In this activity, we are trying to simulate an outbreak of a new infectious disease that our population of people has not been exposed to before. This means that we are starting with a single case, everyone else is susceptible to the disease, and no one is yet immune or recovered. This can for example reflect a situation where an infected person introduces a new disease into a geographically isolated population, like on an island, or even when an infections "spill over" from other animals into a human population. In terms of the initial conditions for our model, we can define:  and .

Remember, the differential equations for the simple SIR model look like this:

Initial number of people in each compartment
and

Parameters:

Remember that in a simple homogenous SIR model, is directly related to the proportion of the population that is susceptible:

The effective reproduction number is highest when everyone is susceptible: at the beginning,  = . At this point in our example, every infected cases causes an average of 4 secondary infections. Over the course of the epidemic,  declines in proportion to susceptibility.

The peak of the epidemic happens when  goes down to 1 (in the example here, after 50 days). As decreases further below 1, the epidemic prevalence goes into decline. This is exactly what you would expect, given your understanding of the meaning of  once the epidemic reaches the point where every infected case cannot cause at least one more infected case (that is, when  < 1), the epidemic cannot sustain itself and comes to an end.

Prevalence is simply the value of at a given point in time. Now we can see that the incidence is the number of new cases arriving in the  compartment in a given interval of time. The way we represent this mathematically is by taking the integral of new cases over a given duration.

For example, if we wanted to calculate the incidence from day 7 to 14,

Assign all the subsequent rows

Tabulate through the DocumentTools