By considering the unique QSD associated with Markovian SIS dynamics on finite strongly connected networks, along with its implications under duality, we have pro- vided meaningful mathematical definitions for both endemic prevalence (quasi-prevalence) and invasion probability (quasi-invasion). Utilising these definitions, we have provided a general statement of the exact relationship between invasion probability and en- demic prevalence at the individual and population level, for any finite undirected net- work of arbitrary heterogeneity (including undirected networks with weighted links and individual-specific recovery parameters). The definitions also generalise to weakly connected networks, and the prevalence-invasion relationship, with slight modification, applies to arbitrary directed networks.
We note that for infinite homogeneous networks, invasion probability (in these cases, the probability of indefinite persistence) from a single initial infected has been shown to be equal to the fraction of the population infected in the upper invariant measure (Grimmett, 2010; Neal, 2008). Furthermore, the relationship between the probability of long-term persistence and quasi-stationary distributions has previously been investi- gated (see Chaterjee and Durrett (2009) and, for the related concept of ‘metastability’, see Schonmann (1985) and Simonis (1996)). However, although the prevalence-invasion relationship follows easily from a combination of the QSD and duality, to our knowledge this is the first general statement of this exact relationship in the context of arbitrary networks. We have thus related two fundamental epidemiological quantifiers in systems where they cannot usually be calculated analytically due to complexity.
It is generally easier to collect empirical data on endemic prevalence rather than directly on invasion risk. In the case of undirected networks, prevalence data can thus be utilised to inform invasion risk. This method echoes Anderson and May’s (1991) estimation of the basic reproductive ratio of measles from the total number
of susceptible individuals in England and Wales (using data from Fine and Clarkson (1982)).
When other infectious agents exhibit qualitatively similar behaviour on the same undirected network, we can expect that the individuals carrying the greatest level of endemic infection are also those most likely to initiate new successful invasions. This lends support to the targeting of high-risk individuals in these systems as an effective strategy for the mitigation and control of emerging epidemics.
Chapter 3
Moment-closure for Markovian
epidemic dynamics on networks
3.1
Introduction
In this chapter we focus on epidemic dynamics where individuals can only be infected once or not at all. Specifically, we consider a Markovian network-based model with a general compartmental structure described as susceptible-exposed-infected-recovered (SEIR) (see, for example, Keeling and Rohani (2007)). This model is the same as the Markovian network-based SIR model, defined in section 1.10, except that after a susceptible individuali∈V receives an infectious contact it must first pass through an ‘exposed’ state, lasting for a period that is exponentially distributed with parameter
αi, before entering the infected state. While in the exposed state individuals do not
make infectious contacts. Thus, with reference to the definition of the simpler SIR version in section 1.10, the Markovian network-based SEIR model can be described by a continuous time Markov chain{σ(t)}, where σ(t) takes values in {S, E, I, R}V, and
with transition rates as in table 3.1.
Table 3.1: Transitions for the Markovian network-based SEIR model
from to at rate
σ:σi=S σi→E Pj∈V Tij1(σj =I)
σ:σi=E σi→I αi
σ :σi =I σi→R γi
For this model there are 4N Kolmogorov forward equations which give a full descrip-
tion of the evolution of the system (given some initial distribution). However, moment closure methods allow us to write down much smaller systems of ordinary differential equations which attempt to capture the evolution of the expected number in each com- partment. For example, the time derivative of the expected number of pairings of a susceptible and an infected individual depends on the expected numbers of connected triples in various states. By approximating the expected number of connected triples
of a given type, in terms of expected numbers of pairs and individuals, a small closed system of equations is obtained (Matsuda et al., 1992; Keeling, 1999; House and Keel- ing, 2010). Similarly, at the individual level, the time derivative of the probability of a connected pair being in a given state depends on the joint probabilities of the triples which it forms with its neighbours. By expressing the joint distribution for such triples in terms of pairs and individuals, a closed system is obtained (Sharkey, 2008).
We will first outline the construction of pair-based moment closure systems (at the population level), with a focus on finite and directed networks. We will adopt the systematic approach to construction, starting at the individual level and then making independence and homogeneity assumptions, given by Sharkey (2008). We will then go on to develop ‘exact’ moment closure systems for the case of tree networks, extending the work of Sharkey et al. (2013) and Kiss et al. (2014) from SIR to SEIR dynamics. We then propose an exact closure theorem, extending a result given by Kiss et al. (2014), which allows us to define exact systems for non-tree networks, and examine the relevance of network structure to the dimensionality of such systems. Finally, we will define hierarchies of approximating moment closure systems, which are non-decreasing in dimensionality, and which start with a pair-based system and end with an exact system (Sharkey and Wilkinson, 2015).