Journal of Mathematical Biology

, Volume 56, Issue 6, pp 743–763

# A stochastic model for head lice infections

• Patricia Stone
• Hilde Wilkinson-Herbots
• Valerie Isham
Article

## Abstract

We investigate the dynamics of head lice infections in schools, by consideringa model for endemic infection based on a stochastic SIS (susceptible-infected-susceptible) epidemic model, with the addition of an external source of infection. We deduce a range of properties of our model, including the length of a single outbreak of infection. We use the stationary distribution of the number of infected individuals, in conjunction with data from a recent study carried out in Welsh schools on the prevalence of head lice infections, and employ maximum likelihood methods to obtain estimates of the model parameters. A complication is that, for each school, only a sample of the pupils was checked for infection. Our likelihood function takes account of the missing data by incorporating a hypergeometric sampling element. We arrive at estimates of the ratios of the “within school” and “external source” transmission rates to the recovery rate and use these to obtain estimates for various quantities of interest.

### Keywords

Head lice Epidemic Stochastic SIS model Transmission dynamics

62M05

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### References

1. 1.
Andersson H. and Djehiche B. (1998). A threshold limit theorem for the stochastic logistic epidemic. J. Appl. Probab. 35: 662–670
2. 2.
Ball F. (1999). Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math. Biosci. 156: 41–67
3. 3.
Ball F. and Lyne O. (2001). Stochastic multitype SIR epidemics among a population partitioned into households. Adv. Appl. Probab. 33: 99–123
4. 4.
Burgess I. (2004). Human lice and their control. Annu. Rev. Entomol. 49: 457–81
5. 5.
Clancy D. (2005). A stochastic SIS infection model incorporating indirect transmission. J. Appl. Probab. 42: 726–737
6. 6.
Counahan M., Andrews R., Buttner P., Byrnes G. and Speare R. (2004). Head lice prevalence in primary schools in Victoria, Australia. J. Paediatr. Child. Health 40: 616–619
7. 7.
Dodd, C.S.: Interventions for treating headlice. Cochrane Database Syst. Rev. (4), CD001165 doi:10.1002/14651858.CD001165.pub2 (2006)
8. 8.
Downs A., Harvey I. and Kennedy C. (1999). The epidemiology of head lice and scabies in the UK epidemiol. Infection 122: 471–477 Google Scholar
9. 9.
Downs A., Stafford K., Hunt L., Ravenscroft J.C. and Coles G. (2002). Widespread insecticide resistance in head lice to the over-the-counter pediculocides in England and the emergence of carbaryl resistance. Br. J. Dermatol. 146: 88–93
10. 10.
Goel N. and Richter-Dyn N. (1974). Stochastic Models in Biology. Academic, New York Google Scholar
11. 11.
Hill N., Moor G., Cameron M.M., Butlin A., Preston S., Williamson M. and Bass C. (2005). Single blind, randomised, comparative study of the Bug Buster kit and over the counter pediculicide treatments against headlice in the UK. BMJ 331: 384–387
12. 12.
Koch T., Brown M., Selim P. and Isam C. (2001). Towards the eradication of head lice: literature review and research agenda. J. Clin. Nurs. 10: 364–371
13. 13.
Kryscio R. and Lefevre C. (1989). On the extinction of the S-I-S stochastic logistic epidemic. J. Appl. Probab. 26: 685–694
14. 14.
Kurtz T. (1970). Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 7: 49–58
15. 15.
Kurtz T. (1971). Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 8: 344–356
16. 16.
Nåsell I. (1996). The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Probab. 28: 895–932
17. 17.
Nåsell I. (1999). On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosci. 156: 21–40
18. 18.
Norden R. (1982). On the distribution of the time to extinction in the stochastic logistic population model. Adv. Appl. Probab. 14: 687–708
19. 19.
Pollett, P.: Diffusion approximations for ecological models. In: Ghassemi F (ed.) Proceedings of the International Congress on Modelling and Simulation, vol. 2 of Modelling and Simulation Society of Australia and New Zealand, Canberra, Australia, pp. 843–848 (2001)Google Scholar
20. 20.
Riggs T. and Koopman J. (2004). A stochastic model of vaccine trials for endemic infections using group randomisation. Epidemiol. Infect. 132: 927–938
21. 21.
Roberts R., Casey D., Morgan D. and Petrovic M. (2000). Comparison of wetcombing with malathion for treratment of head lice in the UK: a pragmatic randomised controlled trial. Lancet 356: 540–544
22. 22.
Ross J., Taimre T. and Pollett P. (2006). On parameter estimation in population models. Theor. Popul. Biol. 70: 498–510
23. 23.
Ross S. (2000). Introduction to Probability Models. Academic, New York
24. 24.
Thomas D., McCarroll L., Roberts R., Karunaratne P., Roberts C., Casey D., Morgan S., Touhig K., Morgan J., Collins F. and Hemingway J. (2006). Surveillance of insecticide resistance in head lice using biochemical and molecular methods. Arch. Dis. Child. 91: 777–778
25. 25.
Vermaak Z. (1996). Model for the control of pediculus humanus capitis. Public Health 110: 283–288
26. 26.
Weiss G. and Dishon M. (1971). On the asymptotic behaviour of the stochastic and deterministic models of an epidemic. Math. Biosci. 11: 261–265
27. 27.
Wierman J. and Marchette D. (2004). Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction. Comput. Stat. Data. Ann. 45: 3–23