Evaluating the probability of silent circulation of polio in small populations using the silent circulation statistic

Figures

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  Fig. 1. A schematic diagram of the model used in this paper. The compartments of the model are: S (naive susceptible), I1 (individuals experiencing their first infection), R (recovered and fully immune from infection), P (partially susceptible to infection), and Ir (individuals experiencing a repeat infection). The diagram was reproduced from Vallejo et al. (2017).

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  Fig. 2. Effect of population size on the silent circulation statistic. The top panels are the silent circulation statistic under NICA (no initial case assumption). The silent circulation statistic under ICA (initial case assumption) is omitted for clarity. The bottom panels are the differential probability of circulation (NICA - ICA) for each population size considered. The right column contains the graphs from the left column restricted between years 3 and 4 to highlight the behavior around the three year threshold. For all plots, the x-axis is the time since the last detected paralytic case (a Δt interval of time).

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  Fig. 3. Effect of modifying the paralytic case detection rate on the silent circulation statistic for the N=5,000 population size. The top panels are the silent circulation statistic under NICA. The silent circulation statistic under ICA is omitted for clarity. The bottom panels are the differential probability of circulation (NICA - ICA) for each detection rate considered. The right column contains the graphs from the left column restricted between years 3 and 4 to highlight the behavior around the three year threshold. For all plots, the x-axis is the time since the last detected paralytic case (a Δt interval of time).

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  Fig. 4. Effect of modifying the paralytic case detection rate on the silent circulation statistic for the N=20,000 population size. The top panels are the silent circulation statistic under NICA. The silent circulation statistic under ICA is omitted for clarity. The bottom panels are the differential probability of circulation (NICA - ICA) for each detection rate considered. The right column contains the graphs from the left column restricted between years 3 and 4 to highlight the behavior around the three year threshold. For all plots, the x-axis is the time since the last detected paralytic case (a Δt interval of time).

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  Fig. 5. Effect of varying contact rates on the silent circulation statistic for the population size N=5,000. The top panels are the silent circulation statistic under NICA. The silent circulation statistic under ICA is omitted for clarity. The bottom panels are the differential probability of circulation (NICA - ICA) for each contact rate considered. The right column contains the graphs from the left column restricted between years 3 and 4 to highlight the behavior around the three year threshold. For all plots, the x-axis is the time since the last detected paralytic case (a Δt interval of time).

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  Fig. 6. Effect of varying contact rates on the silent circulation statistic for the population size N=20,000. The top panels are the silent circulation statistic under NICA. The silent circulation statistic under ICA is omitted for clarity. The bottom panels are the differential probability of circulation (NICA - ICA) for each contact rate considered. The right column contains the graphs from the left column restricted between years 3 and 4 to highlight the behavior around the three year threshold. For all plots, the x-axis is the time since the last detected paralytic case (a Δt interval of time).

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  Fig. 7. Effect of varying waning immunity scenarios on the silent circulation statistic for the population size N=5,000. The top panels are the silent circulation statistic under NICA. The silent circulation statistic under ICA is omitted for clarity. The bottom panels are the differential probability of circulation (NICA - ICA) for each waning immunity scenario considered. The right column contains the graphs from the left column restricted between years 3 and 4 to highlight the behavior around the three year threshold. For all plots, the x-axis is the time since the last detected paralytic case (a Δt interval of time).

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  Fig. 8. Effect of varying waning immunity scenarios on the silent circulation statistic for the population size N=20,000. The top panels are the silent circulation statistic under NICA. The silent circulation statistic under ICA is omitted for clarity. The bottom panels are the differential probability of circulation (NICA - ICA) for each waning immunity scenario considered. The right column contains the graphs from the left column restricted between years 3 and 4 to highlight the behavior around the three year threshold. For all plots, the x-axis is the time since the last detected paralytic case (a Δt interval of time).

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  Fig. 9. The dot and whisker plot displays the mean (dot) and 95% confidence intervals (whiskers) for the time between detected paralytic cases for varying population sizes for two points of measurement: measuring the time between paralytic cases that occur from the start of the simulation (first Eichner & Dietz intercase interval) and measuring the time between paralytic cases that occur from the first explicitly simulated case (first true intercase interval). The black dashed lines represent the expected time between detected paralytic cases at the endemic equilibrium of the related differential equations model for each population size considered, calculated under the assumption of an exponential distribution (i.e. the average time between cases is the reciprocal of the mean rate at which cases occur).

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  Fig. 10. Effect of right censoring on the silent circulation statistic for the N=20,000 population size. The top panel is the silent circulation with right censoring (truncates the simulation at 10 years or elimination, whichever happens first) and without right censoring (simulations end at elimination). The bottom panel presents the differential probability of circulation (the difference between the two curves in the top panel: no right censoring minus right censoring). For both plots, the x-axis is the time since the last detected paralytic case (a Δt interval of time).

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  Fig. 11. Effect of right censoring on the silent circulation statistic accounting for the N=25,000 population size. The top panel is the silent circulation with right censoring (truncates the simulation at 10 years or elimination, whichever happens first) and without right censoring (simulations end at elimination). The bottom panel presents the differential probability of circulation (the difference between the two curves in the top panel: no right censoring minus right censoring). For both plots, the x-axis is the time since the last detected paralytic case (a Δt interval of time).