Put option is a contract to sell some underlying assets in the future with a certain price. On European put options, selling only can be exercised at maturity date. Behavior of European put options price can be modeled by using the Black-Scholes model which provide an analytical solution. Numerical approximation such as binomial tree, explicit and implicit finite difference methods also can be used to solve Black-Scholes model. Some numerical methods are applied and compared with the analytical solution to determine the best numerical method. The results show that numerical approximations using the binomial tree is more accurate than explicit and implicit finite difference method in pricing European put options. Moreover when the value of T is higher then the error obtained is also higher, while the error obtained is lower when the value of N is higher. The value of T and N cause the increase of the computation time. When the value of T is higher the computation time is lower, while computation time is higher if the value of N is higher. Overall, the lowest computation time is obtained by using an explicit finite difference method with an exceptional as the value of T is big and the value of N is small. The lowest computation time is obtained by using a binomial tree method.

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The spread of infectious disease in a heterogeneous area can be grouped as a homogeneous group. The graph theory approach to analyze the spread of infectious disease in the group using a mathematical model. Heterogeneity in a population can be caused by many factors. Within a group can be divided into several homogeneous groups based on clusteritation, such as grouping the population based on age in the spread of infectious diseases. Population heterogeneity can be described as a network where each vertex represents a homogeneous group and an edge (j, i) exists if and only if the disease can be transmitted from group i to group j. The system of mathematical differential equations is formed based on the graph theory approach and the infectios disease distribution compartment diagram. Based on the numerical solution that we have obtained, the rate of change in the exposed population increases as the increasing of the disease transmission. And the rate of change in the infected population increases as the endemic appears.