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Asymptomatic transmission of infectious diseases has been recognized recently in several epidemics or pandemics. There is a great need to incorporate asymptomatic transmissions into traditional modeling of infectious diseases and to study how asymptomatic transmissions shift epidemic dynamics. In this work, we propose a compartmental model with asymptomatic transmissions for waterborne infectious diseases. We conduct a detailed analysis and numerical study with shigellosis data. Two parameters, the proportion p of asymptomatic infected individuals and the proportion k of asymptomatic infectious individuals who can asymptomatically transmit diseases, play major rules in the epidemic dynamics. The basic reproduction number R0 is a decreasing function of parameter p when parameter k is smaller than a critical value while R0 is an increasing function of p when k is greater than the critical value. R0 is an increasing function of k for any value of p. When R0 passes through 1 as p or k varies, the dynamics of epidemics is shifted. If asymptomatic transmissions are not counted, R0 will be underestimated while the final size may be overestimated or underestimated. Our study provides a theoretical example for investigating other asymptomatic transmissions and useful information for public health measurements in waterborne infectious diseases.