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7

Infectious Disease

7.1 Introduction

Infectious diseases have had major impacts on the human condition through- out history. In the mid-fourteenth century, the plague epidemic called “the Black Death” killed about one-third of Europe’s population. European colonists brought diseases to the Americas in the sixteenth century, which killed more Native Americans than guns did. In 1918, a pandemic flu killed about 20 million people, more than were killed in all of World War I. Since 1981, about 40 million people have died of AIDS worldwide.

These numbers show the significance of infectious diseases. Although medical innovations have reduced the impact of some infectious diseases, viruses have been known to gain resistance to vaccines, rendering such in- novations ineffectual. A complementary approach to slowing the spread of infectious disease is the use of mathematical models to understand how dis- eases spread through a population and what the potential impact of various types of interventions could be.

If a new pandemic influenza breaks out would it be best to prohibit international travel, or close all schools, or quarantine people with influenza? When do we know an intervention has sufficiently reduced the spread of the disease? If we have a limited number of vaccines, who do we need to vaccinate to reduce the spread of the disease in the entire population?

In this chapter, we discuss another non-linear population model: a model of infectious diseases spreading within a population. Infectious diseases such as influenza, HIV or measles spread from person to person within the pop- ulation. Models are developed to describe the spread of such diseases and can be used to find effective ways to slow or stop the spread. We present the basic SIR model, referring to susceptible, infectious and recovered subpop- ulations, and calculate the so-called basic reproduction number R0, which is the indicator of whether a disease is epidemic or not.

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86 7. INFECTIOUS DISEASE

Figure 7.1: Asian women trying to protect themselves from getting infected with influenza by wearing masks.

7.2 SIR model

When we develop a model of infectious diseases, we essentially model a structured population. Like chapter 8, we will consider different classes of the population. But now it is not age that makes the difference between those classes; it is the different stages of a disease a person can be in. A person can be healthy and free of the disease. But if the person is not immune to the disease or has not been vaccinated, she is susceptible to getting the disease if she comes in contact with an infected person. Infectious persons are people who are ill and can spread the disease. The third category consists of individuals who are recovered from the disease: they cannot become sick anymore in our basic model.

Of course, different diseases progress differently. For some diseases, say, a cold, you become susceptible again after you recover. Other diseases have major age-related differences. Malaria has a much higher mortality rate among young children, and thus when modeling malaria we have to take that into account. A third difference is that there might be more stages of the disease. For example, a person who does not feel sick yet may already be able to infect others. In this chapter, we will focus only on the basic SIR model, whose flow diagram is shown Figure 7.2.

Figure 7.2: Flow diagram of SIR model.

7.2. SIR MODEL 87

We will now start to construct the model more formally. Consider a population of N individuals. Let us ignore for the moment new births and deaths or migration. We can classify the individuals within the population into three categories:

• St: the susceptible class, which are individuals who may catch the disease but are currently not infected.

• It: the infectious class, which are individuals who are infected with the disease and are currently contagious.

• Rt: The removed class, which are individuals who cannot get the disease, because they either have recovered permanently, are naturally immune, or have died.

The sum of St + It + Rt = N for all t (because we’re ignoring the birth, death, and migration processes for now).

The SIR classification above is a simplification of many real diseases. In many diseases, we may have to include more classes, such as age classes, or classes where individuals are infectious but not ill, or have different stages of the disease. For our purposes, the basic SIR is sufficient.

In the basic SIR, model we assume that individuals are randomly mixing. This means that every individual has equal probability to interact with another individual and be exposed to the disease. This assumption is not always realistic; for various diseases the structure of social networks affect the spread of the disease, a topic that is beyond the scope of the course.

An outbreak of chickenpox at an elementary school is a good example that can be described well by the SIR model. Students get chickenpox by encountering others with the disease. Once they have gotten the disease and recover, they will be permanently immune to the disease. Also the seasonal influenza, the flu that typically appears in winter time, is a good example of a disease approximated by the SIR model. What is the reason that influenza appears in winter months and not during the whole year?

How will a disease spread through a population? The more a susceptible individual encounters other individuals who have the disease, the more likely he/she would be to get infected. Dependent on the disease, infection can be spread through contaminated door-knobs, handshakes, coughs or shared coffee mugs. Assuming random mixing of the individuals, we can say that the higher the product StIt, the more individuals will get infected. (Notice the similarity of this formulation to that for the predation term in the predator- prey model.) This means that with the same number of susceptible people, the more infected people there are, the faster the spread of the disease will be; similarly, with the same number of infected people, the more susceptible people there are, the faster the spread of the disease will be.

Since not all random contacts have the same effect with every disease, we also include a parameter α, the transmission coefficient, as a measure of the

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likelihood that a contact between a susceptible and an infective will result in a new infection. For example, α is high if contact with a contaminated door- knob can spread the disease and low if blood-to-blood contact is required for the disease to spread. Different diseases have different values of α.

We can now model the number of susceptible individuals over time. In the following difference equation, we capture how the number of susceptible people changes because a number of them become infected:

St+1 = St −αStIt (7.1)

The number of susceptible people getting infected, αStIt, will then be added to the number of infective people. At the same time, some infected individuals may recover or die. The removal rate γ measures the fraction of individuals in the infectious class that stops being infectious, and thus moves into the removed class, at every time step. The removal rate differs between diseases. For some diseases, γ is high since people are infectious for only a few days, like a cold. The parameter γ is called the removal rate and can be defined as 1 divided by the average number of days somebody is infectious. Other diseases may leave people infectious for months or years, such as TB or HIV. Putting these elements together, we arrive at the following difference equation for the number of infected people:

It+1 = It + αStIt −γIt (7.2)

The removed class increases in size every time step by the number of individuals who are removed/recovered from the infectious class. We now can define the final difference equation for the removed/recovered class:

Rt+1 = Rt + γIt (7.3)

Before we proceed, let us pause and consider the unit of the transmission coefficient α. For this, let us consider the second difference equation above: It+1 = It + αStIt − γIt. First, note that all three key variables, namely St,It, and Rt, have the same unit of “person” or “individual.” For any two quantities to be added to or subtracted from each other, they must be in the same unit. Therefore, the infection term αStIt also has the unit of “person.” For this to be true, α must have the unit of (1/person).

Now we can start exploring the model. Suppose we have the following situation: there is a population of 100,000 individuals, and 100 individuals got infected. Furthermore, we assume that the rest of the individuals are susceptible. The transmission coefficient is 0.00001 per person and once infected the person is contagious for 10 days on average. This means that α = 0.00001 and γ = 0.1. When we simulate the model in Excel, we get Figure 10.3. To implement this in Excel we need to distinguish 3 columns. Implement the model in Excel and make sure that you can reproduce the

7.3. R0, BASIC REPRODUCTION NUMBER 89

Figure 7.3: Time plots of the numbers of susceptive, infectious and recovered within a population.

results as depicted in Figure 7.3. (Please revisit Chapter 9 for implementing a model in Excel if need be.)

We assumed that 100 people are infected. What would happen if there is no infectious person in the population? Nothing. Hence, you see that this model is mainly focused on the spread of the disease, not how the disease enters the population.

7.3 R0, Basic Reproduction Number In this section, we introduce the basic reproduction number, denoted by R0, pronounced as r naught, which is one of the key metrics in evaluating infectious diseases. The value R0 quantifies how many new cases on average an infected person generates during his infectious period. Knowing this number for a disease helps to define how many people need to be vaccinated in order to slow down an epidemic.

To define R0, we revisit the SIR model and write Equations 7.1, 7.2, and 7.3 in terms of the change of the number of people in the different classes, namely ∆S, ∆I, and ∆R. Recall, for example, that ∆S = St+1−St.

∆S = −αStIt (7.4) ∆I = αStIt −γIt (7.5) ∆R = −γIt (7.6)

If ∆I > 0 the number of infectious individuals is growing, while if ∆I = 0 the size of the infectious class is neither increasing nor decreasing. SIR

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models are often used to find out the seriousness of the disease outbreak. The first step in doing this is to understand the conditions for which ∆I = It(αSt −γ) is positive, zero or negative.

If It = 0 there is no infectious individual in the population, and ∆I will also be 0. For our purposes, we can ignore this situation since we are interested in the case when a disease might be epidemic. In an epidemic disease, ∆I > 0. Whether ∆I will be positive, zero or negative depends on the sign of αSt −γ. We can rewrite ∆I in the following way:

∆I = γIt ( α

γ St − 1

) (7.7)

The expression ∆I < 0 if α γ St is smaller than 1, and ∆I > 0 if αγ St is

greater than 1. Among mathematical epidemiologists, the following expres- sion is used:

R0 = α

γ S0 (7.8)

This is the basic reproductive number of the infectious class. Note that S0 is the initial number of susceptible people, but R0 (usually pronounced “R naught”) is not the initial number of removed individuals—this is just some slightly confusing convention of the notation. If R0 > 1, then ∆I > 0 and an epidemic occurs.

If we write R0 = ( 1γ )(αS0) we can interpret the expression more clearly. αS0 is the number of individuals who become infected by contact with a single ill individual during one time step. ( 1

γ ) is the average duration of

the infection. Thus R0 can be interpreted as the average number of sec- ondary infections that would be produced by one infective in a susceptible population of size S0.

Let’s discuss some examples. Let’s assume a disease with a transmis- sion coefficient α = 0.0001, a recovery rate γ = 0.1 and an initial amount to susceptible individuals equal to 10,000. Then R0 = 0.00010.1 10, 000 = 0.1. This means that the disease is not epidemic. However, if we increase the transmission coefficient to 0.0002, for example because the disease starts to be airborn, then R0 becomes 2 and we will observe an epidemic disease. Without extra policies, the disease will continue to spread through the pop- ulation.

For various infectious diseases, their basic reproduction numbers are available (http://en.wikipedia.org/wiki/Basic_reproduction_number). Measles has an R0 between 12 and 18, which means that an infected person infects 12 to 18 people during the infectious period. SARS has a R0 of 2 to 5, which is the same as HIV. Pandemic influenza has a R0 between 2 and 3.

With those high values of R0, why doesn’t everyone have measles? Well, if we vaccinate people, we reduce S0, and this reduces R0. If enough people

7.3. R0, BASIC REPRODUCTION NUMBER 91

are vaccinated, R0 becomes lower than 1 and the epidemic spread is stopped. So how many people do we need to vaccinate to stop an epidemic? For this, we introduce V , the fraction of the susceptible population that needs to be vaccinated. When the fraction V of the people are vaccinated, the actual number of susceptible people become S0(1 −V ). To determine the value of V , we replace the original number of susceptible people with this S0(1−V ) in the ∆I equation. This leads to

∆I = γIt ( α

γ S0(1 −V ) − 1

) = γIt(R0(1 −V ) − 1) (7.9)

Therefore, ∆I will be less than 0 if R0(1 −V ) − 1 < 0, which is the case if V > 1 − 1

R0 .

Thus for a disease like measles, which has a R0 up to 18, we need to vaccinate 1 − 118 = 94.5% of the people. Thus not everyone needs to be vaccinated to avoid an epidemic. When enough people are vaccinated to stop the spread of an epidemic, we call this herd immunity. Since enough people are immune to the disease, the disease will not spread and is not epidemic within the population.

What is the vaccination rate for a pandemic influenza? How do you evaluate the chances to reach this level of vaccinations? Why is it difficult for a new strain of viruses that become epidemic?

What causes a disease to have a high value of R0? R0 is high if the average duration of the disease is long. For example, HIV infection has a long average duration of infection—many years—which contributes to the high value of its R0. If we can cure individuals faster, R0 will decline and the disease might not be epidemic anymore. If we cannot reduce the average duration, another option in reducing the epidemic spread among the popu- lation is to reduce S0. This can be done by vaccination of a certain fraction of the population. Yet another option is to reduce α; this may be done by quarantining a certain fraction of the infected population, which reduces the likelihood that an infected individual would transmit the disease.

Look at the example of Figure 10.3. At the start of the simulation we could have predicted that an epidemic would spread through the population by calculating the value of R0. R0 is equal to (0.00001)(99900)(10) = 9.99. How many people do we need to vaccinate to derive herd immunity? V > 1− 19.99 = 0.8999 ≈ 0.9. This means that (99900)(0.9) = 89, 910 people need to be vaccinated to achieve herd immunity.

Example. A new computer virus is spreading among university accounts. This virus gains access to a computer when people login via their univer- sity account. The virus damages the computers of the account holders and therefore rapid action needs to be taken. We can formulate the problem as a SIR model. There are 100,000 students, staff and faculty with a univer- sity account, and when the virus was discovered there were already 1000 accounts infected and 2000 accounts recovered from the virus. From the

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data of the first hours of the virus spreading through the system, we know that α = 0.00001 per computer and γ = 0.1. What is the value of R0 and what do we know from this value?

Answer: R0 = αS0γ = (97, 000) 0.00001

0.1 = 0.97 0.1 = 9.7 This means that

the virus is epidemic and will spread fast through the population. Every infection will on average cause about 10 others to be infected.

How many accounts need to be vaccinated to avoid an epidemic (users need to install a patch on their computer)?

Answer: To stabilize the spread, we need to bring R0 lower than 1. To do this we need to “vaccinate” a fraction V of the population, so that the following expression holds:

(1 −V ) αS0 γ ≤ 1, that is (1 −V )R0 ≤ 1 (7.10)

V ≥ 1 − 1 R0

= 1 − 1

9.7 = 89.7% (7.11)

Therefore, about 90% of the susceptible accounts need to be vaccinated to control the spread of the virus. These are 89.7% of 97,000 = 87,009 accounts.

7.4 Summary In this chapter, we introduce the SIR model to analyze the spread of in- fectious diseases. The three equations describe the changes in the numbers of susceptible, infectious and removed individuals within a population. The model also helps us determine whether the disease is epidemic and how many people need to be vaccinated to stop the epidemic (and reach herd immu- nity). This is done by calculating and manipulating the basic reproduction number, R0.