Within-Host and Between-Hosts Epidemic Dynamics: A Journey with Alberto

In the past 20 years several authors have analysed models in which within-host infection dynamics is coupled to between-host transmission. Frequently in such models within-host dynamics was used to build appropriate functions appearing in epidemic models in order to study evolutionary problems (Gilchrist and Sasaki, J. Theor. Biol. 218:289–308, 2002), or to better describe the dynamics of immune boosting and waning (Barbarossa and Röst, J. Math. Biol. 71:1737–1770, 2015). Alberto Gandolfi, Carmela Sinisgalli and I worked at a model (Gandolfi et al., J. Math. Biol. 70:399–435, 2015) in which epidemic dynamics could affect within-host dynamics because the initial viral inoculum depended on the state of the infector, and thus on the overall state of the epidemics; this two-way interaction yields new dynamical patterns, with subsequent epidemic waves in absence of external perturbations. A shortcoming of that model is the fact that once a host is infected, s/he cannot be reinfected at any stage; Alberto was aware of the problem, and had prepared a manuscript draft in which reinfections were allowed at the beginning of an infection, where they can be relevant for the dynamics. Unfortunately, we were not able to complete the work before Alberto’s premature death. Here I show a simple model that takes account the effect of multiple infectious contacts through a system of ODEs with unexpected behaviour, thus showing the potential relevance of the theoretical models analysed with Alberto.