Transcript Mathematical Modelling of Infectious Diseases and Decision

Mathematical Modeling
for understanding and predicting communicable
diseases:
a tool for evidence-based health policies
Antoine Flahault
Geneva, May 20, 2015
Model = simplified miroring of true world
Model = « drawing board »
True world
validation
Understanding
Early warning
Prevision
Observation
calibration
Theory
(models)
 dS j
 1 (1   1 )(Ci ) j   (     ) S j I j  ( S j )

 dt
d
(
E

n)j
 dt   (1   ) S j I j   ( En ) j  (( En ) j )

 d ( Ei ) j
 dt  S j I j  ( 1 2   (1   2 ))( Ei ) j  (( Ei ) j )

 d (Ci ) j   (1   ) S I   (C )  ((C ) )
j j
1
i j
i j
 dt

 dI j   ( E )  (   (1   )) I
n j
j
 dt

 dQ j  I   (1   )( E )   Q
j
2
i j
2 j
 dt
 dU
j

  (1   ) I j   2Q j
 dt
 dV
 j  1 ( 2 ( Ei ) j   1 (Ci ) j )  (V j )
 dt
2
Simulation
of scenarios
Better Understanding
Early Warning
Mathematical Theory
of Communicable
Diseases
Simple Compartmental Model
susceptible
.c
Infectious
1/d
Removed
1927: Kermack & McKendrick
SIR : deterministic formulation
dX
  cXY / N
dt
dY
 cXY / N  (1 / d )Y
dt
dZ
 (1 / d )Y
dt
Threshold Theorem
dY
 cX Y N  (1 d).Y  0 
dt
Epidemic

Probability of transmission
c
Number of contacts per unit of time
d
Duration of Infectious period
R0  cd  1
d.Ln(2)
Td 
R0  1
Basic Reproductive Rate
Doubling time
Ro: Early warning signal for
outbreak
e.g. Seasonal Influenza (Sentinelles, France)
d  4d
d .Ln(2)
d .Ln(2)  Td
Td 
 3d  R 0 
2
R0  1
Td
A doubling of incidence in 3 days =>
Ro > 1 <=> detection of an outbreak
Applying the threshold theorem to
vaccination schedule
Measles: What is optimal age for vaccination?
ln(1 D)  ln(1 A)

T
1 D  1 A
D
A
Duration of protection from maternal antibodies
Mean age of wild cases
Example: Measles in developing countries, D=6 months,
A=18 months, then T = 10 months
(Katzmann & Dietz, 1984)
Ro: « Richter scale » for
communicable diseases?
Measles
Influenza
Smallpox
SARS
Hepatitis B
- High risk groups
- General Population e
Ro= 15 à 20
Ro= 1.4 à 2
Ro= 3
Ro= 2
Ro= 4 à 8.8
Ro= 1.1
SARS at Singapore (Lloyd-Smith, 2005)
Immunization Strategies: heard immunity
– What proportion of population needs to be
immunized to prevent an outbreak?
p  (1  1 R 0 )
Measles
Influenza
Hepatitis B
- High risk groups
- Low risk groups
- Very high risk groups
(R0 = 15-20) p = 93-95%
(R0 = 2-4) p = 50-75%
(R0 = 4)
(R0 = 1.1)
(R0 = 8.8)
p = 75%
p = 10%
p = 89%
Measles in France
1984 - 2004
Governmental
Media campaign
1988
Change in Vacc Schd
2nd dose
2nd dose
at 11-13 yrs at 3 - 6 yrs
Sept. 1996 April. 1998
Source : réseau Sentinelles, Inserm
Modeling: Scenarios simulating
decrease in vaccine coverage
4
4
4
x 10
4
x 10
Linear decrease in 5%
up to 2010
3
Nombre de cas
Nombre de cas
No change
2
1
0
3
2
1
0
1990 2000 2010 2020 2030 2040 2050
4
3
Linear decrease in 10%
up to 2010
2
1
0
4
x 10
1990 2000 2010 2020 2030 2040 2050
4
Nombre de cas
Nombre de cas
4
1990 2000 2010 2020 2030 2040 2050
3
x 10
Linear decrease in 20%
up to 2010
2
1
0
1990 2000 2010 2020 2030 2040 2050
(H. Sarter, 2004)
Agent centered models
Ferguson et al. : the largest simulation on
computer ever published
• Simulation of an 85 Mn population living
in Thailand
• 10 high capacity computers in parallel
• > 1 month of CPU time
Prevention and Control of
pandemic influenza
Basic Reproductive Rate
R0 = .c.d
probability of
transmission
contact
rate
duration of
infectious period
Threshold Theorem, pandemic when R0 > 1
Antivirals (curative, preventive)
Protective masks
Hand washing
Vaccines (when available)
Increase « social distance »
– Quarantine of patients
– Closing schools
– Reduction of transport
Flahault A et coll, Vaccine 2006
Decrease in duration of infectious period (2.6d)
– Antivirals
– Anti coughing
Cauchemez S, Stat Med, 2004
Ferguson N, Nature 2005
Expected pattern of spread of an uncontrolled epidemic Ro=1.5.
(a) Spread of a single simulation. Red = infectives, green = recovered from infection or died.
(b) Daily incidence of infection. Thick blue line = average, grey shading = 95% envelope of incidence timeseries.
Multiple coloured = a sample of realisations.
(c) Root Mean Square (RMS) distance from seed infective of all individuals infected since the start of the epidemic
as a function of time.
(d) Attack rate by age (mean = 33%).
(e) Number of secondary cases per primary case
Understanding better, detection=ok,
but
what about prediction?
Ebola
http://dougrobbins.blogspot.fr/2014/11/charting-2014-ebola-epidemic.html
Ebola
http://dougrobbins.blogspot.fr/2014/11/charting-2014-ebola-epidemic.html
The black swan !
Nassim Nicholas Taleb, 2010
Smallpox (bioterrorism) : modeling
Cumul. No Cases
Duration
235 [190;310] days
Doses of vaccine
5 440 [3 910;6 840]
Isolated people
550 [415;686]
Legrand J, Epidemiol Infect 2004
Number of days after the attack
27
Key role of time to intervention
(in terms of epidemic size)
Cumul. No Cases
Isolation rate (%)
Tracking rate (%)
Reference scenario
28
Time to intervention (days)
To detect and forecast seasonal
influenza
A series of epidemics
Each winter
An average of
2.5 M cases
6 M influenza cases
6000 deaths
Method of Analogues:
Forecasting Time-Series
I(D+1)
1500
weekly ILI
incidence
/ 100 000
F(T+1)
1000
I(D)
I(T)
I(B)
X(B)
500
X(D)
I(B+1)
I(C+1)
I(T-1)
I(D-1)
I(B-1)
X(A)
I(A-1)
I(C)
X(C)
I(A)
X(T)
I(C-1)
I(A+1)
0
0
50
100
150
week number
200
250
300
807
Time-Space Prediction of Influenza
Method of Analogues
Viboud C. et coll. Am J Epidemiol,
2003
Viboud C. et coll., 2003
H1N1pdm: predictions?
R0 < 2
Gen. Time Interval= 3d
And what next?
New-Zealand (May-Oct./2009)
Scenario without immunization
Flahault et al. BMC Inf Dis, August 2009
Chikungunya,
Indian Ocean
2005-2007
Why such a strong second
wave (en 2006)?
Genome sequencing of
chikungunya virus
Indian Ocean : 92 sequences from 89 patients
Date
Sequences
A226
V226
March to June 2005
19
19
0
September to December
27
0
27
January to March 2006
46
6
40
Mutation from A226 to V226 between the 2 waves
Schuffenecker I et al., PLoS Medicine, 2006
Boelle et al., Vect Born Zoon Dis, 2007
2005-13
2005-17
2005-21
2005-25
2005-29
2005-33
2005-37
2005-41
2005-45
2005-49
2006-1
2006-5
2006-9
2006-13
Two waves
but a unique
epidemic force
0
2
4
6
8
10
Reproduction number
Incidence
50000
10000
5000
1000
500
100
50
10
3<R<4
Week
The convergence model
(Source IOM, 2003)
Conclusion
Modelling = Math Epidemiology
Mathematical models
• Allow for targeting epidemiologic surveillance
(observation)
• Help better understanding communicable diseases
• Provide tools for early warning
• Aid public decisions (vaccination, interventions for
prevention and control)
• Help designing/assessing various scenarios for the
future
However they deliver previsions… which remain
previsions!
PhD in Global Health