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A mathematical model for onchocerciasis and its treatment with ivermectin
Thesis (MSc)--Stellenbosch University, 2016
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2016
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| _version_ | 1867613896913715200 |
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| access_status_str | Open Access |
| author | Omondi, Evans Otieno |
| author2 | Nyabadza, Farai |
| author_browse | Nyabadza, Farai Omondi, Evans Otieno |
| author_facet | Nyabadza, Farai Omondi, Evans Otieno |
| author_sort | Omondi, Evans Otieno |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2016 |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/100317 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:43:26.328Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/100317 A mathematical model for onchocerciasis and its treatment with ivermectin Omondi, Evans Otieno Nyabadza, Farai Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences Onchocerciasis Ivermectin Epidemiology -- Backward bifurcation Impulsive differential equations Parasites -- Control -- Mathematical models Epidemiology -- Mathematical models UCTD Thesis (MSc)--Stellenbosch University, 2016 ENGLISH ABSTRACT : Onchocerciasis is a human disease that is caused by the filirial worm Onchocerca volvulus. It occurs close to oxygen rich flowing streams and rivers in the inter-tropical zones. The disease is a serious public health problem and any control interventions must be effective so as to eliminate it from the population. A rational control intervention method is through mass administration of ivermectin. In this study, a non-linear mathematical model is formulated to model the transmission dynamics and spread of onchocerciasis disease. First, a mathematical model is formulated to evaluate the impact of long term mass treatment of onchocerciasis with ivermectin. The model basic reproduction number is computed and the stability analysis presented. The model is found to exhibit a backward bifurcation so that for R0 less than unity is not sufficient to eradicate the disease from the population and the need is to lower R0 to below a certain threshold, Rc0 for effective disease control. Optimal control theory is applied to investigate the control strategies for eliminating onchocerciasis using time dependent controls. The characterization of the optimal control is carried out via the Pontryagin’s Maximum Principle. The simulation results demonstrate that the maximum implementation of vector control, personal protection strategies and enhanced mass treatment with ivermectin strategies in the epidemic are of very critical impacts in the optimal control. This gives a theoretical interpretation to the practical experiences that the insecticide and use of insect repellents are important in the control of onchocerciasis. The model is extended to assess the effectiveness of mass administration using impulsive differential equations since the treatment itself is considered to be impulsive. The basic reproduction number is derived and we establish that the global dynamics are completely determined by the values of R0. The disease-free equilibrium is globally asymptotically stable for basic reproduction number R0 < 1, while for R0 > 1, a unique endemic equilibrium exists and is globally asymptotically stable. We then use the model to determine the threshold for the proportion of infected individuals that reduces the infection in the human (host) population. Finally, we present numerical simulations results. Sensitivity analysis reveals that the most sensitive parameters in the onchocerciasis epidemic are the human transmission contact rate bh , the vector transmission contact rate bv and vector death rate μv. The numerical results show that efficiency of the mass administration of ivermectin contributes a greater impact on the disease dynamics and moves the system towards the disease free state. AFRIKAANSE OPSOMMING : Onchocerciasis is ’n menslike siekte wat veroorsaak word deur die filariese wurm Onchocerca volvulus. Dit kom voor naby suurstofryke vloeiende strome en riviere in intertropiese met ivermektien strategieë in die epidemie van kritiese belang is in die optimale kontrole. Dit gee ’n teoretiese verklaring vir die praktiese ondervinding dat insekgif en die gebruik van (insek) afweermiddels belangrik is in die kontrole van onchocerciasis. Die model word uitgebrei om die effektiwiteit van massa toedien vas te stel deur die gebruik van impulsiewe differensiaalvergelykings aangesien die behandeling self as impulsief beskou word. Die basiese voorplantingsgetal word afgelei en ons bevestig dat die globale dinamika volledig bepaal word deur die waardes van R0. Die siektevrye ewewig is globaal asimptoties stabiel wanneer die voortplantingsgetal R0 < 1, terwyl daar vir R0 > 1 ’n unieke endemiese ewewig bestaan wat globaal asimptoties stabiel is. Ons gebruik dan die model om die drumpel van persentasie van besmette individue te bepaal wat die besmetting in die menslike (gasheer) bevolking verminder. Laastens bied ons uitslae van numeriese simulasies aan. Sensitiwiteitsanalise toon aan dat die mees sensitiewe parameters in die onchocerciasis epidemie die menslike oordragsverhouding bh, die vektoroordragskontakverhouding bv en die vektordoodsverhouding μv is. Die numeriese uitslae toon dat die die effektiwiteit van massa toedien van ivermektien ’n groter invoed op die siekte se dinamika het en die stelsel na die siektevrye toestand beweeg sones. Hierdie siekte is ’n ernstige publieke gesondheidsgevaar en enige kontrole ingrypings moet effektief wees om dit uit die bevolking uit te wis. ’n Rasionele kontrole ingrypingsmetode is die massa toedien van ivermektien. In hierdie studie word ’n nie-lineêre wiskundige model geformuleer om die oordrags- en verspreidingsdinamika van die onchocerciasis siekte te modelleer. Eerstens word ’n wiskundige model geformuleer om die impak van langtermyn massabehandeling van onchoncersiasis met ivermektien te bepaal. Die model se basiese voortplantingsgetal word bereken en die stabiliteitsanalise word aangebied. Die model word bevind om ’n terugwaardse bifurkasie ten toon te stel sodat die voorwaarde R0 kleiner as eenheid nie genoeg is om die siekte uit die bevolking uit te wis nie en dat dit nodig is om R0 te verlaag tot onder ’n sekere drumpel Rc0 vir effektiewe siektekontrole. Optimale kontrole teorie word met tydafhanklike kontroles toegepas om die kontrole strategieë om onchocerciasis uit te wis te bestudeer. Die karakterisering van die optimale kontrole word uitevoer met Pontryagin se Maksimumbeginsel. Die uitslae van die simulasie wys dat die maksimum uitvoering van vektorkontrole, persoonlike beskermingstrategieë en verbeterde massabahandeling 2016-12-22T13:39:45Z 2016-12-22T13:39:45Z 2016-12 Thesis http://hdl.handle.net/10019.1/100317 en_ZA Stellenbosch University xv, 31 pages : illustrations (chiefly colour), maps application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Onchocerciasis Ivermectin Epidemiology -- Backward bifurcation Impulsive differential equations Parasites -- Control -- Mathematical models Epidemiology -- Mathematical models UCTD Omondi, Evans Otieno A mathematical model for onchocerciasis and its treatment with ivermectin |
| title | A mathematical model for onchocerciasis and its treatment with ivermectin |
| title_full | A mathematical model for onchocerciasis and its treatment with ivermectin |
| title_fullStr | A mathematical model for onchocerciasis and its treatment with ivermectin |
| title_full_unstemmed | A mathematical model for onchocerciasis and its treatment with ivermectin |
| title_short | A mathematical model for onchocerciasis and its treatment with ivermectin |
| title_sort | mathematical model for onchocerciasis and its treatment with ivermectin |
| topic | Onchocerciasis Ivermectin Epidemiology -- Backward bifurcation Impulsive differential equations Parasites -- Control -- Mathematical models Epidemiology -- Mathematical models UCTD |
| url | http://hdl.handle.net/10019.1/100317 |
| work_keys_str_mv | AT omondievansotieno amathematicalmodelforonchocerciasisanditstreatmentwithivermectin AT omondievansotieno mathematicalmodelforonchocerciasisanditstreatmentwithivermectin |