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Basic reproduction number for vector-borne diseases

with a periodic vector population

Nicolas Baca¨ er

IRD (Institut de Recherche pour le D´ eveloppement)

Bondy, France

ECDC meeting on chikungunya modeling, 28 April 2008

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N. Baca¨ er (2007) Approximation of the basic reproduction

number for vector-borne diseases with a periodic vector

population. Bull. Math. Biol. 69, 1067-1091.

S(t), E(t), I(t), R(t): humans

S′(t), E′(t), I′(t): mosquitoes

P = S(t) + E(t) + I(t) + R(t)

P′(t) = S′(t) + E′(t) + I′(t)

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dS′

dt

dE′

dt

dI′

dt

dS

dt

dE

dt

dI

dt= δ E(t) − αI(t)

dR

dt

= λ(t) − β S′(t)I(t)

P

− µS′(t)

= β S′(t)I(t)

P

− (γ + µ)E′(t)

= γ E′(t) − µI′(t)

= −β I′(t)S(t)

P

= β I′(t)S(t)

P

− δ E(t)

= αI(t)

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Maximum and minimum temperature, rainfall

(Sainte Marie, La R´ eunion)

JJFFMMAAMMJJJJAASSOONNDD

20052006

15

20

25

30

0

100

200

300

400

500

600

700

800

900

1000

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fixed parameters

latent period in vectors

life expectancy of vectors

latent period in humans

infectious period in humans

time between two bites

population of La R´ eunion

peak of vector population

1/γ

1/µ

1/δ

1/α

1/β

P

φ

7 days

1 month

4 days

7 days

4 days

785,000

2π

12(beginning of February)

P′(t) = P′

P′

P′

0(1 + εcos(ωt − φ))

max= P′

0(1 + ε)

min= P′

0(1 − ε)