For a polycyclic epidemic we can use essentially the same model as for a monocyclic pathogen viewed over several seasons, where instead of repeating the cycle season after season, we have many repeated cycles within the same season. The time step becomes days or weeks instead of years, and since the time steps are no longer necessarily one time unit (years), the time increment is given as .

As a matter of notation, we will use a lower case to represent the quantity of inoculum during the epidemic and a lower case to represent the proportion by which inoculum increases in each time step. The units of correspond to the units of . For example, if the time is measured in days, the units of would be proportion/day.

Inoculum production actually tends to occur irregularly in discontinuous, discrete infection periods of differing lengths, depending on the weather, and the value of would likely be different for each infection period. However, in keeping with our objective of developing the simplest model possible to be useful as a management tool, we will simplify the above model by using uniform time steps and assuming a constant . (Instead of having a that varies according to the environmental conditions, we will use a value of "averaged" over the whole epidemic.) We will first rearrange the above equation to get:

The change in the amount of inoculum in one time step, , is simply the difference between the amount of inoculum at time and the amount of inoculum at time :

Rearranging we get:

Now instead of advancing time in discrete steps, we will advance time continuously, making infinitesimally small:

In this differential equation, is an infinitesimally small change in the quantity of inoculum, and is an infinitesimally small change in time. It tells us that the rate of change of the quantity of inoculum is proportional to the quantity of inoculum at any point in time. Using the calculus, this equation can be integrated to:

This, we see, is the familiar exponential function, where is the initial inoculum, and is the base of the natural (Napierian) logarithm. The instantaneous rate of change in is , the slope of the tangent to the curve at any point.