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

ABSTRACT

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.