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

Lee, Hyunsun

Early contributions to infectious disease modeling were pioneered mostly by public health physicians. The first known result in mathematical epidemiology was to appraise the effectiveness of a controversial inoculation against smallpox in 1760 by a mathematician, Daniel Bernoulli (1766) . In 1889, P. D. En'ko contributed to the development of theoretical basis of epidemiology (Dietz, 1988) , correlating a discrete time model with actual cases of a measles epidemic, and Hamer developed a discrete time model to understand the spread of the measles epidemic in 1906 (Hamer, 1906) . Ross assumed that the rate of a new infection was proportional to the numbers of susceptible and infectious individuals, and developed the ordinary differential equation models of vector-borne diseases to first model malaria in 1911 (Ross, 1911) . However, one of the early triumphs in mathematical epidemiology was the formulation of a simple model by Kermack and McKendrick in 1927 (Kermack & McKendrick, 1932; McKendrick, 1926) . Their predictions explained the behavior of countless epidemics, including one of the worst epidemics in history, known as the Great Plague of London, that dicimated more than 15% of London's population in 1665e1666. The Kermack and McKendrick epidemic model is a standard SIR model, consisting of three compartments that are Susceptible (S), Infected (I) and Recovered (R) , and assumes that the sizes of the compartments are large enough so that each compartment is assumed to be homogeneous, or at least there is homogeneous mixing in each subgroup if the population is stratified by activity levels. The compartment model has been often used, modified and applied to different epidemic models (Brauer & Castillo-Chavez, 2011; Hethcote, 2000; Keeling & Rohani, 2008) , as well as other phenomena in social networking, viral marketing, sensor networking, etc. However, at the beginning stage of a disease outbreak, there are a relatively small number of infected individuals, and the transmission of the infection is stochastic rather than homogeneous where the individual contacts between members of the population are distinguishable and traceable. To describe the early stage of an epidemic, stochastic branching process, first formulated by Galton and Watson in 1874, was used to model the reproduction of a population from generation to generation. It was also used to study the extinction of family names by Galton and Watson. Gets et al. (Lloyd-Smith and SchreiberGetz, 2006) introduced a natural generalization of the basic reproductive ratio that was critical in controlling disease outbreak. In the stochastic branching process model, the major interest is placed on finding the probability that the disease eventually dies out, i.e., the probability of ultimate extinction, during the transmission.