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

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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.

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Data Availability

Not applicable.

Code availability

The code is available at https://github.com/celestevallejo/polio under the Metapopulation_model folder. The C++ folder contains the simulation model code and the code to calculate the silent circulation statistic using model output. The Model_output folder contains zipped files of all simulation model output used to generate the figures or the silent circulation statistic. The Plotting_scripts folder contains all R scripts used to generate the figures in the manuscript. The SC_statistic_output folder contains zipped files of all output generated from the silent circulation C++ code in the C++ folder.

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Acknowledgements

This research was partially supported by a National Science Foundation Grant Number DMS 1440386 (CV) and by NIH/National Institute of General Medical Sciences Grant U54 GM111274 (TJH).

Author information

Authors and Affiliations

  1. Mathematical Biosciences Institute, The Ohio State University, Columbus, OH, 43210, USA

    Celeste Vallejo

  2. Depart of Infectious Disease Epidemiology and Centre for Mathematical Modelling of Infectious Disease, London School of Hygiene and Tropical Medicine, Keppel Street, London, WC1E 7HT, UK

    Carl A. B. Pearson

  3. South African DSI-NRF Centre of Excellence in Epidemiological Modelling and Analysis (SACEMA), Stellenbosch University, Jonkershoek Road, Stellenbosch, 7600, South Africa

    Carl A. B. Pearson

  4. University of Michigan, Ann Arbor, MI, 48109, USA

    James S. Koopman

  5. Department of Biology, University of Florida, Gainesville, FL, 32611, USA

    Thomas J. Hladish

  6. Emerging Pathogens Institute, University of Florida, Gainesville, FL, 32608, USA

    Thomas J. Hladish

Authors
  1. Celeste Vallejo
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  2. Carl A. B. Pearson
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  3. James S. Koopman
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  4. Thomas J. Hladish
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Correspondence to Celeste Vallejo.

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5 Appendix

5 Appendix

1.1 5.1 Intercase Interval Definition

The initial intercase interval has been defined as either the time between the start of the simulation and the first simulated paralytic case (referred to as the initial case assumption (ICA) in Vallejo et al. (2019)) (Eichner and Dietz 1996; Kalkowska et al. 2012) or as the time between the first two explicitly simulated paralytic cases (referred to as the non-initial case assumption (NICA)) (Vallejo et al. 2019). Vallejo et al. (2019) explored the consequence of the ICA in small populations (25000 and smaller). They determined that defining the first interparalytic case interval as the time between the start of the simulation and the first paralytic case had the effect (in most cases) of estimating a higher probability of silent circulation when compared to the NICA. This effect decreased with an increase in population size. In this paper the population size considered is large enough such that either definition of the first interparalytic case interval is appropriate (see Fig. 9). In any case, we believe it is more realistic for observation of the system to begin after a paralytic case had been detected, rather than at the exact moment of detection. Therefore, we define all interparalytic case intervals as between two explicitly simulated paralytic cases (or the NICA).

Fig. 10
figure 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 (Color figure online)

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1.2 5.2 Initial Patch Population Distribution

See Fig. 10.

1.3 5.3 Intercase and Extinction Interval Distribution in Single Populations

See Fig. 11.

Fig. 11
figure 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

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1.4 5.4 Intercase and Extinction Interval Distribution for the Multi-patch Model with Interpatch Movement

See Fig. 12.

Fig. 12
figure 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

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1.5 5.5 The Probability of Silent Circulation in Heterogeneous Patches

See Fig. 13.

Fig. 13
figure 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 \(\times \) 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.

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Vallejo, C., Pearson, C.A.B., Koopman, J.S. et al. Effect of Population Partitioning on the Probability of Silent Circulation of Poliovirus. Bull Math Biol 84, 62 (2022). https://doi.org/10.1007/s11538-022-01014-6

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