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