Stochastic and spatio-temporal analysis of the Middle East Respiratory Syndrome outbreak in South Korea, 2015

Lee, Hyunsun

where bðtÞ is the average number of newly infected individuals whom one infectious patient will produce per unit time when infected for total time t, and FðtÞ is the survival function, i.e., the probability that a newly infected individual remains infectious for at least time t. The function bðtÞFðtÞ is called the reproduction function. This approach allows us to use a time dependent bðtÞ instead of a fixed constant rate b. The stochastic branching process model is often used to describe the beginning stage of a disease outbreak when the number of infected patients remains significantly smaller than the entire population. The network of individual contacts and transmission is more focused in this approach, and a network diagram of patients, consisting of two major objects: vertices and edges, is often used to visualize it. Vertices represent infected patients in our study, and edges are contacts between the members that cause transmission of the disease. We assume that every contact through each edge leads to an infection. The orientation of transmission always flows from a generation to a higher generation but never flows reversely, which means that the previous generation cannot be infected again by any subsequent generation. The degree of a vertex is the number of edges connected to the vertex in the network. For a random node with a vertex degree k, the excess degree of the vertex is defined as the number of contacts infected by the current vertex. Since the current vertex has the k connected neighbor vertices including the source node who infected the disease to the current node, and a patient cannot be infected again from its descendants as we assumed earlier, then the excess degree of the current node is ðk À 1Þ. The distribution of vertex degree is essential in the description of disease transmission since it determines how fast or slow the disease spreads or dies out.