Effect of Population Partitioning on the Probability of Silent Circulation of Poliovirus

Abstract

Polio can circulate unobserved in regions that are challenging to monitor. To assess the probability of silent circulation, simulation models can be used to understand transmission dynamics when detection is unreliable. Model assumptions, however, impact the estimated probability of silent circulation. Here, we examine the impact of having distinct populations, rather than a single well-mixed population, with a discrete-individual model including environmental surveillance. We show that partitioning a well-mixed population into networks of distinct communities may result in a higher probability of silent circulation as a result of the time it takes for the detection of a circulation event. Population structure should be considered when assessing polio control in a region with many loosely interacting communities.

Keywords: Asymptomatic transmission; Markov model; Metapopulation; Poliovirus.

Fig. 9

A comparison of the silent circulation statistic curves with the initial case assumption (defining the time between the start of the simulation and the first simulated paralytic case as an intercase interval; ICA) and without the initial case assumption (defining only the time between explicitly simulated paralytic cases as an intercase interval; NICA) for the 16 patches of 4k scenario. Note that in this paper NICA was used

Fig. 10

Box plots demonstrating the distribution of starting values for each compartment after the 50-year burn-in period compared to the endemic equilibrium value obtained by solving the related system of differential equation represented by the solid red horizontal line. Note that the extinction dynamics are highly influential in determining the starting conditions. Even the 64k population is not large enough to reproduce equilibrium-like conditions

Fig. 11

Cumulative distribution function (CDF) of intercase (time between detected paralytic cases, (A) and extinction (time between the last detected paralytic case and extinction, (B) intervals for single populations

Fig. 12

Cumulative distribution function (CDF) of intercase (time between paralytic cases, (A) and extinction (time between the last detected case and extinction, (B) intervals for the multi-patch model with an interpatch movement rate of 0.1 per year. Lighter, more transparent, lines represent the value of the quantity in the absence of movement to use for comparison

Fig. 13

Comparison of the probability of silent circulation between evenly distributed patch populations and heterogeneous patch distributions visualized using the silent circulation statistic (A), the differential comparison to the 1 × 64k population (B), and the odds ratio (C). The probability differential (B) is calculated by subtracting the probability of silent circulation in the partitioned populations from that of the large 64k population. Negative values indicate that the partitioned populations have a higher probability of silent circulation. Values less than one in the odds ratio plot (C) indicate that the 64k population is less likely to have continued silent circulation compared to the partitioned populations. The inset plot shows the curves restricted to between 2.5 and 3.5 years since a paralytic case. The mixed population distributions are represented by dashed lines.

Fig. 1

A schematic diagram of the model used in this paper, modified from Vallejo et al. (2019). The compartments of the model are: S (naive susceptible), I₁ (first infection with the virus), V (fully vaccinated against infection), R (recovered and fully immune from infection), P (partially susceptible to infection), and I_r (reinfected). The transmission term (β(t)) is time dependent to incorporate seasonal forcing. Movement between patches (represented by the dashed lines) is reciprocal to maintain patch sizes

Fig. 2

Effect of population partitioning on the probability of silent circulation visualized using the silent circulation statistic (A), the probability differential (B), and the odds ratio (C). The probability differential (B) is calculated by subtracting the probability of silent circulation in the partitioned populations from that of the large 64k population. Negative values indicate that the partitioned populations have a higher probability of silent circulation. Values less than one in the odds ratio plot (C) indicate that the 64k population is less likely to have continued silent circulation compared to the partitioned populations. The inset plots expand the y-axis scale to show behavior between 2.5 and 3.5 years since a paralytic case was observed

Fig. 3

Cumulative distribution of intercase (time between detected paralytic cases, (A) and extinction (time between the last paralytic case and extinction, (B) intervals for isolated populations. Intercase intervals beyond 3 years are very rare, while some extinction intervals can last more than 3 years

Fig. 4

Effect of interpatch movement on the probability of silent circulation given a Δt interval of time since the last detected paralytic case in the 4 × 16k population visualized using the silent circulation statistic (A), the differential comparison to the 1 × 64k population (B), and the odds ratio (C). The inset plot shows the curves restricted to between 2.5 and 3.5 years since a paralytic case. A movement rate (α) of 0 indicates that the 4 patches are isolated from each other. For the nonzero movement rates, the value indicates the rate at which one individual initiates movement, but with the assumption of reciprocated movement between the sink and the source patch, two individuals move when one initiates

Fig. 5

Effect of interpatch movement on the probability of silent circulation given a Δt interval of time since the last detected paralytic case in the 16 × 4k population visualized using the silent circulation statistic (A), the differential comparison to the 1 × 64k population (B), and the odds ratio (C). The inset plot shows the curves restricted to between 2.5 and 3.5 years since a paralytic case. A movement rate (α) of 0 indicates that the 16 patches are isolated from each other. For the nonzero movement rates, the value indicates the rate at which one individual initiates movement, but with the assumption of reciprocated movement between the sink and the source patch, two individuals move when one initiates

Fig. 6

Comparison of the probability of silent circulation in partitioned populations with a reciprocated movement rate of 0.1 per year visualized using the silent circulation statistic (A), the probability differential (B), and the odds ratio (C). The probability differential (B) is calculated by subtracting the probability of silent circulation in the partitioned populations from that of the large 64k population. Negative values indicate that the partitioned populations have a higher probability of silent circulation. Values less than one in the odds ratio plot (C) indicate that the 64k population is less likely to have continued silent circulation compared to the partitioned populations. Lighter, more transparent, lines represent the value of the quantity in the absence of movement to use for comparison. The inset plot focuses on behavior between 2.5 and 3.5 years since a paralytic case was observed

Fig. 7

Comparison of the effect of vaccination on the probability of silent circulation in the 1 × 64k (dotted lines) and the 4 × 16k population with movement rate 0.1 per year (solid lines) visualized using the silent circulation statistic (A), the probability differential (B), and the odds ratio (C). The probability differential (B) is calculated by subtracting the probability of silent circulation in the partitioned populations from that of the large 64k population. Negative values indicate that the partitioned populations have a higher probability of silent circulation. Values less than one in the odds ratio plot (C) indicate that the 64k population is less likely to have continued silent circulation compared to the partitioned populations. The inset plot focuses on behavior between 2.5 and 3.5 years since a paralytic case was observed

Fig. 8

Comparison of the effect of utilizing environmental surveillance on the probability of silent circulation in the 1x64k (dotted lines) and the 4 × 16k population with movement rate 0.1 per year (solid lines) visualized using the silent circulation statistic (A), the probability differential (B), and the odds ratio (C). The probability differential (B) is calculated by subtracting the probability of silent circulation in the partitioned populations from that of the large 64k population. Negative values indicate that the partitioned populations have a higher probability of silent circulation. Values less than one in the odds ratio plot (C) indicate that the 64k population is less likely to have continued silent circulation compared to the partitioned populations. A detection event is defined as detection through either a paralytic case or through ES