This textbook has focused, so far, on models of outbreaks. In each model, a disease has emerged, then peaked, then faded away. Yet many diseases remain in a population over a long period of time, passing from person to person in such a way that the disease remains in the population across many years. This chapter addresses such diseases.
Begin in Exploration 12.1 by using the skills you have been developing to think about how you might construct two different types of model showing endemic illness, that is, an illness that remains in a population long-term rather than peaking and fading.
Exploration12.1.A Disease that Remains Long-term in a Population.
Think through each of the following scenarios. Use your experience building compartmental models to develop a new compartmental model for each of the new scenarios described.
Assume a disease has Susceptible, Infectious, and Removed phases, and that people in the Removed compartment lose immunity over time. These people then go back to the Susceptible compartment. This ongoing renewal of the Susceptible population allows the disease to last beyond just an initial peak of infectious people. Draw a possible compartmental diagram and differential equations for such a model, and draw a graph sketching how the long-term population values might look. This exploration is not asking you to develop Python code to draw the graph, but instead to hand-draw your best guess.
Next, assume a disease has Susceptible, Infectious, and Removed phases, and that people in the Removed compartment do not lose immunity over time. Instead, new people enter the Susceptible compartment due to births or immigration, and people depart all three compartments due to deaths or emigration. Draw a possible compartmental diagram and differential equations for such a model, and draw a graph sketching how the long-term population values might look. This exploration is not asking you to develop Python code to draw the graph, but instead to hand-draw your best guess.
Section12.1Comparing Endemic Illnesses with Epidemics
This chapter focuses on models of endemic illness. By contrast, so far this semester, we have studied epidemic illnesses, also called epidemics or outbreaks. With epidemics, the Infectious population reaches its peak, drops to 0, and the epidemic ends. With endemic illness, the illness persists over time in the population.
A wording note: while we may refer to “an epidemic”, we never refer to “an endemic”. Instead, we can talk about “endemic illness” or “endemic disease”: the word “endemic” always describes something, but “endemic” does not refer to the event itself.
Consider the compartmental diagram shown in Figure 12.3, depicting an illness in which people in the Removed compartment can lose immunity and return to the Susceptible compartment. Use the questions below the diagram to think through the details of this model.
The compartmental diagram for an SIRS model, having compartments for populations of Susceptible, Infectious, and Removed or Recovered people. Arrows point from S to I, from I to R, and from R to S.
Sometimes the R compartment in a model includes only people who have recovered from illness, meaning the R could stand for Recovered. Alternatively, sometimes the R compartment includes recovered people as well as others, such as those who have died, in which case we typically say that R stands for Removed. Write a reasonable interpretation for the R compartment shown in Figure 12.3.
The formula \(\alpha R\) describes movement from the R compartment to the S compartment. The value of \(\alpha\) indicates how quickly people move from R to S.
In this model, people enter the S compartment from the R compartment, unlike in, say, the SIR model from Figure 4.4, where people can only depart the S compartment. This means the S population may decrease less quickly, or can remain at some nonzero population value, or can increase. These possibilities mean we may not run out of Susceptibles as a way of ending the illness, so it is possible for the Infectious compartment to never reduce to 0 people.
While we could still call the R compartment “Removed”, it may be more precise to say that R here means “Recovered”. This indicates that everyone in the R compartment has recovered from illness and is still alive, which is important when we say that people in the R compartment are able to move to the Susceptible compartment where they may be infected again.
Now consider the compartmental diagram shown in Figure 12.5, depicting an infectious illness in a population where people enter and exit the population. Use the questions below the diagram to think through the details of this model.
The compartmental diagram for an SIR model with demographics, having compartments for populations of Susceptible, Infectious, and Removed or Recovered people. Arrows point from S to I and from I to R. An arrow points into S from the outside, and arrows point out from all three compartments.
This model allows people to enter the population by births or immigration, and allows people to leave the population by deaths or emigration. Comment on what it may mean for the timespan of such a model, in comparison with the timespan of an outbreak model in which we leave out these options for people to enter or exit the population.
The formula \(\Pi\) indicates that people enter the S compartment at some constant rate. The formulas \(\mu S\text{,}\)\(\mu I\text{,}\) and \(\mu R\) indicate the people leave the S, I, and R compartments at a rate proportional to how many people are in those compartments. These formulas also show that the proportion of people leaving each compartment is the same proportion, rather than different proportions for different compartments.
In this model, people enter the S compartment due to births or immigration, unlike in, say, the SIR model from Figure 4.4, where people can only depart the S compartment. This means the S population may decrease less quickly, or can remain at some nonzero population value, or can increase. These possibilities mean we may not run out of Susceptibles as a way of ending the illness, so it is possible for the Infectious compartment to never reduce to 0 people.
While we could still call the R compartment “Removed”, it may be more precise to say that R here means “Recovered”. This indicates that everyone in the R compartment has recovered from illness and is still alive. This distinguishes people in the R compartment from those people who leave the R compartment through the formula \(\mu R\text{.}\)
