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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)

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Varying t0, P′

(t0February, βP′max/P = 1.2/week, P′

minand P′maxfor δ E(t) to fit the incidence

min/P′max= 6%)

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

100

200

300

400

500

0

JJFFMMAAMMJJJJAASSOONNDD

20052006

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Different ways of getting R0≃ 3.4

Linearization near the disease-free situation

where S(t) = P and S′(t) = P′(t)

de′

dt

di′

dt= γ e′(t) − µi′(t)

de

dt= β i′(t) − δ e(t)

di

dt= δ e(t) − αi(t)

= β P′(t)i(t)

P

− (γ + µ)e′(t)

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Method 1: Floquet

dX

dt

=

−(γ + µ)

γ

0

0

00

0

β

P

P′(t)

R0

0

0

−α

−µ

β

0

−δ

δ

X(t)

X(0) = 14,4

spectral radius ofX(T) = 1 ,T = 2π/ω

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Method 2: “McKendrick”

e′(t,0)

i′(t,0)

e(t,0)

i(t,0)

=

?∞

0

00

0

0

0

0

β P′(t)

P

e−αx

γe−(γ+µ)x

0

0

0

0

0

βe−µx

0

δe−δx

e′(t − x,0)

i′(t − x,0)

e(t − x,0)

i(t − x,0)

dx

R0ψ(t) =

?∞

0

K(t,x)ψ(t − x)dx

ψ(t + T) = ψ(t)

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Method 3: cyclic + Fourier + sinusoidal

R0

z0

− 1 = 2ℜ

ε2/4

R0

z1

− 1 −

ε2/4

− 1 −ε2/4

R0

z2

···

zk=

β2(P′

0/P)γ δ

(α + kiω)(µ + kiω)(γ + µ + kiω)(δ + kiω)

R0 ≃

ε→0z0+ε2

2ℜ

z0z1

z0− z1

Remark: R0≃ 3.4 but z0≃ 3.9.

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Varying the time of introduction of the disease

0

1

01234

0

1

0

1

01234

0

1

0

1

01234

0

1

0

1

01234

0

1

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Suppose R0= 0.95: small epidemics still possible

0

0.0002

01234

0

1

0

0.0002

01234

0

1

0

0.0002

01234

0

1

0

0.0002

01234

0

1

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Determistic vs stochastic models

dπk

dt

= a(t)(k−1)πk−1(t)−[a(t)+b(t)]k πk(t)+b(t)(k+1)πk+1(t)

πk0(t0) = 1

π0(t) =

1 −

e−?t

t0a(τ)e−?t

t0(b(τ)−a(τ))dτ

1 +?t

τ(b(σ)−a(σ))dσdτ

k0

R0=

?T

0a(τ)dτ/

?T

0b(τ)dτ

If R0< 1 then π0(t) → 1 as t → ∞.

If R0> 1 then π0(t) → f(t0,k0) as t → ∞.

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References

• N.B., S. Guernaoui: The epidemic threshold of vector-borne diseases

with seasonality - The case of cutaneous leishmaniasis in Chichaoua,

Morocco. J. Math. Biol. 53, 421–436 (2006)

• N.B.: Approximation of the basic reproduction number R0 for vector-

borne diseases with a periodic vector population, Bull. Math. Biol. 69,

1067–1091 (2007)

• N.B., R. Ouifki: Growth rate and basic reproduction number for popula-

tion models with a simple periodic factor. Math. Biosci. 210, 647–658

(2007)

• N.B., X. Abdurahman: Resonance of the epidemic threshold in a periodic

environment. To appear in J. Math. Biol.

• N.B.:

population models. Submitted to Bull. Math. Biol.

A simple formula for the sensitivity analysis of periodic matrix