Chapter 8 Isolation

Goals

Learn strategies for modeling isolation, that is, the separation of Infectious people from the rest of the population, in order to prevent them from spreading the disease to others
Revisit how to use a model’s parameter-based formula for \(\mathcal{R}_0\) when computing biologically-feasible values for the transmission parameter \(\beta\text{,}\) as this computation requires additional thought when isolation is part of the model
Consider that isolation may be complete or incomplete, and develop modeling methods for each scenario

Section 8.1 Introduction

This chapter functions, along with Chapter 6 and Chapter 7, as a sort of trilogy of methods for making models more realistic and for building public health interventions into models. As with vaccines, isolation of the infected is a public health approach used widely for the reduction of disease spread. As with incubation periods, isolation in a model can help that model to better fit data because the model more fully replicates what is happening in the world.

As we proceed through this chapter, we will consider complete isolation, when Infectious people have no contact at all with Susceptibles. We will also consider incomplete isolation, when Infectious people significantly reduce their contact with the general population but still have some chance of infecting Susceptibles. An example of incomplete isolation is staying home when sick, yet still interacting somewhat with roommates, family members, or others who share living space or deliver food. We will additionally spend time on thinking about how to use our parameter-based formula for \(\mathcal{R}_0\) in isolation models, and we will distinguish between the outbreak interventions of isolation and quarantine.

Exploration 8.1. What if People Choose to Isolate?

In this activity, use the knowledge you have been building about mathematical modeling to develop two possible mathematical models, one for complete isolation and one for incomplete isolation. We set these models in fictional small, remote towns, where there are not a lot of visitors to town or travel of town residents elsewhere. We therefore focus the model just on the town’s residents. We also focus this model on an outbreak of a fictional “Disease X”, for which \(\mathcal{R}_0 \approx 4\) and the Infectious period is approximately 8 days.

(a)

First, consider “Town A”, which has 6,000 residents, and imagine a scenario of complete isolation. That is, suppose that as soon as people know they are sick with Disease X, they isolate completely, having no contact with anyone else. Fortunately, Town A has found a way to deliver food and supplies, and to otherwise care for, people who are sick. Unfortunately, people are contagious with Disease X two days before they know they are sick, so they cannot isolate yet during those two days.

Write an SIR model for this outbreak, without (yet) including isolation. (Do not include vaccine or incubation period in your model.) Determine all parameter values for your model.
Adjust your SIR model to incorporate isolation, which in Town A is complete as soon as Infectious individuals know they are sick. Show both the differential equations and compartmental diagram for your adjusted model. Determine all parameter values for this adjusted model.

(b)

Next, consider “Town B”, which has 8,000 residents, and think of ways to model incomplete isolation. Suppose that Disease X appears in Town B, but Town B’s people do not have a way to care for or deliver supplies to people who are sick. This means that friends, family members, care workers, or others interact somewhat with the Infectious who know they are sick. Therefore, isolation is not complete, though individuals do reduce their interactions with others as soon as they know they are sick, that is, typically two days after first becoming contagious.

Adjust your original SIR model, which did not have isolation, to incorporate the incomplete isolation described for Town B. (Do not include vaccine or incubation period in your model.) Show both the differential equations and a compartmental diagram for your adjusted model. Determine all parameter values for this adjusted model.
Discuss the differences between your models for complete and incomplete isolation, and discuss the reasons for your modeling choices.

Section 8.2 Complete Isolation: Models and Parameter Estimation

Part (a) of Exploration 8.1 seeks a possible model of complete isolation, in which people truly and completely isolate for some portion of their contagious period. There are multiple ways to model this. Activity 8.2 describes two possible approaches for you to consider.

Activity 8.2.

Examine the two models described in the images and captions of Figure 8.3 and Figure 8.4. 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.

described in detail following the image — Figure 8.3. A compartmental diagram showing one possible approach to modeling complete isolation. Infectious people move to a new compartment, labeled J for Isolation, since I is already used for Infectious. Parameters \(\epsilon\) and \(\eta\) have values determined by how long people typically remain in each of the compartments I and J, respectively.

Use these two models to respond to the prompts below.

Compare and contrast the two models. Identify and describe any advantages or disadvantages you perceive for either model.
Write the system of differential equations for each model.
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 8.3? And what would be the values of \(\beta\) and \(\widetilde{\gamma}\) in Figure 8.4?

Answer.

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.
For the model in Figure 8.3, 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 8.4, 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*}
For both models, the value of \(\beta\) should not change.

For the model in Figure 8.3, 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 8.4, set \(\widetilde{\gamma} = 1/3\text{,}\) for a similar reason to setting \(\epsilon = 1/3\) for the model in Figure 8.3. 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.

The models in Activity 8.2 are two common ways to represent complete isolation. In both models, everyone isolates, and people in isolation do not spread disease at all.

Let us take a moment to discuss the parameter values computed for the models in Figure 8.3 and Figure 8.4. As with \(\gamma\) in Chapter 4 and \(\kappa\) in Chapter 7, the values for parameters \(\widetilde{\gamma}\text{,}\) \(\epsilon\text{,}\) and \(\eta\) are determined by the length of time spent in a compartment. Specifically, each of these parameters has a value that is the reciprocal of the time spent in that compartment, and we make sure the units of time match. For example, if we are using the model in Figure 8.3 and expect the typical person to isolate for seven days, and if our unit of time in the model is days, then \(\eta = 1/7\text{.}\) However, if we have a different unit of time, then we adjust, in a way similar to that shown in the first exercise in For Further Thought 7.3.

We also address the expectation that real-life biological information about a disease outbreak should correspond with the parameter-based formula for \(\mathcal{R}_0\) that we first studied in Chapter 5. Historical values of \(\mathcal{R}_0\) take into account that people may have interacted less with others while sick, but do not include widespread intervention, such as coordinated isolation campaigns. Here are steps you can take when determining all parameter values in a model with such intervention, with help from the \(\mathcal{R}_0\) formula.

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).
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.
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 8.3 and Figure 8.4. 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.
In Section 8.3, 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 8.3 Incomplete Isolation, and Quarantine vs. Isolation

Incomplete isolation can take multiple forms. One possibility is that everyone tries to isolate when they know they are sick, but they cannot completely isolate, for many possible reasons: they may share living space with people who are not sick; they may require care from someone who is not sick; they may not have any way to get food without at least some interaction with people who are not sick; and there could be yet other reasons. Another possibility is that some people are contagious but do not isolate. They may know they are sick and choose not to isolate, or with some illnesses, people can have no symptoms yet still be contagious. Think through these possibilities in the following activity.

Activity 8.5.

Figure 8.6 and Figure 8.7 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.

Use the questions below to explain your thoughts about the models in Figure 8.6 and Figure 8.7.

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 8.6 and Figure 8.7 might you prefer for representing this situation? Why is this your choice?
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 8.6 and Figure 8.7 might you prefer for representing this situation? Why is this your choice?
Discuss the transmission term \(\beta S (I + \delta J)\) in Figure 8.6. 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 \(\delta\text{.}\)
Discuss the transmission term \(\beta S (I_1 + I_2)\) in Figure 8.7. Explain why both \(I_1\) and \(I_2\) appear in this term, and state what form of incomplete isolation this model shows. Then consider including \(J\text{,}\) as we did in Figure 8.6. How might you change the transmission term in Figure 8.7 to include \(J\text{?}\) If \(J\) is included, then the model shows two forms of incomplete isolation, represented by compartment \(I_2\) and the transmission term involving \(J\text{.}\) Identify what the real-world interpretation is for each of these types of incomplete isolation.

Answer.

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 8.6 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 8.6 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.
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 8.7. 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 8.7 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.
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 8.3. 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.)
In this model, both \(I_1\) and \(I_2\) represent infectious people who are not isolating, so both compartments contribute to new infections. This model represents incomplete isolation because the population \(I_2\) does not isolate. Alternatively, we could create a model in which the people isolating in the J compartment are unable to fully isolate. In this case, we could write the transmission formula as \(\beta S (I_1 + I_2 + \delta J)\text{,}\) using our ideas about \(\delta\) from the previous part of this activity. Such a model contains two forms of incomplete isolation: the population \(I_2\) does not isolate at all, and the population \(J\) can still spread the disease, though perhaps to a lesser extent than the population \(I_2\text{.}\)

Notice that the interpretation of the Isolation compartment J is changes throughout the models in Activity 8.2 and Activity 8.5. This is an example showing why we should be careful, whenever building a model, to describe the meaning of each compartment, and to define precisely each population variable and each parameter.

You have now worked with two different models for complete isolation, in Activity 8.2, and two different models for incomplete isolation, in Activity 8.5. As you have been seeing in Chapter 6 on vaccines, Chapter 7 on incubation periods, and this chapter on isolation, there are a variety of ways to expand on an SIR model to address different aspects of an outbreak. As you move forward from here, more exercises and activities will give you freedom to choose how to build your models.

Before we move on to such an activity, let us take a moment to address a vocabulary topic that commonly occurs when studying isolation, namely the difference between isolation and quarantine. According to the U.S. Department of Health and Human Services, isolation “separates sick people with a contagious disease from people who are not sick”. By contrast, quarantine “separates and restricts the movement of people who were exposed to a contagious disease to see if they become sick. These people may have been exposed to a disease and do not know it, or they may have the disease but do not show symptoms.” It can be more challenging to enact quarantine, and to model quarantine, because we often have less information about which people have been exposed to a disease, in comparison to the information we have about who is actually sick. Seeing symptoms makes it easier to determine when people are sick. For this reason, we emphasize isolation models. However, it is possible to include quarantine in models, most often by designating a separate compartment for quarantined individuals.

As an additional note: the previous paragraph used the word “exposed” as it often appears in articles and websites about epidemiology. In the E compartment we introduced in Chapter 7, E stands for “Exposed-and-incubating”, which has a different meaning than just “exposed”. The term “exposed” means a person has been near someone who was contagious with an illness: such a person may or may not then get that illness themself. Our modeling term “Exposed-and-incubating” indicates that a person has indeed contracted the illness, but is not yet contagious themself, so has not moved into the Infectious compartment. It is important to keep reminding ourselves of this difference in meaning. (Beware: if you read published articles about mathematical epidemiology, authors often use the word “exposed” when they mean what we are calling “exposed-and-incubating”. This is common in the literature of mathematical epidemiology, and it will help you as a reader to be aware of the context.)

Section 8.4 Practice with an Isolation Model

We have now worked with both complete and incomplete isolation, as well as both SIR and SEIR models. In Activity 8.8, continue to build isolation models while also considering when you might start with the SIR or SEIR model. After, in For Further Thought 8.5, continue to compare SIR and SEIR models, while also making choices about what kind of isolation to model.

Activity 8.8.

Each academic year, many college campuses experience one or more outbreaks of influenza, also referred to as flu. According to the Centers for Disease Control and Prevention (CDC), flu symptoms may include fever, chills, cough, aches, runny or stuffy nose, and more. Also according to the CDC, influenza spreads via tiny droplets when people sneeze or cough or talk, as well as when people touch surfaces that have the virus on them and then touch parts of their face such as the mouth, nose, and eyes.

Search online for information about the incubation period of flu, how long flu is contagious, and typical \(\mathcal{R}_0\) values for past flu outbreaks. At the time of this writing, this ``About Influenza’’ page from the CDC and this article from the National Institutes of Health (NIH) are helpful sources. The NIH article distinguishes between the basic reproduction number \(\mathcal{R}_0\) and the effective reproduction number, which the NIH article designates as \(\mathcal{R}\text{.}\) Similar to our work with \(\mathcal{R}_{\text{eff}}\) in Chapter 6, \(\mathcal{R}\) refers to the average number of new infections caused by a single infectious individual in a population that has some underlying level of immunity.

Use the idea of a campus outbreak to think about (at least) two ways to model isolation, guided by the prompts below.

To model the spread of flu, choose either the SIR model as described in Chapter 4 or the SEIR model as described in Chapter 7. Use the online facts you have gathered to decide whether or not to include an incubation period in your model. Be sure to explain your reasoning. (There is not a right or wrong answer here: either choice can make sense, yet it is important to justify your decision using facts about the flu.) Provide the range of parameter values for each parameter in your model, and cite your sources. Remember to use a model without isolation, along with the basic reproduction number \(\mathcal{R}_0\text{,}\) when computing \(\beta\) via formula (5.1) or (7.3). Then, in the steps that follow, use this value of \(\beta\) when building a mathematical model that includes isolation.
In many cases, people with the flu know they are sick and have the opportunity to isolate. Let us suppose that every person in an outbreak isolates after they have been sick for one day, and that they isolate perfectly. (This is unlikely in reality, but we can model this to see the outcome.) Update your model from step 1 accordingly, changing only the value of \(\gamma\) (which relates to the time spent in the \(I(t)\) population) so that people move more rapidly from compartment I to compartment R.
Try your model from step 2 in Python, using either the Python code for an SIR model, as in Activity 5.5, or the Python code for an SEIR model, as in Activity 7.6. Suppose there are 2500 students on campus, and update all parameters and time ranges in the model as needed. Feel free to try just one initial Infectious case, or a few (perhaps 2-10) initial Infectious cases, as it is possible for multiple people to become Infectious simultaneously in an influenza outbreak, perhaps because they all encountered the same Infectious person off-campus, or perhaps because some social event led to multiple initial infections within a very short time frame.

First try your baseline model (from step 1) for influenza without isolation, and then try your influenza model with its updated \(\gamma\) value (from step 2) to represent the complete isolation of Infectious people beginning one day after symptoms appear. Describe your results.

Answer.

There truly is no right or wrong decision here. Influenza is often modeled using the SIR model because the incubation period can be relatively short: the linked CDC page says influenza is most contagious for three days. However, the incubation period is not zero, and may be important to include when modeling a real campus outbreak and trying to estimate the timeline of the outbreak’s trajectory, and the CDC page also tells us that influenza can be contagious from one day before symptoms appear until five to seven days after symptoms appear, meaning a total of six to eight days. In summary: either model is a reasonable choice, and the decision rests on the modeler’s goals and the questions being asked when modeling.

The contagion numbers listed above suggest that, without isolation, we may choose from a range of possible \(\gamma\) values for our model, with \(1/8 \leq \gamma \leq 1/3\text{.}\) The Virginia Department of Health tells us that \(1 \leq \mathcal{R}_0 \leq 2\text{.}\) Putting these ranges of values and our population size of 2500 into our formula for \(\mathcal{R}_0\) tells us that \(0.00005 \leq \beta \leq 0.000267\text{.}\) (Be sure to confirm these numbers, using the correct \(\mathcal{R}_0\) for the model you are using, and showing with \(\gamma\) and \(\mathcal{R}_0\) values used to compute the least and greatest values for \(\beta\text{.}\))
If everyone knows they are sick, and everyone isolates perfectly after one day of illness, then we should set \(1/2 \leq \gamma \leq 1/1\) or, more succinctly, \(1/2 \leq \gamma \leq 1\text{,}\) for a complete isolation model in which the time unit is days. The range of \(\gamma\) values is due to the possibility that flu can be contagious one day before symptoms begin, which would mean \(\gamma = 1/2\text{,}\) or the possibility that contagion can begin close to when symptoms begin, which would mean \(\gamma = 1\text{.}\)
This is open-ended. Explore in Python, and make as many observations as you can.

For Further Thought 8.5 For Further Thought

1.

Revisit Activity 8.8, starting with an SIR model of the type described in Chapter 4 and introducing incomplete isolation. Incomplete isolation can take multiple forms; for this exercise, use one of the forms described in Section 8.3.

Describe the kind of incomplete isolation you are including in your model. This requires partially rewriting the fictional outbreak in Activity 8.8: perhaps people do not always know if they have the flu, or they do not isolate perfectly, or there may be other reasons for incomplete isolation. Construct the compartmental diagram and differential equations you will use to represent your described form of incomplete isolation.
Determine all parameter values or ranges for your model. Cite official sources wherever possible, including links to any web pages, the title of each web page used, the name of the publisher of each web page, and the date you most recently accessed the web page. Since you are deciding the kind of incomplete isolation, and since this is a fictional outbreak, there may be one or more parameter values that are based on your decisions and not on official sources. Indicate this wherever it is relevant.
Use Python to show results of your SIR model, with and without incomplete isolation. Describe at least three ways the model changes when incomplete isolation is included, in comparison to when incomplete isolation is not included.

2.

Revisit Activity 8.8, starting with an SEIR model of the form described in Chapter 7 and introducing incomplete isolation. Incomplete isolation can take multiple forms; for this exercise, use one of the forms described in Section 8.3. If you are also completing exercise 1 in For Further Thought 8.5, use the same form of incomplete isolation.

Describe the kind of incomplete isolation you are including in your model. This requires partially rewriting the fictional outbreak in Activity 8.8: perhaps people do not always know if they have the flu, or they do not isolate perfectly, or there may be other reasons for incomplete isolation. Construct the compartmental diagram and differential equations you will use to represent your described form of incomplete isolation.
Determine all parameter values or ranges for your model. Cite official sources wherever possible, including links to any web pages, the title of each web page used, the name of the publisher of each web page, and the date you most recently accessed the web page. Since you are deciding the kind of incomplete isolation, and since this is a fictional outbreak, there may be one or more parameter values that are based on your decisions and not on official sources. Indicate this wherever it is relevant.
Use Python to show results of your SEIR model, with and without incomplete isolation. Describe at least three ways the model changes when incomplete isolation is included, in comparison to when incomplete isolation is not included.
If you also completed exercise 1 in For Further Thought 8.5, discuss the differences between using an SIR model with isolation and using an SEIR model with isolation. As part of your discussion, provide at least one advantage for using the SIR model, and at least one advantage for using the SEIR model.

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