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Of lice and math: using models to understand and control populations of head lice.

Laguna MF, Laguna MF, Risau-Gusman S - PLoS ONE (2011)

Bottom Line: In the case of treatments, we study the difference in performance that arises when they are applied in systematic and non-systematic ways.It is shown that this parameter can be tuned to obtain collective infestations whose characteristics are compatible with what is given in the literature on real infestations.For both cases we assess the impact of several collective strategies of treatment.

View Article: PubMed Central - PubMed

Affiliation: Consejo Nacional de Investigaciones Cientficas y Técnicas and Centro Atómico Bariloche, Bariloche, Río Negro, Argentina. lagunaf@cab.cnea.gov.ar

ABSTRACT
In this paper we use detailed data about the biology of the head louse (pediculus humanus capitis) to build a model of the evolution of head lice colonies. Using theory and computer simulations, we show that the model can be used to assess the impact of the various strategies usually applied to eradicate head lice, both conscious (treatments) and unconscious (grooming). In the case of treatments, we study the difference in performance that arises when they are applied in systematic and non-systematic ways. Using some reasonable simplifying assumptions (as random mixing of human groups and the same mobility for all life stages of head lice other than eggs) we model the contagion of pediculosis using only one additional parameter. It is shown that this parameter can be tuned to obtain collective infestations whose characteristics are compatible with what is given in the literature on real infestations. We analyze two scenarios: One where group members begin treatment when a similar number of lice are present in each head, and another where there is one individual who starts treatment with a much larger threshold ("superspreader"). For both cases we assess the impact of several collective strategies of treatment.

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Evolution of lice colonies.Average number of lice of a colony that is started at day  by a female that had her last moult 10 days before, as a function of time. Symbols represent averages taken over  populations whereas full lines represent the theoretical predictions. The inset shows the first days of one of these populations. Here, as in the rest of the figures, the error bars represent the standard error of the mean, and data whose error bars are not shown have standard errors lower than symbol size.
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pone-0021848-g001: Evolution of lice colonies.Average number of lice of a colony that is started at day by a female that had her last moult 10 days before, as a function of time. Symbols represent averages taken over populations whereas full lines represent the theoretical predictions. The inset shows the first days of one of these populations. Here, as in the rest of the figures, the error bars represent the standard error of the mean, and data whose error bars are not shown have standard errors lower than symbol size.

Mentions: For the two sets of parameters used, the intrinsic growth rate [25], i.e. the largest eigenvalue of the corresponding evolution matrix, is: for the ES model and for the TL model. Fig. 1 shows the average evolution of a population initiated by a 10-day old single female (i.e. a female which has undergone her last moult 10 days ago). The time-dependence of the population is calculated using evolution matrices and by means of numerical simulations. In the case of the simulations, we plot the average values obtained from realizations which start with the same initial condition (one 10-day old female). The figure shows that there is a very good agreement between theory and the average of simulations, which confirms that the algorithm is a stochastic version of the model. We also studied the evolution of colonies starting with different initial conditions, and found no substantial differences with the case shown in Fig. 1. In all our calculations we have assumed that the first female does not need to be fertilized by a male: it has been reported that females can lay eggs during several days after being fertilized [26], and it has even been suggested that a single mating could be enough to achieve lifetime fertility [27]. The figure shows that after the first month the number of adults grows rapidly from hundreds to thousands of individuals. In real populations living in human heads, however, it is well known that the average number of live lice is typically about [2] (although there are records of individuals with hundreds of adult lice [28], [29]).


Of lice and math: using models to understand and control populations of head lice.

Laguna MF, Laguna MF, Risau-Gusman S - PLoS ONE (2011)

Evolution of lice colonies.Average number of lice of a colony that is started at day  by a female that had her last moult 10 days before, as a function of time. Symbols represent averages taken over  populations whereas full lines represent the theoretical predictions. The inset shows the first days of one of these populations. Here, as in the rest of the figures, the error bars represent the standard error of the mean, and data whose error bars are not shown have standard errors lower than symbol size.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3140471&req=5

pone-0021848-g001: Evolution of lice colonies.Average number of lice of a colony that is started at day by a female that had her last moult 10 days before, as a function of time. Symbols represent averages taken over populations whereas full lines represent the theoretical predictions. The inset shows the first days of one of these populations. Here, as in the rest of the figures, the error bars represent the standard error of the mean, and data whose error bars are not shown have standard errors lower than symbol size.
Mentions: For the two sets of parameters used, the intrinsic growth rate [25], i.e. the largest eigenvalue of the corresponding evolution matrix, is: for the ES model and for the TL model. Fig. 1 shows the average evolution of a population initiated by a 10-day old single female (i.e. a female which has undergone her last moult 10 days ago). The time-dependence of the population is calculated using evolution matrices and by means of numerical simulations. In the case of the simulations, we plot the average values obtained from realizations which start with the same initial condition (one 10-day old female). The figure shows that there is a very good agreement between theory and the average of simulations, which confirms that the algorithm is a stochastic version of the model. We also studied the evolution of colonies starting with different initial conditions, and found no substantial differences with the case shown in Fig. 1. In all our calculations we have assumed that the first female does not need to be fertilized by a male: it has been reported that females can lay eggs during several days after being fertilized [26], and it has even been suggested that a single mating could be enough to achieve lifetime fertility [27]. The figure shows that after the first month the number of adults grows rapidly from hundreds to thousands of individuals. In real populations living in human heads, however, it is well known that the average number of live lice is typically about [2] (although there are records of individuals with hundreds of adult lice [28], [29]).

Bottom Line: In the case of treatments, we study the difference in performance that arises when they are applied in systematic and non-systematic ways.It is shown that this parameter can be tuned to obtain collective infestations whose characteristics are compatible with what is given in the literature on real infestations.For both cases we assess the impact of several collective strategies of treatment.

View Article: PubMed Central - PubMed

Affiliation: Consejo Nacional de Investigaciones Cientficas y Técnicas and Centro Atómico Bariloche, Bariloche, Río Negro, Argentina. lagunaf@cab.cnea.gov.ar

ABSTRACT
In this paper we use detailed data about the biology of the head louse (pediculus humanus capitis) to build a model of the evolution of head lice colonies. Using theory and computer simulations, we show that the model can be used to assess the impact of the various strategies usually applied to eradicate head lice, both conscious (treatments) and unconscious (grooming). In the case of treatments, we study the difference in performance that arises when they are applied in systematic and non-systematic ways. Using some reasonable simplifying assumptions (as random mixing of human groups and the same mobility for all life stages of head lice other than eggs) we model the contagion of pediculosis using only one additional parameter. It is shown that this parameter can be tuned to obtain collective infestations whose characteristics are compatible with what is given in the literature on real infestations. We analyze two scenarios: One where group members begin treatment when a similar number of lice are present in each head, and another where there is one individual who starts treatment with a much larger threshold ("superspreader"). For both cases we assess the impact of several collective strategies of treatment.

Show MeSH
Related in: MedlinePlus