ with rate $\lambda (x).$ Note that the full history of $\{N(u),0\le u\le t\}$ can be equivalently specified in terms of $\{T_1 ,T_2 ,\ldots ,T_{N(t)} ,N(t)\}.$ Then, in accordance with (7), the survival function of an organism given the shocks history is

$$\begin{aligned} P(T>t\,|\,N(u),0\le u\le t)= & {} P(T>t\,|T_1 ,T_2 ,\ldots ,T_{N(t)} ,N(t))\nonumber \\= & {} \exp \left\{ {-\int _0^t {\mu _0 (u)du} } \right\} \exp \left\{ {-\eta \int _0^t {N(u)du} } \right\} \nonumber \\= & {} \exp \left\{ {-\int _0^t {\mu _0 (u)du} } \right\} \exp \left\{ {-\sum _{i=1}^{N(t)} {\eta (t-T_i )} } \right\} .\nonumber \\ \end{aligned}$$

(11)

Observe that the joint distribution of $(T_1 ,T_2 ,\ldots ,T_{N(t)} ,N(t))$ is given by

$$\begin{aligned}&f_{T_1 ,T_2 ,\ldots ,T_{N(t)} ,N(t)} (t_1 ,t_2 ,\ldots ,t_n ,n)\nonumber \\&\quad =\lambda (t_1 )\exp \left\{ {-\int _0^{t_1 } {\lambda (u)du} } \right\} \lambda (t_2 )\exp \left\{ {-\int _{t_1 }^{t_2 } {\lambda (u)du} } \right\} \ldots \nonumber \\&\qquad \times \,\lambda (t_n )\exp \left\{ {-\int _{t_{n-1} }^{t_n } {\lambda (u)du} } \right\} \exp \left\{ {-\int _{t_n }^t {\lambda (u)du} } \right\} \nonumber \\&\quad =\left( {\prod _{i=1}^n {\lambda (t_i )} } \right) \exp \left\{ {-\int _0^t {\lambda (u)du} } \right\} , \quad 0\le t_1 \le t_2 \le \cdots \le t_n \le t, \quad n=0,1,2,\ldots , \nonumber \\ \end{aligned}$$

(12)

where, $\exp \left\{ {-\int _0^{t_1 } {\lambda (u)du} } \right\}$ represents the probability that there is no shock in $(0,t_1 )$, then $\lambda (t_1 )$ represents the probability that a shock occurs at $t_1$, then $\exp \left\{ {-\int _{t_1 }^{t_2 } {\lambda (u)du} } \right\}$ represents that the probability that there is no shock in $(t_1 ,t_2 )$, then $\lambda (t_2 )$ represents the probability that a shock occurs at $t_2$, and so forth.

Therefore, combining (11) and (12), integrating out $T_1 ,T_2 ,\ldots ,T_n$ and using the properties of the NHPP, the joint distribution of $\{T>t,N(t)\}$ is obtained as

$$\begin{aligned}&P(T>t,N(t)=n)\nonumber \\&\quad =\exp \left\{ {-\int _0^t {\mu _0 (u)du} } \right\} \exp \Bigg \{ -\int _0^t {\lambda (u)du} \Bigg \}\nonumber \\&\qquad \times \int _0^t {\ldots } \int _0^{t_3 } {\int _0^{t_2 } } \prod _{i=1}^n {\lambda (t_i )} \exp \left\{ {-\eta (t-t_i )} \right\} dt_1 dt_2\ldots dt_n\nonumber \\&\quad =\exp \left\{ {-\int _0^t {\mu _0 (u)du} } \right\} \exp \left\{ {-\int _0^t {\lambda (u)du} } \right\} \frac{\left( {\int _0^t {\exp \{-\eta (t-x)\}\lambda (x)dx} } \right) ^{n}}{n!},\nonumber \\ \end{aligned}$$

(13)

where, in calculating the multiple integrals, we used the following property: for any function $\varphi (x)$,

$$\begin{aligned} \int _0^t {\ldots } \int _0^{t_3 } {\int _0^{t_2 } } \prod _{i=1}^n {\varphi (t_i )} dt_1 dt_2 \ldots dt_n =\frac{\left( {\int _0^t {\varphi (x)dx} } \right) ^{n}}{n!}, \end{aligned}$$

which can be directly obtained by integrating with respect to $t_1 ,t_2 ,\ldots ,t_n$, ‘sequentially’.

From (13),

$$\begin{aligned} P(T>t)= & {} \exp \left\{ {-\int _0^t {\mu _0 (u)du} } \right\} \nonumber \\&\times \exp \Bigg \{-\int _0^t {\lambda (u)du} \Bigg \}\sum _{n=0}^\infty {\frac{\left( {\int _0^t {\exp \{-\eta (t-x)\}\lambda (x)dx} } \right) ^{n}}{n!}}\nonumber \\= & {} \exp \left\{ {-\int _0^t {\mu _0 (u)du} } \right\} \exp \Bigg \{ -\int _0^t {\lambda (u)du} \Bigg \}\nonumber \\&\times \exp \left\{ {\int _0^t {\exp \{-\eta (t-x)\}\lambda (x)dx} } \right\} . \end{aligned}$$

(14)

Finally, from (13) and (14), we have

$$\begin{aligned} P(N(t)=n\,|\,T>t)= & {} \frac{\left( {\int _0^t {\exp \{-\eta (t-x)\}\lambda (x)dx} } \right) ^{n}}{n!} \nonumber \\&\times \exp \Bigg \{-\int _0^t {\exp \{-\eta (t-x)\}\lambda (x)dx} \Bigg \}, \quad n=0,1,2,\ldots .\qquad \nonumber \\ \end{aligned}$$

(15)