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Pattern of tick aggregation on mice: larger than expected distribution tail enhances the spread of tick-borne pathogens.

Ferreri L, Giacobini M, Bajardi P, Bertolotti L, Bolzoni L, Tagliapietra V, Rizzoli A, Rosà R - PLoS Comput. Biol. (2014)

Bottom Line: Moreover, we found that the tail of the distribution significantly changes with seasonal variations in host abundance.Specifically, we found that the epidemic threshold and the prevalence equilibria obtained in epidemiological simulations with PL distribution are a good approximation of those observed in simulations feed by the empirical distribution.Moreover, we also found that the epidemic threshold for disease invasion was lower when considering the seasonal variation of tick aggregation.

View Article: PubMed Central - PubMed

Affiliation: Computational Epidemiology Group, Department of Veterinary Sciences, University of Torino, Torino, Italy; Applied Research on Computational Complex Systems Group, Department of Computer Science, University of Torino, Torino, Italy.

ABSTRACT
The spread of tick-borne pathogens represents an important threat to human and animal health in many parts of Eurasia. Here, we analysed a 9-year time series of Ixodes ricinus ticks feeding on Apodemus flavicollis mice (main reservoir-competent host for tick-borne encephalitis, TBE) sampled in Trentino (Northern Italy). The tail of the distribution of the number of ticks per host was fitted by three theoretical distributions: Negative Binomial (NB), Poisson-LogNormal (PoiLN), and Power-Law (PL). The fit with theoretical distributions indicated that the tail of the tick infestation pattern on mice is better described by the PL distribution. Moreover, we found that the tail of the distribution significantly changes with seasonal variations in host abundance. In order to investigate the effect of different tails of tick distribution on the invasion of a non-systemically transmitted pathogen, we simulated the transmission of a TBE-like virus between susceptible and infective ticks using a stochastic model. Model simulations indicated different outcomes of disease spreading when considering different distribution laws of ticks among hosts. Specifically, we found that the epidemic threshold and the prevalence equilibria obtained in epidemiological simulations with PL distribution are a good approximation of those observed in simulations feed by the empirical distribution. Moreover, we also found that the epidemic threshold for disease invasion was lower when considering the seasonal variation of tick aggregation.

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Comparison among fittings of distributions of ticks per host with different functions.Left: Kolmogorov-Smirnov statistic between subsets of data above  and the fitting models on these subsets. Vertical dotted lines represent the optimum value of  for different models (NB: magenta; PoiLN: green; PL: cyan). For the NB and PoiLN models the optimum is observed for , i.e. on the entire data set, while for the PL model the optimum is reached for . Center: goodness-of-fit p-value of fitting models on data larger than or equal to . As suggested by Clauset and collaborators [31] for p-value greater than 0.1 (horizontal line) the fitting model is a good description of the data. For NB the GOF is low (p<10−3), suggesting the inappropriateness of the NB model in describing the data. The GOF of the PoiLN indicates that the model is appropriate only for large value of , thus simultaneously with large values of KS and therefore pointing out the low performance of the model. The PL fits should not be rejected for values of  larger than  concurrently with the lowest value of KS. Right: Log-likelihood Ratio (LLR) test with Vuong's sign interpretation. Negative (positive) values suggest the alternative model NB (red) or PoiLN (blue) distributions are (are not) favoured in describing values larger than  when compared to PL. The horizontal line shows the sign threshold. Full marks show statistically significant tests (p<0.05) while empty marks refer to non significant tests (p>0.05).
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pcbi-1003931-g003: Comparison among fittings of distributions of ticks per host with different functions.Left: Kolmogorov-Smirnov statistic between subsets of data above and the fitting models on these subsets. Vertical dotted lines represent the optimum value of for different models (NB: magenta; PoiLN: green; PL: cyan). For the NB and PoiLN models the optimum is observed for , i.e. on the entire data set, while for the PL model the optimum is reached for . Center: goodness-of-fit p-value of fitting models on data larger than or equal to . As suggested by Clauset and collaborators [31] for p-value greater than 0.1 (horizontal line) the fitting model is a good description of the data. For NB the GOF is low (p<10−3), suggesting the inappropriateness of the NB model in describing the data. The GOF of the PoiLN indicates that the model is appropriate only for large value of , thus simultaneously with large values of KS and therefore pointing out the low performance of the model. The PL fits should not be rejected for values of larger than concurrently with the lowest value of KS. Right: Log-likelihood Ratio (LLR) test with Vuong's sign interpretation. Negative (positive) values suggest the alternative model NB (red) or PoiLN (blue) distributions are (are not) favoured in describing values larger than when compared to PL. The horizontal line shows the sign threshold. Full marks show statistically significant tests (p<0.05) while empty marks refer to non significant tests (p>0.05).

Mentions: The probability distribution of tick burden on mice was skewed and showed a heavy tail. The best fit of the NB distribution was obtained on the largest available subsets of data, i.e. with , see left panel of Figure 3. In this setting, the MLE method estimated ( confidence intervals (CI) ) and (95%CI = ). However, the GOF of the NB distribution was very low for any value of , see central panel of Figure 3, thus giving evidence for rejecting the hypothesis of the NB functional form. Similarly, the best fit of PoiLN distribution was achieved on the largest subsets of data, (, see left panel of Figure 3). In this case the estimated parameters were (CI = ) and (CI = ). The GOF of the PoiLN, central panel of Figure 3, suggested that PoiLN was acceptable only for . However, for , the KS statistic displayed values that were too large to consider the PoiLN distribution appropriate for describing real data.


Pattern of tick aggregation on mice: larger than expected distribution tail enhances the spread of tick-borne pathogens.

Ferreri L, Giacobini M, Bajardi P, Bertolotti L, Bolzoni L, Tagliapietra V, Rizzoli A, Rosà R - PLoS Comput. Biol. (2014)

Comparison among fittings of distributions of ticks per host with different functions.Left: Kolmogorov-Smirnov statistic between subsets of data above  and the fitting models on these subsets. Vertical dotted lines represent the optimum value of  for different models (NB: magenta; PoiLN: green; PL: cyan). For the NB and PoiLN models the optimum is observed for , i.e. on the entire data set, while for the PL model the optimum is reached for . Center: goodness-of-fit p-value of fitting models on data larger than or equal to . As suggested by Clauset and collaborators [31] for p-value greater than 0.1 (horizontal line) the fitting model is a good description of the data. For NB the GOF is low (p<10−3), suggesting the inappropriateness of the NB model in describing the data. The GOF of the PoiLN indicates that the model is appropriate only for large value of , thus simultaneously with large values of KS and therefore pointing out the low performance of the model. The PL fits should not be rejected for values of  larger than  concurrently with the lowest value of KS. Right: Log-likelihood Ratio (LLR) test with Vuong's sign interpretation. Negative (positive) values suggest the alternative model NB (red) or PoiLN (blue) distributions are (are not) favoured in describing values larger than  when compared to PL. The horizontal line shows the sign threshold. Full marks show statistically significant tests (p<0.05) while empty marks refer to non significant tests (p>0.05).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4230730&req=5

pcbi-1003931-g003: Comparison among fittings of distributions of ticks per host with different functions.Left: Kolmogorov-Smirnov statistic between subsets of data above and the fitting models on these subsets. Vertical dotted lines represent the optimum value of for different models (NB: magenta; PoiLN: green; PL: cyan). For the NB and PoiLN models the optimum is observed for , i.e. on the entire data set, while for the PL model the optimum is reached for . Center: goodness-of-fit p-value of fitting models on data larger than or equal to . As suggested by Clauset and collaborators [31] for p-value greater than 0.1 (horizontal line) the fitting model is a good description of the data. For NB the GOF is low (p<10−3), suggesting the inappropriateness of the NB model in describing the data. The GOF of the PoiLN indicates that the model is appropriate only for large value of , thus simultaneously with large values of KS and therefore pointing out the low performance of the model. The PL fits should not be rejected for values of larger than concurrently with the lowest value of KS. Right: Log-likelihood Ratio (LLR) test with Vuong's sign interpretation. Negative (positive) values suggest the alternative model NB (red) or PoiLN (blue) distributions are (are not) favoured in describing values larger than when compared to PL. The horizontal line shows the sign threshold. Full marks show statistically significant tests (p<0.05) while empty marks refer to non significant tests (p>0.05).
Mentions: The probability distribution of tick burden on mice was skewed and showed a heavy tail. The best fit of the NB distribution was obtained on the largest available subsets of data, i.e. with , see left panel of Figure 3. In this setting, the MLE method estimated ( confidence intervals (CI) ) and (95%CI = ). However, the GOF of the NB distribution was very low for any value of , see central panel of Figure 3, thus giving evidence for rejecting the hypothesis of the NB functional form. Similarly, the best fit of PoiLN distribution was achieved on the largest subsets of data, (, see left panel of Figure 3). In this case the estimated parameters were (CI = ) and (CI = ). The GOF of the PoiLN, central panel of Figure 3, suggested that PoiLN was acceptable only for . However, for , the KS statistic displayed values that were too large to consider the PoiLN distribution appropriate for describing real data.

Bottom Line: Moreover, we found that the tail of the distribution significantly changes with seasonal variations in host abundance.Specifically, we found that the epidemic threshold and the prevalence equilibria obtained in epidemiological simulations with PL distribution are a good approximation of those observed in simulations feed by the empirical distribution.Moreover, we also found that the epidemic threshold for disease invasion was lower when considering the seasonal variation of tick aggregation.

View Article: PubMed Central - PubMed

Affiliation: Computational Epidemiology Group, Department of Veterinary Sciences, University of Torino, Torino, Italy; Applied Research on Computational Complex Systems Group, Department of Computer Science, University of Torino, Torino, Italy.

ABSTRACT
The spread of tick-borne pathogens represents an important threat to human and animal health in many parts of Eurasia. Here, we analysed a 9-year time series of Ixodes ricinus ticks feeding on Apodemus flavicollis mice (main reservoir-competent host for tick-borne encephalitis, TBE) sampled in Trentino (Northern Italy). The tail of the distribution of the number of ticks per host was fitted by three theoretical distributions: Negative Binomial (NB), Poisson-LogNormal (PoiLN), and Power-Law (PL). The fit with theoretical distributions indicated that the tail of the tick infestation pattern on mice is better described by the PL distribution. Moreover, we found that the tail of the distribution significantly changes with seasonal variations in host abundance. In order to investigate the effect of different tails of tick distribution on the invasion of a non-systemically transmitted pathogen, we simulated the transmission of a TBE-like virus between susceptible and infective ticks using a stochastic model. Model simulations indicated different outcomes of disease spreading when considering different distribution laws of ticks among hosts. Specifically, we found that the epidemic threshold and the prevalence equilibria obtained in epidemiological simulations with PL distribution are a good approximation of those observed in simulations feed by the empirical distribution. Moreover, we also found that the epidemic threshold for disease invasion was lower when considering the seasonal variation of tick aggregation.

Show MeSH
Related in: MedlinePlus