A New Algorithm to Extract the Time
Dependent Transmission Rate from
Infection Data
Hao Wang
University of Alberta
MATH 570
from model
Infection
data
smooth
interpolation
Birth data
Algorithm
 (t )
FT to check
dominant
frequencies
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
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).
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.
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
Weekly measles cases in
seven UK cities: 1948-58
Why ???
Changes in atmospheric conditions ?
1. Cholera outbreaks follow monsoons
in south Asia.
2. Lower absolute humidity in Winter
causes expelled virus particles to
persist in the air for long periods.
(Shaman and Kohn, PNAS, 2008)
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)
Our goal: understand the seasonal
dependence through studying the
time-dependent transmission rate.
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.
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.
Basic SIR transmission model
ν
Basic SIR transmission model
Infectives recover (with permanent immunity)
at rate ν. Thus the duration of infection is 1/ν.
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.''
Time dependent transmission
coefficient
For many acute IDs, the transmission
coefficient is time dependent.
We will consider (t).
SIR model with time dependent
transmission rate
Inverse problem
Given smooth f(t) > 0 defined on [0,
T], and  > 0, does there exist (t) >
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)
This result clearly shows a
serious danger in overfitting
transmission models
Explicit Inversion formula
Works provided denominators  0
Underdetermined inverse
problem
Inversion requires that
and
There are infinitely many solutions.
Instead of
the actual necessary&sufficient
condition is
f ' (t)  νf(t)  0
because of equation (2) (for I)
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
Interpolation
Two artificial examples
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.
Derivation
We require I(t)=f(t)
Solve (2)
for S(t)
and plug
into (1)
Derivation, continued
Bernoulli equation - has closed form
solution
Bernoulli equation
The change of coordinates x = 1/
transforms this nonlinear ODE into
a linear ODE
Solution
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
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.
Weakly measles cases in
seven UK cities: 1948-58
Aggregated UK measles data
Notice pronounced biennial and annual
spectral peaks
What is driving the biennial cycle?
Modeling pre-vaccination
measles transmission
SEIR model
with vital
rates
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.
Extended recovery algorithm
Parameterizing the measles
transmission model
We chose parameters for measles from
Anderson & May:
=52/year =52/12/month,
a=52/year=52/12/month,
=1/70/year=1/70/12/month,
where 1/ is the period of infectiousness, 1/a is
the latent period,  is the birth rate.
(0) from measles modeling literature
Recovered (t) with constant birth
Extended recovery algorithm
with historical birth rates
All other steps in the algorithm remain the same except
in Step 4:
UK births from 1948-57
Recovered (t) with actual births
 (t )
 (0)  140
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).
 (t )
 (0)  60
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.
Comparison with Haar (t)
Summer Low points are consistent.
Earn et al., Science, 2000
Cities test (constant B.R.)
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
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)
Coauthors
Howie Weiss
Mark Pollicott
(Georgia Tech, US)
(Warwick, UK)