Generalized Nets in Medicine: An Example of Telemedicine for People with Diabetes

Abstract

In the present paper, an overview of the Generalized Nets (GNs) models in medicine and telecare/telehealth is given. The apparatus of GNs has been used in the modelling of physiological processes, diagnostics of diseases, organisational and administrative processes in hospitals. Recently, in a series of papers, GNs have been used to model telecare/telehealth services. On the basis of these models, a GN model of telemedicine for patients with diabetes is proposed. The sensors included in the model are blood pressure monitor, weight scale, pulse oximeter and blood glucose monitor. Smart filtering of false positive alarm messages is included which reduces the number of events for which the health care person has to take a decision. The GN model can be used to develop a decision support tool for telemedicine for people with diabetes.

Appendix: Short Remark on Generalized Nets

GNs [5, 9] are extensions of Petri Nets [73]. They are defined in a way that is principally different from the ways of defining the other types of Petri nets. We shall first give an example of a GN and make remarks about the notation. A GN is shown in Fig. 2. The places are marked with

. Each part of the net which looks like the one shown on Fig. 3., is called transition (more precisely graphic structure of transition). Transition’s conditions are denoted by

. GNs, like other nets, contain tokens which are transferred from place to place. Every token enters the net with an initial characteristic. During each transfer, the token receives new characteristics. So, they accumulate their “history”. This is the first essential difference with the other types of Petri nets.

Fig. 2

Generalized net

Fig. 3

Transition

Every GN-place has at most one arc entering and at most one arc leaving it. The places with no entering arcs are called input places for the net $(l_{1}, l_{2}$ on Fig. 2.) and those with no leaving arcs are called output places $(l_{14}$ and $l_{15}$ on Fig. 2.). The input places are always at the transition’s left, and the output places are always at the transition’s right side. When tokens enter the input place of a transition, it becomes potentially fireable and at the moment of their transfer towards the transition’s output places, it is being fired. The transition becomes active at a given time-moment and remains active up to another predefined moment.

The second basic difference between GNs and the ordinary Petri nets is the “place—transition” relation. Here, transitions are objects of a more complex nature. A transition may contain m input and n output places where $m, n \ge 1$.

The third basic difference is related to the time during which the GN functions. The time can be determined from some global time-scale and in this case the net is not invariant about the time-parameters. When we have GN models of some (different, but connected) processes that flow in parallel at time, we can use many time-scales or a single one, accounting the moments of the separate events in the processes. In the present form of the GN-definition, time is discrete. It increases with discrete steps. We can see the status of the GN model in each current time-moment.

Formally, every transition is described by a seven-tuple:

$$ Z = \langle L', L'', t_{1}, t_{2}, r, M, \Box \rangle , $$

where:

(a) $L'$ and $L''$ are finite, non-empty sets of places (the transition’s input and output places, respectively); for the transition in Fig. 3 these are

$$ L'=\lbrace l'_{1},l'_{2},\dots ,l'_{m}\rbrace $$

and

$$ L''= \lbrace l''_{1},l''_{2},\dots ,l''_{n} \rbrace ; $$

(b) $t_{1}$ is the current time-moment of the transition’s firing;

(c) $t_{2}$ is the current value of the duration of its active state;

(d) r is the transition’s condition determining which tokens will transfer from the transition’s inputs to its outputs. Parameter r has the form of an IM:

$$r=\begin{array}{c|c} &{} l''_{1} \dots l''_{j} \dots l''_{n} \\ \hline l'_{1} &{} \\ \vdots &{} r_{i,j} \\ l'_{i} &{} (r_{i,j} \, \, - \text{ predicate }) \\ \vdots &{} (1 \le i \le m, 1 \le j \le n) \\ l'_{m} &{} \end{array} \,\,; $$

where $r_{i,j}$ is the predicate which expresses the condition for transfer from the ith input place to the jth output place. When $r_{i,j}$ has truth-value “true”, then a token from the ith input place can be transferred to the jth output place; otherwise, this is impossible;

(e) M is an IM of the capacities of transition’s arcs:

$$M=\begin{array}{c|c} &{} l''_{1} \dots l''_{j} \dots l''_{n} \\ \hline l'_{1} &{} \\ \vdots &{} m_{i,j} \\ l'_{i} &{} (m_{i,j} \ge 0 - \text{ natural } \text{ number } \text{ or } \infty ) \\ \vdots &{} (1 \le i \le m, 1 \le j \le n) \\ l'_{m} &{} \\ \end{array} \, \,; $$

(f) $\Box$ is called transition type and it is an object having a form similar to a Boolean expression. It may contain as variables the symbols that serve as labels for transition’s input places, and it is an expression constructed of variables and the Boolean connectives $\wedge$ and $\vee$ determining the following conditions:

$$ \begin{array}{lcl} \wedge (l_{i_{1}}, l_{i_{2}}, \dots ,l_{i_{u}}) &{} - &{} \text{ every } \text{ place } l_{i_{1}},l_{i_{2}},\dots ,l_{i_{u}} \text{ must } \text{ contain } \text{ at } \text{ least }\\ &{} &{} \text{ one } \text{ token },\\ \vee (l_{i_{1}},l_{i_{2}},\dots ,l_{i_{u}}) &{} - &{} \text{ there } \text{ must } \text{ be } \text{ at } \text{ least } \text{ one } \text{ token } \text{ in } \text{ the } \text{ set } \text{ of } \text{ places }\\ &{} &{} l_{i_{1}}, l_{i_{2}},\dots , l_{i_{u}}, \text{ where } \lbrace l_{i_{1}},l_{i_{2}},\dots , l_{i_{u}} \rbrace \subset L'. \end{array} $$

When the value of a type (calculated as a Boolean expression) is “true”, the transition can become active, otherwise it cannot.

The ordered four-tuple

$$ E=\langle \langle A,\pi _{A},\pi _{L},c, f,\theta _{1},\theta _{2}\rangle , \langle K,\pi _{K},\theta _{K} \rangle ,\langle T, t^{0}, t^{*} \rangle , \langle X, \varPhi , b \rangle \rangle $$

is called a Generalized Net if:

(a) A is a set of transitions (see above);

(b) $\pi _{A}$ is a function giving the priorities of the transitions, i.e., $\pi _{A} : A \rightarrow \mathcal{N}$;

(c) $\pi _{L}$ is a function giving the priorities of the places, i.e., $\pi _{L}:L\rightarrow \mathcal{N}$, where

$$ L=pr_{1} A \cup pr_{2} A $$

and obviously, L is the set of all GN-places;

(d) c is a function giving the capacities of the places, i.e., $c: L\rightarrow \ \mathcal{N}$;

(e) f is a function that calculates the truth values of the predicates of the transition’s conditions;

(f) $\theta _{1}$ is a function giving the next time-moment, for which a given transition Z can be activated, i.e., $\theta _{1} (t) = t'$, where $pr_{3}Z = t, t' \in [T, T + t^{*}]$ and $t \le t'$; the value of this function is calculated at the moment when the transition terminates its functioning;

(g) $\theta _{2}$ is a function giving the duration of the active state of a given transition Z, i.e., $\theta _{2} (t) = t'$, where $pr_{4}Z = t \in [T, T + t^{*} ]$ and $t' \ge 0$; the value of this function is calculated at the moment when the transition starts functioning;

(h) K is the set of the GN’s tokens. In some cases, it is convenient to consider this set in the form

$$ K = \bigcup _{l\in Q^{I}} K_{l}, $$

where $K_{l}$ is the set of tokens which enter the net from place l, and $Q^{I}$ is the set of all input places of the net;

(i) $\pi _{K}$ is a function giving the priorities of the tokens, i.e., $\pi _{K} : K \rightarrow \mathcal{N}$;

(j) $\theta _{K}$ is a function giving the time-moment when a given token can enter the net, i.e., $\theta _{K}(\alpha )= t$, where $\alpha \in K$ and $t \in [T, T + t^{*} ]$;

(k) T is the time-moment when the GN starts functioning; this moment is determined with respect to a fixed (global) time-scale;

(l) $t^{0}$ is an elementary time-step, related to the fixed (global) time-scale;

(m) $t^{*}$ is the duration of the GN functioning;

(n) X is a function which assigns initial characteristics to every token when it enters input place of the net;

(o) $\varPhi$ is a characteristic function that assigns new characteristics to every token when it makes a transfer from an input to an output place of a given transition;

(p) b is a function giving the maximum number of characteristics a given token can receive, i.e., $b: K \rightarrow \ N$.

For the algorithms of transition and GN functioning the reader can refer to [9].