Heterogeneous population dynamics and scaling laws near epidemic outbreaks

Abstract

In this paper, we focus on the influence of heterogeneity and stochasticity of the population on the dynamical structure of a basic susceptible-infected-susceptible (SIS) model. First we prove that, upon a suitable mathematical reformulation of the basic reproduction number, the homogeneous system and the heterogeneous system exhibit a completely analogous global behaviour. Then we consider noise terms to incorporate the fluctuation effects and the random import of the disease into the population and analyse the influence of heterogeneity on warning signs for critical transitions (or tipping points). This theory shows that one may be able to anticipate whether a bifurcation point is close before it happens. We use numerical simulations of a stochastic fast-slow heterogeneous population SIS model and show various aspects of heterogeneity have crucial influences on the scaling laws that are used as early-warning signs for the homogeneous system. Thus, although the basic structural qualitative dynamical properties are the same for both systems, the quantitative features for epidemic prediction are expected to change and care has to be taken to interpret potential warning signs for disease outbreaks correctly.

Anticipating epidemic transitions with imperfect data

“…For mathematical reasons, proposed EWS for disease emergence have assumed access to regular recordings ("snapshots") of the entire infectious population [8][9][10][11][12][13]. However, epidemiological data are typically aggregated into periodic case reports subject to reporting error.…”

“…Combined, they can make each pathogen's emergence seem idiosyncratic. In spite of this apparent particularity, there is a recent literature on the possibility of anticipating epidemic transitions using model-independent metrics [6][7][8][9][10][11][12][13][14]. Referred to as early-warning signals (EWS), these metrics are summary statistics (e.g.…”

“…A major obstacle to deploying early-warning systems is the type of data available to calculate the EWS. Theoretical predictions assume the data will be sequential recordings (or "snapshots") of the true number of infected in the population through time [8][9][10][11][12][13]. In this paper we refer to this as snapshots data.…”

Epidemics on networks with heterogeneous population and stochastic infection rates

“…Hence, the rate coefficients, β(ω) and δ(ω), as well as the unknowns x i (t, ω)s, are assumed to be random variables on the probability space (Ω, F , P), for a given σ-algebra F and a probability measure P on it. However realistic heterogeneous parameter distributions are not constant in time but could be considered as additional dynamical variables [20]. Thus, in order to model the fluctuations in time of the parameters, we consider white noise [42], a natural starting point for the case when the functional form and properties of the stochastic process are not known ( [20,11,12]).…”

“…We need to consider also that the effective diffusion occurs through individual people, that may differ in many different aspects (genetics, biology or social behavior), thus the parameters characterizing the infection (say, the rate of infection or the rate of recovery) have a variety among the population and, in most cases they cannot be fixed a priori: only their statistical properties are known. A short overview on works in literature that consider heterogeneous populations can be found in [20,21].…”