Epidemic dynamics, interactions and predictability of enteroviruses associated with hand, foot and mouth disease in Japan

Figure 1.. Weekly time-series data and wavelet analysis, 1982–2015. (a) Raw virologic counts of EV-A71. (b) Wavelet power spectrum of square-root-transformed EV-A71 (x-axis is time (year), y-axis is the period (in years), colour is the power spectrum, strong to weak (yellow–red gradient)). (c) Raw counts of virologic CV-A16. (d) Wavelet power spectrum of square-root-transformed CV-A16. (e) Raw counts of syndromic HFMD. (f) Raw counts of virologic CV-A6 (available from 2000).

Figure 2.. Two-serotype TSIR model compartments. (a) Each individual starts out susceptible ( class) to both serotypes i and j, and becomes infected at a seasonal rate proportional to the transmission rate . (b) Upon infection ( class) with serotype i, the individual immediately becomes cross-protected ( class) against infection with serotype j. (c) The individual permanently recovers ( class) from infection with serotype i during the next time step (here, one week), and remains cross-protected against infection with serotype j for duration . (d) Cross-protection is lost after time steps, and the individual is once again susceptible to infection with serotype j (but is permanently immune against serotype ).

Figure 3.. Deterministic one-serotype TSIR output for EV-A71 and CV-A16, 1997–2015. (a) values for EV-A71 (x-axis is week of year). (b) Observed time series (black) against predicted model fit (green) for EV-A71 (x-axis is time (year), y-axis is weekly number of cases). (c) values for CV-A16. (d) Observed time series (black) against predicted model fit (red) for CV-A16. Parameter values in table 1.

Figure 4.. Empirical signatures of (asymmetric) interactions between EV-A71 and CV-A16, 1982–2015. (a) Annual detections of raw EV-A71 (y-axis) against raw CV-A16 (x-axis). (b) Centre of gravity (COG, in weeks) of yearly EV-A71 epidemics stratified by size of yearly CV-A16 epidemics. (c) COG of yearly CV-A16 epidemics stratified by size of yearly EV-A71 epidemics. (d) Skewness of yearly EV-A71 epidemics stratified by size of yearly CV-A16 epidemics. (e) Skewness of yearly CV-A16 epidemics stratified by size of yearly EV-A71 epidemics.

Figure 5.. Deterministic two-serotype TSIR output for EV-A71 and CV-A16, 1997–2015. (a) values for EV-A71 (x-axis is week of year). (b) Observed time series (black) against predicted model fit (green) for EV-A71 (x-axis is time (year), y-axis is weekly number of cases). (c) values for CV-A16. (d) Observed time series (black) against predicted model fit (red) for CV-A16. (e) Observed data (x-axis) against model-predicted time series (y-axis) for EV-A71, aggregated to four-week bins. Fitted line from simple linear regression without an intercept and 95% confidence interval is shown in green, and the line is shown in black. (f) Observed time series (black) against predicted out-of-sample model fit (blue) for EV-A71, 2007–2015, fitted to training data from 1997 to 2006. (g) Observed data (x-axis) against model-predicted time series (y-axis) for CV-A16, aggregated to four-week bins. Fitted line from simple linear regression without an intercept and 95% confidence interval is shown in red, and the line is shown in black. (h) Observed time series (black) against predicted out-of-sample model fit (purple) for CV-A16, 2007–2015, fitted to training data from 1997 to 2006. Parameter values in table 1.

Figure 6.. Spectral densities of simulated time series from the two-serotype TSIR model, under varying levels of cross-protection after CV-A16 infection. (a) Spectral density of log-transformed EV-A71 time series (fill colour), as a function of the duration of cross-protection after CV-A16 infection (x-axis, in weeks) and the period of EV-A71 (y-axis, in years). (b) Spectral density of log-transformed CV-A16 time series (fill colour), as a function of the duration of cross-protection after CV-A16 infection (x-axis, in weeks) and the period of CV-A16 (y-axis, in years). Pink grid cells indicate spectral densities that are not significant, at a threshold defined by the 2.5th quantile of the white noise spectrum. The duration of cross-protection after EV-A71 infection was fixed at the estimated value of = 8 weeks. The estimated values of cross-protection from table 1 are marked by diamonds along the x-axis: EV-A71 in green, CV-A16 in red.