A New Algorithm to Extract the Time Dependent Transmission Rate from Infection Data

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A New Algorithm to Extract the Time Dependent
Transmission Rate from Infection Data

• Hao Wang
• University of Alberta
• MATH 570

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from model
Infection data
Algorithm
FT to check dominant frequencies
smooth interpolation
Birth data
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Outline of talk

1. Very brief sales pitch for utility of
   mathematical models of transmission of IDs in
   populations
2. Seasonality/periodicity of IDs
3. The time-dependent transmission rate
   (coefficient) of an ID, and a new method to
   estimate it from infection data.
4. Application to Measles - detection of 1/yr cycle
   and 3/yr cycle

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Introduction

• Molecular studies have revolutionized our
  understanding of the causes and mechanisms of
  IDs.
• However, the quantitative dynamics of pathogen
  transmission is less understood.
• Large scale transmission experiments (e.g.,
  influenza transmission in ferrets) are useful to
  understand the transmission dynamics, but are
  usually impractical (economic and ethical
  reasons).

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Mathematical models

• Thus we need indirect methods to study the
  transmission dynamics of an ID in a population.
• Mathematical models are a powerful tool.
• Mathematical models can include virology,
  immunology, viral and host genetics, and behavior
  sciences.
• Main way to estimate key epidemiological
  parameters from data.

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Seasonal dependence of occurrence of acute IDs

• Many IDs exhibit seasonal cycles of infection (Is
  this true for animal IDs?)
• Influenza, pneumococcus, rotavirus, etc. peak in
  Winter
• RSV, measles, distemper (some animal IDs), etc.
  peak in Spring
• Polio peaks in Summer

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Weekly measles cases in seven UK cities 1948-58
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Why ???

• Changes in atmospheric conditions ?
• Cholera outbreaks follow monsoons in south Asia.
• Lower absolute humidity in Winter causes expelled
  virus particles to persist in the air for long
  periods.
• (Shaman and Kohn, PNAS, 2008)

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Why ???

• Prevalence of pathogen ?
• Virulence of pathogen ?
• Behavior of host ?
• (e.g., kids have no school during summer and
  Christmas, and have fewer contacts)
• Mechanism assumed by most measles modelers
  (evidence?)
• Seasonal dependence in host susceptibility?
• (Scott F. Dowell, Emerging Infectious Diseases,
  2001)

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Our goal understand the seasonal dependence
through studying the time-dependent transmission
rate.
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Transmission rate of an ID

• An effective contact is any kind of contact
  between two individuals such that, if one
  individual is infectious and the other
  susceptible, then the first individual infects
  the second.
• The transmission rate of an ID in a given
  population is the of effective contacts per
  unit time.
• The transmission rate is the rate at which
  susceptibles become infected.

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How to estimate transmission rate from infection
data?

• Almost all authors use an SIR-type mathematical
  model.
• SIR models assume homogeneous or mass-action
  mixing of infectives and susceptibles.
• If the number of susceptibles/infectives doubles,
  so does the number of new infectives.

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Basic SIR transmission model
?
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Basic SIR transmission model
Infectives recover (with permanent immunity) at
rate ?. Thus the duration of infection is 1/?.
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• In their textbook Infectious Diseases of
  Humans, Anderson and May stated
• ... the direct measurement of the transmission
  coefficient ? is essentially impossible for most
  infections. But if we wish to predict the changes
  wrought by public health programmes, we need to
  know the transmission coefficient.''  

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Time dependent transmission coefficient

• For many acute IDs, the transmission coefficient
  is time dependent.
• We will consider ?(t).

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SIR model with time dependent transmission rate
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Inverse problem

• Given smooth f(t) gt 0 defined on 0, T, and ? gt
  0, does there exist ?(t) gt 0 in SIR model such
  that I(t) f(t) for 0 t T?

We prove YES, with mild necessary and sufficient
condition
Mark Pollicott, Hao Wang, and Howie Weiss.
Extracting the time-dependent transmission rate
from infection data via solution of an inverse
ODE problem, Journal of Biological Dynamics, Vol.
6 509-523 (2012)
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• This result clearly shows a serious danger in
  overfitting transmission models

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Explicit Inversion formula
Works provided denominators ? 0
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Underdetermined inverse problem

• Inversion requires that

and
There are infinitely many solutions.
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Instead of
the actual necessarysufficient condition is
because of equation (2) (for I)
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But infection data is discrete

• First apply your favorite smooth interpolation
  method (spline, trig, rational, etc. ) to
  smooth the data and then apply the inversion
  formula

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Interpolation
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Two artificial examples
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Robustness

• Simulations show that the recovery algorithm with
  any reasonable interpolation method is robust
  with respect to white noise up to 10 of the data
  mean, as well as the number and spacing of sample
  points.

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Derivation
We require I(t)f(t)
Solve (2) for S(t) and plug into (1)
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Derivation, continued
Bernoulli equation - has closed form solution
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Bernoulli equation
The change of coordinates x 1/? transforms
this nonlinear ODE into a linear ODE
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Solution
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Application to Measles

• Respiratory system disease caused by
  paramyxovirus.
• Spread through respiration.
• Highly contagious.
• R0 12-18
• Virus causes Immuno-suppression
• Characteristic measles rash
• Infectivity from 2-4 days prior, until 2-5 days
  following onset the rash
• average incubation period of 14 days

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Measles mortality

• Mortality from measles for otherwise healthy
  children in developed countries 0.3.
• In developing countries with high rates of
  malnutrition and poor healthcare, mortality has
  been as high as 28
• According to WHO, in 2007 there were 197,000
  measles deaths worldwide.

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Weakly measles cases in seven UK cities 1948-58
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Aggregated UK measles data
Notice pronounced biennial and annual spectral
peaks
What is driving the biennial cycle?
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Modeling pre-vaccination measles transmission
SEIR model with vital rates
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Measles ?(t) from transmission modeling literature
All measles modelers assume that ?(t) is solely
determined by school mixing, and choose ?(t) to
be pure sine function or Haar function with one
year period.
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Extended recovery algorithm
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Parameterizing the measles transmission model

• We chose parameters for measles from
• Anderson May
• ?52/year 52/12/month,
• a52/year52/12/month,
• ?1/70/year1/70/12/month,
• where 1/? is the period of infectiousness, 1/a is
  
• the latent period, ? is the birth rate.
• ?(0) from measles modeling literature

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Recovered ?(t) with constant birth
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Extended recovery algorithm with historical birth
rates
All other steps in the algorithm remain the same
except in Step 4
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UK births from 1948-57
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Recovered ?(t) with actual births
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With corrected data
To test the robustness of our spectral peaks, we
incorporate the standard correction factor of
92.3 to account for the underreporting bias in
the UK measles data (with estimated mean
reporting rate 52, note that 92.3 is computed
from 1/0.52 - 1).
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Recall, all measles transmission models assume
?(t) is solely determined by school mixing, and
choose ?(t) to be pure sine function or Haar
function with one year period. The period
1/3 year seems related to internal events (three
big holidays in UK) within each year.
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Comparison with Haar ?(t)
Summer Low points are consistent.
Earn et al., Science, 2000
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Cities test (constant B.R.)
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Final Comments and Open Problems

• Study statistical properties of the estimator for
  ?(t)
• The idea can be applied to almost any ID
  transmission model (waning immunity, indirect
  transmission mode, more classified groups, etc.)
• Apply to other data sets
• Stochastic version of the algorithm
• Examine why different UK cities have quite
  different dominant frequencies

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Key references

1. Bailey (1975)
2. Deitz (1976)
3. Schwartz and Smith (1983)
4. Anderson and May (1992)
5. Bolker and Grenfell (1993)
6. Keeling and Grenfell (1997)
7. Rohani, et al. (1999)
8. Earn et al. (2000)
9. Keeling et al. (2001)
10. Finkenstadt and Grenfell (2002)
11. Bauch and Earn (2003)
12. Dushoff, et al.(2004)

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Coauthors
Mark Pollicott (Warwick, UK)
Howie Weiss (Georgia Tech, US)