0:09

In this session, we will discuss

some basic concepts in infectious disease modeling.

A model is simply our understanding

of how things work in the real world.

In our everyday lives, we rely on models

to make decisions.

For example, when we try to decide

what kind of transportation to take

to go from one place to another,

we inevitably have some model in mind

about the time and cost associated

with each available option.

The model could be based on

our own personal experience or some kind

of knowledge base such as Google Map.

In essence, we develop models for understanding

and predicting the behavior of systems

that we’re interested in.

To this end, models are often approximations

or simplifications of real-world systems

because, as the eminent evolutionary biologist

and geneticist John Maynard Smith said,

“Describing complex, poorly-understood reality

with a complex, poorly-understood model

is not progress.”

For example, in drug development, animal models

are often used to learn about human

biological pathways, though it is obvious that

observations made in animal models

are not generalizable always to humans.

Mathematical models use a bottom-up approach

to integrate assumptions about the underlying processes

and empirical data collected from the real world

to describe the behavior of the system.

The most well-known examples

of mathematical models are those

we learned in high school physics.

For example, the mathematical model

based on Newton’s law of motion

and empirical measurements of mass and gravity

has been a cornerstone for understanding

and predicting the movement of objects on the planet.

Another familiar but much more complex example

is weather forecast, which serves

as a good learning model for what infectious

disease modeling is trying to do:

To develop robust mathematical models

that can explain and predict the behavior

of systems in nature.

Let us now look at a simple example

of epidemic models known as the SIR model.

Consider a hypothetical epidemic scenario where

(i) a novel pathogen invades a population

that has no immunity against it;

(ii) infected individuals gain permanent immunity

against reinfection after recovery;

and (iii), the timescale of the epidemic is much faster

than changes in population demographics

so that we can ignore the effect

of births and immigration.

3:01

The SIR model makes the following

simplifying assumptions regarding disease transmission:

First, at any given time t, each person belongs

to one of three groups: susceptible,

infected, or removed.

The susceptible group corresponds to those

who are susceptible, and have not yet been infected.

The infected group corresponds to those

who are currently infected and spreading the disease.

And the removed group corresponds to those

who have either died or recovered from the disease.

3:50

Note that the disease prevalence at time t

is simply I(t).

The second assumption is that the rate of infection

for a susceptible person at any given time t

is proportional to the disease prevalence.

Let’s call the proportional constant beta,

which can be interpreted as the rate

of making contacts that conduce infection

if the contact is made between a susceptible

and an infected person.

We will learn more about contact patterns

from Professor Marc Lipsitch

later in this course.

The rate of infection per susceptible has

a special name - force of infection.

The third assumption is that infected people

are infectious for D days on average

after which they would recover or die

and can no longer infect other people.

This is the same as saying that

infected people recover at rate 1/D.

Let us now look at what the epidemic

dynamics look like under these assumptions.

Consider an arbitrary time point t and

a small time-step delta_t.

The time-step should be small

compared to the average infectious duration.

For example, if the average infectious duration

is three days, delta_t should be smaller than

half a day or so.

In the SIR model, the number of susceptibles

at time t plus delta_t is simply the number

of susceptibles at time t, minus the number

of infections that occur between time t

and t plus delta_t.

Under assumption two, the number of infections

between time t and t plus delta_t is equal to

S(t) which is the number of people

subject to the risk of infection, times

beta I(t), which is the force of infection,

times delta_t, which is the length of time

that we are considering.

Similarly, the number of infected people

at time t plus delta_t is equal to the number

of infected people at time t plus

the number of infections that occur

between time t and t plus delta_t,

minus the number of recoveries that occur

during the same time interval.

The number of recoveries is simply

I(t), the number of infected people

that may recover, times 1/D,

which is the rate of recovery, times delta_t.

These two simple equations together

specify the epidemic dynamics in the SIR model.

Starting with the initial conditions

at time zero, for example 100 infected people

seeded into the population of 1,000,000,

we can use these equations iteratively

to simulate the epidemic.

The resulting epi-curve has the hallmarks

of a typical acute epidemic.

The epidemic grows exponentially at the beginning,

then slows down and peaks, and finally fades out.

7:02

The most important feature of dynamical

epidemic models is that they explicitly

take into account the effect of herd immunity

in disease transmission.

Herd immunity is one of the fundamental lessons

in infectious disease epidemiology.

It refers to the phenomenon that immunity

of an individual not only protects

the individual himself from infection,

but also reduces the risk of infection

for everyone in the population.

Let us illustrate the mechanism of herd immunity

using the following example.

Suppose the transmission chain

of the epidemic looks like this

after the primary case has been seeded

into the population.

Now let’s look at how the transmission chain

will change if some of the people are immune

before the primary case is seeded,

for example, with a vaccine

that provides 100 percent protection from infection.

In addition to the vaccinated individuals,

three unvaccinated individuals who would have

been infected in the original transmission chain

are now indirectly protected from infection.

Therefore, they have been protected by herd immunity.

8:19

A corollary of herd immunity is that

not everyone in the population

has to be immunized in order to prevent

an infectious disease from taking foothold

and becoming widespread in the population.

Let us illustrate this principle

with the SIR model.

Suppose we vaccinate a proportion p

of the population with a vaccine

that provides 100 percent protection from infection.

This proportion p is called vaccine coverage.

Vaccination reduces the susceptible proportion

from 1 to 1 minus p, which means that

vaccination reduces the reproductive number

from R0 to (1-p) times R0.

Because an epidemic cannot take off

when the reproductive number is smaller than one,

we can prevent an epidemic as long as

the vaccine coverage is above 1-1/R0

This minimal coverage required to prevent an epidemic

is called the critical vaccination coverage.

For example, if R0 is 2, we can in theory

prevent an epidemic if we vaccinate

at least 50 percent of the population.

To summarize, in this session,

we have discussed some basic concepts

in infectious disease modeling.