The various chapters so far have introduced the basics of process-algebraic reasoning, including essential concepts such as recursion, parallel composition, and abstraction, and several extensions of this basic framework with time, data, and state. This one-but-last chapter introduces two more extensions, to reason about priorities and probabilities, and elaborates briefly on mobility and variants of parallel composition in Sections 11.3 and 11.4.

Priorities

In order to specify certain applications, it is useful to be able to restrict the non-determinism in process descriptions, by allowing certain actions to have priority over others in a choice context. A priority mechanism has proved itself useful in the following circumstances:

1. (i) when describing interrupts and disrupts, where the normal execution of a system is pre-empted by an event that has priority;

2. (ii) when giving semantics to certain features of programming languages such as interrupt or error handling mechanisms;

3. (iii) when timing is involved, when some events may not happen prematurely and other events may need to happen as soon as possible (maximal progress);

4. (iv) when describing scheduling algorithms.

This section introduces a priority mechanism in the equational and operational frameworks introduced in earlier chapters. Assume that certain actions have priority over other actions. This is expressed by assuming there is some (irreflexive) partial ordering ≺ on the set of actions A. For simplicity, consider this partial ordering to be fixed. This means the priority is static.

This book about process algebra improves on its predecessor, written by Jos Baeten and Peter Weijland almost 20 years ago, by being more comprehensive and by providing far more mathematical detail. In addition the syntax of ACP has been extended by a constant 1 for termination. This modification not only makes the syntax more expressive, it also facilitates a uniform reconstruction of key aspects of CCS, CSP as well as ACP, within a single framework.

After renaming the empty process (∈) into 1 and the inactive process (δ) into 0, the axiom system ACP is redesigned as BCP. This change is both pragmatically justified and conceptually convincing. By using a different acronym instead of ACP, the latter can still be used as a reference to its original meaning, which is both useful and consistent.

Curiously these notational changes may be considered marginal and significant at the same time. In terms of theorems and proofs, or in terms of case studies, protocol formalizations and the design of verification tools, the specific details of notation make no real difference at all. But by providing a fairly definitive and uncompromising typescript a major impact is obtained on what might be called ‘nonfunctional qualities’ of the notational framework. I have no doubt that these nonfunctional qualities are positive and merit being exploited in full detail as has been done by Baeten and his co-authors. Unavoidably, the notational evolution produces a change of perspective. While, for instance, the empty process is merely an add on feature for ACP, it constitutes a conceptual cornerstone for BCP.

Sequential composition

In the process theory BSP(A) discussed in Chapter 4, the only way of combining two processes is by means of alternative composition. For the specification of more complex systems, additional composition mechanisms are useful. This chapter treats the extension with a sequential-composition operator. Given two process terms x and y, the term x · y denotes the sequential composition of x and y. The intuition of this operation is that upon the successful termination of process x, process y is started. If process x ends in a deadlock, also the sequential composition x · y deadlocks. Thus, a pre-requisite for a meaningful introduction of a sequential-composition operator is that successful and unsuccessful termination can be distinguished. As already explained in Chapter 4, this is not possible in the theory MPT(A) as all processes end in deadlock. Thus, as before, as a starting point the theory BSP(A) of Chapter 4 is used. This theory is extended with sequential composition to obtain the Theory of Sequential Processes TSP(A). It turns out that the empty process is an identity element for sequential composition: x · 1 = 1 · x = x.

The process theory TSP

This section introduces the process theory TSP, the Theory of Sequential Processes. The theory has, as before, a set of actions A as its parameter. The signature of the process theory TSP(A) is the signature of the process theory BSP(A) extended with the sequential-composition operator.

Introduction

In this chapter we investigate the behaviour of two classical epidemic models on general graphs. We consider a closed population of n individuals, connected by a neighbourhood structure that is represented by an undirected, labelled graph G = (V, E) with node set V = {1, …, n} and edge set E. Each node can be in one of three possible states: susceptible (S), infective (I) or removed (R). The initial set of infectives at time 0 is assumed to be non empty, and all other nodes are assumed to be susceptible at time 0. We will focus on two classical epidemic models: the susceptible-infected-removed (SIR) and susceptible-infected-susceptible (SIS) epidemic processes.

In what follows we represent the graph by means of its adjacency matrix A, i.e. aij = 1 if (i, j) ∈ E and aij = 0 otherwise. Since the graph G is undirected, A is a symmetric, non-negative matrix, all its eigenvalues are real, the eigenvalue with the largest absolute value ρ is positive and its associated eigenvector has non-negative entries (by the Perron–Frobenius theorem). The value ρ is called the spectral radius. If the graph is connected, as we shall assume, then this eigenvalue has multiplicity one, the corresponding eigenvector is strictly positive and is the only one with all elements non-negative.

Introduction

So far we have dealt with microscopic models of interaction and epidemic propagation. If we are interested in macroscopic characteristics, such as the time before a given fraction of the population is infected, a simpler analysis is often possible in which we can identify deterministic dynamic models, specified by differential equations, that reflect accurately the dynamics of the original system at a macroscopic level. Such macroscopic description is referred to as mean-field approximation.

Differential equations (macroscopic models) and Markov processes (microscopic models) are the basic models of dynamical systems in deterministic and probabilistic contexts, respectively. Since the analysis, both mathematical and computational, of differential equations is often more feasible and efficient, it is of interest to understand in some generality when the sample paths of a Markov process can be guaranteed to lie, with high probability, close to the solution of a differential equation.

We shall provide generic results applicable to all such contexts. In what follows we approximate certain families of jump processes depending on a parameter n usually interpreted as the total population size, and we approximate certain jump Markov processes as the parameter n becomes large. It is worth mentioning that the techniques presented here can be applied to a wide range of problems such as epidemic models, models for chemical reactions and population genetics, as well as other processes.

Introduction

The Reed–Frost model is a particular example of an SIR (susceptible-infectiveremoved) epidemic process. It is one of the earliest stochastic SIR models to be studied in depth, because of its analytical tractability. In the general SIR model, the population initially consists of healthy individuals and a small number of infected individuals. Infected individuals encounter healthy individuals in a random fashion for a given period known as the infectious period and then are removed and cease spreading the epidemic. Alternatively, in the context of rumour spreading, healthy individuals correspond to nodes that ignore the rumour whereas infected individuals are nodes that initially hold the rumour and actively pass it on to others. Removed individuals correspond to nodes that cease spreading the rumour, or stiflers.

The Reed–Frost epidemic corresponds to a discrete-time version of the SIR model where the infectious period lasts one unit of time. Another commonly used model assumes that infectious periods are independent and identically distributed (i.i.d.) according to an exponential distribution, so that the system evolves as a continuous-time Markov process. This continuous-time SIR model is amenable to the analysis presented in Chapter 5 whereby the dynamics of the Markovian epidemic process is approximated by the solution of a set of differential equations.

The basic version of the Reed–Frost model is as follows. A set of n individuals is given, indexed by i ∈ {1, …, n}.

Model and motivation

In Chapter 8 we tried to understand the impact of a network's topology on the behaviour of epidemics. In the present chapter, we focus on the role played by the initial condition in determining the size of the epidemic. Moreover, we adopt a different viewpoint, taking an algorithmic perspective. That is to say, we address the following question: given a set of individuals that form a network, how should one choose a subset of these individuals, of given size, to be infected initially, so as to maximise the size of an epidemic? The idea is that by carefully choosing such nodes we could trigger a cascade of infections that will result in a large number of ultimately infected individuals.

This problem finds its motivation in viral marketing. In this context, limited advertising budget is available for the purpose of convincing a small number of consumers (i.e. the size of the set of initial infectives) of the merits of some product. Such consumers may in turn convince others, and the aim is to maximise the ultimate reach of the advertisement by leveraging such “contaminations”.

We address this problem by considering the following version of the Reed–Frost epidemic. We assume that the network is described by a directed graph G. The potentially infected individuals constitute the set V ≔ {1, …, n}.