Mathematical Modelling of the Spread and Control of Onchocerciasis in Tropical Countries : Case Study in Nigeria

and Applied Analysis 3 4. Mathematical Formulation of the Model From the model transfer diagram (Figure 2), we have the following set of equations:


Introduction
Onchocerciasis is a country-wide public health problem in Nigeria.The present estimates had suggested that 7-10 million Nigerians are infected with Onchocerca volvulus, and approximately 40 million are at risk of the disease [1].More than 120,000 cases of Onchocerciasis related blindness have been reported [2][3][4], and thousands of people suffer from disabling complications of this disease [5].Onchocerciasis, which is the world's second-leading infectious cause of blindness, is a parasitic disease caused by the filarial nematode, Onchocerca volvulus.It is termed as "river blindness" because it is spread by the bite of the blackfly vector that breads near oxygen-high fast-flowing streams and rivers people rely on for washing, drinking, and farming which results in depopulation of the fertile river valleys.The lifespan of blackflies is short, lasting only 2-3 weeks.The disease affects rural communities in Nigeria and is the major cause of blindness and skin disease in endemic areas with serious socioeconomic effects.It has been identified by the World Health Organization as one of the neglected tropical diseases (NTD) [6,7], "Neglected" because they are not mentioned in the millennium Development Goals.This implied that they are not usually included among the important development discussions, and as such they do not receive adequate attention or funding.
Figure 1 explains the life cycle of the parasite Onchocerca volvulus.
Onchocerca volvulus is transmitted to humans through the bite of a blackfly which introduces immature larva forms of the parasite (infective larvae) into their human host.
The larvae migrate to the subcutaneous tissue where they undergo further development and form skin nodules as they mature into adult worms (macrofilaria).The adult worms mate and produce millions of microscopic larvae (microfilaria).This continues for years of the adult worm's life.Although the infection with larvae begins immediately, the disease may not become apparent in an individual for months to years.In most individuals, it develops slowly in the skin, while some patients may show symptoms initially with eye problems, ocular involvement, and general debilitation.
The blackflies absorb the microfilariae as they feed during the day, which further undergo developmental stages within the blackflies into infective larvae, which are then transmitted to the next human victim.
A study was carried out in Imo River Basin, Imo State, Southeastern Nigeria, to assess the prevalence and intensity of L1 larvae.
Human stages Subcutaneous tissues.
Adults in subcutaneous nodule.
Adults produce unsheathed microfilariae that typically are found in skin and lymphatics of connective tissues, but also occasionally in peripheral blood, urine, and sputum.microfilaria of Onchocerca volvulus in the area [9].The survey coverage was about 91.8% of the study population and it was found by EC.Uttah found that thirty-seven percent of those examined was positive for Onchocerca volvulus microfilariae (39.2% of males and 34.9% of females).The microfilariae (mf) prevalence increased steadily with increasing age to reach 70.4% in the oldest age group.Going towards the northern part of Nigeria, the endemicity of human onchocerciasis was assessed by Anosike and Onwuliri [10] in eight rural communities at risk, in Ningi Local Government Area, Bauchi State, Nigeria, between July 1990 and March 1991.Of the 1536 subjects skin-snipped, 334 (21.7%) were positive for Onchocerca volvulus microfilariae.
Wogu and Okaka investigated the overall prevalence of onchocerciasis in Okpuje, Owan West L.G.A. of Edo State, Nigeria.The figure recorded in their study of 47.5% compares favourably with the 48.6% prevalence reported by Edungbola and Asaolu [11] in Kwara State, Nigeria, but higher than the 26.9% recorded by Nwaorgu et al. [12] in an onchocerciasis mesoendemic area of Enugu State, Nigeria.However, Akinbo and Okaka [13] reported a high overall prevalence (83%) of the disease in Ovia North East L.G.A., Edo State, Nigeria [14].These reports suggest that we need a well-thought mathematical formulation of a simple and appropriate model for the spread and control of this disease.

Model Formulation
In this section, we define the parameters used in the model below as follows:

Model Diagram
Figure 2 shows the model diagram which consists of five sets of ordinary differential equations highlighting the rate of change with respect to time of human susceptible compartment   ; human latent (incubation) period compartment   ; human infective compartment   ; vector susceptible compartment   ; and the vector infective compartment   .

Mathematical Formulation of the Model
From the model transfer diagram (Figure 2), we have the following set of equations: To find the steady state, we assume that the compartments   ,   ,   ,   , and   do not depend on time.Therefore, We now solve the differential equations   / =   / =   / =   / =   / = 0 to find the value for   ,   ,   ,   , and   .From (4), From (5), From (1), But From (2), Lastly, from (3), We have that provided that  2  +  3  >  4 .Therefore, the disease free equilibrium (DFE) is given as and the endemic equilibrium is given as

Stability Analysis
To assess the stability of the (DFE), we solve the model characteristics equation using the Jacobean matrix formulated from ( 1), ( 2), ( 3), (4), and ( 5) above as follows: Applying the eigenvalue we have det Hence, we obtain from −( From From −( 4 + ) = 0, From ( 5 + ) = 0, From ( + ) = 0, Given that the results of  1 ,  2 ,  3 ,  4 , and  5 are all negative and considering the principles of linearized stability which says that the systems are asymptotically stable if and only if all the eigenvalues of the system have negative real parts, we say that the disease-free equilibrium (DFE) is asymptotically stable.

Application of the Homotopy Decomposition Method (HDM)
In this section, we apply the HDM [15][16][17] to solve these mathematical sets of equations representing the onchocerciasis disease model as follows: We then assume that the solutions of the above integral equations can be put in the following form, for  ∈ (0, 1), Replacing the above expressions in the integral equations, we obtain the following expressions: We compared the terms of the same power of  and obtained the following: (32) We now integrate the above equations to obtain the following: Abstract and Applied Analysis 7 We therefore use the following theoretical parameters  1 = 0.5,  2 = 0.5,  3 = 0.5,  4 = 0.5,  5 = 0.5,  = 1,  = 1,  = 10,  = 10,  = 1,  = 1, and  = 1 to examine the behavior of the five different compartmental numerical solutions in the system.Figure 3 shows the behavior of the numerical solution.

Conclusion
Onchocerciasis model was examined and the mathematical formulation underpinning the spread and control of this disease has been proposed on one hand.On the other hand, we made use of some new analytical methods called the homotopy decomposition method (HDM) to derive the solution of the resulting set of (1) to (5).The numerical results were presented to test the efficiency and the accuracy of the method used.The mathematical analysis of the resulting set of (1) to ( 5) above revealed that the stability analysis of the five roots of the characteristics equations has all negative real part.In particular, given that the stability analysis shows that the DFE is asymptotically stable, it would be possible that the parasite Onchocerca volvulus be eradicated in a finite time.On the other hand, the numerical solution confirmed that as the number of humans and vectors in the susceptible compartments reduces, the number of infective humans and vectors in the infectious compartment increases.The numerical solutions in both cases displayed the biological behaviour of the real world situation.Therefore, in order to build a healthy and a sustainable population, treatment with Ivermectin/Mectizan drugs and a more hygienic environment that does not allow the breeding and survival of blackflies could lead to a total eradication of the disease.

Figure 3 :
Figure 3: Numerical solution of the model.
1 is the contact rate between   and   ;  2 is the contact rate between   and   ;  3 is the contact rate between   and   ;  4 is the contact rate between    5 is the contact rate between   and   ;  is the transmission rate from   to   ;  and  are the proportions of Ivermectin/Mectizan treatment that cured the patient;  is the rate of movement of humans from the infective   compartment into the human susceptible   compartment;  is the rate of movement of human from the human infective   compartment into the latent (incubation) period   compartment.