Allman E.S., Rhodes J.A. Mathematical Models in Biology: An Introduction
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296 Infectious Disease Modeling
a. For the values α = .000145 and γ = .1428, draw the nullclines for
the SIR model, as well as arrows suggesting directions of orbits.
b. Draw the nullclines and arrows suggesting directions of orbits for
the general SIR model with parameters α and γ .
7.2.10. Recall the quarantine model of Problem 7.1.6 of the last section. If a
fraction q of infectives is successfully quarantined, in terms of q, α,
and γ what is the resulting relative removal rate? What is the effect
of increasing q on the resulting relative removal rate? How does this
show that a large q may prevent an epidemic?
7.2.11. As discussed in the text, to model a real disease using the SIRmodel,
we can estimate the parameter γ from data on the length of time indi-
viduals are typically infectious. The parameter α is harder to estimate
for real populations and diseases.
Explain how you could estimate the value of the parameter α for
a disease if you had sufficient data on the number of susceptibles and
infectives in a population at various times throughout the course of a
previous epidemic. Discuss how such data might be collected.
7.3. Variations on a Theme
Once we have understood the basic SIR model, it’s not hard to make modifi-
cations to produce models that might be more appropriate for other diseases.
Here, we will consider two basic variations that capture different dynami-
cal behavior for diseases with different characteristics. We will also consider
modeling large populations by tracking fractions of populations in each class,
rather than absolute numbers.
The SI and SI S models. For some infectious diseases, there is no re-
moved class. For instance, it might be that infective individuals simply cannot
recover. This leads to the SI model:
Susceptibles → Infectives .
The SI model assumes that S + I = N and that once individuals enter the
infective class, they remain there for the duration of the model. For instance,
the spread of head lice in a human population with no means of effective
treatment would fit the SI framework.
7.3. Variations on a Theme 297
The equations for the SI model are
S
t+1
= S
t
− αS
t
I
t
,
I
t+1
= I
t
+ αS
t
I
t
.
Compare these equations with those of the SIR model. How do they
differ?
How would you expect graphs of S
t
and I
t
vs. time to look for the SI
model?
Another way there could be no removed class is if once infectives recover,
they are again at risk for contracting the disease. Since recovering from many
diseases does not confer immunity on the former sufferer, an SIS model – in
which individuals pass from the infective class back into the susceptible class –
gives a more appropriate description of them. Syphilis and gonorrhea, for
example, can be treated with antibiotics, but a patient, once cured, may become
reinfected. The common cold is a disease that most of us get repeatedly.
Schematically, the SIS model is:
Susceptibles Infectives ,
and its equations are:
S
t+1
= S
t
− αS
t
I
t
+ γ I
t
,
I
t+1
= I
t
+ αS
t
I
t
− γ I
t
.
Compare these equations to those of the SIR model. How do they
differ?
How would you expect graphs of S
t
and I
t
vs. time to look for the SIS
model?
In the problems at the end of the section, you will explore these models
in more detail. In particular, you will find that the SIS model can lead to a
constant, yet nonzero number of infectives in a population. In this situation,
we say the disease is endemic.
Think of a few different infectious diseases you are familiar with, such
as leprosy, mononucleosis, and tuberculosis. Of the three basic models,
SIR, SI,orSIS, which if any is most appropriate for each disease?
Contact rate and contact number. Infectious disease models are often
formulated a bit differently than we did above, using proportions of the pop-
ulation instead of absolute numbers. For large populations in particular, this
298 Infectious Disease Modeling
may be a natural approach, because the precise size of the population may
not be known, or even really matter. Such a formulation also allows us to
replace the transmission coefficient α with a new parameter that has a more
meaningful biological interpretation.
Working with the SIR model as an example, let’s set
s =
S
N
, i =
I
N
, r =
R
N
,
and rebuild the model using proportions. Notice now that
s + i + r = 1,
because we are measuring fractions of the population. We will continue to
assume there are no births or immigration so that the total population size N
remains constant, even though N will not appear in our equations.
In this setting, a similar thought process as used in the SIR model lead to
formulas for s, i, r, the change in proportions of the three classes. The
sir equations are:
s =−βs
t
i
t
,
i = βs
t
i
t
− γ i
t
,
r = γ i
t
,
where β is called the contact rate.
Before we explain the interpretation of β, we should note that, if we use
an sir model and want to introduce a specific total population size N ,wecan
recover SIR data by simply multiplying the sir formulas by N . For example,
the net change in number of susceptibles is
S = (s)N =−βsi N =−(βi )(sN) =−(βi )S.
Use this to show β = N α by replacing i with
I
N
.
In the equation S =−α S
t
I
t
, both S
t
and I
t
are measured in “individ-
uals.” What are the units of the transmission coefficient α?
In the equation s =−βs
t
i
t
, both s and i denote “fractions of a pop-
ulation,” and so have no units. What are the units of the contact rate
β?
Let’s try to understand the equation S =−(βi )S more fully. By the term
“contact,” we will mean an interaction between individuals that is sufficient
for disease transmission. As infectious disease spreads to a susceptible only
from contact with an infective, not from contact with a healthy or immune
individual, the factor βi in the equation must give the average number of
7.3. Variations on a Theme 299
contacts an individual has with infectives during a single time step. Because i
denotes the fraction of the population in the infective class, we can interpret β
as the average number of contacts that an individual has during a single time
step. This means β measures contacts between anyone, whether infective or
not, that would have caused an infection if one of the individuals was infective
and the other susceptible.
Let’s turn this explanation around to make sure it is clear. The contact
rate β is defined as the average number of contacts an individual experiences
during one time step. Then βi represents the average number of contacts
with infectives that an individual experiences in a single time step. To get
the number of susceptibles that fall ill from contact with infectives during a
single time step, we multiply by the number of susceptibles S and obtain the
expression βiS.
The value of β of course depends on the particulars of the disease under
study. For example, because chickenpox is highly contagious, it seems plau-
sible that an elementary school child might have contact, sufficient to spread
chickenpox, with four other children throughout the course of a day. In this
case, we would take β = 4 and t = 1 day. For a less contagious disease, β
might be a small number such as .02.
Another important value is the contact number σ , the average number
of contacts of a typical infective during the entire infectious period. This is
a “per-infective” measurement. For the sir model above, we multiply the
contact rate β by the mean infectious period
1
γ
to find
σ =
β
γ
.
The contact number σ is closely related to the basic reproduction number
R
0
. This is not surprising because both are measures of the number of sec-
ondary cases of illness produced by one infective introduced into a wholly
susceptible population. In the problems, you will work out the relationship
between R
0
and σ , and see why they are essentially the same as long as S
0
is
almost as big as N .
Immunization strategies. A population can be protected from disease in
many ways. For example, the number of susceptible individuals can be re-
duced through immunizations, the contact rate can be reduced through quaran-
tines or public health campaigns, or the removal rate can be increased through
better medical treatment of the sick. In short, a society might try to change
any of the parameters or initial conditions of the model describing the disease.
300 Infectious Disease Modeling
Vaccinations, when available, are an attractive way to control disease dy-
namics, but the private and public risks of immunization must be balanced.
One of the main goals of any immunization program is to achieve herd im-
munity, (i.e., to ensure that no epidemic can take place even if a few cases
of the disease are present). However, individuals receiving vaccinations are
usually at some small risk of adverse effects. Thus, vaccination programs
have goals that benefit the public welfare, possibly at the expense of a few
unlucky individuals.
As a result, even immunization policies that are understood mathematically
to be capable of preventing epidemics can be controversial. In the 1970’s, con-
cern over the safety of the pertussis (whooping cough) vaccine sparked public
debate in Great Britain. Although the World Health Organization Smallpox
Eradication Program (1967–1980) was wildly successful, the United States
and Great Britain stopped routine vaccinations for smallpox in 1971, a short
time after the initiation of the program. Disease surveillance in these countries
indicated that more people were dying from complications arising from vac-
cination than would have died from smallpox itself. Moreover, discontinuing
smallpox vaccinations had economic benefits; public health costs diminished.
Before public health policy is set, the likely outcome of proposed vacci-
nation programs must be well understood so that tradeoffs in terms of public
and private good can be weighed. Thus, estimating what level of vaccination
confers herd immunity for a particular disease in a particular society is an
essential first step.
As an example, consider a large population at risk for a disease modeled
well by the sir equations. We would like to ensure that, regardless of the size
i
0
of the fraction of the population that was infective, i
t
never increases. Thus,
we want
i = βs
t
i
t
− γ i
t
< 0.
But, βs
t
i
t
− γ i
t
= (βs
t
− γ )i
t
and since i
t
≥ 0, this means we want
βs
t
− γ<0, or s
t
<
γ
β
=
1
σ
.
In other words, 1/σ is the threshold value determining whether disease
spreads. If the fraction of the population that is susceptible can be brought
below 1/σ then an epidemic cannot occur. The fraction of the population that
must be vaccinated successfully to ensure herd immunity is thus 1 − 1/σ .
Of course, using this result for a disease in the real world requires estimating
σ from epidemiological data. Because of uncertainties, it would also be wise
to aim for immunizing a larger fraction of the population. Whether such a goal
7.3. Variations on a Theme 301
is achievable depends on many social and economic factors, but the model
identifies the target.
Studying realistic immunization issues requires using more complicated
models. Disease dynamics often are different among different age and social
groups, and so often each of the s, i, and r groups must be broken into
subgroups. (Compulsory school attendance, for instance, can have a large
effect on disease transmission.) A model might break the population into
several groups by age, sex, or other factors, and be used to determine which
groups should be targeted in an immunization campaign.
Social and medical considerations are crucial. A vaccination campaign
successful in one country may be a failure in another due either to different
disease dynamics or to differing social acceptance of the program. It may,
in practice, be impossible to vaccinate a high enough percentage of the pop-
ulation in an overcrowded city or country to avert epidemics or gain herd
immunity from highly infectious diseases like measles. The best realistic pol-
icy may be to allow citizens to catch measles at a young age, when there are
few complications, and gain disease-conferred immunity.
Different strategies might also be equally successful. For instance, the
United States and Great Britain have adopted different vaccination programs
for rubella. Rubella is not a life-threatening or dangerous disease in general,
but if a pregnant woman becomes infected her infant may suffer from a serious
condition know as congenital rubella syndrome (CRS). Thus, ensuring the
immunity of women of childbearing age is the primary goal of any program.
In the United States, all children are routinely vaccinated against rubella as
part of their MMR shot at around 15 months. In Great Britain, children are
allowed to contract rubella while young. Only those girls who have failed to
gain disease-conferred immunity are vaccinated at around age 12.
Problems
7.3.1. Use a computer program such as sir or twopop to study the SI
model. Use a variety of values of α and N . For each choice, examine
the behavior of the SI model for a variety of values of S
0
and I
0
.
Describe your observations.
7.3.2. Investigate the SI model by doing the following:
a. Solve for all equilibria (S
∗
, I
∗
). Are these biologically reasonable?
b. In a phase plane, draw nullclines and arrows suggesting orbit di-
rections. What does this tell you about the dynamics of a disease
modeled by an SI model?
302 Infectious Disease Modeling
7.3.3. The analysis of the SI model is made easier if we note that S
t
+ I
t
=
N is constant, so we can substitute the formula S
t
= N − I
t
into the
formula for I
t
and only track the number of infectives.
a. Do this and find a formula for I
t+1
in terms of I
t
.
b. Now use a computer program such as onepop to investigate
the model. Compare your simulations to the ones obtained with
twopop.
7.3.4. Use a computer program such as sir or twopop to explore the
SIS model. Vary the parameters α, γ , N , S
0
, and I
0
. Describe your
observations.
7.3.5. Investigate the SIS model by doing the following:
a. Solve for all equilibria (S
∗
, I
∗
). Are these biologically reasonable?
An equilibrium with I
∗
> 0 represents the endemic occurrence of
a disease. Can an SIS disease be endemic?
b. What do the phase plane, nullclines, and orbit directions tell you
about the dynamics of a disease modeled by an SIS model?
7.3.6. The analysis of the SIS model is also made easier if we note
that because S
t
+ I
t
= N is constant, we can substitute the formula
I
t
= N − S
t
into the formula for S
t
and only track the number of
susceptibles.
a. Do this and find a formula for S
t+1
in terms of S
t
.
b. Now use a computer program such as onepop to investigate
the model. Compare your simulations to the ones obtained with
twopop.
c. Can you find parameter values for which the approach to an en-
demic equilibrium is oscillatory? For which the equilibrium is not
approached?
7.3.7. For the SIR model, the threshold value ρ plays an important role.
a. Is there an analogous threshold value for the SI model? If so, find
it. If not, explain why there is not.
b. Is there an analogous threshold value for the SIS model? If so,
find it. If not, explain why there is not.
7.3.8. For the SIR model, the basic reproduction number R
0
plays an im-
portant role. How should you define R
0
for the SIS model?
7.3.9. Suppose a disease is modeled well by the SI framework. Explain
how new medical developments might make the model invalid, so
that either an SIS or SI R model would be needed instead.
7.3. Variations on a Theme 303
7.3.10. The dynamics of SISmodel can be understood in terms of the logistic
model of Chapter 1.
a. Show the SIS model can be expressed as
I = (α N − γ )I
1 − I /
N −
γ
α
.
b. Use part (a) and your knowledge of the logistic model to determine
the equilibrium, and hence a possible endemic level of the disease.
What interpretation does this give to γ/α?
c. Use part (a) and your knowledge of the logistic model to determine
a condition on the parameters of the model that ensures a stable
equilibrium.
d. Use part (a) and your knowledge of the logistic model to deter-
mine a condition on the parameters of the model that ensures an
oscillatory approach to the stable equilibrium. Do you think such
behavior is likely to occur naturally?
7.3.11. Both R
0
for the SIR model and σ for the sir model are interpreted
as measures of the number of secondary cases of illness produced
by one infective introduced into a wholly susceptible population. To
understand their relationship better:
a. Express σ in terms of the SI R parameters α, γ , and N .
b. From your answer to part (a), compute a simple expression for
R
0
/σ .
c. If a population is mostly susceptible initially, so S
0
≈ N , what will
the value of R
0
/σ be?
d. Which is larger: R
0
or σ ?
7.3.12. For the SIR model, we saw that ρ =
γ
α
was a threshold value for
S
0
, determining whether an epidemic would occur or not. What is
the analogous threshold value for s
0
in the sir model? Explain your
reasoning.
7.3.13. Suppose that an sir-modeled infectious disease in a certain population
has an estimated contact rate of 0.1 and a removal rate of 0.05.
a. What is the contact number for this disease?
b. Assuming a vaccination is developed that is 100% effective, what
percentage of the population should be immunized to achieve herd
immunity?
c. Assuming the vaccination is only 90% effective (so only 9 of
every 10 vaccinations confer immunity on the recipient), what
304 Infectious Disease Modeling
percentage of the population should be immunized to achieve herd
immunity?
7.3.14. For the SIR model with a fixed total population size N , how many
individuals must be immunized to confer herd immunity? Express
your answer in terms of the parameters of the model.
7.3.15. Suppose that, for an sir disease, we estimate the contact number is
σ = 0.5. Then, according to the formula developed in the text, we
should try to vaccinate a fraction 1 −
1
.5
=−1 of the population to
prevent an epidemic. Since a negative fraction of the population does
not make sense, explain what this must mean.
7.3.16. Recall the approach to disease control through quarantine introduced
in Problem 7.1.6.
a. Formulate an sir model with a fraction q of the infectives prevented
from infecting others through quarantine.
b. In terms of the parameters of your model, find the threshold value
for s
0
determining whether an epidemic will occur or not.
c. In terms of the other parameters of the model, find the smallest
value of q that will prevent epidemics from occuring for any s
0
.
7.3.17. Why is there no point in asking what percentage of the population
should be vaccinated to confer herd immunity for an SIS disease?
7.3.18. For some diseases, like tetanus and rabies, vaccination of an individual
in no way contributes to achieving herd immunity.
a. What are the features of the way these diseases are transmitted that
are responsible for this?
b. Using your understanding of such diseases, how would you design
a reasonable vaccination policy for them?
7.3.19. Discuss the characteristics that an infectious disease and its vaccina-
tion might exhibit in order for a voluntary vaccination program to be
worthwhile. What circumstances might make a mandatory program
justified?
7.3.20. One infectious disease model that takes into account births and deaths
from natural causes, as well as disease-related deaths, is given by the
equations:
s =−βs
t
i
t
+ pµi
t
+ γ i
t
,
i = βs
t
i
t
+ (1 − p)µi
t
− (µ + ν)i
t
− γ i
t
,
where β is the contact rate, γ is the removal rate, µ is the birth and
death rate due to natural causes (which are assumed to be equal), ν
7.3. Variations on a Theme 305
is a disease-related death rate, and p is the probability that an infant
born to an infective is disease free.
a. Explain the meaning of each term in these modeling equations.
b. This model allows births and deaths. Does it assume that the total
population size will remain constant?
c. Would you call this a modified SI model, a modified SIS model,
or a modified SIR model? Explain.
d. The equations above do not explicitly have any terms describing
deaths of susceptibles due to natural causes, or births of infants of
susceptibles. Why are they not there?
7.3.21. Many of the common childhood diseases must be modeled by a more
general infectious disease model, known by the acronym MSEIR.
It allows for births and deaths, and in addition to S, I , and R uses
two more classes, M and E. If a pregnant woman has immunity,
either disease-conferred or from vaccination, then some antibodies
are transferred across the placenta, and a newborn inherits passive
immunity from its mother. This immunity is temporary, lasting pos-
sibly as long as 1 year after birth. Newborn infants with protection
from maternal antibodies are placed in the M class and pass into
the susceptible class after passive immunity lapses. The exposed
class E consists of those who are infected but have not yet en-
tered the infectious period where they may transmit the disease to
others.
a. Make a schematic diagram for the MSEIRmodel that shows trans-
fer into and out of the various classes, including births and deaths.
For a challenge, write down possible equations for an MSEIR
model.
b. The childhood diseases measles, mumps, and rubella are modeled
well with an MSEIR model, whereas an SI R or SEIR model
fails to capture key dynamics of their transmission. In the United
States, the recommended age for infants to receive their first dose
of measles, mumps, rubella vaccination (MMR) is between 12
and 15 months of age. Explain why the MMR vaccination age
recommendation reflects the importance of including an M class
when realistically modeling these childhood diseases.
Projects
1. Investigate an SIR model that also takes into account births and deaths.
For many diseases, the continual introduction of newborn susceptibles is