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University of Toronto St. George
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TRN125Y1
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Caroline Barakat
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Trinity College Courses

TRN125Y1

Caroline Barakat

Fall

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C07 5: Disease Modelling
Date: Oct 17, 2012
Slide 2: Lecture Outline
Infectious Disease Modelling
o SIR Model
o Basic reproduction number R0
o 2 states to show why ppl use modelling:
Endemic steady state
Critical vaccination coverage
o Advanced models
With continuous age structure
Stochastic models where something happens out of the blue
Network models
o Model use
General reasons
Chronic Disease Modelling
o Macrosimulation
o Microsimulation
Slide 3: Intro – Infectious Disease Modelling
Tool for the analysis of policy options- to come up with good policy options, use mathematical
modelling – constantly developed
Reasons why mathematical modelling became so popular:
o 1) Complexity of infectious disease dynamics
Complexity not simple to understand and they are diff for diff diseases
Transmission of infectious diseases involves contact between individual and
someone else, interaction between age, enviro, agents, ease of transmission
a system that is pretty complex and has many diff dimensions/dynamics
Virulence can change, enviro can change, contact can change and it is this
complexity that makes modelling so good to use
Using model doesn’t mean you have to address that question all over again –
can just change that one factor (ex: population became 10 million) and put that
into model and it will spit out new answer
To simplify complexity of diseases
o 2) Not possible to decide on different interventions just by pure reasoning
Look @ the effectiveness which is dependent on diff factors
Ex) if you increase vaccination rates for the flu, the incidence rate will go down
by X factor
Trying to get the possible interventions which will lead to the best option –
based on the model – can decide on the best intervention
o 3) Process requires quantitative estimates – gives us indication of what we can do
1 C07 5: Disease Modelling
Date: Oct 17, 2012
In order to get sound cost-effective analysis, the impact of diff interventions on
the prevalence/incidence is best if it is quantified
Ex) let’s increase vaccination rates for the flu by 95% - what does that mean in
terms of incidence/prevalence/admission to hospital wrt flu events?
Put those #s in – and get results li0e R – so modelling gives quantitative
estimate that is needed in order to make sound policies and for proper
intervention
So, you want to know how changing one input impacts the output; want to make decisions
based on sound science; want to look @ various interventions
Slide 4: What is involved?
Mechanistic description of the transmission of infection between 2 individuals = modelling
o A system to tell us how infection is happening, developing a way to find out how things
are proceeding and if things change, what would happen
o This description makes it possible to describe the time-evolution of an epidemic in math
terms
o Ex) if population increases at some point, what does that have to do with the epidemic –
how will it increase? Trying to put evolution of disease in mathematical term – can
connect the individual level of transmission with the population level description
o If you know that R0 for the flu is 20 and you want to bring it down to 10 – what should
we do in terms of population? So, connecting individual level process to population level
description of incidence and prevalence of an infectious disease
Connect the individual-level process of transmission with a population-level description
Detail requirements of all the dynamic processes that contribute to disease transmission
o Every disease = diff and dynamic processes behind each disease is diff some are
symptomatic, some asymptomatic; some are host-related, some enviro-related
o So, detailed info can be used in getting that mechanistic description of transmission
Slide 5: Why is developing a mathematical model impt?
2 main reasons for importance of developing math model:
o Model helps focus thoughts on essential processes involved in shaping the epidemiology
of an infectious disease
Based on those dynamic processes and when you have a model in place – can
see what is impt (does death rate, birth rate, contact matter?) in the
epidemiology of infectious diseases – what variables are most important
It also reveals the parameters that are most influential and amenable for control
Can identify which of the factors in the model can be changed and can be a
target for control & intervention
Ex) if we reducing smoking by 50%, can reduce cancer by 15%; can only
know this if we manipulate variables in the equation
o Reveals the parameters that are most influential and amenable for control
2 C07 5: Disease Modelling
Date: Oct 17, 2012
Slide 6: What has fuelled the applicability of mathematical disease modelling? – certain events that
fuelled: SARS (incubation periods, travelling patterns, density etc; can all be inputs to the model),
H1N1, smallpox (huge)
Bernoulli – 18 century used disease modelling to estimate impact of smallpox vaccination on
life expectancy
Hamer in 1906 used modelling to argue that an epidemic can come to an end without all
susceptible ppl having the infection/not exposed to the disease
Kermack and Mckendrick (1927) – basis for SIR modelling and R0
Based on your knowledge, which diseases required or make use of mathematical modelling?
o HIV/AIDS in 1980s, SARS, and diff pandemics (infectious diseases)
Slide 7: SIR Model – all elements for the transmission of a disease are in this simple model
Basic assumption: human population is subdivided into 3 groups:
o Susceptible persons (S)
o Infected persons (I)
o Removed persons (R)
Maybe because they’ve acquired resistance to an infection or they’re no longer
part of this infectious disease
Movements in and out of these groups via:
o Birth
o Death
o Transmission of infection
o Recovery
Movements:
o So you have people who are susceptible who get the infection – so people from S group
move to the I group and assuming that they now have this acquired immunity they’ll
move from I group to R group
o Movements occur also via birth = flow into the susceptible group; immunity from
breastfeeding (antibodies that mom has); nevertheless, everyone who is born is
susceptible to certain infections
o Death is flow out from all groups:
Ppl who are susceptible may die
Ppl who are infected may die
Ppl who are removed may die
o If a transmission of infection occurs mvt from S to I group
o Recovery mvt from I group to recovery; and if it’s something like the flu, can be back
in the S group again (because flu mutates) or can be thought of as new infection
altogether
Slide 8:
3 C07 5: Disease Modelling
Date: Oct 17, 2012
Transmission between compartments is governed by rates
SIR model = interpreted in terms of rates; all transmission between those compartments happen
based on rates
o So, how many ppl per unit of time go into susceptible group, how many in infected
group and how many ppl in recovery group
o Anytime looking @ rate of change of some variable with respect to time – this refers to
a rate
o Rate = looking @ change of a variable y wrt time R = dy/dt
Birth rate = v, but if a fraction of vaccinated newborns are brought in there, then
birth rate = (1-p)v where p = the # of newborns are vaccinated
o Ppl who are vaccinated fall out of that equation; (1-p)v is the rate that’ll be going in the
equation
Death rate = µ
Recovery rate = ƴ
Key element is the term describing the transmission of infection; how is the transmission of
infection happening? And it happens according to a rate known as β aka mass action term
o β = transmission of infection and it is related to two diff factors:
K= contact rate
Q = probability of transmission
o Β = Kq
So we have:
o Birth rate
o Death rate
o Recovery rate
o The rate of transmission of infection (Beta = Kq)
Things depend on population density or prevalence in a population
o Force of infection, λ = βI
Infection can have higher force if the population is pretty dense
It measures the risk of a susceptible person to become infected with certain
disease per unit time – depends on the # of ppl that are infected in the
population
o Where N = total population size, λ = βI / N
Where I is the rate of infection occurring
o So, the force of infection is the mass action term times the rate of ppl being infected
and then the force of infection will be divided by the total population gives us
indication of how strong is that infection affecting the population
So, this is all wrt to the terms
Now, look at the rate of change of the susceptible individuals, rate of change of infected and
rate of change of the recovered individuals
o 1) rate of change of susceptible individuals
4 C07 5: Disease Modelling
Date: Oct 17, 2012
V(1-P) – have certain birth rate and some individuals are getting vaccinated and

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