Article

Modeling the transmission dynamics and control of lymphatic filariasis (Wuchereria bancrofti parasites) in Papua New Guinea

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Abstract

A deterministic model was developed and analyzed to investigate the dynamics of lym-phatic filariasis disease through mosquitoes in four stages of human population based on the microfilariae and antibody levels. The stages are functions of transmission rates, reproduction, mortality, drug chemotherapy, bednet usage and insecticide spraying. The survival of the infected mosquitoes (vector) depends on the size of the load of microfilariae engorged.This disease is significantly a public health problem in many countries, especially Papua New Guinea (PNG) where the level of transmission of wuchereria bancrofti parasite and body manifestation of the disease register among the highest in the world. Data from Papua New Guinea are used to estimate some of the parameters in the model that was analyzed based on a system of six ordinary differential equations. It was established that the disease free equilibrium was locally asymptotically stable. The model was also found to be globally asymptotically stable when the basic reproduction number was less than unity. It was also revealed that the disease invaded the endemic population at R 0 = 1 near the disease free equilibrium point. Results in our simulations indicated a combination of drug chemotherapy, use of bed-nets and insecticide residual spraying as most efficient measures in controlling the disease in Papua New Guinea.

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  • Pe Simonsen
  • Em Pederssen
  • Rt Rwegoshora
  • Mn Malacela
  • Ya Derua
  • Sm Magesa
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  • P Sapak
  • Melrose
  • Pawa
  • Wynd
  • Leggat
  • T Taufa
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Lymphatic Filariasis Control in Tanzania: Effect of Repeated Mass Drug Administration with Ivermectin and Albendazole on Infection and Transmission. tex-titPLoS Negl Trop Dis
  • P E Simonsen
  • E M Pederssen
  • R T Rwegoshora
  • M N Malacela
  • Y A Derua
  • S M Magesa
Simonsen, PE, Pederssen EM, Rwegoshora RT, Malacela MN, Derua YA, Magesa SM. Lymphatic Filariasis Control in Tanzania: Effect of Repeated Mass Drug Administration with Ivermectin and Albendazole on Infection and Transmission. tex-titPLoS Negl Trop Dis. 2010 June; 4(6): e696.
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  • M J Taylor
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