Title: A New Algorithm to Extract the Time Dependent Transmission Rate from Infection Data
1A New Algorithm to Extract the Time Dependent
Transmission Rate from Infection Data
- Hao Wang
- University of Alberta
- MATH 570
2from model
Infection data
Algorithm
FT to check dominant frequencies
smooth interpolation
Birth data
3Outline of talk
- Very brief sales pitch for utility of
mathematical models of transmission of IDs in
populations - Seasonality/periodicity of IDs
- The time-dependent transmission rate
(coefficient) of an ID, and a new method to
estimate it from infection data. - Application to Measles - detection of 1/yr cycle
and 3/yr cycle
4Introduction
- 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).
5Mathematical 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.
6Seasonal 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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8Weekly measles cases in seven UK cities 1948-58
9Why ???
- 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)
10Why ???
- 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)
11Our goal understand the seasonal dependence
through studying the time-dependent transmission
rate.
12Transmission 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.
13How 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.
14Basic SIR transmission model
?
15Basic SIR transmission model
Infectives recover (with permanent immunity) at
rate ?. Thus the duration of infection is 1/?.
16- 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.''
17Time dependent transmission coefficient
- For many acute IDs, the transmission coefficient
is time dependent. - We will consider ?(t).
18SIR model with time dependent transmission rate
19Inverse 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)
20- This result clearly shows a serious danger in
overfitting transmission models
21Explicit Inversion formula
Works provided denominators ? 0
22Underdetermined inverse problem
and
There are infinitely many solutions.
23Instead of
the actual necessarysufficient condition is
because of equation (2) (for I)
24But 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
25Interpolation
26Two artificial examples
27Robustness
- 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.
28Derivation
We require I(t)f(t)
Solve (2) for S(t) and plug into (1)
29Derivation, continued
Bernoulli equation - has closed form solution
30Bernoulli equation
The change of coordinates x 1/? transforms
this nonlinear ODE into a linear ODE
31Solution
32Application 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
33Measles 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.
34Weakly measles cases in seven UK cities 1948-58
35Aggregated UK measles data
Notice pronounced biennial and annual spectral
peaks
What is driving the biennial cycle?
36Modeling pre-vaccination measles transmission
SEIR model with vital rates
37Measles ?(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.
38Extended recovery algorithm
39Parameterizing 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
40Recovered ?(t) with constant birth
41Extended recovery algorithm with historical birth
rates
All other steps in the algorithm remain the same
except in Step 4
42UK births from 1948-57
43Recovered ?(t) with actual births
44With 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).
45Recall, 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.
46Comparison with Haar ?(t)
Summer Low points are consistent.
Earn et al., Science, 2000
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48Cities test (constant B.R.)
49Final 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
50Key references
- Bailey (1975)
- Deitz (1976)
- Schwartz and Smith (1983)
- Anderson and May (1992)
- Bolker and Grenfell (1993)
- Keeling and Grenfell (1997)
- Rohani, et al. (1999)
- Earn et al. (2000)
- Keeling et al. (2001)
- Finkenstadt and Grenfell (2002)
- Bauch and Earn (2003)
- Dushoff, et al.(2004)
51Coauthors
Mark Pollicott (Warwick, UK)
Howie Weiss (Georgia Tech, US)