Infectious disease modelling —some data challenges

Birgitte Freiesleben de Blasio Dept. of Infectious Disease Epidemiology,

Norwegian Institute of Public Health

/Dept. of Biostatistics, University of Oslo

Outline

• Infectious disease modelling

– Compartmental models SIR

– Data challenges: who acquires infection from whom?

• Example:

– Effects of vaccines and antivirals during the 2009 H1N1 pandemic in Norway

– Data: surveillance data, vaccine uptake, antivirals

Compartmental models: Susceptible-Infected-Recovered (SIR)

• 1927 Kermarck & McKendrick

• SIR model

• Threshold

– R_0 basic reproductive number

S I R

Suscept. Infected Recovered

W.O.Kermack A.G.McKendrick

1898-1970 1876-1948

γ λ

dS/dt = - β I S

dI/dt = β I S – γ I

dR/dt = γ I

Population N=S+I+R

Initial conditions (S(0),I(0),R(0))

Susceptible (S) Infected (I) Removed(R)

SIR model: differential equations

= β I

SIR model: epidemic output

Basic reproductive number R_0 intuitively ….

expected number of secondary infections arising from a single infected individual during the infectious period in a fully susceptible population

R_0 = p * c * D

p: transmission probability per exposure

c: number of contacts per time unit

D: duration of infectiousness

R_0 > 1 epidemic

R_0 = 1 endemicity

R_0 < 1 die out

Basic reproductive number R_0

• Crit. vaccination coverage to prevent epidemic 1-1/R0

• Exponential growth rate in early epidemic

• Peak prevalence of infected

• Final proportion of susceptible

2 1.778 1.556 1.333 1.111 0.8889 0.6667 0.4444 0.2222 0

0

Time

Pro

po

rtio

n o

f p

op

ula

tio

n

R0=2 R0=3

R0=5 R0=10

0.2

0.4

0.6

0.8

1

R_0 for some selected infections

Infection R_0

Varicella 10-12

Measles 16-18

Rotavirus 16-25

Smallpox 3-10

Spanish flu 2.0 [1.5 – 2.8]

Seasonal

influenza

1.3 [0.9 – 1.8]

A H1N1 swine

flu 2009

1.2 – 1.5 Smallpox,

Bangladesh 1973

Erradicated 1979

Interventions

• Reduce R0

R_0 = p * c * D

p: transmission probability per exposure (masks, condoms)

c: number of contacts per time unit (school closure)

D: duration of infectiousness (antivirals)

• Reduce the proportion of susceptible

– vaccination

Data challenges: who’s acquiring infection from whom ?

• Directly transmitted infections require

contact between individuals

• Knowledge about contact patterns is a necessary for accurate model predictions

• Which contacts are important for the spread of infectious diseases? (household, work school, random encounters …)

Social mixing, WAIFW matrix estimation of R0

• Social mixing matrix C

– Contact rates c_ij= m_ij/pop_i

– Transmission matrix Beta; beta_ij=q* c_ij

• Next generation matrix G

G = (N*D/L)*Beta

• Basic reproductive number

R0 = Max(Eig(G))

Contact structure data

• Naive approach: assume homogeneous mixing

• Surveys (POLYMOD)

– Large scale empirical data

• Simulation of virtual society

– Inferring structure from socio-demographic data

POLYMOD STUDY: Smoothed Contact Matrices Based on Physical Contacts

Mossong J, Hens N, Jit M, Beutels P, et al. (2008) Social Contacts and Mixing Patterns Relevant to the Spread of Infectious

Diseases. PLoS Med 5(3): e74. doi:10.1371/journal.pmed.0050074

http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.0050074

Relative change in R0 from the week to the weekend for all contacts and close contacts '*' indicating a significant relative change in R0.

All contacts Close contacts

Country

Number of participant

s in weekend vs week

Total No. Relative

Change in R0

95% Bootstrap

CI.

Relative Change in

R0

95% Bootstrap

CI.

BE 202/544 746 0.78* 0.64, 0.94 0.88* 0.86, 0.93 DE 266/1041 1307 1.02 0.83, 1.21 1.03 0.68, 1.39 FI 283/716 999 0.78 0.73, 1.16 0.88 0.85, 1.18

GB 258/710 968 0.88* 0.69, 0.90 0.95* 0.74, 0.97 IT 226/614 840 0.80* 0.63, 0.82 0.79* 0.68, 0.99 LU 205/788 993 0.74* 0.70, 0.74 0.88* 0.66, 0.89 NL 68/189 257 0.78* 0.59, 0.79 0.79* 0.62, 0.81 PL 280/722 1002 0.77* 0.66, 0.89 0.84* 0.71, 0.86

Hens et al. BMC Infectious Diseases 2009 9:187 doi:10.1186/1471-2334-9-187

Epidemic curves showing the prevalence of symptomatic infections for unmitigated pandemic versus implementing a 12-week school closure with R0=1.5, 2.0 and 2.5.

Simulated Social Contact Matrices based on demographic data

• Households

– Frequencies of house size and type, age of household members by size

• School, work

– Rates of employment/inactivity and school attendance by age, structure of educational system, school and workplace size distribution

• General population

– Random contacts

Example: Household

Simulated Social Contact Matrices based on demographic data Fumanelli et al. Plos Comp. Biol. 2012

FLU:

30% households

18% schools

19% workplaces

33% general contacts in the pop.

Example: Estimating the effect of vaccination and antivirals during

the 2009 H1N1 pandemic in Norway

Timeline of 2009 pandemic in Norway

• Influenza-like-illness (ILI) rate : weekly % of patients with ILI

• Sentinel: 200 GPs throughout the country (15% of the population)

ILI data, purchase of antivirals and vaccine uptake (week 40-week 2)

VACCINE

ANTIVIRALS

ILI

SEIR model (susceptible-exposed-infected-recovered)

• Population 4.86 million

• Age-structured 0-14y (18.9%)

15-64y (66.4%)

65+y (14.7%) 60% assumed immune

• POLYMOD mixing matrix (European study)

• Symptomatic: 65% children; 55% adults

Vaccination

• Pandemrix – Adjuvanted vaccine: improve immunogenecity

– Rapid strong response

• Random vaccination within age groups – Daily data on # vaccinated

– Effect of vaccination: Delay of 7 days

• Effect of vaccination – Susceptibility VE_sus : 0.8 (<65y) ; 0.55 (65+y)

– Infectiousness VE_inf : 0.15

– Disease VE_d : 0.6

– Duration of infectious stages reduced by 1 day

Antivirals

Tamiflu (Relenza)

– Prophylactic (to avoid infection) 10 days

– Treatment of symptoms 5 days

• Effect?

Model fitting

Estimated model parameters

Antiviral

use

p_use

Suscept.

children

(rel)

Infect.

children

(rel)

R_0

RSS

50.0% 1.051 1.207 1.371 1.1721

37.5% 1.058 1.175 1.388 1.1915

25.0% 1.059 1.178 1.392 1.2319

Conclusion

• Mathematical modelling of the spread of infectious diseases is an important tool for planning and preparedness

• Accurate characterization of the structure of social contacts is key element

• Registry data are vital to inform the models