1. Epidemiology and Global Health
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Structure in the variability of the basic reproductive number (R0) for Zika epidemics in the Pacific islands

  1. Clara Champagne  Is a corresponding author
  2. David Georges Salthouse
  3. Richard Paul
  4. Van-Mai Cao-Lormeau
  5. Benjamin Roche
  6. Bernard Cazelles  Is a corresponding author
  1. IBENS, UMR 8197 CNRS-ENS Ecole Normale Supérieure, France
  2. CREST, ENSAE, Université Paris Saclay, France
  3. Institut Pasteur, Unité de Génétique Fonctionnelle des Maladies Infectieuses, France
  4. URA 3012, France
  5. Institut Louis Malardé, France
  6. UPMC/IRD, France
Research Article
Cite this article as: eLife 2016;5:e19874 doi: 10.7554/eLife.19874
21 figures, 7 tables and 5 additional files

Figures

Graphical representation of compartmental models.

Squared boxes and circles correspond respectively to human and vector compartments. Plain arrows represent transitions from one state to the next. Dashed arrows indicate interactions between humans and vectors. (a) Pandey model (Pandey et al., 2013). HS susceptible individuals; HE infected (not yet infectious) individuals; HI infectious individuals; HR recovered individuals; σ is the rate at which HE-individuals move to infectious class HI; infectious individuals (HI) then recover at rate γ; VS susceptible vectors; VE infected (not yet infectious) vectors; VI infectious vectors; V constant size of total mosquito population; τ is the rate at which VE-vectors move to infectious class VI; vectors die at rate μ. (b) Laneri model (Laneri et al., 2010). HS susceptible individuals; HE infected (not yet infectious) individuals; HI infectious individuals; HR recovered individuals; σ is the rate at which HE-individuals move to infectious class HI; infectious individuals (HI) then recover at rate γ; implicit vector-borne transmission is modeled with the compartments κ and λ; λ current force of infection; κ latent force of infection reflecting the exposed state for mosquitoes during the extrinsic incubation period; τ is the transition rate associated to the extrinsic incubation period.

https://doi.org/10.7554/eLife.19874.003
Results using the Pandey model.

Posterior median number of observed Zika cases (solid line), 95% credible intervals (shaded blue area) and data points (black dots). First column: particle filter fit. Second column: Simulations from the posterior density. Third column: R0 posterior distribution. (a) Yap. (b) Moorea. (c) Tahiti. (d) New Caledonia. The estimated seroprevalences at the end of the epidemic (with 95% credibility intervals) are: (a) 73% (CI95: 68–77, observed 73%); (b) 49% (CI95: 45–53, observed 49%); (c) 49% (CI95: 45–53, observed 49%); (d) 39% (CI95: 8–92). See Figure 4.

https://doi.org/10.7554/eLife.19874.004
Results using the Laneri model.

Posterior median number of observed Zika cases (solid line), 95% credible intervals (shaded blue area) and data points (black dots). First column: particle filter fit. Second column: Simulations from the posterior density. Third column: R0 posterior distribution. (a) Yap. (b) Moorea. (c) Tahiti. (d) New Caledonia. The estimated seroprevalences at the end of the epidemic (with 95% credibility intervals) are: (a) 72% (CI95: 68–77, observed 73%); (b) 49% (CI95: 45–53, observed 49%); c) 49% (CI95: 45–53, observed 49%); d) 65% (CI95: 24–91). See Figure 5.

https://doi.org/10.7554/eLife.19874.005
Infected and recovered humans evolution during the outbreak with Pandey model.

Simulations from the posterior density: posterior median (solid line), 95% and 50% credible intervals (shaded blue areas) and observed seroprevalence (black dots). First column: Infected humans (HI). Second column: Recovered humans (HR). (a) Yap. (b) Moorea. (c) Tahiti. (d) New Caledonia.

https://doi.org/10.7554/eLife.19874.013
Infected and recovered humans evolution during the outbreak with Laneri model.

Simulations from the posterior density: posterior median (solid line), 95% and 50% credible intervals (shaded blue areas) and observed seroprevalence (black dots). First column: Infected humans (HI). Second column: Recovered humans (HR). (a) Yap. (b) Moorea. (c) Tahiti. (d) New Caledonia.

https://doi.org/10.7554/eLife.19874.014
Posterior distributions.

Pandey model, Yap island.

https://doi.org/10.7554/eLife.19874.015
Posterior distributions.

Pandey model, Moorea island.

https://doi.org/10.7554/eLife.19874.016
Posterior distributions.

Pandey model, Tahiti island.

https://doi.org/10.7554/eLife.19874.017
Posterior distributions.

Pandey model, New Caledonia.

https://doi.org/10.7554/eLife.19874.018
Posterior distributions.

Laneri model, Yap island.

https://doi.org/10.7554/eLife.19874.019
Posterior distributions.

Laneri model, Moorea island.

https://doi.org/10.7554/eLife.19874.020
Posterior distributions.

Laneri model, Tahiti island.

https://doi.org/10.7554/eLife.19874.021
Posterior distributions.

Laneri model, New Caledonia.

https://doi.org/10.7554/eLife.19874.022
Correlation plot of MCMC output.

Pandey model, Yap island.

https://doi.org/10.7554/eLife.19874.023
Correlation plot of MCMC output.

Pandey model, Moorea island.

https://doi.org/10.7554/eLife.19874.024
Correlation plot of MCMC output.

Pandey model, Tahiti island.

https://doi.org/10.7554/eLife.19874.025
Correlation plot of MCMC output.

Pandey model, New Caledonia.

https://doi.org/10.7554/eLife.19874.026
Correlation plot of MCMC output.

Laneri model, Yap island.

https://doi.org/10.7554/eLife.19874.027
Correlation plot of MCMC output.

Laneri model, Moorea island.

https://doi.org/10.7554/eLife.19874.028
Correlation plot of MCMC output.

Laneri model, Tahiti island.

https://doi.org/10.7554/eLife.19874.029
Correlation plot of MCMC output.

Laneri model, New Caledonia.

https://doi.org/10.7554/eLife.19874.030

Tables

Table 1

Parameter estimations for the Pandey model. Posterior median (95% credible intervals). All the posterior parameter distributions are presented in Figures 69 .

https://doi.org/10.7554/eLife.19874.006
Pandey modelYapMooreaTahitiNew Caledonia
Population sizeH689216,200178,100268,767
Basic reproduction numberR03.2 (2.4–4.1)2.6 (2.2–3.3)2.4 (2.0–3.2)2.0 (1.8–2.2)
Observation rater0.024 (0.019-0.032)0.058 (0.048-0.073)0.060 (0.050-0.073)0.024 (0.010-0.111)
Fraction of population involvedρ74% (69–81)50% (48–54)50% (46–54)40% (9–96)
Initial number of infected individualsHI(0)2 (1–8)5 (0–21)329 (16–1047)37 (1–386)
Infectious period in human (days)γ-15.2 (4.1–6.7)5.2 (4.1–6.8)5.2 (4.1–6.7)5.5 (4.2–6.8)
Extrinsic incubation period in mosquito (days)τ-110.6 (8.7–12.5)10.5 (8.6–12.4)10.5 (8.6–12.6)10.7 (8.9–12.5)
Mosquito lifespan (days)μ-115.6 (11.7–19.3)15.3 (11.4–19.1)15.1 (11.2–19.0)15.4 (11.6–19.4)
Table 2

Parameter estimations for the Laneri model. Posterior median (95% credible intervals). All the posterior parameter distributions are presented in Figures 1013.

https://doi.org/10.7554/eLife.19874.007
Laneri modelYapMooreaTahitiNew Caledonia
Population sizeH689216,200178,100268,767
Basic reproduction numberR02.2 (1.9–2.6)1.8 (1.6–2.0)1.6 (1.5–1.7)1.6 (1.5–1.7)
Observation rater0.024 (0.019–0.033)0.057 (0.047–0.07)0.057 (0.049–0.069)0.014 (0.010–0.037)
Fraction of population involvedρ73% (69–78)51% (47–55)54% (49–59)71% (27–98)
Initial number of infected individualsHI(0)2 (1–10)9 (1–28)667 (22–1570)82 (2–336)
Infectious period in human (days)γ-15.3 (4.1–6.6)5.3 (4.1–6.7)5.2 (4.1–6.7)5.4 (4.1–6.8)
Extrinsic incubation period in mosquito (days)τ-110.7 (8.8–12.7)10.6 (8.6–12.6)10.5 (8.5–12.5)10.8 (8.9–12.8)
Table 3

Details of the observation models for seroprevalence

https://doi.org/10.7554/eLife.19874.008
IslandDateStandard deviationObserved seroprevalence
ΛHRobs
Yap2007-07-291505005 (Duffy et al., 2009)
Moorea2014-03-283250.49 × 16200 = 7938 (Aubry et al., 2015b)
Tahiti2014-03-2835620.49 × 178100 = 87269 (Aubry et al., 2015b)
Table 4

Prior distributions of parameters. 'Uniform[0,20]' indicates a uniform distribution in the range [0,20]. 'Normal(5,1) in [4,7]' indicates a normal distribution with mean five and standard deviation 1, restricted to the range [4,7].

https://doi.org/10.7554/eLife.19874.009
ParametersPandey modelLaneri modelReferences
R02squared basic reproduction numberUniform[0, 20]Uniform[0, 20]assumed
βVtransmission from human to mosquitoUniform[0,10].assumed
γ-1infectious period (days)Normal(5,1) in [4,7]Normal(5,1) in [4,7](Mallet et al., 2015)
σ-1intrinsic incubation period (days)Normal(4,1) in [2,7]Normal(4,1) in [2,7](Nishiura et al., 2016b; Bearcroft, 1956; Lessler et al., 2016)
τ-1extrinsic incubation period (days)Normal(10.5,1) in [4,20]Normal(10.5,1) in [4,20](Hayes, 2009; Chouin-Carneiro et al., 2016)
μ-1mosquito lifespan (days)Normal(15,2) in [4,30].(Brady et al., 2013; Liu-Helmersson et al., 2014)
ρfraction of population involvedUniform[0,1]Uniform[0,1]
Initial conditions (t=0)Pandey modelLaneri model
HI(0)infected humansUniform[10-6,1]NUniform[10-6,1]N
HE(0)exposed humansHI(0)HI(0)
HR(0)recovered humans00
infected vectorsVI(0)=Uniform[10-6,1]HL(0)=Uniform[10-6,1]N
exposed vectorsVE(0) = VI(0)K(0)=L(0)
Local conditionsYapMooreaTahitiNew CaledoniaReferences
robservation rateUniform[0,1]Uniform[0,1]Uniform[0,0.3]Uniform[0,0.23](Mallet et al., 2015; DASS, 2014)
Hpopulation size6,89216,200178,100268,767(Duffy et al., 2009; Insee, 2012, 2014)
Table 5

Square root of the number of secondary cases after the introduction of a single infected individual in a naive population. Median and 95% credible intervals of 1000 deterministic simulations using parameters from the posterior distribution.

https://doi.org/10.7554/eLife.19874.010
Pandey modelLaneri model
Yap3.1 (2.5–4.3)2.2 (1.9–2.6)
Moorea2.6 (2.2–3.3)1.8 (1.6–2.0)
Tahiti2.4 (2.0–3.2)1.6 (1.5–1.7)
New Caledonia2.0 (1.8–2.2)1.6 (1.5–1.7)
Table 6

Sensitivity analysis in Pandey model. Tahiti island. 1000 parameter sets were sampled with latin hypercube sampling (LHS), using 'lhs' R package (Carnell, 2016). On each parameter set, the model was simulated deterministically in order to explore variability in parameters without interference with variations due to stochasticity. PRCC were computed using the 'sensitivity' R package (Pujol et al., 2016).

https://doi.org/10.7554/eLife.19874.011
ParametersDistributionSeroprevalencePeak intensityPeak date
Model parameters
R02Uniform[0,20]0.870.90−0.55
βVUniform[0,10]−0.66−0.730.35
γ-1Uniform[4,7]−0.250.100.20
σ-1Uniform[2,7]−0.03−0.100.15
τ-1Uniform[4,20]−0.05−0.070.06
μ-1Uniform[4,30]−0.56−0.700.49
Initial conditions
HI(0)Uniform[2.10-5,0.011]0.05−0.020.02
VI(0)Uniform[10-4,0.034]0.11−0.00−0.26
Fraction involved and observation model
ρUniform[0.46,0.54]0.470.15−0.03
rUniform[0.048,0.072]−0.040.030.05
Table 7

Sensitivity analysis in Laneri model. Tahiti island. 1000 parameter sets were sampled with latin hypercube sampling (LHS), using 'lhs' R package (Carnell, 2016). On each parameter set, the model was simulated deterministically in order to explore variability in parameters without interference with variations due to stochasticity. PRCC were computed using the 'sensitivity' R package (Pujol et al., 2016).

https://doi.org/10.7554/eLife.19874.012
ParametersDistributionSeroprevalencePeak intensityPeak date
Model parameters
R02Uniform[0,20]0.620.93−0.50
γ-1Uniform[4,7]0.010.620.15
σ-1Uniform[2,7]−0.03−0.540.21
τ-1Uniform[4,20]−0.03−0.700.47
Initial conditions
HI(0)Uniform[10-5,0.015]0.050.02−0.32
L(0)Uniform[2.10-5,0.004]0.050.00−0.16
Fraction involved and observation model
ρUniform[0.49,0.59]0.800.340.02
rUniform[0.048,0.068]−0.010.01−0.02

Additional files

Supplementary file 1

Codes for the implementation of each model.

https://doi.org/10.7554/eLife.19874.031
Supplementary file 2

Data file, Yap island.

Number of cases per week and number of immune individuals at the end of the epidemic (Duffy et al., 2009).

https://doi.org/10.7554/eLife.19874.032
Supplementary file 3

Data file, Moorea island.

Number of cases per week and number of immune individuals at the end of the epidemic (Mallet et al., 2015; Aubry et al., 2015b).

https://doi.org/10.7554/eLife.19874.033
Supplementary file 4

Data file, Tahiti island.

Number of cases per week and number of immune individuals at the end of the epidemic (Mallet et al., 2015; Aubry et al., 2015b).

https://doi.org/10.7554/eLife.19874.034
Supplementary file 5

Data file, New Caledonia.

Number of cases per week (DASS, 2014).

https://doi.org/10.7554/eLife.19874.035

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