# Mathematical Models of Isolation and Quarantine

The recent emergence of severe acute respiratory syndrome (SARS) has
drawn attention to the strategies of isolation and quarantine (I&Q) as
a method of disease control. The fundamental dilemma associated with the implementation
of I&Q is how to predict the population-level efficacy of individual quarantine:
Which and how many individuals need to be quarantined to achieve effective
control at the population level? Although some forms of I&Q have proven
effective in SARS,^{1}^{,2} they
are not appropriate for all infectious diseases. Diseases like varicella,
for which costs of quarantine may be high (many work and school days are lost
when noninfected contacts are kept at home) and the return minimal (a relatively
mild disease is avoided), require a different approach. Furthermore, in some
cases, I&Q may be not only costly but harmful. An I&Q policy for varicella,
in the long run, may actually increase the average age (and therefore the
severity) of first infection. Using I&Q to control rubella in China could
actually lead to higher levels of disease because under the current system
(of no control), about 97% of the population has rubella antibodies obtained
from direct exposure to infectious individuals.^{2} Such
a level of natural immunity would be impossible to accomplish under the current
effective US and Canadian vaccination policies.

Mathematical modeling can help determine when I&Q are the best strategies for disease control as well as how they might affect short- and long-term disease dynamics. Mathematical modeling offers ways of integrating population-level knowledge based on previous epidemics with available individual and population data to predict the outcomes of several alternative scenarios. This kind of mathematical epidemiology is particularly well suited to problems for which formal experimentation is impossible for logistical or ethical reasons. In these situations, mathematical models can play a role in planning and experimental design in epidemiology, ecology, and immunology.

Mathematical disease modeling is an attempt to fit empirical data to abstract processes. Decisions must always be made about which variables to exclude from the model. Although inclusion of more variables (for example, the baseline health status of every individual) would make the model more accurate, such models would be impossibly complex. The balance between predictive power and its level of detail depends on the questions the model is intended to answer. Variables that can influence the outcome of I&Q policies include the number of contacts an infected person has per unit of time, the probability of infection per contact, and the proportion of the population that is vaccinated, quarantined, isolated, or educated to avoid infection. In general, it is difficult to judge the effects and interactions of these variables at the population level. A simple model is usually mathematically "tractable" (ie, easy to manipulate and to calculate outcomes for), thus allowing the entire range of possible outcomes to be studied. Addition of detail and complexity can make models more accurate, but this also complicates their mathematics. Current computer technology, however, allows studies of extremely detailed models.

We developed a model to predict whether I&Q could stop the spread
of SARS in greater Toronto, Ontario.^{1}^{,2} We
limited the time frame of the model to the duration of a single SARS outbreak.
The simplicity of the question and the simplicity of our assumptions reduced
the amount of data required to test the SARS model. Our "simple" SARS model
has about 11 parameters. Obviously, completely accurate and specific predictions
from such a model were impossible, but the model was able to illustrate the
power of I&Q as control measures. The model predicted that these policies
would help and showed how dramatically they could reduce the size of a SARS
outbreak (by a factor of 1000). These results agreed with actual observations.^{1}^{,2} Models can provide rapid estimates
of the impact of control strategies even before data from other areas are
available (before epidemic spread occurs) and when experimental data may be
incomplete or inaccurate.

Mathematical models are frequently represented by a box diagram showing the categories of persons the model addresses (boxes), the movement between categories as the disease progresses (arrows), and the mathematical rates at which this movement occurs (formulas on arrows). An example for the SARS model is provided below. There are 5 categories of persons: susceptible (S), capable of catching the disease; exposed (E), infected with the disease but in a latent state; infectious (I), capable of infecting others but undiagnosed; infectious and diagnosed (J); and recovered (R).

In this model, γ_{i} represents
the (per capita) recovery rate (ie, the movement from one of the infected
categories, I or J, to recovered, R); δ is the (per capita) SARS-induced
death rate (ie, the movement from I or J out of the system); α(*t*) is the time-dependent (per capita) diagnosis rate (ie,
the movement from I to J), and *l*(*t*), β, and *q* are coefficients representing
estimates of how much contact exists between persons and how infectious they
are. These coefficients therefore determine how quickly movement from S to
E occurs. *l*(*t*) is the time-dependent
reduction in infectiousness from isolation, β is the transmission coefficient
(the expected number of contacts per unit of time per person that result in
an infection), and *q* measures the relative infectiousness
of individuals who are in a latent state.

A fundamental concept of mathematical epidemiology is that a "threshold"
can be identified. The basic reproductive number,
R_{0}, a dimensionless quantity, introduced in the early 20th century by
Ross,^{3} and Kermack and McKendrick,^{4}^{,5} estimates the average number
of secondary infections generated by a typical infectious individual with
a given infection. The theory is based on the understanding that a basic reproductive
number greater than 1 generates an increasing number of infected persons and
results in an epidemic outbreak, while no epidemic will emerge if R_{0} is less than 1. Important factors in controlling a disease's spread
can be identified by examining their effect on R_{0}. One of the most
important contributions in mathematical epidemiology has been to show that
the most important factor in any I&Q or treatment campaign is the speed
of response. Irrespective of other measures, the longer a case goes undiagnosed,
the more likely it will be that the infected individual will be able to spread
the disease before he or she is treated; hence, the more likely that R_{0} will be greater than 1. Mathematical epidemiology provides a way of
studying and characterizing these influences in a systematic way. But devising
strategies to decrease R_{0} is not the only issue. Finding realistic
diagnostic or isolation strategies quickly can be difficult and costly. Theoretical
and mathematical epidemiologists have frequently assumed a perfect world (a
world in which there are no complicating variables or random variability in
behavior), where response times are not particularly relevant, where perfect
isolation is possible, and where individuals are all well informed and compliant
with government policy. These scenarios are simpler to model. However, it
is impossible to maintain perfect I&Q strategies in the real world. The
often irrational but predictable social aspects associated with disease transmission
and control have a significant impact. Epidemiologists are beginning to incorporate
more of these elements into their models, thanks to the increased availability
of powerful and inexpensive computers. The impact of the SARS model in Toronto
provides an example of this.^{1}^{,2} The
addition of more realistic elements, however, often requires that researchers
make unrealistic assumptions, particularly about the nature of human interactions
(social dynamics). Hence, every theoretical and numerical result needs to
be observed cautiously until the underlying model assumptions are verified.

*Washington Post.*May 3, 2003:A7.

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