Table 6.4: Relation between the degree of periodical variation and average transmission rate in Simulation 1 and 2.
Jakarta Colombo Simulation 1: β1/β0 0.16 0.22 Simulation 2: β1˜/β0 0.46 0.44
Table 6.4 shows that the relationship between periodical variation and the average trans-mission rate leads to similar results in Jakarta and Colombo. These also allow conclusions about the behavior of the mosquito population. Provided the size of the vector population can be modeled byM(t) =M0eχωsin(ω(t+ϕ/52)), the term
eχωsin(ω(t+52ϕ))≈1 +χ ωsin
ω t+ ϕ
52
describes the seasonal variation of the mosquito population. As shown in Section 6.4 in Simulation 1 the fraction ββ1
0 approximately corresponds to the size of the expression χω. Since
χωsin (ω(t+ϕ/52))
≤0.22 applies for ωχ = ββ1
0, we can conclude that the size of the mosquito population varies by a maximum of approximately 25%. In Simulation 2 the term
β1 β0
Z t−η521 t−η2
52
pc(τ)dτ < β˜1 β0
corresponds to a time–dependent expression χ(t)ω withχ:D →R+0. Using the result from Table 6.4 we get
χ(t)
ω sin (ω(t+ϕ/52))
< 0.46 and consequently a variation of at most about 50%.
The transmission rate graphs show the periodicity associated with the dynamics within disease transmission, see Tables 6.6–6.7. The influence of the cut–off parametercis clearly visible. Using the values forβ0 and β1 respectively ˜β1 , restricting intervals for the Basic Reproduction NumberR0(t) can now be determined from equation (6.17), see Table 6.5.
Table 6.5: Limiting intervals to the order of magnitude of the Basic Reproduction Number R0(t) =β(t)/α+µin Jakarta and Colombo.
Jakarta Colombo Simulation 1 [1.5,2.1] [1.4,2.2]
Simulation 2 [1.1,3.0] [0.7,1.8]
It can be seen that in Simulation 1 very similar limits exist for both locations. However, there are larger differences in the second simulation. It is also noticeable here that in Colombo the value can temporarily fall below the barrierR0 = 1 with maximum fluctua-tions.
Table 6.6: Results of Simulation 1 and 2 based on the data sets from Jakarta. The total population size is assumed to be N = 10154584. The following parameters are fixed: α, µ, ω, η1.
Parameters β0 β1 c η2 ϕ γ α µ ω η1
Simulation1 46.71 7.47 / / 10.41 0.26 26 1/69 2π / Simulation2 54.01 84.69 15.77 6.97 9.12 0.34 26 1/69 2π 6
20090 2010 2011 2012 2013 2014 2015 2016 2017 5
10 15 20
per 105 inhabitants
Jakarta (Simulation 1)
Id(t) I(t) (fitted)
20090 2010 2011 2012 2013 2014 2015 2016 2017 5
10 15 20
per 105 inhabitants
Jakarta (Simulation 2)
Id(t) I(t) (fitted)
S0= 5552245 I0 = 2079 S0 = 4652792 I0= 1772 R0 = 4600260 J = 114.13 R0 = 5500020 J = 102.63
2009 2010 2011 2012 2013 2014 2015 2016 2017 40
45 50 55
(t)
Jakarta Transmission Rate (Simulation 1)
2009 2010 2011 2012 2013 2014 2015 2016 2017 40
50 60 70 80
(t)
Jakarta Transmission Rate (Simulation 2)
β(t) =β0+β1sin (ω(t+ϕ/52)) β(t) =β0+β1Rt−
η1 52
t−η522 pc(τ)dτsin ω t+52ϕ
Table 6.7: Results of Simulation 1 and 2 based on the data sets from Colombo. The total population size is assumed to be N = 1538671. The following parameters are fixed: α, µ, ω, η1.
Parameters β0 β1 c η2 ϕ γ α µ ω η1
Simulation1 47.71 10.39 / / 10.60 0.37 26 1/75 4π / Simulation2 32.54 99.64 9.45 9.80 9.35 0.33 26 1/75 4π 9
20090 2010 2011 2012 2013 2014 2015 2016 10
20 30
per 105 inhabitants
Colombo (Simulation 1)
Id(t) I(t) (fitted)
20090 2010 2011 2012 2013 2014 2015 2016 10
20 30
per 105 inhabitants
Colombo (Simulation 2)
Id(t) I(t) (fitted)
S0= 838311 I0 = 321 S0 = 1238470 I0 = 302 R0 = 700039 J = 63.80 R0 = 299899 J = 55.44
2009 2010 2011 2012 2013 2014 2015 2016
35 40 45 50 55 60
(t)
Colombo Transmission Rate (Simulation 1)
2009 2010 2011 2012 2013 2014 2015 2016
20 30 40 50
(t)
Colombo Transmission Rate (Simulation 2)
β(t) =β0+β1sin (ω(t+ϕ/52)) β(t) =β0+β1Rt−
η1 52
t−η522 pc(τ)dτsin ω t+52ϕ
6.6.2 Prediction quality of the model
With regard to the intention to use control methods to reduce the spread of disease, the predictive quality of the model plays a major role. In the following, the parameters included inu= (β0, β1, c, η2, ϕ, γ, S0, R0)T are fitted up to a time ˆt∈ D. The model sub-sequently uses these parameters and makes a prediction to the end timet1. In Simulation 3 we use the available rain data and in Simulation 4 the average rainfall data of previous years within the prognosis interval, see Tables 6.9 and 6.10. To give more weight to current than to past data we introduce a weight functionH:D →R+ with
H(t) =w·exp − t−ˆt2
2σ2
! +z .
The parametersw= 50,σ = 524 and z= 1 are selected so that the period of the last four weeks before ˆtis weighted considerably more strongly. Hence, we solve the minimization problem
minu J(u) = min
u
1 maxt∈DId
Z ˆt t0
H(t)
γI(t)−Id(t)2
dt+ kuk
N 2
.
In the following simulations the end time ˆt of the parameter fit is chosen so that in the previous 4–8 weeks the number of dengue cases increased significantly. In practice, this optimization should be constantly updated. Additionally we calculate based on the L1 norm
E1= Z ˆt+1
ˆt
γI(t)−Id(t)
maxt∈[ˆt,t+1ˆ ]Id(t)dt and E2 = 1 t1−ˆt
Z t1
ˆt
γI(t)−Id(t) maxt∈[ˆt,t1]Id(t)dt and additionally in Colombo
E3 = 2 Z ˆt+1/2
ˆt
γI(t)−Id(t) maxt∈[ˆt,ˆt+1/2]Id(t)dt .
These values are used to determine the deviation of the model in relation to the corre-sponding maximum value within the data. Although the forecast for the coming season is in the foreground, the model also reveals tendencies in the following years.
In Jakarta, the respective forecasts for the following year apply well to both simulations.
The relation of the predicted peak to the previous one is accurately reflected. The course of the following years is also determined by the model. In some years the forecasts for the coming season are slightly better with the average rain data of the previous years in Simulation 4 than with the actual ones. However, the long–term predictions clearly show better results with the real rain data, see Tables 6.8 and 6.9. In comparison, the simulations in Colombo show greater difficulties in making accurate forecasts. It is noticeable that the model in the shortened time periods
t0,ˆt
of the parameter fit can be better adapted to the dengue data than over the full time scale D, e.g. for ˆt = 2013−1week. Due to the half–yearly frequency of the peaks and their strongly fluctuating intensities, the forecasts for the coming half–year are much better than for the entire following year or even the following years. The half–yearly short–term predictions provide useful values which reflect the correct relation to the previous dengue eruptions. Beyond this period, the model becomes inaccurate, e.g. for ˆt = 2012−1week. In terms of short–term forecasting, the actual rainfall data in Simulation 3 delivers better results. In contrast, the long–term prediction is better with the averaged rain data in Simulation 4, see Tables 6.8 and 6.10.
The t–test concerning the residuals r =γI−Id shows in Jakarta as well as in Colombo thatr is not normally distributedN(0, σ2) in most cases [22].
Table 6.8: Medians of the numerical deviations between model and dengue data.
Simulation 3 Simulation 4 Jakarta Colombo Jakarta Colombo E˜1 0.22 0.26 0.22 0.19 E˜2 0.24 0.27 0.63 0.23 E˜3 / 0.23 / 0.24