The front of the epidemic spread and first passage percolation

In this paper we establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert 2013 and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad, Hooghiemstra 2012, when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour Reinert 2013 from bounded degree graphs to general sparse random graphs with degrees having finite third moments as the number of vertices tends to infinity. We also study the epidemic trail between the source and typical vertices in the graph. This connection to first passage percolation can be also be used to study epidemic models with general contagious periods as in Barbour Reinert 2013 without bounded degree assumptions.Comment: 14 page

Weighted distances in scale-free preferential attachment models

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a non-negative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear in Random Structures and Algorithm

First passage percolation on the Newman-Watts small world model

The Newman-Watts model is given by taking a cycle graph of n vertices and then adding each possible edge $(i,j), |i-j|\neq 1 \mod n$ with probability $\rho/n$ for some $\rho>0$ constant. In this paper we add i.i.d. exponential edge weights to this graph, and investigate typical distances in the corresponding random metric space given by the least weight paths between vertices. We show that typical distances grow as $\frac1\lambda \log n$ for a $\lambda>0$ and determine the distribution of smaller order terms in terms of limits of branching process random variables. We prove that the number of edges along the shortest weight path follows a Central Limit Theorem, and show that in a corresponding epidemic spread model the fraction of infected vertices follows a deterministic curve with a random shift.Comment: 29 pages, 4 figure

Defining an epidemiological landscape that connects movement ecology to pathogen transmission and pace-of-life

Pathogen transmission depends on host density, mobility and contact. These components emerge from host and pathogen movements that themselves arise through interactions with the surrounding environment. The environment, the emergent host and pathogen movements, and the subsequent patterns of density, mobility and contact form an ‘epidemiological landscape’ connecting the environment to specific locations where transmissions occur. Conventionally, the epidemiological landscape has been described in terms of the geographical coordinates where hosts or pathogens are located. We advocate for an alternative approach that relates those locations to attributes of the local environment. Environmental descriptions can strengthen epidemiological forecasts by allowing for predictions even when local geographical data are not available. Environmental predictions are more accessible than ever thanks to new tools from movement ecology, and we introduce a ‘movement-pathogen pace of life’ heuristic to help identify aspects of movement that have the most influence on spatial epidemiology. By linking pathogen transmission directly to the environment, the epidemiological landscape offers an efficient path for using environmental information to inform models describing when and where transmission will occur

Explosiveness of Age-Dependent Branching Processes with Contagious and Incubation Periods

We study explosiveness of age-dependent branching processes describing the early stages of an epidemic-spread: both forward- and backward process are analysed. For the classical age-dependent branching process $(h,G)$, where the offspring has probability generating function $h$ and all individuals have life-lengths independently picked from a distribution $G$, we focus on the setting $h = h_{\alpha}^L$, with $L$ a function varying slowly at infinity and $\alpha \in (0,1)$. Here, $h^L_{\alpha}(s) = 1 - (1-s)^{\alpha} L(\frac{1}{1-s}),$ as $s \to 1$. For a fixed $G$, the process $(h^L_{\alpha},G)$ explodes either for all $\alpha \in (0,1)$ or for no $\alpha \in (0,1)$, regardless of $L$. Next, we add contagious periods to all individuals and let their offspring survive only if their life-length is smaller than the contagious period of their mother: a forward process. An explosive process $(h^L_{\alpha},G)$, as above, stays explosive when adding a non-zero contagious period. We extend this setting to backward processes with contagious periods. Further, we consider processes with incubation periods during which an individual has already contracted the disease but is not able yet to infect her acquaintances. We let these incubation periods follow a distribution $I$. In the forward process $(h^L_{\alpha},G,I)_{f}$, every individual possesses an incubation period and only her offspring with life-time larger than this period survives. In the backward process $(h^L_{\alpha},G,I)_{b}$, individuals survive only if their life-time exceeds their own incubation period. These two processes are the content of the third main result that we establish: under a mild condition on $G$ and $I$, explosiveness of both $(h,G)$ and $(h,I)$ is necessary and sufficient for processes $(h^L_{\alpha},G,I)_{f}$ and $(h^L_{\alpha},G,I)_{b}$ to explode.Comment: References adde