The time between virus entry and the start of viral production (i.e., ecliptic phase) is widely distributed. However, so far, many mathematical models were constructed by ordinary differential equations (ODE) and implicitly assumed the ecliptic phase is distributed exponentially, although some mathematical models considered non-exponential distribution. This simplification of the ecliptic phase might miss several important features underlying virus infection dynamics. Therefore, multi-scale modeling which could include non-exponential distribution is important and recently has been receiving a lot of attention.

In this study, we were interested in quantifying and modeling the ecliptic phase during virus infection. One simple mathematical approach describing this phase is modeling by delay differential equations (DDE). However, unfortunately, the distribution of the ecliptic phase was poorly understood. Here, from cell culture experiments with simian/human immunodeficiency virus (SHIV), we quantified the distribution of the ecliptic phase and found that the phase obeys gamma distribution. Based on this experimental result, we made a mathematical model including the distribution by DDE and estimated viral parameters from previously published experimental data. Our analyses showed that modeling the ecliptic phase affected estimation of the virus infection rate (therefore, basic reproductive number) but not the virus production rate and the death rate of infected cells.