The relationships between message passing, pairwise, Kermack–McKendrick and stochastic SIR epidemic models

Abstract

We consider a very general stochastic model for an SIR epidemic on a network which allows an individual’s infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time . We specialise these results to a homogeneous special case where the graph (network) is symmetric. The message passing system here reduces to just four equations. We prove that cycles in the network inhibit the spread of infection, and derive important epidemiological results concerning the final epidemic size and threshold behaviour for a major outbreak. For Poisson contact processes, this message passing system is equivalent to a non-Markovian pair approximation model, which we show has well-known pairwise models as special cases. We show further that a sequence of message passing systems, starting with the homogeneous one just described, converges to the deterministic Kermack–McKendrick equations for this stochastic model. For Poisson contact and recovery, we show that this convergence is monotone, from which it follows that the message passing system (and hence also the pairwise model) here provides a better approximation to the expected epidemic size at time than the Kermack–McKendrick model.

Impact of the infectious period on epidemics

“…We allow arbitrarily distributed exposed and infectious periods, heterogeneous contact processes between individuals, and heterogeneity in susceptibility and infectiousness. Many previously studied models such as , pairwise [12,13], message passing [10] and spatial models [4,14] are identical to, consistent with, or approximations of, special cases of the stochastic model which we examine here [15]. We show how our conclusions apply to these well-known models.…”

“…Note that, in both cases, making these changes to (9) does not alter the probability, as it is represented in the message passing system, that a given infected individual will make an infectious contact to a given neighbour before recovering (this is the quantity in (10) and (11)). It is then straightforward, following section IV of [10] and the proof of Theorem 4 in [15], that lim t→∞ S (i) mes (t) is also unaltered for all i ∈ V. On these grounds, we expect the bounds and approximations to be good. Additionally, replacing S As an example, if contact processes are Poisson such that the ω ji are exponentially distributed with parameters β ji > 0, we can then conveniently obtain the lower bounds via delay differential equations (DDEs)…”

“…We can now build on these results in order to write down systems of equations which are simpler to solve and which provide rigorous lower and upper bounds, and approximations, for S that i is susceptible at time t, and an upper bound on R (i) mes (t) is also an upper bound on the probability that i is recovered/vaccinated at time t. This follows since S (i) mes (t) is a lower bound on the former probability while R (i) mes (t) is an upper bound on the latter [10,13,15]. Such bounds provide a 'worst case scenario' [10] and an upper bound on the expected final size of the epidemic.…”

Pairwise approximation for SIR -type network epidemics with non-Markovian recovery

“…Potentially a more natural extension of our model would be to relax the assumption of Markovian transmission, and this is indeed a more promising future research direction. We note that Wilkinson et al [46] have independently derived a similar pairwise model by starting from the message passing sytem [13] Several different approaches exist to model non-Markovian epidemics on networks. These are largely guided by the choice of model and variables to be tracked.…”

Prediction of the COVID-19 outbreak based on a realistic stochastic model

“…See for instances, ; ; ; for pioneering works, and ; Zhang and Wang (2013); ; for recent developments. See also Wilkinson et al (2017) and the references therein for a brief review.…”