The SIRI Model in Epidemiology: Pertussis and MalariaEPIDEMIOLOGY ORGANISM-centered biology...

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



Models establish bridges between levels of organisation

RecoveredSusceptible Infected

infection recovery



SIR model – totally protective immunity

e

e e e

S I RR0 I



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.





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.



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





Pertussis - resurgence





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



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



Pertussis - resurgence

0 2 4 6 8 10 12 14 16 Basic reproduction number, R0



Increasing disease with decreasing transmission



Model equations

)(

)(

)(

)()1(

222

1101

22110

000

µτσλ

µτλ

µασλττµ

µλαµ

+−=

+−=

++−++=

+−+−=

IRdtdI

ISdtdI

RIIvdtdR

SRvdtdS



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



Malaria – data from sub-Saharan Africa



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



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



Malaria – sensitivity to reporting rate



Malaria – sensitivity to infectively reduction



Malaria – intervention design



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”



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).



Demographic andGeographicalInformation

A tool for public health policy making José Lourenço - IGC



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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The SIRI Model in Epidemiology: Pertussis and MalariaEPIDEMIOLOGY ORGANISM-centered biology...

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