The problem of detrending when analysing potential indicators of disease elimination

Highlights • We derive indicators of disease eradication so that control efforts may cease.• Detrending is necessary to analyse single timeseries data and is difficult to achieve.• Detrending using the mean of even a few simulations of the same process works well.• Metapopulation models suggest a promising solution to the problem of detrending.


Deviation of the Fokker-Planck equation for SIS dynamics:
For the SIS model without metapopulations, the mean-field equations are given by dφ dt = βφ(1 − φ) − γφ where φ = I N . The following deviation follows from van Kampen (Chapter 8 and 10). The linear noise approximation for the discrete infectious state I is given by: (1) The general form of the master equation for the SIS model based on the transition probabilities given in Table 1 The master equation can be written using step operators which act on an arbitrary function of n, defined as Ef (n) = f (n + 1) and E −1 f (n) = f (n − 1).
Define a new probability distribution function Π by P (I, t) = Π(ζ, t). The derivative of the probability distribution function with respect to t, is needed for deriving the continuous space master equation.
Combining equations 3 and 4 together, we can write down the continuous space master equation: t) and substitute the linear approximation We collect powers of N in equation 5 and substitute the mean-field deterministic approximation as N → ∞ ( macroscopic description which ignores fluctuations). This results in the linear Fokker-Planck equation for this system: The solution for the analytical variance can be deduced from the following equations, At steady state when dφ dt = 0 and dV dt = 0, we obtain φ * = 1 − γ β and V * = 1 3 Deviation of the Fokker-Planck equation for metapopulation dynamics:

Derivation for 4 populations
We write the transition probabilities as given in Table 3 in terms of the step operators. For example, the transition probability of an susceptible individual in population 1 becoming infected can be written as T (I 1 , ...|I 1 − 1, ...) = E I 1 T (I 1 + 1, ...|I 1 ). The total population size was taken to be 20, 000 and we divide this equally (depending on the number of subpopulations M 2 ) assuming that the size of each subpopulation, N M , is the same. The master equation for M 2 = 4 subpopulations on a lattice, P (I 1 , I 2 , I 3 , I 4 , t), is the probability observing I infectives at time t, The step operators are defined for each subpopulation i, depending on the linear noise approximation: Similarly to the simple SIS model, we define a new probability distribution function Π by P (I 1 , I 2 , I 3 , I 4 , t) = Π(ζ 1 , ζ 2 , ζ 3 , ζ 4 , t). The derivative of the probability distribution function with respect to t, Substitute the master equation (Equation 8) and step operators into the above equation. Evaluating The multivariate Fokker-Planck Equation is fully described in terms of matrices A and B, where B is symmetric and positive definite. If both A and B are constant matrices then the solution is Gaussian (linear Fokker-Planck Equation), We assume that the mean-field solution is the same for all populations, φ i = φ. Collecting 6 terms in Equation 9 into the format of the FPE (Equation 10), we arrive at: A ij = ρ, if i and j are adjacent, , if i and j are adjacent,

Derivation for general M 2 subpopulations
The general master equation for M 2 subpopulations, P ({I i , i ∈ [1, M 2 ]}, t), is the probability observing I infectives at time t and depends on the number of neighbours, N i , of each subpopulation i where the degree of each subpopulation i (total number of neighbours) is given by d i ,

Define a new probability distribution function Π by
The derivative of the probability distribution function with respect to t, Then following a similar analysis as above (when M 2 = 4), We assume that the mean-field solution is the same for all populations, φ i = φ.
Then it follows in the form of the Fokker-Planck Equation 10: A ij = ρ, if i and j are adjacent,

ROC curve analysis
The ROC curves below were calculated using the Kendall's tau rank correlation coefficient. Kendalltau coefficient is a test of statistical significance that is widely used in the literature for early warning signals of critical transitions. The statistic was calculated on a moving window for 50 * M 2 realisations of NExt and Ext, for M 2 subpopulations. We calculated the Kendall-tau coefficient for each realisation as measure if the increasing trend in the variance and CV is statistically significant. We evaluated this for our simulations up to a variety of endpoints: t = 400 to t = 450. We considered different endpoints since the dynamics (and therefore the increasing trend) of NExt and Ext are the same up until R 0 = 1 at t = 400.