A Discrete Model for the Evolution of Infection Prior to Symptom Onset

We consider a 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 four phases: non-infectious asymptomatic phase, 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 in discrete time where linear transitions are based on geometric (main model) or negative-binomial (enhanced model) probability distributions. The whole system is reduced to a single non-linear renewal equation. Moreover, after linearization, at least two meaningful 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 study the evolution of infection transmission before and after symptom onset. Provided that individuals can develop symptoms and die from the disease, we take disease-induced mortality as a measure of virulence and it is assumed to be positively correlated with a weighted average transmission rate. According to our findings, transmission rate of the infection is always higher in the symptomatic phase yet under a suitable condition, most of the infections take place prior to symptom onset.