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TRN125Y1
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Caroline Barakat
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TRN125Y1

Caroline Barakat

Winter

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HLTC07: Patterns of Health, Disease, and Injury
Lecture 5: Disease Modelling
Lecture Outline
Infectious Disease Modelling
SIR model
Basic reproduction number R o
Endemic steady state and Critical vaccination coverage
Advanced models
Model use
Chronic disease modelling
Macrosimulation
Microsimulation
Intro - Infectious Disease Modelling
Tool for the analysis of policy options
o To come up with good policy options, use mathematical modelling – constantly developed
o INCREASE VACCINATION RATE, DOES THAT LEAD TO GOOD RESULTS
Reasons why mathematical modelling became so popular:
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, environmental, agents, ease of transmission --> a system that is pretty
complex and has many diff dimensions/dynamics
Virulence can change, environment 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
2. Not possible to decide 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
3. Process requires quantitative estimates
For cause and effects
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
What is involved?
Mechanistic description of the transmission (between different people)
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
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
o Putting individual in the population level
Detail requirements of all the dynamic processes (great and knowledge and processes)
o Every disease = different and dynamic processes behind each disease is diff --> some are symptomatic,
some asymptomatic; some are host-related, some environment-related
o So, detailed info can be used in getting that mechanistic description of transmission everything that
contributes to disease transmission
Why is developing a mathematical model important?
Model helps focus on essential processes involved in shaping the epidemiology of an infectious disease
o Based on those dynamic processes and when you have a model in place – can see what is important
(does death rate, birth rate, contact matter?) in the epidemiology of infectious diseases – what
variables are most important
o It also reveals the parameters that are most influential and amenable for control
o Can identify which of the factors in the model can be changed and can be a target for control &
intervention
o Ex) if we reducing smoking by 50%, can reduce cancer by 15%; can only know this if we manipulate
variables in the equation
Reveals the parameters that are most influential and amenable for control
o You want to know if we work more on treatment does that mean the pop greats better?
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)
th
Bernoulli - 18 century
o 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
o Without all susceptible people having the infection/not exposed to the disease; pathogen and virulence
did not increase
o Based on various knowledge of disease
Kermack and Mckendrick (1927) - basis for SIR modelling and R o
o Provided the basis for the SIR modelling
Based on your knowledge, which diseases require or make use of mathematical modelling?
o HIV/AIDS in 1980s, SARS, and different pandemics (infectious diseases)
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)
Recovered or removed from the system
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 Birth rate π, ; if p=fraction of vaccinated newborns, then birth rate=(1-p)π
o 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 People 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, γ
ϐ , mass action term where Β = κq; κ=contact rate, q=probability of transmission
o 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
Things depend on population density or prevalence in a population
Force of infection, λ = βI
o Infection can have higher force if the population is pretty dense
o It measures the risk of a susceptible person to become infected with certain disease per unit time –
depends on the # of people that are infected in the population
Where N = total population size, λ = βI / N
o Where I is the rate of infection occurring
o So, the force of infection is the mass action term times the rate of people 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 with respect 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 individual
dS/dt = π(1-p) – ϐS(I/N) - µS
dI/dt = ϐS (I/N) - γI - µI
dR/dt = πp + γI - µR
0 ≤ p ≤ 1
N = S + I + R
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
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
Death is flow out from all groups:
o People who are susceptible may die
o People who are infected may die
o People who are removed may die
If a transmission of infection occurs → movement from S to I group
Recovery → movement 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 1) Rate of change of susceptible individuals
o V(1-P) – have certain birth rate and some individuals are getting vaccinated and
o they come out of that (1-P)
o So the rate at which people go into susceptible group is the birth rate minus the rate @ which they get
infected minus those that die out from the system
o So, you have people being born and going into system, taking out people who have been vaccinated
and people who are dying out from susceptible group and the people who are moving from susceptible
group to infected group (middle term refers to those individuals)
2) Rate of change of the individuals in the infected group per time – same idea

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