Simulations on a two-age-class Poisson model

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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.

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Fig. 5.8 MeanT_I→J^S (solid blue line) andT_J→I^S (solid black line) values for a range of within- and between-group transmission rates, as estimated from 100 ensembles of 800 epidemics for each value ofzandk, simulated using an individual-based SIR model. The shaded bands provide the approximate 95% confidence intervals of the STE estimates. The left-hand plot depicts the STE for the relative rate matrix given in Eq 5.44, which transitions smoothly from completely decoupled age groups to mean-field dynamics where within- and between-group transmission rates are equal. The right-hand plot depicts the STE for the relative rate matrix given in Eq 5.45, which transitions smoothly from mean-field dynamics to strong forcing from groupI. In the left-hand plot, the STE increases steadily withz, and theI→JandJ→I STE estimates overlap, correctly capturing the symmetric coupling between age groups that increases withz. In the right-hand plot, the STE correctly identifies statistically significant forcing from groupI to groupJ askincreases.

wherei_t and j_t are case counts, andλ_mn,t is the infection rate from age classnto classmat (discrete) timet. Unlike in the SIR model in 5.2.3, the infection ratesλ_mn,t are allowed to vary in time.

For the simulations, the length of the time steps is assumed to coincide with the generation interval for the disease. This way, the next-generation matrix is simply

NGM_t=λλλ_t=

"

λ11,t λ12,t

λ_21,t λ_22,t

#

. (5.48)

For a given matrixrrrwhere elementr_{i j} denotes the relative rate of infection from group j to groupi, and a given reproduction numberR, the next-generation matrix may be expressed as

λλλ_t= R

ρrrr (5.49)

whereρis the dominant eigenvalue of the matrixrrr. As before, this ensures that the dominant eigenvalue of the next-generation matrix is equal to the reproduction number, and that the relative magnitudes of the elements of the next-generation matrix match the relative magnitudes of the elements ofrrr.

Epidemics are simulated by placing a single initial infected individual in either groupI or groupJ with probability 0.5, and then simulating the numbers of infected individuals in each group for subsequent weeks according to draws from the Poisson distributions given in Eq 5.46-5.47. For the first eight time steps, R is fixed at 1.5. After the eighth time step,R is decreased to 0.8 for the rest of the epidemic. This yields outbreaks of similar size as the ones simulated using the individual-based SIR model presented in §5.2.3 (compare Figs 5.7 and 5.9). Only outbreaks in which at least 400 people become infected are recorded. Five simulations from this two-age-class Poisson-type model are depicted in Fig 5.9.

To check consistency with the individual-based SIR model, 100 ensembles of 800 epidemics each were simulated for the rate matrices given in Eq 5.44 and Eq 5.45, withz andkranging from 0 to 1 in steps of 0.1. The mean STE estimates from these simulations are depicted in Fig 5.10 (compare with Fig 5.8). The same trends hold: theI→JandJ→I STE estimates overlap and increase aszincreases from 0 to 1, and theI→JSTE quickly dominates over theJ→ISTE askincreases from 0 to 1.

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GroupIGroupJ

Fig. 5.9 Five simulations from the two-age-class Poisson-type simulation algorithm, Eqs 5.46-5.47. Each column depicts the output for a single epidemic simulation, separated into case counts from groupI(top) and groupJ (bottom). The reproduction numberRis 1.5 for the first eight weeks of the outbreak, and then drops to 0.8 for the rest of the epidemic. 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→Itransmission rate.

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STE(bits)

Fig. 5.10 MeanT_I^S_→J (solid blue line) andT_J→I^S (solid black line) values for a range of within- and between-group infection rates, as estimated from 100 ensembles of 800 epidemics for each value of z and k between 0 and 1 in steps of 0.1, simulated using the two-age- class Poisson-type model (Eq 5.46-5.47). The shaded bands provide the approximate 95%

confidence intervals for the STE estimates. The left-hand plot depicts the STE for the relative rate matrix given in Eq 5.44, which transitions smoothly from completely decoupled age groups to mean-field dynamics where within- and between-group transmission rates are equal. The right-hand plot depicts the STE for the relative rate matrix given in Eq 5.45, which transitions smoothly from mean-field dynamics to strong forcing from groupI. The trends in these plots match those in Fig 5.8, showing agreement between STE estimates when using either the individual-based SIR model or the more computationally efficient Poisson-type model.

5.2.5 Simulations on a four-age-class Poisson model with variable re-