Approximate Master Equations for Dynamical Processes on Graphs

Abstract

Abstract. We extrapolate from the exact master equations of epidemic dynamics on fully connected graphs to non-fully connected by keeping the size of the state space N + 1, where N is the number of nodes in the graph. This gives rise to a system of approximate ODEs (ordinary differential equations) where the challenge is to compute/approximate analytically the transmission rates. We show that this is possible for graphs with arbitrary degree distributions built according to the configuration model. Numerical tests confirm that: (a) the agreement of the approximate ODEs system with simulation is excellent and (b) that the approach remains valid for clustered graphs with the analytical calculations of the transmission rates still pending. The marked reduction in state space gives good results, and where the transmission rates can be analytically approximated, the model provides a strong alternative approximate model that agrees well with simulation. Given that the transmission rates encompass information both about the dynamics and graph properties, the specific shape of the curve, defined by the transmission rate versus the number of infected nodes, can provide a new and different measure of network structure, and the model could serve as a link between inferring network structure from prevalence or incidence data.

Network inference from population-level observation of epidemics

“…Consistent with results in 27 , we notice that, although estimated curves are distinct for different network classes, they all share some common features: specifically, they all satisfy and exhibit a single maximum in [0, N ]. Perhaps the most important features that change between the three distinct network classes are the flatness and skewness of the curves (see Fig.…”

“…Here, we take a different route and choose to use a surrogate model to represent the evolution of the count of infected nodes in the population. A reasonable candidate for this is a Birth-and-Death process (BD), see 27 , characterised by only equations and free parameters, the rates of infection and recovery, that need to be tuned to best represent the exact model. Whilst the rates of recovery are network independent and known exactly, the rates of infection in the surrogate model are more challenging to define.…”

PDE approximation of large systems of differential equations

“…In Figure 1 the solution of system (14) is compared to the solution of the PDE (15) when τ = γ = 0.5, the latter was plotted using the Fourier method with the first 40 eigenfunctions. The first 40 eigenvalues were determined by using Newton's method within each interval given above, and then we solved equation (17) restricted to the first 40 variables.…”

“…for t ∈ [0, T ]. The system (14) was solved with MATLAB's ode45 solver, while the partial differential equation with MATLAB's pdepe solver. The results of the comparison are shown in Fig.…”

“…Analysing the mean field approximation for the expected number of infected nodes in an epidemic process on a large network we were led to a first order PDE approximation in our previous work Bátkai et al [7]. In a recent paper, in which regular random, Erdős-Rényi, bimodal random, and Barabási-Albert graphs are studied, we have shown that a suitable choice of the coefficients in the master equation leads to an ODE approximation with tridiagonal transition rate matrices (see Nagy, Kiss and Simon [14]). Our study in this paper aims at approximating dynamic processes on networks, for which the transition rate matrix of the underlying Markov chain has a tridiagonal structure (with a possible extension to similar matrices).…”