5.2 Symbolic transfer entropy and epidemiological processes
5.2.5 Simulations on a four-age-class Poisson model with variable re- porting ratesporting rates
5.2.5 Simulations on a four-age-class Poisson model with variable re-
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Group1Group2Group3Group4
Fig. 5.11 Five simulations from the 4-age-class Poisson-type model (Eq 5.46-5.47) with relative rate matrixrgiven in Eq 5.53. Columns correspond to a single epidemic simulation, separated into case counts for the four age groups. Grey lines depict the full case counts simulated from the Poisson model, and black lines depict the case counts obtained by using a Binomial observation model (Eq 5.52) with a reporting rate ofci=0.5 for all age groups.
Note the different vertical scales.
To explore how reporting rates influence the STE, two scenarios are considered. First, we consider how varying the reporting rate equally across all age groups affects the identifiability of true differences in transmission rate. Second, we consider how different reporting rates between age groups affects the STE, when the true transmission rates are all equal.
For the first scenario, consider a relative rate matrix of form
rrr=
1 2 1 1 1 4 1 1 1 2 1 1 1 1 1 1
(5.53)
so that the within-group transmission rate for group 2 is four times the baseline transmission rate, and the group 2 to group 1 and group 2 to group 3 transmission rates are double the baseline transmission rate. If the four age groups correspond to infants, children, adults, and elderly, then this would match a scenario with high transmission among children and intermediate transmission between children and infants and between children and adults.
To study how varying reporting rates uniformly across all age classes affects the ability of the STE to identify children as the dominant transmitters of disease, 100 ensembles of
800 epidemics each were simulated for reporting ratescibetween 0.1 and 1 in steps of 0.1, with equal reporting rates across all age groups. Fig 5.12 depicts the STE from each age group to every other age group as a function ofci. Even for reporting rates as low as 0.1, the STE values from group 2 are higher than those from any other group. As the reporting rates increase, the differences become more pronounced. The left-hand plot in Fig 5.13 depicts the mean pairwise STE values withci=0.5 in all age groups, which is essentially a vertical slice from the plots in Fig 5.12 atci=0.5. Matrix 5.56 provides the mean STE values used to make the left-hand plot in Fig 5.13, with their 95% confidence intervals. There is significantly elevated transmission from group 2 to all other groups.
For the second scenario, with mean-field transmission dynamics and reporting rates that differ by age, the relative rate matrix is
rrr=
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
(5.54)
so that all within- and between-group rates of transmission are equal. Roughly following the age-structured reporting rates reported by Biggerstaffet al.(2012) [22], the reporting rate vector is fixed at
ccc= (0.4,0.6,0.4,0.4) (5.55) corresponding to elevated reporting rates in group 2 (children). To be more consistent with the reporting rates in Biggerstaffet al.(2012) [22], the reporting rate for group 1 (infants) should also be approximately 0.6, but for illustrative purposes we for now consider elevated reporting in just one age group. A more realistic scenario, with elevated reporting rates for both infants and children, is considered later in the 12-age-class simulations.
To estimate how age-variable reporting rates affect the STE, 100 ensembles of 800 epidemics each were simulated using the rate matrix Eq 5.54 and the reporting rates Eq 5.55.
The pairwise STE between each age group was calculated for each ensemble. The right-hand plot in Fig 5.13 depicts the pairwise mean STE values, and matrix 5.57 provides the mean values and 95% confidence intervals. A 60% reporting rate in group 2 and a 40% reporting rate in all other age groups yields significantly higher STE estimates from group 2 than from any other age group. However, the first scenario, with elevated transmission from group 2 but constant 50% reporting across all age groups, yields even higher STE estimates from
Group 1
0 0.2 0.4 0.6 0.8 1 0
0.05 0.10 0.15
0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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Group 2
0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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Group 3
0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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Group 4 Group 1
0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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Group 2
0 0.2 0.4 0.6 0.8 1 0
0.05 0.10 0.15
0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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Group 3
0 0.2 0.4 0.6 0.8 1 0
0.05 0.10 0.15
0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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0 0.2 0.4 0.6 0.8 1 0
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Group 4 Reporting rate(c)
Reporting rate(c)
Reporting rate(c)
Reporting rate(c)
Reporting rate(c)
Reporting rate(c)
Reporting rate(c)
Reporting rate(c)
Reporting rate(c)
Reporting rate(c)
Reporting rate(c)
Reporting rate(c)
STE(bits)STE(bits)STE(bits) STE(bits)STE(bits)STE(bits) STE(bits)STE(bits)STE(bits) STE(bits)STE(bits)STE(bits)
Fig. 5.12 Mean pairwise STE values (solid lines) with 95% confidence intervals (shaded bands) for 100 ensembles of 800 simulated epidemics, for reporting ratesccc(horizontal axis) between 0 and 1 in steps of 0.1. The vertical axis corresponds to the estimated STE, in units of bits. Reporting rates for the simulations are uniform across all age groups. The relative rate matrix that specifies within- and between-group transmission rates is given by Eq 5.53. The plot in rowiand column jdepicts the STE from group jto groupi. Even with reporting rates near 0.1, group 2 is correctly identified as the primary driver of transmission.
As reporting rates increase, the dominant transmission from group 2 becomes clearer. The estimated STE increases with reporting rate for all age groups, but more quickly for group 2 than for the other age groups. According to Biggerstaffet al.(2012), true reporting rates for ILI in the US during the 2009 pandemic were between 0.4 and 0.6.
group 2. For comparison, the mean pairwise STE values for the two scenarios are depicted side-by-side in Fig 5.13.
012.0(11.1,12.9)7.9(7.1,8.5)6.2(5.5,6.8) 6.5(5.9,7.2)06.5(5.9,7.2)5.1(4.5,5.7) 8.0(7.2,8.7)12.0(11.2,12.9)06.2(5.5,6.8) 8.5(7.7,9.2)11.4(10.5,12.1)8.5(7.8,9.1)0
×10−2 (5.56) MeanSTEvalues(95%CI)forsimulatedepidemicswithstrongtransmissive forcingfromgroup2(relativeratematrixgivenbyEq5.53)andwith50%reporting forallagegroups.Thei,jth entrygivestheSTEfromgroupjtogroupi.TheSTE valuesfromgroup2toallotheragegroupsaresignificantlyhigherthantheSTE valuesfromanyotheragegroup.Themeanvaluesarealsodepictedintheleft-hand plotofFig5.13.
08.1(7.2,8.8)6.2(5.7,6.8)6.2(5.7,6.7) 5.9(5.5,6.5)05.9(5.3,6.5)6.0(5.3,6.6) 6.2(5.6,6.9)8.1(7.4,8.9)06.2(5.6,6.7) 6.2(5.5,6.7)8.1(7.4,8.7)6.2(5.7,6.7)0
×10−2 (5.57) MeanSTEvalues(95%CI)forsimulatedepidemicswithconstantwithin-and between-grouptransmissionrates(relativeratematrixgivenbyEq5.54),andwith 60%reportingratefromgroup2and40%reportingrateforallotheragegroups. Thei,jth entrygivestheSTEfromgroupjtogroupi.TheSTEvaluesfromgroup 2toallotheragegroupsaresignificantlyhigherthantheSTEvaluesfromanyother agegroup,thoughlowerthanthevaluesinmatrix5.56obtainedfromtheexplicitly forcedsimulations.Themeanvaluesarealsodepictedintheright-handplotofFig 5.13.
0 6.5 8.0 8.5
12.0 0 12.0 11.4
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0 5.9 6.2 6.2
8.1 0 8.1 8.1
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Fig. 5.13 Mean pairwise STE values in bits, multiplied by a factor of 102, for 100 ensembles of 800 epidemics using relative rate matrix Eq 5.53 and 50% reporting rate for all age groups (left), and using relative rate matrix Eq 5.54 with 60% reporting rate in group 2 and 40%
reporting rate in all other age groups (right). A box in rowiand column j corresponds to the mean STE from group jto groupi, with darker shades corresponding to higher STE. In both scenarios, the STE from group 2 to all other age groups is elevated. The elevation is higher for the scenario with explicitly higher transmission rates from group 2 (left) than for the scenario with equal transmission rates and elevated reporting in group 2 (right). Mean values and confidence intervals are given in matrices 5.56 and 5.57.