The basic reproduction number R_{0} is a standard indicator of infection control in epidemiology. Although R_{0} has been studied extensively for deterministic epidemic models, it has not been formulated accurately for models adopting network structures. Here, we extend a four-compartment model that includes commonly used epidemic models to a Markov process on networks. By examining the Markov process in detail, we derive a canonical formula for R_{0} involving two probability values. Numerical calculations show that the derived formula is a better approximation than the conventional formula when the network is very sparse. We propose this as a standard formula for controlling infections that can only be transmitted through intimate contact, where contacts between individuals can be described as a sparse network.