A rigorous approach to investigating common assumptions about disease transmission

Abstract

Changing scale, for example, the ability to move seamlessly from an individual-based model to a population-based model, is an important problem in many fields. In this paper, we introduce process algebra as a novel solution to this problem in the context of models of infectious disease spread. Process algebra allows us to describe a system in terms of the stochastic behaviour of individuals, and is a technique from computer science. We review the use of process algebra in biological systems, and the variety of quantitative and qualitative analysis techniques available. The analysis illustrated here solves the changing scale problem: from the individual behaviour we can rigorously derive equations to describe the mean behaviour of the system at the level of the population. The biological problem investigated is the transmission of infection, and how this relates to individual interactions.

Appendix: WSCCS

The Weighted Synchronous Calculus of Communicating Systems (WSCCS) is one of many process algebras based on Milner’s seminal Calculus of Communicating Systems (CCS) (1980). For our work, the main difference between CCS and WSCCS is the addition of weighted, or probabilistic, choice. A full introduction to the notation of WSCCS can be found in Tofts’ introductory paper (1994). Here, we use examples from our models to give a brief overview of the WSCCS notation. Biological models in WSCCS utilise various different types of agents. Complex behaviours within the system can be constructed by having agents interact with each other and by allowing different simple behaviours to happen at different stages.

Consider the agent S3, which is written

$$ S3\mathop = \limits^{{\rm def}}p_{\rm d}.\surd:0 + (1-p_{\rm d}).\surd:S1. $$

This says that S3 will become the agent S1 with weight (1 − p _d), or become the null agent 0 with weight p _d. In each case it performs the action $\surd,$ which can be thought of as representing a step of time, though this is not equal to any particular unit of time and tick actions need not be uniformly distributed in time. It is convenient to consider agents which can perform only tick actions as representing a probabilistic choice between the different outcomes. In general weights in WSCCS define the relative frequencies of the different outcomes but for probabilistic agents such as S3 we choose the weights to be between 0 and 1 with the sum of the weights equal to 1.

Another type of agent which must be considered is a parallel agent. The probabilistic agent

$$ I1 \mathop = \limits^{{\rm def}} p_{\rm b}.\surd:I2\times Nb2 + p_{\rm c}.\surd:I2\times Trans + (1-p_{\rm b}-p_{\rm c}).\surd:I2 $$

can become the agent I2 and also the parallel agents

$$ I2\times Nb2, $$

(7)

and

$$ I2\times Trans. $$

(8)

The agent in (7), which consists of an I2 agent in parallel with an Nb2 agent, represents an infected individual giving birth to a newborn individual. On the other hand the agent in (8), which consists of an I2 agent in parallel with a Trans agent, represents two different aspects of an infected individual’s behaviour (absorbing an infectious contact and making an infectious contact). In the next stage the Trans agent will always become the null agent 0 so that the infected individual is once more represented by a single agent.

The agent

$$ Popn \mathop = \limits^{{\rm def}} S1\{s\}\times I1\{i\}\times R1\{r\}\lceil\{\surd\} $$

is another type of parallel agent, which defines an initial population with which to analyse the system, consisting of s S1 agents, i I1 agents and r R1 agents. The $\lceil\{\surd\}$ forces communicating agents to cooperate on all actions other than $\surd.$

Interaction between agents forms an important part of agent behaviour in WSCCS. For example the agent Trans,

$$ Trans \mathop = \limits^{{\rm def}} \omega.\overline{infect}:I3 + 1.\surd:I3, $$

will either perform the $\overline{infect}$ action or perform a tick, in either case becoming I3. The special weight ω (which is larger than any real number weight) means that this agent must perform the $\overline{infect}$ action when available. However, the restrictions on the actions which are possible in Popn mean that Trans can only perform the $\overline{infect}$ action if an agent is available to perform a complementary action. For example the agent S2,

$$ S2 \mathop = \limits^{{\rm def}} \omega. infect :SI3 + 1.\surd:S3 $$

can perform the action infect which would allow Trans to perform $\overline{infect}.$ If this happens the S2 agent will become an SI3 agent, indicating that it has made an infectious contact. Otherwise the S2 agent will perform a tick and become an S3 agent. The action infect is thought of as the input action while $\overline{infect}$ is the output action.

The probabilities with which agents interact is influenced by the number of agents of the same type in the population, as well as the number of agents available to perform a complementary action. For example if there are no agents in the population available to perform an $\overline{infect}$ action then an S2 agent would be unable to perform the input action and would be forced to perform a tick action.

Our models are designed in such a way that interaction and probabilistic choice happen in different stages. This means that behaviours in the models happen over multiple consecutive stages with behaviour being averaged over these stages to represent one time step in the MFEs. By separating probabilistic choice and communication it is easier to write the models and correctly determine the MFEs for the system.

Previous work introduced notation for functional parameters in WSCCS (McCaig 2007; McCaig et al. 2009). The models considered here make use of this notation to incorporate density-dependent behaviour. For example the probability of birth in both models is density dependent:

$$ p_{\rm b} \mathop = \limits^{{\rm prob}} p_{{\rm b}_0}-k(\lfloor S1\rfloor+\lfloor I1\rfloor+\lfloor R1\rfloor), $$

(9)

where $\lfloor S1\rfloor, \lfloor I1\rfloor$ and $\lfloor R1\rfloor$ are respectively the numbers of S1, I1 and R1 agents present in the population. When deriving MFEs functional parameters are incorporated by substituting the functional form of the parameter into the equations. For (9) this means substituting for

$$ \begin{aligned} p_{\rm b} &= p_{{\rm b}_0}-k(S1_t +I1_t+R1_t), \\ &= p_{{\rm b}_0}-kN_t. \\ \end{aligned} $$