Mathematical Epidemiology and More: A Course on Compartmental Models

Section 8.1 Complete Isolation: Models and Parameter Estimation

Activity 8.2.

Examine the two models described in the images and captions of Figure 8.1 and Figure 8.2. Each model is a possible way to adapt an SIR model, as first described in Chapter 4, to represent the concept of complete isolation. Pay attention to the compartments, arrows, and parameter value descriptions.

Use these two models to respond to the prompts below.

1. Compare and contrast the two models. Identify and describe any advantages or disadvantages you perceive for either model.
2. Write the system of differential equations for each model.
3. Suppose that an outbreak without isolation would use a standard SIR model having parameter values \(\beta = 0.005\) and \(\gamma = 1/10\text{.}\) Further suppose that for that same outbreak, with isolation, people isolate three days after becoming Infectious. What would be the values of \(\beta\text{,}\) \(\epsilon\text{,}\) and \(\eta\) for the model in Figure 8.1? And what would be the values of \(\beta\) and \(\widetilde{\gamma}\) in Figure 8.2?

Answer.

1. This is an open-ended question. Possible answers may include preferring the simplicity of the second model, or preferring the detail in the first model of tracking how many people are isolating rather than removed. Responses may also refer to preferences for computing \(\epsilon\) and \(\eta\) separately, or for computing only \(\widetilde{\gamma}\text{.}\) There are many additional possible responses.
2. For the model in Figure 8.1, here is the system of differential equations:
   \begin{align*} \frac{dS}{dt} \amp = -\beta S I \\ \frac{dI}{dt} \amp = \beta S I - \epsilon I \\ \frac{dJ}{dt} \amp = \epsilon I - \eta J \\ \frac{dR}{dt} \amp = \eta J. \end{align*}
   For the model in Figure 8.2, here is the system of differential equations:
   \begin{align*} \frac{dS}{dt} \amp = -\beta S I \\ \frac{dI}{dt} \amp = \beta S I - \widetilde{\gamma} I \\ \frac{dR}{dt} \amp = \widetilde{\gamma} I. \end{align*}
3. For both models, the value of \(\beta\) should not change.
   For the model in Figure 8.1, we notice that \(\gamma = 1/10\) indicates a ten-day contagious period for the disease. We split up this time into the Infectious and Isolated compartments. Since people stay in the Infectious compartment for three days, we set \(\epsilon = 1/3\text{,}\) using the ideas from (4.1), (4.2), (7.1), and (7.2). This leaves seven days for people to stay in the Isolated compartment, to add up to their total of ten days, and so we set \(\eta = 1/7\text{.}\)
   For the model in Figure 8.2, set \(\widetilde{\gamma} = 1/3\text{,}\) for a similar reason to setting \(\epsilon = 1/3\) for the model in Figure 8.1. This reduces the length of time people spend in the Infectious compartment, and we consider these people to be Removed for the rest of the time they are contagious.

1. When using the parameter-based formula for \(\mathcal{R}_0\) to compute the value of parameter \(\beta\) for an outbreak, start with a model that does not include intervention by large numbers of individuals or by an entire community. For instance, to model an outbreak using either of the strategies in Activity 8.2, start with a standard SIR model, shown in (4.3), (4.4), and (4.5), for computing \(\beta\text{.}\) The parameter-based formula corresponding to this model is (5.1).
2. The reason for starting with a model without intervention is this: \(\mathcal{R}_0\) is, as we learned, the average number of new infections caused by one infectious person in an otherwise entirely susceptible population. Interventions, such as vaccines or widespread isolation, change the ability of a disease to infect large numbers of people. Historical values of \(\mathcal{R}_0\) computed from past outbreaks do not include such interventions.
3. Once you have computed a biologically sound value or range of values for \(\beta\) using an intervention-free version of your model, along with that model’s parameter-based formula for \(\mathcal{R}_0\text{,}\) you can then incorporate isolation in your model. Possible models include those shown in Figure 8.1 and Figure 8.2. Do not change the value of \(\beta\) you have already computed, but do incorporate appropriate values of \(\widetilde{\gamma}\) or \(\epsilon\) and \(\eta\text{.}\) Computing parameter values in this way will allow your models to show the effects of isolation as intervention.
4. In Section 8.2, we will show other ways to model isolation. In those models too, estimate \(\beta\) first, in an appropriate corresponding model without intervention. Then adjust that model to include intervention, but do not change \(\beta\text{.}\)

Section 8.2 Incomplete Isolation, and Quarantine vs. Isolation

Activity 8.3.

Figure 8.3 and Figure 8.4 represent two possible models of incomplete isolation. Each expands on the SIR model first shown in Chapter 4. Think carefully through the two models, considering when one model or the other might be a good choice for modeling isolation.

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A compartmental diagram showing one possible approach to modeling incomplete isolation. In this model, all Infectious people pass through an Isolation compartment, labeled J, before moving to the Removed compartment. Notice that the arrow from compartment S to compartment I, representing transmission of the illness, depends on both I and J, in contrast to the transmission terms shown in Activity 8.2. The arrow from compartment S to compartment I is labeled with the formula \(\beta S(I+\delta J)\text{.}\) The arrow from compartment I to compartment J is labeled epsilon I. The arrow from compartment J to compartment R is labeled eta J.

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A compartmental diagram showing a second possible approach to modeling incomplete isolation. This model allows Infectious people to move either directly to the Removed compartment or first to an Isolation compartment, labeled J, and then to the Removed compartment. As in the first approach shown in this activity, transmission depends on both I and J. The arrow from compartment S to compartment I is labeled with the formula \(\beta S(I+\delta J)\text{.}\) The arrow from compartment I to compartment R is labeled gamma I. The arrow from compartment I to compartment J is labeled epsilon I. The arrow from compartment J to compartment R is labeled eta J.

Use the questions below to explain your thoughts about the models in Figure 8.3 and Figure 8.4.

1. Imagine an illness primarily affects children, who require hands-on care and may be near siblings or other people despite being sick. Suppose that everyone who gets this illness experiences symptoms, and the symptoms are distinctive in a way that makes it clear the person has this illness. (Measles is an example of this kind of illness.) In this example, suppose everyone tries to isolate when their symptoms appear, but given the nature of the illness, there are still interactions with some relatives or other caregivers. Which of the models in Figure 8.3 and Figure 8.4 might you prefer for representing this situation? Why is this your choice?
2. Suppose a residential campus experiences an outbreak of influenza. Some students experience high fevers, severe achiness, and other relatively extreme symptoms. These students rest and mostly isolate, but may have roommates or may need to interact with others while acquiring food or medicine. Other students have only a low fever and mild symptoms. These students continue with their daily routines, not resting much more than usual and not isolating. Which of the models in Figure 8.3 and Figure 8.4 might you prefer for representing this situation? Why is this your choice?
3. Discuss the transmission term \(\beta S (I + \delta J)\text{.}\) Explain why both \(I\) and \(J\) appear in this term. Then decide what values for \(\delta\) make the most sense. The role of \(\delta\) is to scale the effect of \(J\text{,}\) since people in the Isolated compartment typically have less chance to transmit illness than people in the Infectious compartment. This information should help you identify possible values for \(\delta\text{.}\)

Answer.

1. First and most importantly: there is not a single right answer to this question. Any answer that is well-reasoned works.
   The author of this text would consider using a model like the one shown in Figure 8.3 in this situation. The reason is that all or nearly all people with such an illness would isolate, so we do not have anyone moving from I to R without passing through compartment J. Since families or other caregivers are likely to interact with sick children, there is a nonzero chance of disease spread, and historically, such illnesses have definitely spread in this way.
   An isolation model may or may not be needed here. Using a model such as that in Figure 8.3 may be most useful when isolation is implemented in a notably expansive or thorough way. A modeler should think about how much isolation is happening, what the modeling goals are, and whether a simpler model, such as SIR, might provide enough information.
2. Again: there is not a single right answer to this question. Reasoning and justification are crucial to any response.
   One suggested answer is a model like that shown in Figure 8.4. This model provides a J compartment to represent the students who do their best to isolate. However, these students may interact with roommates or may receive food deliveries or briefly emerge from isolation to go get food or medicine. Such interactions give a small but nonzero chance for influenza transmission, and the \(\delta J\) in the transmission term, multiplied by \(\beta S\text{,}\) represents this chance. The model in Figure 8.4 also shows that some students do not isolate at all. These students stay in compartment I while they are contagious, and they move to compartment R only after their Infectious time period ends.
3. The transmission term can also be written \(\beta S I + \delta (\beta S J)\text{.}\) (Check the algebra to confirm this yourself.) The \(\beta S I\) portion is the same transmission term from our underlying SIR model. The other term has a similar form, \(\beta S J\text{,}\) indicating transmission from the J compartment to the S compartment, but it is multiplied by a scaling factor \(\delta\text{.}\) The value of \(\delta\) is typically between \(0\) and \(1\text{.}\) A value of \(\delta = 0\) would indicate no transmission from J to S, which was the case in our complete isolation model shown in Figure 8.1. A value of \(\delta = 1\) would indicate that transmission from the Isolation compartment is as likely as from the Infectious compartment, in which case we may have no reason to include an Isolation compartment. For values of \(\delta\) between \(0\) and \(1\text{,}\) the larger the value of \(\delta\text{,}\) the more likely it is for people in isolation to transmit the disease, yet it is still less likely than if they were not isolating at all. (In theory, \(\delta\) could be larger than \(1\text{,}\) meaning people in compartment J would be more likely to transmit disease. While this seems unlikely when J means isolation, there could perhaps be a different outbreak and a different model in which some group of people is more contagious than those in the regular Infectious compartment, or there could be a behavior-based reason why one group is more more likely to transmit an illness than others.)

Section 8.3 Practice with an Isolation Model