Simulations on a two-age-class SIR model

Finally, consider the case in which the within-group transmission rate for group I is quadruple the transmission rate from groupJ to groupI, and the transmission rate from groupI to groupJ is double the transmission rate from groupJ to groupI. The within-group transmission rate for groupJ is equal to the transmission rate from groupJ to groupI. That is,

rrr=

"

4 1 2 1

#

. (5.41)

This corresponds to strong driving of transmission from groupI. Again, the contextual STE is calculated using the relative rate matrix Eq 5.41 forR∈[0.6,1.5]and j_t−1∈ {5, . . . ,50}, with i_t−1 fixed at 25. The contour plot in Fig 5.6 depicts T_I→J,t−1^S −T_J→I^S _,t−1 under this scenario. TheI→Jcontextual STE is dominant throughout the parameter space, and is most dominant wheni_t−1> j_t−1and R>1. Here, the contextual STE correctly identifies that groupIdominates transmission for the full range of epidemiologically-feasible parameter values.

↑ 25

10 20 30 40 50

0.6 0.8 1.0 1.2 1.4

jt-1

R

ΔSTE

0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

Fig. 5.6 Difference in contextual STE,∆STE=T_I→J,t−1^S −T_J→I,t−1^S , for j_t−1between 5 and 50 andRbetween 0.6 and 1.5, withi_t−1fixed at 25. Units for the vertical scale are in bits.

The within-group rate of infection for groupIis quadruple the within-group rate of infection for groupJ, and the infection rate from groupI to groupJ is double that from groupJ to groupI (see Eq 5.41). TheI→J contextual STE dominates throughout of the parameter space (∆STE>0 everywhere), especially when j_t−1<i_t−1=25 andR>1.

Table 5.1 Infection rates for the two-age-class SIR model

Transition Rate

X_I →Y_I β₁₁X_IY_I/N+β₁₂X_IY_J/N X_J→Y_J β₂₁X_JY_I/N+β₂₂X_JY_J/N

Y_I →Z_I γY_I

Y_J →Z_J γY_J

The next-generation matrix for this model is NGM=1

γ

"

β₁₁p_I β₁₂p_I β₂₁p_J β₂₂p_J

#

(5.42) where p_I =N_I/Nand p_J=N_J/N. The within- and between-group transmission dynamics therefore depend on the infection ratesβ_mnand the relative population sizesN_I andN_J.

Let rrrbe a matrix in which the i,j^th entry gives the relative rate of infection from age class jto age classi. For a given basic reproduction number R₀ and recovery rateγ, the transmission rates can be calculated:

βββ =

"

β₁₁ β₁₂ β₂₁ β₂₂

#

= R₀γ

ρ diag(1/p_I,1/p_J)·rrr (5.43) whereρ is the dominant eigenvalue ofrrr. This yields a next-generation matrix with dominant eigenvalue equal toR₀, in which the proportional differences between the terms match those of the rate matrixrrr.

Table 5.2 lists the parameter values used for the individual-based SIR simulations. The basic reproduction numberR₀is fixed at 1.5, which is consistent with estimated values of R₀for 2009 pandemic influenza [122, 257]. The average time to recovery 1/γ is 3.5 days, which is in line with estimates of the infectious period for 2009 pandemic influenza [257].

The population sizesN_I andN_J are small enough to ensure clearly stochastic dynamics (the dynamics become nearly deterministic as the population sizes increase), and are in line with the population sizes of the smaller age groups in the IMS-ILI dataset.

Table 5.2 Parameter values for the two-age-class SIR model

Parameter Value Description Units

R₀ 1.5 Basic reproduction number people

γ 1/3.5 Recovery rate 1/day

N_I 400 Number of individuals in age groupI people N_J 400 Number of individuals in age groupJ people

Simulations are implemented using the Gillespie algorithm, starting with one infected individual in age groupI. SinceR₀is relatively low, there is a high chance of early epidemic die-out, so only outbreaks that last for at least 12.5 weeks are recorded. Once an outbreak is simulated, infections are binned into half-week intervals. Poisson noise is added to each bin

at a rate of 0.5% of the population size to simulate background non-influenza ILI. This rate is just below the 0.6% ILI ratio cutoff used in Goget al.(2014) [91] to define out-of-season, low ILI activity weeks. Fig 5.7 depicts five simulated epidemic time series from this model.

To study how STE detects the transition from completely decoupled age groups to mean- field dynamics with equal within- and between-group transmission rates, consider a relative rate matrix of form

rrr=

"

1 z z 1

#

(5.44) withz∈[0,1]. For each value ofzbetween 0 and 1 in steps of 0.1, one hundred ensembles of 800 outbreaks each are simulated. The epidemic time series are then symbolised using symbols of lengthm=3. We do not consider longer symbol lengths because estimating the necessary joint and conditional probabilities becomes impractical. Withm=4, there are 24 possible symbols, which begins to defeat the purpose of symbolising to reduce the size of the state space. To my knowledge,m=4 is the largest symbol length that has been considered in practice, and that was using a dataset of millions of Twitter tweets [24].

For each of the 100 ensembles,T_I^S_→J andT_J→I^S are estimated using the relative symbol frequencies in the 800 symbolised time series. This gives 100 STE estimates for each value of z. The left-hand plot in Fig 5.8 depicts the mean STE and 95% confidence interval in both directions (I →J and J→I) as a function of z. TheI→J (blue) and J→I (black) STEs overlap for all values ofz, correctly identifying the balanced influence between the age groups. The STE is near zero forz=0 and increases steadily to approximately 0.06 bits asz approaches 1, correctly identifying the increasing degree of coupling between the two age groups aszincreases.

Next, consider a rate matrix of form rrr=

"

1+3k 1 1+k 1

#

(5.45) withk∈[0,1]. Whenk=0, this is equivalent to the previous rate matrix, Eq 5.44, withz=1.

Whenk=1, the within-group transmission rate for age class I is four times the baseline transmission rate, and the transmission rate from groupI to groupJ is twice the baseline transmission rate. This captures a continuum between the mean-field scenario and a scenario with strong forcing from age classI. As before, 100 ensembles of 800 epidemics each are simulated for values ofkbetween 0 and 1 in steps of size 0.1. For each ensemble, the STE fromItoJand fromJtoIis calculated. The right-hand plot in Fig 5.8 depicts the mean STE and 95% confidence interval in both directions forkbetween 0 and 1. Fork=0, the mean

0 5 10 15 20 25 0

10 20 30 40

0 5 10 15 20 25 0

10 20 30 40

0 5 10 15 20 25 0

10 20 30 40

0 5 10 15 20 25 0

10 20 30 40

0 5 10 15 20 25 0

10 20 30 40

0 5 10 15 20 25 0

10 20 30 40

0 5 10 15 20 25 0

10 20 30 40

0 5 10 15 20 25 0

10 20 30 40

0 5 10 15 20 25 0

10 20 30 40

0 5 10 15 20 25 0

10 20 30 40

GroupIGroupJ

Fig. 5.7 Five simulations from the two-age-class individual-based SIR model, implemented using the Gillespie algorithm. Each column depicts the output from a single epidemic simulation, separated into case counts for groupI(top) and groupJ(bottom). Vertical axes have units of case counts, and horizontal axes have units of half weeks, which is also assumed to be the length of the disease’s generation interval. Transition rates are given in Table 5.1, with parameter values in Table 5.2. The transmission rates are specified by the relative rate matrix 5.45 withk=1, for which age groupI (upper row) has quadruple the within-group transmission rate as groupJ, and for which theI→Jtransmission rate is twice theJ→I transmission rate.

STE is approximately 0.06 bits, the same as it is forz=1. Askincreases,T_I→J^S increases steadily andT_J→I^S decreases steadily, correctly identifying the increasing forcing from group Ito groupJ.