Stability Analysis of a Mathematical Model for Onchocerciaisis Disease Dynamics

: In this work, we propose a Deterministic Mathematical Model that Combines Infectious but not Blind and Infectious Blind Compartments for Onchocerciasis Transmission and Control. Onchocerciasis is usually the term used to describe river blindness, it is a disease that causes blindness, and the second largest cause of blindness after trachoma. It mainly affects the eyes and the skin. The equilibrium states of the model are obtained. The disease free equilibrium state is analysed for stability; the condition for its stability is obtained as an inequality constraint on the parameters. Results shows that although, a 60% treatment coverage rate of infected and infectious blind individuals only is better than 80% treatment coverage rate of infected but not blind individuals only. Also, all the four control strategies reduce the effective reproduction number below unity. A 40% coverage rate of fumigation and treatment of infectious but not blind is better than a 40%coverage rate of fumigation only. It further reveals that a 30% coverage rate of fumigation and treatment of infectious blind is better than 80%coverage rate of fumigation only or fumigation and treatment of infected but not blind only. We are able to show that disease free equilibrium and endemic equilibrium exists and are both locally and globally stable, and we computed the c R of the model and showed that it is a parameter to test for stability, we also use the Jacobi stability technique to show that disease free equilibrium and endemic equilibrium are both locally and globally stable. The sensitivity analysis results shows that the most sensitive parameter is ρ while the least sensitive is v µ ,

Onchocerciasis is usually the term used to describe river blindness, it is a disease that causes blindness, and the second largest cause of blindness after trachoma. It mainly affects the eyes and the skin. It is caused by the parasitic filarial nematode Onchocerca volvulus and is transmitted by the bites of Simulium blackflies Center for Disease Control and World Health Organisation (CDC, 2013;WHO, 1995). Onchocerciasis is often referred to as River Blindness because the blackfly laid its eggs attached to rocks and vegetation submerged in fast flowing, highly oxygenated rivers and streams where larval and pupa stages develop before transformation to the adult. This result leads to high prevalence of eye disease in villages located along fast flowing rivers where the blackfly breeds. (World Health Organization, 1995) Despite numerous strategies employed by various national and international organization to eradicate the disease, recent surveillance data from the World Health Organization (WHO) and Onchocerciasis Control Programme (OCP) reveal that more than 17.7 million are infected globally. Onchocerciasis is also endemic in Western and Central Africa and also in Central and Southern America.
Most of the cases of blindness caused by Onchocerciasis are found in sub-Sahara Africa (outside the areas covered by Onchocerciasis Control Programme; OCP, (WHO, 1995). Although Onchocerciasis is not a disease that leads to death, there is clear evidence that blindness may cause social and economic complications which may lead to early death because sight is light. Onchocerciasis is particularly prevalent in tropical Africa and parts of tropical America; (Basáñez 2002). More recent estimates by World Health Organisation (WHO, 1995) mention more than 17.7 million infected, 500000 visually impaired and another 270000 blind. About 99% of infected persons are in Africa and 11% in Nigeria and it is more prevalence in Mubi Village in Adamawa State where about 89% of the entire village suffers one form of blindness or the other. This hyper-endemic earns the community the village of the blind. The occurrence is found in Yemen and some countries in South America. Onchocerciasis is locally transmitted in thirty countries of Africa, 13 foci in the Americas (Mexico, Guatemala, Ecuador, Colombia, Venezuela,Brazil) and in Yemen. These Countries are classified as Meso and Hyper endemic by the OCP Countries categories. In West Africa, the fear of infection is one of the major causes of migration from fertile riverine areas into sub marginal lands, which results in over cultivation and low productivity (Basáñez et al. 2006) BAKO, DU; AKINWANDE, NI; ENAGI, AI; KUTA, FA; ABDULRAHMAN, S During the last decade, Jimmy and Horst (2003), Basáñez and Ricardez -Esquinca (2001), Jibrin and Ibrahim (2011), Ikechukwu and Thomas (2014), Shaib et al (2015), Abdon and Rubayyi (2015) and Hugo (2013) have designed mathematical models on Onchocerciasis (river blindness). Considering the works of the afore-mention authors, the study at hand is an improvement on the cited models above in that it includes; The latent, the infectious blind and recovered classes The treatment of actively infected individuals and blind as control parameters Incorporating the fumigation parameter Loss of immunity after recovered.

MATERIALS AND METHODS
Model Formulation: A mathematical model for the transmission dynamics and control of Onchocerciasis was developed, improving on the existing models as explained in the literature review by incorporating the infectious but not blind and the blind compartments varying population size (birth rate not equal to death removal rate), and standard incidence.
in the biological feasible region: System (1)

It can be seen that all solutions of the system starting in Ω
which can be shown to be positively invariant.
Effective Reproductive number ( ) c R One of the most important concern in the analysis of epidemiological models is the determination of the asymptotic behaviour of their solution which is usually based on the stability of the associated equilibria. These models typically consist of disease free equilibrium and at least one endemic equilibrium. The local stability is determined based on a threshold parameter known as basic reproduction number c R this represents the average number of  A better widely accepted and used method for finding ( ) c R that reflect its biological meaning is the next generation operator approach described by Diekmann and Heesterbeek (2000) and subsequently analysed by Van de Driessche and Watmough (2002). Using this technique we obtained the effective reproductive number, ( ) c R of the system (1) which is the spectral radius ( ) ρ of the next generation matrix, ( ) Then,

RESULT AND DISCUSSION
In this section, we presented some numerical simulation to monitor the dynamics of the model for various values of the associated effective reproductive number in order to confirm our analytic results on the global stability of the disease free equilibrium and the effect of different control strategies.   It reveals that any of the 5 control strategies have positive effects in controlling Onchocerciasis but not all can lead to stable disease free state. ( ) 1 c R ≤ , or treatment of infectious but not blind only or treatment of infectious blind only at any coverage rate does not have much impact on the control of onchocerciasis; the disease continue to persist even though there is a decrease in morbidity.

Fig 2:
reveals that the four control strategies reduce the effective reproductive number below unity. Although, a 40% coverage rate of fumigation and treatment of Infectious but not blind is better than a 40% coverage rate of fumigation only. It further reveals that a 30% coverage rate of fumigation and treatment of Infectious but not blind is better than 80% coverage rate of fumigation only or fumigation and treatment of Infectious but not blind only.
Numerical Simulations: Next, we used numerical simulations to further confirm and extend the results earlier obtained. . This shows that more of the Infectious individuals should be treated with Ivermectin annually, since 75% treatment rate of Infectious but not blind reduces the morbidity to a disease free equilibrium in the first three (3) years. Hence, it is advice that 75% treatment will reduce onchocerciasis to a disease free equilibrium state. Conclusion: We presented a new mathematical model for the stability analysis for onchocerciasis transmission dynamics, incorporating the infectious but not blind, the infectious blind individuals and the fumigation parameter. Our analysis reveals that the four control strategies reduce the effective reproductive number below unity. Although, a40% coverage rate of fumigation and treatment of Infectious but not blind is better than a 40% coverage rate of fumigation only. It further reveals that a 30% coverage rate of fumigation and treatment of Infectious but not blind is better than 80% coverage rate of fumigation only or fumigation and treatment of Infectious but not blind only.