Harris [12] showed that the contact process, and equivalently Markovian network-based SIS dynamics, can be fruitfully represented as follows: Each member of the population is assigned its own positive real number line or ‘time line’. Next, a Poisson point process with intensity is placed on the time line corresponding to each individual where . Following Grimmett [31], such points are called ‘points of cure’. Then, for each ordered pair of individuals , such that , we place arrows going from 's time line to 's time line according to a Poisson point process with intensity . These arrows are called ‘arrows of infection’. Under such a representation, the probability that there is at least one path from 0 on a time line corresponding to an individual in subset to on the time line corresponding to individual is equal to the probability that will be infected at time when only the members of are initially infected (paths take into account the direction of time and the directions of the arrows of infection and are blocked by the points of cure). An example of this kind of representation, for a fully connected network of three individuals, is given in figure 1.

Assume that we have a network , with an associated vector of recovery parameters . Now, graphically representing Markovian SIS dynamics on this network, and adopting the notation of Harris [19], let (with ) be the set of individuals such that if and only if there is at least one path from 0 on a time line corresponding to an individual in to on 's time line (see figure 1 for an example of a path). We will use to denote the appropriate probability measure. Thus, the probability that at least one member of a subset will be infected at time , given that only the members of subset are initially infected, i.e. at , can be expressed as .

The property of ‘duality’ pertaining to Markovian SIS dynamics (and the contact process) can be expressed as follows:(2)where , is an arbitrary weighted contact network and is the transpose of . Note that for undirected networks which simplifies the relationship and, in this case, the process is said to be ‘self-dual’.

To understand equation 2, it is simplest to firstly consider the case where is undirected and where the subsets contain a single individual; and . Then the probability of existence of a path from at time 0 to at time is expressed by the left hand side of equation 2. With reference to figure 1, reversing the arrow of time (turning the diagram upside down), we see that the probability that a path exists from individual at time 0 to at time is the same calculation since the Poisson point processes are the same in reverse. This is expressed by equation 2.

When the network possesses an asymmetry, the reverse process needs to be computed on the transposed network to be equivalent since the direction of infection is reversed. More generally, the probability that there exists at least one path from an individual in set at time 0 to an individual in set at time is equal to the probability that there is at least one path from an individual in set at time 0 to an individual in set at time in the transposed network, and this statement is represented by equation 2.

An immediate problem with finite systems is that there are no genuine stationary distributions corresponding to endemic infection because the long-term behaviour is always guaranteed extinction (of the infectious agent). However, from a practical point of view, it is sometimes possible to obtain something like the endemic stationary distribution; figure 2a illustrates this clearly for our example network (File S1) (figure 2b illustrates the probabilistic nature of the invasion process which is discussed in subsequent sections). While a genuine stationary distribution does not exist, it is clear that we can still measure something similar to endemic prevalence through stochastic simulation since ultimate extinction is, in this case, very unlikely on short time scales.

From a theoretical perspective, the situation can be made precise. Our system is finite and Markovian with a single absorbing state (extinction of the infection). Also, if the network under consideration is strongly connected such that infection can be transmitted, via some route, from any individual to any other individual, then the transient states form a single commuting class. In this case, if we initiate the system in a transient state and condition on the survival of the infection, then the system tends to a unique distribution referred to as the quasi-stationary distribution (QSD) [32]–[34]. Let us denote by an arbitrary subset of the population represented by a strongly connected network , and let be the vector of individual-specific recovery parameters. We can now unambiguously define to be the probability that at least one individual in subset of the network is infectious in the QSD. In the notation of the graphical representation introduced earlier, we can write:(3)since .

It can be argued that the quasi-stationary distribution has practical relevance (i.e. is a good ‘representation’ of the endemic situation [34]) if the rate of convergence to this distribution, when conditioning on non-absorption, is rapid compared to the rate at which the system decays to inevitable absorption when it is ‘initiated’ in the QSD [32]. This can often be the case for SIS dynamics for which, according to Nåsell [34], ‘it is easy to find examples where the expected time to extinction even for a rather small population exceeds the age of the universe’. More specifically, Simonis [35] has shown that the contact process on large but finite multi-dimensional homogeneous square lattices, where the initial state is all-infected, will be near to the upper invariant measure of the corresponding infinite process, restricted to the finite set, for most of its lifetime (assuming the corresponding infinite process is supercritical).

Let us consider the following quantities for an arbitrary strongly connected network and arbitrary :

Note that in the limit as quantity 3 is equal to , the probability that at least one member of is infected in the QSD.

In figure 3a, the way in which these three quantities may vary with respect to time is illustrated for a scenario in which the QSD has practical relevance. In such a scenario, the quantifier is able to capture the value at which initially ‘plateaus’ before its slow decay to zero.

We have defined in terms of the process which occurs when all individuals are initially infected. We could, however, define in terms of the process which occurs when only the members of are initially infected since the QSD, as defined by Daroch and Seneta [32], [33], is independent of initial conditions. Nonetheless, the process which occurs when all individuals are initially infected is unique in that it has the maximum expected time to extinction across all possible initial states (this follows from (with - see, for example, Grimmett [31]). Therefore, this is the non-conditioned process which is most appropriately described by the QSD.

In the context of finite contact networks, invasion probability has not been given a rigorous theoretical definition (see Nåsell [36] for a discussion of the threshold phenomenon in the stochastic SIS model). For dynamics with a genuine stationary endemic distribution, which (in this context) can only exist when the population size is infinite, the probability of invasion can be defined as the probability that the infection will persist indefinitely. For finite populations, this is complicated because we have to distinguish between infections that fail to invade and infections which successfully invade but then subsequently die out.

In this section we show that the quantifier of invasion probability, which is as equally meaningful and relevant as our quantifier for endemic quasi-prevalence, for outbreaks initiated on the members of a subset of a strongly connected network (with associated vector ) can be defined as:(4)Note that the definition of implies that quasi-invasion is certain when all individuals are initially infected. Our definition corresponds to the intuitive notion of invasion, i.e. a ‘large’ outbreak, or the ‘attainment’ of the QSD.

Let us now consider the following quantities:

It follows from duality that the three quantities, 4, 5 and 6, are all equal respectively to the three quantities, 1, 2 and 3, provided that we transpose . Note also that, in the limit as , quantity 6 is equal to .

Quantity 4 denotes the survival probability up to time . We see in figure 3b that this ‘plateaus’ in exactly the same way as quantity 1 for the transposed network (figure 3a). This plateau, which is captured by quantity 6 in the limit as , i.e. , corresponds to the ‘achievement’ of the QSD and thus with successful quasi-invasion.

Our quantifier of invasion probability can be generalised as:(5)where is a transient stochastic state in which the infection is present. is the probability of survival to time given that the system is initiated in state , and is the probability of survival to time given that the initial state is that which maximises the expected time to extinction. In this form, the definition becomes applicable to other Markovian infection dynamics (on strongly connected networks) which permit endemic behaviour, e.g. susceptible-infected-removed-susceptible (SIRS) dynamics. It is the existence of a unique QSD which enables our definition to capture the probability of invasion in the same way as for SIS dynamics. Note that the definition of quasi-prevalence can also be generalised to any infection model with a unique QSD.

Our main result can be stated as the following (prevalence-invasion) relationship:(6)for any subset of a weighted and strongly connected contact network , conditional on Markovian SIS dynamics.

Equation 6 can be re-written as:(7)which holds because of the property of duality.

Note that for a single individual we have:(8)that is, the probability of quasi-invasion from a given individual is equal to the probability that it is infected in the QSD (in the transposed network). By summing over all and dividing by we get(9)where is the probability of quasi-invasion given that the infection is seeded on a single individual selected uniformly at random, and is the average fraction of the population that are infected in the QSD. An implication of the global-level relationship is that, for (strongly connected) directed networks, reversing the transmission processes (transposing ) will result in an interchange between these two quantifiers without affecting the ‘stability’ of the quasi-stationary behaviour, i.e. the rate at which the system decays to inevitable extinction when ‘initiated’ in the QSD is the same for and its transpose. This can be understood by observing that . Also note that , i.e. given the infection is initiated by individual , the probability that is infectious at any is the same for and its transpose.

For the case where is undirected (), the relationship implies that:(10)and for a single individual :(11)and globally:(12)

The probability that individual is infectious in the QSD is equal to the proportion of time for which is infected after the system has ‘reached’ this distribution, assuming the infection does not die out. Therefore, we measure this as(13)where is a large number of simulated consecutive global events which occur when the probabilities for the system states obey the QSD. is the simulated time between the event and the event, and if is in the infectious state between these events but is zero otherwise. The total simulated time is denoted by . Obviously, the larger we can make , the more exact our measurement becomes. Notice that the events corresponding to a change in the state of individual , and thus to a change in the value of , represent a very small fraction of the total global simulated events (assuming the population size is not extremely small).

In practice, we run the simulation for a sufficiently long time such that the system is likely to be described by the QSD before we start to compute . For example, were our simulation to produce an infectious time line similar to the one in figure 2a we would only include the data from after, say, in our computation. Note that in every simulation there is always the possibility of extinction, even after the QSD has been ‘reached’. However, we can safely discard any post-extinction data since the quantity we are interested in is conditioned on non-extinction. This method of measurement is only valid for systems in which quasi-stationary behaviour can be easily identified i.e. systems for which the QSD has significant practical relevance.

In measuring the probability of quasi-invasion through stochastic simulation, the key requirement is separating major outbreaks from minor outbreaks. Therefore, we look for dichotomised behaviour in relation to the time until extinction by carrying out large numbers of simulations, each initiated in the same stochastic state. For example, we can measure as the fraction of simulations in which an outbreak initiated by individual persists for some sufficiently large number of infection events, such that a clear distinction can be made between the simulations in which long-term persistence occurs and the ones in which it does not. This is because corresponds to the value at which initially plateaus. Without observation of this dichotomised behaviour it may not be possible to classify every simulation. For example, the histogram of infection events per simulation in figure 2b enables us to clearly identify the simulations which exhibit long-term persistence, even though we have only tracked each simulation to the 300th infection event i.e. . If such a histogram cannot be produced from the simulations, can be increased in order to try and ‘uncover’ the dichotomised behaviour.

In general, so long as this kind of dichotomised behaviour is found, the practical issues which emerge in measuring quasi-invasion probability by stochastic simulation for the SIS framework are minor. In other words, this method of measurement is valid, and easily carried out, in systems where significantly plateaus i.e. systems where has significant practical relevance.

By varying a scalar multiplier of a network matrix we can attempt to investigate infections of varying transmissibility spreading through the same population. Figure 4 illustrates our theoretical results, via this method of investigation, for a single individual in our example network (File S1), clearly showing the relationship between quasi-invasion probability and endemic quasi-prevalence. We supply these numerical results to illustrate the prevalence-invasion relationship. We do of course obtain the same qualitative behaviour for any strongly connected network as proved in ‘Theoretical Results’.

To obtain measurements of for individual , 100 simulations were run for each of 20 different multipliers of , and in each simulation the first 10 million events, out of 11 million, were discarded. Each simulation was initiated with sufficient infected individuals such that the probability of early extinction was negligible. As extinction never occurred, none of the data was discarded.

To obtain measurements of , 1 million simulations were run for each of the same 20 multipliers. In each simulation, the infection was initiated on individual and the process was tracked for a maximum of 500 infection events.

An undirected and homogeneously weighted square lattice of 25 individuals was investigated (see figure 5). Due to the small population size, the probability of extinction on a relatively short timescale was significant, even when starting from all-infected. This network enables us to illustrate the numerical measurement of our quantifiers in a scenario where the QSD has less practical relevance, i.e. where endemic quasi-stationary behaviour and dichotomised persistence are not ‘recognisable’ phenomena. In this case, we can compute () by directly measuring at increasing time points and then estimating its convergent value. Thus, for two different global transmission parameters (0.8, 0.5), and two different initial states (all-infected, one infected), 1 million simulations were allowed to run up to some specific point in simulated time (the global recovery parameter was always set to 1). For each simulation, the time at which extinction occurred was recorded so that the probability of survival up to increasing points in time could be measured.

In figure 5a, our quantifier is able to capture an important feature of the model, i.e. the value at which plateaus before its inevitable decay to zero. Figure 5b gives an example of a scenario where, although our quasi-invasion and quasi-prevalence quantifiers are clearly defined, their practical relevance is less obvious. This is because the transmission parameter was sufficiently low such that early extinction was the dominant behaviour.

Through duality, we can approximate by measuring at increasing points in time (as in figure 5) in order to estimate the value at which it may initially plateau. This could, in certain circumstances, be much more efficient than trying to establish global quasi-stationary behaviour and then computing the proportion of time for which the infection is present in . Conversely, if we wish to approximate , for all , it may be more computationally efficient to first establish global quasi-stationary behaviour in the transposed network and then measure the proportion of time each individual spends infected.