Percolation on a spatial network with individual heterogeneity as a model for disease spread among animal host populations

Abstract

Recent efforts to directly or indirectly observe animal contact networks reflect an increasing awareness that the spread of infectious diseases, and their control, can be critically affected by the contact structure of the host population. This has long been realised for sexually-transmitted diseases of humans but has also recently been shown for seasonal influenza, where only casual contact, or indirect contact, is required for transmission. Many animals, though by no means all, are radically less mobile than humans and their social contact networks are consequently spatial networks where each host can be assigned a location in space and the rate of contact between two hosts depends on the distance between them. Such contact networks have received little attention and are poorly understood. Here we present a model for animal contact networks that allows for both spatial constraints and individual heterogeneity: let x i be the sociability of host i, let k ij be the rate of contact between hosts i and j, and let s ij be the Euclidean distance between i and j, then κ ij = λ 2e -λsijx ix j ; where λ determines the scale over which the spatial constraints operate. To study the transmission of an infectious agent on large scale realisations of these network models we use long-range percolation. That is, we relate the probability of an open, directed edge between i and j, given i is infected, to the contact rate between the two hosts by p ij = 1 - e -vκijτi, where v is the probability of actual transmission given contact occurs and τ i is the infectious period of host i. We define outbreaks in terms of spatial spread: occurring when the distance between a newly infected host and an initially infected host exceeds certain milestones. The preliminary results presented here show that for at least this model of contacts (i) the stochasticity in model behaviour is a combination of the spatial constraints and the variance of the xi rather than just the variance of the xi, and (ii) stronger spatial constraints on contact rates tends to nullify the effect of individual heterogeneity to increase pathogen spread. We would hence argue that tailored methods of network analysis are needed that can quantify the role of space in determining contact rates between animal hosts. At both a national and global scale, public health and veterinary authorities periodically face challenges from new pathogens that arise in wildlife and livestock populations. These can pose a threat to human health, such as in the case of recent and recurrent outbreaks of Hendra virus, or, in the case of the Devil Facial Tumour Disease, they may threaten native species. Our preparedness and ability to control such pathogens will be markedly improved as we better understand the contact networks of animal hosts, and the implications of their structure for the spread of infectious disease.