We consider a discrete-time between-host model for a single epidemic outbreak of an infectious disease. According to the progression of the disease, hosts are classified in regard to the pathogen load. Specifically, we are assuming the following phases: non-infectious asymptomatic phase, \textbf{infectious asymptomatic phase} (key-feature of the model where individuals show up mild or no symptoms), infectious symptomatic phase and finally an immune phase.
The system takes the form of a non-linear Markov chain assuming that transitions are based on geometric or negative-binomial probability distributions. The whole system is reduced to a single non-linear renewal equation. Moreover, after linearization, two natural definitions of the Basic Reproduction number arise: firstly as the expected secondary asymptomatic cases produced by an asymptomatic primary case, and secondly as the expected number of symptomatic individuals that a symptomatic individual will produce. We define a measure of virulence as the disease-induced mortality, provided that individuals can develop symptoms, and we assume to be correlated with a weighted average transmission rate. We have found that transmission rate is higher in the symptomatic phase, but however the accumulated number of infections is higher in the asymptomatic phase.