Epidemic Spread on Weighted Networks

Citation: Kamp C, Moslonka-Lefebvre M, Alizon S ( Epidemic Spread on Weighted Networks Christel Kamp 0 Mathieu Moslonka-Lefebvre 0 Samuel Alizon 0 Christophe Fraser, Imperial College London, United Kingdom 0 1 Paul-Ehrlich-Institut, Federal Institute for Vaccines and Biomedicines , Langen, Germany, 2 INRA , UR 0341 Mathe matiques et Informatique Applique es, Jouy-en-Josas, France , 3 AgroParisTech, F-75005 Paris, France, 4 Laboratoire MIVEGEC (UMR CNRS 5290, IRD 224, UM1, UM2), Montpellier , France The contact structure between hosts shapes disease spread. Most network-based models used in epidemiology tend to ignore heterogeneity in the weighting of contacts between two individuals. However, this assumption is known to be at odds with the data for many networks (e.g. sexual contact networks) and to have a critical influence on epidemics' behavior. One of the reasons why models usually ignore heterogeneity in transmission is that we currently lack tools to analyze weighted networks, such that most studies rely on numerical simulations. Here, we present a novel framework to estimate key epidemiological variables, such as the rate of early epidemic expansion (r0) and the basic reproductive ratio (R0), from joint probability distributions of number of partners (contacts) and number of interaction events through which contacts are weighted. These distributions are much easier to infer than the exact shape of the network, which makes the approach widely applicable. The framework also allows for a derivation of the full time course of epidemic prevalence and contact behaviour, which we validate with numerical simulations on networks. Overall, incorporating more realistic contact networks into epidemiological models can improve our understanding of the emergence and spread of infectious diseases. - Contact structure between hosts is known to have a key influence on disease spread [1]. A striking result is for instance that the more heterogeneous the contact network is, i.e. the higher the variance in the number of contacts per individual, the more rapid the initial disease spread. One way to capture contact structure is to use a network [2]. Such contact networks are typically described by a square binary adjacency matrix, where each term on the ith line and jth column can take the value 0 or 1 to indicate respectively the absence or the presence of a contact between individuals i and j. Contact networks are widely used because they possess several convenient properties, one of which being that the dominant eigenvalue of the adjacency matrix is an indicator of the initial propagation speed of an infectious disease spreading on this network [3,4]. The main limitation of contact networks is that their exact shape is often difficult to infer. This is why there is a continuous effort to predict disease spread from network summary statistics that are easier to estimate, such as the distribution of the number of contacts (degrees). For instance, the number of secondary infections generated by a typical infected host in a fully susceptible population, i.e. the basic reproductive number R0 [1], scales with the ratio of the second moment Sk2T and first moment (mean) SkT of the distribution in the number of contacts k. This result holds both for static networks (denoted Rstat) [5] as well as for fully 0 mixed, dynamic networks (denoted R0mix) [6,7] with where s2k~Sk2T{SkT2 is the variance of the distribution of the number of contacts. The static case corresponds to networks in which the identity of contacts is fixed (as approximatively seen in sexual contact networks) and the fully mixed dynamic case corresponds to a situation in which individuals update their contacts dynamically in a fully mixed fashion within the population (as approximatively seen in airborne infections). Rstat and Rmix represent the lower and upper bounds of the 0 0 basic reproductive ratio [8] for SIR epidemics on random networks if individuals transmit the infection at a rate b and recover from the infection at a rate c. On both static and dynamic heterogeneous networks with a large or even diverging variance in the distribution of the number of contacts, epidemics die out only for very small or even vanishing transmission rates b. One of the typical key assumptions epidemiological models on networks make to obtain such elegant expressions for R0 is that the transmission rate is the same between all pairs of individuals. This is materialized by the fact that all the edges of the contact matrix have a weight of 0 or 1. This is known to be a simplifying assumption [9]. A well-studied example related to infectious diseases is that of sexual contact networks, where the number of sex acts per unit of time is not constant in all partnerships [1012]. More generally, the number of interaction events (which correspond to potential transmission events) may vary among contact pairs and is likely to decrease with the number of contacts Understanding how infectious diseases spread has public health and ecological implications. The contact structure between hosts strongly affects this spread. However, most studies assume that all types of contacts are identical, when in reality some individuals interact more strongly than others. This is particularly striking for sexual-contact networks, where the number of sex acts is not identical for all partnerships. This heterogeneity in activity can either speed up or slow down epidemic spread depending on how strongly the individuals number of contacts coincides with their activity. There are two limitations to current frameworks that can explain the lack of studies on weighted networks. First, analytical results are difficult to obtain, which requires numerical simulations. Second, inferring weighted networks from survey data is extremely difficult. Here, we present a novel framework that allows to alleviate these two limitations. Building on configuration type network epidemic approaches, we manage to capture disease spread on weighted networks from the distribution of the number of contacts and distribution of the number of interaction events (e.g. sex acts). This allows us to derive analytical estimates for the epidemic threshold and the rate of spread of the disease. It also allows us to readily incorporate survey data, as illustrated in this study with data from the National Survey of Sexual Attitudes and Lifestyles (NATSAL) carried out in the UK. an individual has (see also Figure S3). Simplifying the reality is commendable but the problem is that tampering with the weighting of the network has been shown to lead to the loss of important epidemiological properties of heterogeneous unweighted networks, such as the low value of the epidemiological threshold or the negative correlation between the epidemiological threshold value and network size [13]. To summarize, although contact networks appear to be scale free in structure, they might not exhibit the properties o (...truncated)