The SIRI Model in Epidemiology: Pertussis and MalariaEPIDEMIOLOGY ORGANISM-centered biology MOLECULAR and CELL biology mathematical models computer models mathematical models computer
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The SIRI Model in Epidemiology:Pertussis and Malaria
Gabriela Gomes
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
EPIDEMIOLOGY
ORGANISM-centered biology
MOLECULAR and CELL biology
mathematical modelscomputer models
mathematical modelscomputer models
animal models
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Models establish bridges between levels of organisation
RecoveredSusceptible Infected
infection recovery
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
SIR model – totally protective immunity
e
e e e
S I RR0 I
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Endemic threshold – R0 = 1
IISRdtdI
eSISRedtdS
−=
−−=
0
0
The basic reproduction number, R0
The expected number of secondary cases arising from a single infectious individual in a totally susceptible population.
6Rubella (England and Wales)
15Rubella (Gambia)
17Measles
4SmallpoxR0Disease
R0 depends on the disease and the population.
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
S I R
(1-v)e ve
R0 Ie e e
Vaccination reduces the susceptibility pool
IISRdtdI
eSISRevdtdS
−=
−−−=
0
0)1(
Partial immunity induces a reinfection threshold, R0 = 1 / σ, above which the prevalence of infection is very high and insensitive to vaccination.
σ: reduction in the risk of infection due to partial immunity.
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Gomes MGM, Franco AO, Gomes MC, Medley GF 2004 The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy. Proc. R. Soc. Lond. B 271, 617-623.
Gomes MGM, Rodrigues P, Hilker FM, Mantilla-Beniers NB, Muehlen M, Paulo ACAS, Medley GF 2006 The potential of post-exposure interventions in global tuberculosis control (submitted).
Rodrigues P, Rebelo C, Gomes MGM 2007 Resistant tuberculosis: A reinfection model. Theoretical Population Biology 71, 196-212.
Tuberculosis – geographical variability
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Pertussis - resurgence
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Águas R, Gonçalves G, Gomes MGM (2006) Pertussis: Increasing disease as a consequence of reducing transmission. Lancet Infectious Diseases 6, 112-117.
S R I2I1λ σ λ
α
τ1
τ2
Ricardo Águas - IGC
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Águas R, Gonçalves G, Gomes MGM (2006) Pertussis: Increasing disease as a consequence of reducing transmission. Lancet Infectious Diseases 6, 112-117.
S R I2I1λ σ λ
α
τ1
τ2
Ricardo Águas - IGC
B(1-v)ρ
vρ
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Pertussis - resurgence
0 2 4 6 8 10 12 14 16 Basic reproduction number, R0
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Increasing disease with decreasing transmission
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Model equations
)(
)(
)(
)()1(
222
1101
22110
000
µτσλ
µτλ
µασλττµ
µλαµ
+−=
+−=
++−++=
+−+−=
IRdtdI
ISdtdI
RIIvdtdR
SRvdtdS
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Age-dependent model equations
)(
)(
)(
)(
2222
11011
22110
0000
µτσλ
µτλ
µασλττµ
µµλα
+−=∂∂
+∂∂
+−=∂∂
+∂∂
++−++=∂∂
+∂∂
++−=∂∂
+∂∂
IRaI
tI
ISaI
tI
RIIvaR
tR
vSRaS
tS
Boundary conditions:
S0(t,0) = µ R(t,0) = I1(t,0) = I2(t,0) = 0
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Malaria – data from sub-Saharan Africa
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Malaria – data from Sub-Saharan Africa
Age in years Age in years
Prop
ortio
nsPr
opor
tions
Prop
ortio
nsPr
opor
tions
Siaya
0
0,05
0,1
0,15
0,2
0,25
0 1 2 3 4 5 6 7 8 9 10
Mponda
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0 1 2 3 4 5 6 7 8 9 10
Foni Kansala
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0,18
0 1 2 3 4 5 6 7 8 9 10
Ifakara
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0 1 2 3 4 5 6 7 8 9 10
Chonyi
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0 2 4 6 8 10
Kilifi
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0 1 2 3 4 5 6 7 8 9 10
Chonyi
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0 2 4 6 8 10
Ifakara
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0 1 2 3 4 5 6 7 8 9 10
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Águas R, White LJ, Snow RW, Gomes MGM 2007 Prospects for sustainable malaria control in sub-Saharan areas of mesoendemic transmission (submitted).
Malaria – sustainability of interventions
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Malaria – sensitivity to reporting rate
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Malaria – sensitivity to infectively reduction
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Malaria – intervention design
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Theoretical EpidemiologyTheoretical Epidemiology
Advanced Courses and Workshops:
http://sites.igc.gulbenkian.pt/ggomes/meetings.phpFirst Summer School on “Mathematics in Biology and Medicine”Advanced Course on “Tuberculosis: Scientific Basis for Control”First Workshop on “Pathogen Diversity and Disease Epidemiology”Science in Society Workshops - “Gripenet”Advanced Course on “Infectious Disease Epidemiology”Second Summer School on “Mathematics in Biology and Medicine”Second Workshop on “Pathogen Diversity and Disease Epidemiology”
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Epidemic monitoring and forecast
• Fast
• Transparent
• Uniform
Marquet RL, Bartelds AIM, van Noort SP, Koppeschaar CE, PagetJ, Schellevis FG, van der Zee J 2006 An internet.based monitoring of influenza-like illness (ILI) in the general population of the Netherlands during the 2003-2004 influenza season. BMC Public Health 6, 242.
van Noort SP, Lourenço J, Rebelode Andrade H, Muehlen M, Gomes MGM 2007 Internet-based surveillance of ILI performs iniformly in the Netherlands, Belgium and Portugal (submitted).
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
Demographic andGeographicalInformation
A tool for public health policy making José Lourenço - IGC
Theoretical EpidemiologyTheoretical Epidemiology
Reinfection ThresholdReinfection ThresholdApril 2005 April 2005 –– March 2009March 2009
parameter estimation
real-time monitoring systems
predictions
demographicgeographic
data
user friendly
interface
FluSpread simulation package
policy maker
modeller
SLIR modelAge structureGeo structure
Immune profile
FluSpread A tool for public health policy making
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