Heterogeneity in susceptible–infected–removed (SIR) epidemics on lattices

Abstract

The percolation paradigm is widely used in spatially explicit epidemic models where disease spreads between neighbouring hosts. It has been successful in identifying epidemic thresholds for invasion, separating non-invasive regimes, where the disease never invades the system, from invasive regimes where the probability of invasion is positive. However, its power is mainly limited to homogeneous systems. When heterogeneity (environmental stochasticity) is introduced, the value of the epidemic threshold is, in general, not predictable without numerical simulations. Here, we analyse the role of heterogeneity in a stochastic susceptible -infected -removed epidemic model on a two-dimensional lattice. In the homogeneous case, equivalent to bond percolation, the probability of invasion is controlled by a single parameter, the transmissibility of the pathogen between neighbouring hosts. In the heterogeneous model, the transmissibility becomes a random variable drawn from a probability distribution. We investigate how heterogeneity in transmissibility influences the value of the invasion threshold, and find that the resilience of the system to invasion can be suitably described by two control parameters, the mean and variance of the transmissibility. We analyse a two-dimensional phase diagram, where the threshold is represented by a phase boundary separating an invasive regime in the high-mean, low-variance region from a non-invasive regime in the low-mean, high-variance region of the parameter space. We thus show that the percolation paradigm can be extended to the heterogeneous case. Our results have practical implications for the analysis of disease control strategies in realistic heterogeneous epidemic systems.

Epidemics spread in heterogeneous populations

“…The condition when the epidemic outbreak is observed can be deducted from transmissibility [11,12]. The transmissibility Ψ (P I , P R ) is a probability that an epidemic is transmitted from an infected individual to a susceptible one, i.e.…”

“…At the same time the connectivity among individuals is fixed and limited to the von Neuman neighborhood only. The model studied here is similar to ones considered in [11,12] and [14,15]. In contrast to [11,12], within the studied model, the time is discrete, resulting in the replacement of infection and recovery rates by appropriate probabilities per time step.…”

“…Finally, using percolation theory, it has been also shown that the increase of the average transmissibility increases the invasion probability. Contrary to the average transmissibility, an increase in the standard deviation of transmissibility reduces the invasion probability and decreases the epidemic size [11,12] showing again that heterogeneity can decrease epidemic size.…”

“…e.g. susceptibility or infectivity, follow continuous distributions [8][9][10][11][12][13]. Some of such heterogeneous models, by means of nonlinear transformations [9], can be mapped into homogeneous models.…”

Effects of local and global network connectivity on synergistic epidemics

“…This heterogeneity is annealed, i.e. it varies with time, as opposed to quenched heterogeneity which is well studied for epidemics [40][41][42][43][44][45][46][47], and its effect on the spread of epidemics is not obvious. In addition, transmissibilities of sufficiently close donor-recipient pairs have common nodes in their neighbourhoods and are not independent from each other if transmission is synergistic (since the infection history of close donor-recipient pairs neighbourhoods can overlap).…”