The following approximation was made to derive estimates of R(t). Supposing that the number of new infections at calendar time t is I(t), the transmission dynamics are described by the renewal equation using R(t) [48, 49] : 1898 1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 Year where ω(τ) is the probability density function of the generation time. The right-hand side of (1) represents secondary transmissions at calendar time t, which are determined by the number of those who were infected at time t − τ, I(t − τ), and the magnitude of secondary transmissions at time t, R(t). Since the data in the present study were recorded only monthly, the equation has to be simplified to comply with discrete points of time data. From the beginning of the history of mathematical modeling of smallpox in the late 19th century [50] , cases tended to be modeled by generation, the idea of which is applied as follows. Given the number of cases in generation i, I i , the expected number of cases in generation i + 1, E(I i+1 ) is given by