Additional file I

Electronic Supplementary Material:

SVEQIAHRModel Details and SupplementaryData

Ying-Hen Hsieh

Department of Public Health and Center of Infectious Disease Epidemiology Research, ChinaMedicalUniversity, Taichung, Taiwan

Correspondence to: Dr. Ying-Hen Hsieh, Department of Public Health, China Medical University, Taichung, Taiwan 40402. Tel: 886-4-22053366x6109, Fax: 886-4-22078539, email: .

1. SVEQIAHRModel

The model flowchart is given in Fig. 1, where the subscript i denotes the ith age group. The model is a system of 12×nordinary differential equations,with n age-groups and i=1,2,…,n. The time unit is in days. The model equations are as follows:

(1)

Thefrequency-dependentincidencerates forthe susceptible and vaccinated individuals of age group i, ,aregiven by the respective formulae, (2a)

(2b)

where is the total number of contacts by an individual of age group i at time t.

The model variables are described as follows:

: number of susceptible individuals of the ith age group at time t;

: number of vaccinated individuals of the ith age group at time t;

: number of exposed (infected) individuals of the ith age group at time t;

: number of exposed (infected) vaccinated individuals of the ith age group at time t;

: number of quarantined infected individuals of the ith age group at time t;

: number of infective individuals of the ith age group at time t;

: number of vaccinated infective individuals of the ith age group at time t;

: number of asymptomatic (subclinical) infective individuals of the ith age group at time t;

: number of hospitalized (treated) individuals of the ith age group at time t;

: number of recovered and immune individuals of the ith age group at time t;

: number of death individuals of the ith age group at time t;

cij: contact rate of an individual of ith group with an individual of jth group;

: per contact transmission probability of a susceptible individual of ith group by an infective of jth group;

πi: age-specific vaccine efficacy for age group i.

The other model parameters are listed in Table 1.

2. Reproduction Numbers

First we consider the case at the initial stagewith no intervention (vaccination, quarantine, home withdrawal, etc.), and thus obtain the following equations:

(3)

wherei=1,2,…,n, and.

For brevity, we also redefine to be the former , to be the former , to be the former , to be the former , and the equations are simplified to

(4)

Denoting , the basic reproduction number R0(see, e.g., Diekmann et al. [8] orvan den Driessche and Watmough [9]) for disease-free equilibrium (DFE)is

(5)

where

, . (6)

2.1. Effective Reproduction Number with Interventions

Again, for brevity, the equations are simplified to

(7)

We have the following effective reproduction numbers:

(i)The effective reproduction number with interventions other than vaccination RE is

, where

, (8).

(ii) For model with vaccination but no other nonpharmaceutical interventions, . We have

(9)

The effective reproduction number with vaccination onlyover the time period [0, T], RVis:

, where

(10)

with

,

and

(11)

(iii) The effective reproduction numberwith vaccination RVE over the time period [0, T], is

, where

with

(13)

Table A1. 2005 Taiwan population age structure.

Age groups / Age / population / %
1 / 0-2 / 641095 / 2.82
2 / 3-5 / 809663 / 3.56
3 / 6-7 / 550564 / 2.42
4 / 8-14 / 2257727 / 9.92
5 / 15-21 / 2286293 / 10.04
6 / 22-64 / 14008237 / 61.52
7 / ≥ 65 / 2216804 / 9.74
Total / - / 22770383 / 100.00

Table A2. Contact matrix [cij] used in simulation, obtained from [10] by adjusting for the difference in age distribution between Netherlands 1986 and Taiwan 2005.cij is the daily number of contacts for each individual in age group i with individuals in age group j.

Age group / 0-2 / 3-5 / 6-7 / 8-14 / 15-21 / 22-64 / ≥ 65
0-2 / 26.64 / 26.64 / 4.97 / 4.26 / 6.39 / 8.88 / 2.84
3-5 / 26.44 / 26.44 / 4.78 / 4.50 / 6.47 / 8.72 / 2.53
6-7 / 4.96 / 4.96 / 37.64 / 28.54 / 5.79 / 6.62 / 2.07
8-14 / 4.34 / 4.34 / 28.34 / 24.10 / 12.30 / 7.36 / 2.22
15-21 / 6.37 / 6.37 / 5.98 / 12.15 / 23.40 / 9.46 / 3.49
22-64 / 8.76 / 8.76 / 6.47 / 7.40 / 9.14 / 7.17 / 5.66
≥ 65 / 2.67 / 2.67 / 2.05 / 2.26 / 3.49 / 5.65 / 10.37

References

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  8. Diekmann O, Heesterbeek JAP, Metz JAJ:On the definition and the computation ofthe basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations.J Math Biol 1990, 28:365–382.
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Fig. S1. Daily observed and predicated mortality for age group 6corresponding to the case (last row in bold in Table 2 and Fig. 4).

Fig. S2. Daily observed and predicated mortality for age group 7 corresponding to the case (last row in bold in Table 2 and Fig. 4).

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