A New Algorithm to Extract the Time Dependent Transmission Rate from

Infection Data

Hao WangUniversity of Alberta

MATH 570

AlgorithmInfection data

)(tfrom model

smoothinterpolation

FT to checkdominantfrequencies

Birth data

Outline of talk

1. Very brief sales pitch for utility of mathematical models of transmission of IDs in populations

2. Seasonality/periodicity of IDs3. 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 datavia 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

There are infinitely many solutions.

and

Instead of

the actual necessary&sufficientcondition is

0' νf(t)(t)fbecause 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 exceptin Step 4:

UK births from 1948-57

Recovered (t) with actual births

)(t

140)0(

With corrected dataTo 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

60)0(

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 1/3year seems related to internalevents (three big holidays in UK) within each year.

Comparison with Haar (t)

Earn et al., Science, 2000

Summer Low points are consistent.

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 references1. 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

Mark Pollicott

(Warwick, UK)

Howie Weiss

(Georgia Tech, US)