Computational Modelling of the HIV Epidemic

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Computational Modelling of the HIV Epidemic

• Dr Alan Matthews
• School of Physics
• University of KwaZulu-Natal
• E-mail matthewsa_at_ukzn.ac.za

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Basic Epidemiological Model
Not at risk
infected
new infection
uninfected, susceptible
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Scientific Computing

• Science (Latin scire know) Knowledge
  systematically organised by objective principles.
• Know (Old English cnawan ken in sight) have
  in the mind
• System (Greek sustema set up) set of
  connected parts
• Organised orderly structure
• Objective external to the mind
• Principles (primus first, foundation, source)
• Computing (compute calculate count a
  number/quantity)
• Calculate (Latin calculus small stone in an
  abacus)
• Simulate (Latin similis like) produce a
  computer model of a process/system
• Model (Latin modus measure way, method) to
  represent (stand in for). Simplified, often
  mathematical (abstract, in the mind, set of
  rules) description, consisting of ideas (seen in
  the mind)
• Reality nature, physical world, born, not made
  by humans

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Modelling

• Abstract Model reality represented by set of
  ideas (seen in the mind)
• Computational Model real system represented by
  quantities (numbers)
• Model should be as accurate (careful) and precise
  (exact, close) as possible.
• Often, a model is a simplified representation.
• The model is based on a set of assumptions, with
  an algorithm (set of rules) to compute how the
  system evolves with time.
• Parameters are the constants of the model.
  Variables change (e.g. with time)
• Some models are simply mathematical
  representations (equations, graphs, tables) of a
  system at a given time (e.g. a smooth function
  that approximates a set of noisy data)
• A complicated model is not necessarily better
  than a simple model. If little is known, it is
  better not to build in unfounded features, and to
  keep the model simple.

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Modelling Process
Mathematical and/or Algorithmic Representation
Computer code
Model of System and Process
Numerical output
Comparison with data
Hypothesis
Insight
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Epidemic Growth
new infection
new infection
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Random Mixing

• Simplest assumption is random contact in a
  homogeneous way
• i.e. an individual comes into contact with other
  individuals randomly, with equal probability
• Example
• 30 of the population is infected. An uninfected
  (susceptible) individual makes 100 contacts with
  other individuals.
• There is no pattern or preference that determines
  who these individuals are, nor even that
  increases probability of contact.
• On average, 30 contacts are with infected
  individuals.
• For a given individual, there will be statistical
  variations from 30 (as for a sample).
• This assumption may work well for flu or TB, but
  HIV transmission is not a random, homogenous
  process.
• However, on average the process could be random
  within certain limits.
• For the purposes of modelling, the assumption of
  random, homogenous transmission is the simplest,
  and does not build in false knowledge (although
  it might not be a good model of reality)

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Sexual Networks
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Macro vs Micro

• Macro-model is group-based
• Models presented here are macro-models
• Processes are averaged over a large group of
  individuals
• Usually deterministic (i.e. definite outcome
  given a particular initial state) but can have
  stochasticity (randomness) built in to model
  chance variations
• Micro-model is individual-based
• Processes are for contacts between individuals
• Can keep track of partnerships and children
• Usually stochastic, since events occur with given
  probabilities
• Useful for research, if knowledge of parameters
  and patterns exists

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Macro / Group-based Models

• SIR Model
• Susceptible (S)
• Infected (I)
• Recovered (R does not apply to HIV-AIDS)
• SIU Model
• In HIV epidemic there is no recovered group.
• But many models have an unaffected group U

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Calculation of Growth
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Infection Probability
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Simplest Model
Single group All at risk No male/female
distinction No deaths or births Initial
prevalence 1 r 0.001
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Computer Programming Languages
Fortran 95 Old language, fast, widely used in
science and engineering. Originally not
Object-Oriented, Fortran 90 versions incorporate
many O-O features. Simpler than C. Compiled
language. C - Powerful, modern, verstaile, O-O
language, widely used in all fields. Low-level,
usually needs some experience in programming.
Compiled language. Java Like C, but somewhat
simpler, and adapted for web-based use.
Interpreted language, platform-independent. Python
a modern O-O language with good graphics
support Delphi Essentially O-O Visual Pascal. A
complied language. Visual Basic A Microsoft
platform, with lots of in-built Windows-friendly
routines. Excel An application rather than a
programming language. Very widely available and
easy to use. Much can be done with Excel, but it
becomes slow to run and clumsy to program if the
program becomes complex. Matlab A scientific
computing package, with built-in routines,
graphics and interpreted program scripts.
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OneSexModel in Excel
C6F6D2A2
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OneSexModel in C

• double dt1, Q0.5, Initial_Population100 //
  parameters
• double Susceptible, Infected // variables
• int step, nstep30 // time-stepping index and
  max. value
• prevalence 0.01 // set initial prevalence
  (parameter)
• Infected prevalence Initial_Population
• Susceptible Initial_Population Infected
• for(step1 stepltnstep step)
• NewInfections dtQprevalenceSusceptible
• Infected NewInfections
• Susceptible - NewInfections
• prevalence Infected /(Infected Susceptible)
• coutltltstepltlt ltltSusceptibleltlt
  ltltInfectedltltendl

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Epidemiological and Demographic Model

• HIV-AIDS epidemic in a population
• HIV spreads by sexual contact among adults
• Children infected at birth and by breast-feeding
• Population ages
• Calculate AIDS and non-AIDS deaths
• Calculate births

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EPP and Spectrum

• UNAIDS / WHO Model for general use
• EPP - Estimation and Projection Package
• download from www.unaids.org/resources/epidemiolo
  gy/epi_softwaretools
• EPP (Similar to OneSexDemog-SIU-Model)
• calculates an adult prevalence vs time curve
• uses four parameters
• fits the curve to HIV prevalence data
• Spectrum
• uses EPP curve to estimate demographic structure
  and impact of the HIV epidemic
• e.g. HIV prevalence by age, population size

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ASSA Model

• ASSA Actuarial Society of South Africa
• ASSA2002 AIDS and Demographic Models
• Download from
• www.assa.org.za, ASSA2002 AIDS Model
• Designed for South Africa may be adapted to
  other countries
• More complex than EPP

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Demography

• Birth rate
• Fertility
• Add up babies born to women of different ages
• Non-AIDS death rate
• Probability of death, depending on age and sex
• AIDS death rate
• Probability of death from AIDS, depends on time
  since infection
• Median time to death is 5 - 10 years
• Migration, in and out of population

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ASSA Model
Births
Young children 0 - 13
Four adult (14-59) risk groups
NOT not at risk
RSK medium risk
STD high risk
PRO high risk
Old Adults 60 - 90
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Mixing Patterns
Males
Females
PRO (1)
PRO (1)
STD (20)
STD (20)
RSK (30)
RSK (30)
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Epidemic Curve
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SA Population 1985
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2015
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Analysing a Model

• Sensitivity Analysis which parameters, when
  changed, cause a large variation in output
  values?
• Exploration of parameter space what are all the
  possible outcomes for a reasonable range of
  parameter values?
• Uncertainty intervals what is the range of
  output consistent with uncertainties in the
  input?
• Does the model reproduce known data? (to within
  some approximation) e.g. back projections
• Does the model correctly represent the underlying
  processes of the system being studied?
• Does the model predict unknown results?

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Parameters

• In the OneSexDemog-SIU-Model we have three
  epidemiological parameters
• Infection rate Q
• Initial prevalence p0
• Proportion of recruits (teenagers) who join the
  at-risk group W
• And three demographic parameters
• Recruitment rate (depends on birth rate) ß
• AIDS death rate ?AIDS
• Non-AIDS death rate ?Non-AIDS

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UC Berkeley Projection Interval

• Vary 2 parameters (RSK, STD) Proportions
• Range of each 1 - 98 over 21 steps
• Constraint sum lt 98
• PRO and NOT Proportions 1
• Minimise sum of least squares fit to ANC data as
  function of Female (RSK, STD) Male Partners
• Keep runs which fit inside 1 error bars
• 441 runs , 210 runs kept

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Comparison with Data

• Data
• HIV prevalence (e.g. ante-natal clinics)
• Mortality statistics
• For population, by subgroup, by sex and by age
• Unless mechanisms and patterns are well known,
  the model produces curves with a distinctive
  shape but varying scales
• The curves are calibrated by adjusting free or
  low-precision parameters to obtain the best fit
  to data
• A model should match data to within a
  scientifically justified approximation
• i.e. a complex computer program with impressive
  graphs is not a model if it does not approximate
  reality
• Different models of the same process should agree
  

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Values and Limitations of Models

• Human system is complex to model
• But a lot is known biology, statistics,
  surveys
• Also, logic imposes constraints
• Model is an approximate representation
• Because a model does not precisely predict
  (especially not the future) does not mean it has
  no value
• Model represents the realistic extent of our
  knowledge
• Model does not give the final answer nature
  does, through observational data
• Models are an aid to thought and analysis,
  automating complex hypotheses, and providing a
  tool for testing
• Models provide insights that might not have been
  obtained without the model

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Model constrains possibilities
Excluded possibilities
Model
Truth
Data
Logic
Computation