This document discusses infectious disease modeling and some of the data challenges involved. It provides an overview of compartmental SIR models and the basic reproductive number R0. Accurately characterizing social contact patterns is important for modeling disease spread. The document presents an example of estimating the effects of vaccines and antivirals during the 2009 H1N1 pandemic in Norway using surveillance data in an age-structured SEIR model informed by social contact patterns. The model was fitted to data on influenza-like illness, antiviral use, and vaccine uptake.

1. 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

2. 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

3. 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

4. γλ
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

5. SIR model: epidemic output

6. 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

7. 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
21.7781.5561.3331.1110.88890.66670.44440.22220
0
Time
Proportionofpopulation
R0=2R0=3
R0=5R0=10
0.2
0.4
0.6
0.8
1

8. 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

9. 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

10. 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 …)

11. 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))

12. Contact structure data
• Naive approach: assume homogeneous mixing
• Surveys (POLYMOD)
– Large scale empirical data
• Simulation of virtual society
– Inferring structure from socio-demographic data

13. 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

14. 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

15. 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.

16. 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

17. Example: Household

18. 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.

19. Example: Estimating the effect of
vaccination and antivirals during
the 2009 H1N1 pandemic in
Norway

20. 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)

21. ILI data, purchase of antivirals and vaccine
uptake (week 40-week 2)
VACCINE
ANTIVIRALS
ILI

22. 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

23. 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

24. Antivirals
Tamiflu (Relenza)
– Prophylactic (to avoid infection) 10 days
– Treatment of symptoms 5 days
• Effect?

25. Model fitting

26. 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

27. 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