Quarantine Extension & Incubation Period Extension

H1N1 Flu Investigation

This document describes two extensions to the calculus investigation into the spread of the H1N1 Flu. The first of these extensions incorporates the quarantining of some or all flu victims. The second extension retains the quarantine modification but also takes into account an incubation period for the H1N1 flu, during which a victim is asymptomatic but is capable of spreading the disease.

Neither of these extended models will be solvable analytically. Therefore, after studying the models, you should use Euler’s method to see the effects of the models.

Before working on either of these extensions, you should already understand the “basic” model (also called the “SIR” model, for “Susceptible, Infected, and Recovered”) for the spread of the H1N1 flu, which is summarized below.

In this model the variables are:

· , the number of people susceptible to the flu,

· , the number of people infected with the flu,

· , the number of people who have recovered from the flu (and are now immune), and

· , the number of people who have died from the flu;

and the constants (parameters) are:

· , reflecting the rate at which the flu spreads from person to person,

· , reflecting the rate at which people recover from the flu, and

· , reflecting the proportion of flu victims who die from the disease.

The following sections develop the two new models by expanding on the basic model. Depending on your experience with mathematical modeling, you may find the development hard or easy to follow. If you find it hard to follow, you can still explore how these two models are different from the basic model and also how the new models’ parameters affect the solution. There are Excel files posted on the web along with this document that implement the models discussed below.

If you are able to follow the development of these models without too much difficulty (some difficulty is normal!), then you may want not only to explore the affects of the parameters, but also to enrich the models even further. At the end of this document are some suggestions for ways you might want to do that. You may have ideas of your own as well.

Quarantine Model

We’ll now enrich the basic model by assuming that some or all of the infected population can be quarantined. This means that they do not leave their home (or hospital room, or other quarantine location) and their contact with other people is kept to a minimum.

Let us suppose that it is possible to quarantine everyone who is infected. And let us suppose that every quarantined person comes into contact with 5 uninfected people. These are perhaps family members or nurses or others needed to attend to the sick person’s needs. We will say that these five people are “attending” the infected people and we will call them “attendants.”

If the number of infected people was very small, then the number of attendants would be , as illustrated by the graphic below, in which each infected person is represented by a solid circle and each uninfected person is represented by an open circle.

In this graphic, the number of infected people is so small relative to the population size that there is very little chance of anyone attending more than one infected person. Thus, if there are three infected people (as in the graphic) then there must be attendants.

However, if the number of infected people is fairly large, then it is possible that some of the attendants may in fact be attending more than one infected person. That is illustrated in the graphic below, in which the open circles enclosed in dotted lines represent attendants to more than one infected person.

In our “basic” model, it was possible for the disease to spread from any infected person to any uninfected person. Now, however, the disease can only spread from infected people to attendants. It is therefore important for us to count how many attendants there are. Furthermore, since attendants may be either susceptible or recovered (and therefore now immune) we will also want to count how many attendants are susceptible.

Counting the susceptible attendants

We will begin counting the susceptible attendants by noting that the attendants must come from the populations and : the susceptible and the recovered. They cannot come from either or (obviously) . Thus there are “candidate attendants.”

Let’s suppose that we pick a person at random from the group of candidate attendants. What is the probability that that person is in fact an attendant? If the number of infected people were very small then the probability would be , because there would be attendants in all, and candidate attendants. But what if the number of infected people is large, as in the second graphic on the previous page? In that case, would overcount the number of attendants.

To avoid counting attendants twice, we’ll consider instead the chance that a randomly selected candidate attendant is not attending any infected person.

If we consider any particular infected person, then the chance that a randomly selected candidate attendant is not one of his attendants is . And if we assume that being one person’s attendant is more or less independent from being any other person’s attendant[1], then the chance that a randomly selected candidate attendant is not the attendant of any infected person is .

Since being the attendant of at least one person is the complement (or “opposite”) of being no one’s attendant, then the probability of a randomly selected candidate attendant being an actual attendant to at least one person must be .

Finally, we are interested in counting the number of attendants who are susceptible. If we assume that a susceptible person and a recovered person are equally likely to be an attendant, then the above probability applies to any randomly selected person in the susceptible group, which contains people. Thus, the number of susceptible attendants must be:

Note that we have replaced the number five with a new parameter, , which represents the number of attendants required for each infected person.

Incorporating the quarantine into the basic model

The basic model included several components that will not change with the incorporation of quarantine, such as the recovery or death of infected people. The component that will change is the rate at which susceptible people become infected. In the basic model, this rate was . Since the quarantine model restricts the spread only to susceptible attendants, then we need to replace in that expression with the number of susceptible attendants that we derived in the last section. That leads us to the following quarantine model:

Partial quarantine

Next we consider the fact that it is generally not possible to quarantine every infected person. Let us suppose that in fact the proportion of infected people who are actually quarantined is . For example, if it were possible to quarantine 90% of all infected people, then .

There are two things in the quarantine model that we need to change to allow for the partial quarantine. First, recall that we modeled the probability that a randomly selected candidate attendant was not attending anyone as

. The exponent appeared because there were infected people and thus people who required attendants. But if only a fraction of those infected people are quarantined, then only a fraction of them will require attendants. The chance of being an attendant to no one is therefore , and the chance of being an attendant to someone must be . Multiplying this by will, as before, estimate the number of susceptible attendants.

Second, incorporating the partial quarantine requires us also to consider the fact that if the quarantine is partial, then there must be infected people who are exposed to everyone. The disease will spread from these people to the susceptible people in the usual way. There are infected people who are not quarantined, so the rate at which they are spreading the disease to the susceptible people will be .

This leads to the following partial quarantine model:

Incubation period

Now we will consider extending our model to incorporate an “incubation period” for the H1N1 flu. An incubation period refers to a period of time during which an infected person is asymptomatic but is able to spread the disease to others.

Since the asymptomatic infected people cannot be easily identified, they also cannot be quarantined. For this reason, we will need to divide the infected people into two groups: symptomatic and asymptomatic. We will let , where and represent, respectively, the numbers of symptomatic infected people and asymptomatic infected people.

There are three things we must do to incorporate the incubation period into our earlier partial quarantine model:

· We must be sure that only the symptomatic infected people are quarantined.

· We must include the asymptomatic infected people among those who can spread the virus to the general public.

· We must include a new differential equation that reflects the transition of people from the asymptomatic group to the symptomatic group.

The first two of these are accomplished by replacing with in the quarantine components of the model, and with in the non-quarantine components.

The third item can be accomplished simply by having people move from the group to the group at a constant rate, just as in the basic model they move from the group to the group and the group at constant rates. We’ll let be the rate at which asymptomatic people become symptomatic. That leads to the following partial quarantine and incubation model:

Explorations: part one

The explorations suggested in this section do not require you to thoroughly understand the development of the models on the previous pages.

Posted on the web along with this document are two Excel files that implement the partial quarantine model and the partial quarantine with incubation period model. The first of these includes two parameters that are not in the basic model:

· is the number of attendants who come into contact with each quarantined infected person.

· is the fraction of infected people who are quarantined.

The other includes a third parameter in addition to those two:

· is the rate at which asymptomatic people become symptomatic. Roughly speaking, is the fraction of asymptomatic people who become symptomatic each day.

Each of the following suggested explorations may be done by modifying the Excel document(s) and seeing how the resulting solution changes.

These are not questions with right and wrong answers, and you should feel liberated, not constrained by them. The purpose of exploration is not to find out the “right” answers to questions, but to discover relationships between mathematical models and their solutions, and perhaps also to test real-world conjectures about how to protect the general population from the spread of the H1N1 flu.

· Is quarantining at any level effective at preventing some people from ever becoming infected?

· If it is not the case that any will effectively protect some people from becoming infected, then is there a threshold level above which the quarantining will succeed in protecting some of the population from infection?

· If your answer to the previous question was “yes”, then how does that threshold relate to ?

· If you use the incubation period model, how does the threshold level (if there is one) related to ?

If you were a policy expert, would you recommend quarantining H1N1 victims? Perhaps only under certain circumstances?

Explorations: part two

The explorations suggested in this section require further modifications to the mathematical models presented in this document. The modifications are not necessarily complex, but they do require an understanding of the models and what their components mean.

· Masks may be used to decrease the rate of spread of infection (represented by in our models), but more effective masks are very expensive. Suppose that all attendants to quarantined people wear highly effective masks that reduced by 97%, while the rest of the population (or just some of them) wears cheaper masks that reduced by 50%. Will that reduce the number of people who eventually contract the flu? Are the cheaper masks useful at all? Is it worth it to try to get everyone to wear a mask, even if it reduces by only 50%?

· Suppose some fraction of the population can become “carriers”: people who are infected and can transmit the disease, but are permanently asymptomatic. Does the existence of such people alter any strategies you might otherwise implement to reduce the number of people who are eventually infected?

· The models described in this document assume that both susceptible and recovered people are equally likely to be attendants. Suppose that attendants are preferentially selected from among the recovered (and therefore immune) rather than the susceptible. Is that an effective way to protect the population? (You will want your model to allow for the fact that selecting attendants from among recovered people is harder when there are fewer recovered people than when there are more.)