The effects of heterogeneity on stochastic cycles in epidemics

Models of biological processes are often subject to different sources of noise. Developing an understanding of the combined effects of different types of uncertainty is an open challenge. In this paper, we study a variant of the susceptible-infective-recovered model of epidemic spread, which combines both agent-to-agent heterogeneity and intrinsic noise. We focus on epidemic cycles, driven by the stochasticity of infection and recovery events, and study in detail how heterogeneity in susceptibilities and propensities to pass on the disease affects these quasi-cycles. While the system can only be described by a large hierarchical set of equations in the transient regime, we derive a reduced closed set of equations for population-level quantities in the stationary regime. We analytically obtain the spectra of quasi-cycles in the linear-noise approximation. We find that the characteristic frequency of these cycles is typically determined by population averages of susceptibilities and infectivities, but that their amplitude depends on higher-order moments of the heterogeneity. We also investigate the synchronisation properties and phase lag between different groups of susceptible and infected individuals.


S1 Linear-noise approximation
Carrying out the system-size expansion for the model with heterogeneity is tedious, but straightforward and follows the lines of 1 . The final outcome is the linear-noise approximation in Eqs. (11). The variables η i and ν a , represent Gaussian noise, with no correlation in time, but with potential correlation between the different noise variables at equal time. These noise variables can be decomposed as where, broadly speaking, each term on the right-hand side represents one possible type of event in the microscopic model. For example, u ia relates to spontaneous infection of a susceptible individual of type S i , resulting in a newly infective of type I a . Similarly, v iab represents an event in which an individual of type S i is infected by an individual of type I a , and the newly infected is of type I b . The variable w a relates to a recovery event of an individual of type I a , death of susceptible S i and simultaneous birth of susceptible S k is reflected by x ik ; death of an individual of type I a and simultaenous birth of susceptible S i is described by y ai , and finally death of a recovered individual and simultaneous birth of susceptible S i , by z i . The signs on the right-hand-side in Eqs. (S1) reflect the fact that each of these events may either increase or reduce the number of individuals of type S i and I a , respectively. Each of the noise variables on the right-hand-side of Eqs. (S1) are uncorrelated in time, and they have no cross-correlations. Within the LNA their variances are set by the corresponding reaction rates at the deterministic fixed point, i.e. we have Using the shorthand introduced in Eqs. (15), we then find which are needed for the computation of the PSDs. S1/S6

S2 Calculation of power spectra
We start from the result in Eqs. (14) in Sect. 3.2: (S4) As an illustration let us now compute the power spectrum of B. To keep equations manageable, we define and so we write the Fourier transform ofB as where f i , β a , η i and ν a are all functions of ω. We then find The notation * denotes complex conjugation. Substituting the noise correlators from Eqs. (S3), and, using Eq. (S6), we find which is the PSD of B, as also reported in Eq. (16) in the main text.

S2/S6
Following the same process, we can compute the PSD for the remaining quantities, X, I and S. We do not report all details, but only the final results The power spectra of fluctuations for the individual subgroups of infectives and susceptibles are found as

S3 Phase Lag
In order to explore the the phase lag we use the so-called complex coherence function, CCF ij , between subgroups i and j, defined as where x i and P are functions of ω.
For i = j this is in general a complex-valued function (of ω). The argument of CCF ij , given by is known as the phase spectrum; it describes the phase-lag between the time series x i (t) and x j (t) 2 . The cross spectra of the population in the susceptible classes normalized with respect to the total population (x i = n i /N ) is given by This can be written as where we introduced the notation From these we obtain the phase lag as which yields the theoretical lines in Fig. 9a.
To explore the phase lag between the susceptible subgroups when normalized by the total susceptible population (x i = n i /N S), we first need to compute the cross-spectra of the renormalized signals P x i x j (ω). As in Section 3.1, we start from the ansatz We then have and so (after expanding in 1/ √ N ) In Fourier space this turns into For the cross spectra we then find which can be rewritten as From these, we can find the phase-lag as This expression was used to obtain the analytical predictions shown in Fig. 9b. Mean susceptibility at birth.
(2) X Aggregate susceptibility of the population.
(3) B Total 'infective power' in the population. (3) Fraction of susceptible individuals in subgroup i in the limit of infinite system size.