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

Authors: Ikechukwu Chiwueze Oguoma; Thomas Mbah Acho
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Onchocerciasis, also known as river blindness and Robles disease, is a parasitic disease caused byOnchocerca volvulus, a nematode (roundworm), and it is endemic in tropical countries like Nigeria. The objective of this paper is to discuss the mathematical formulation underpinning the spread and control of this disease on one hand. On the other hand, we make use of some new analytical methods to derive the solution of the resulting set of equations. The numerical results are presented to test the efficiency and the accuracy of both methods. The techniques used for solving these problems are friendly, very easy, and less time consuming. The numerical solutions in both cases display the biological behaviour of the real life situation.


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## Body

## 1. Introduction

Onchocerciasis is a country-wide public health problem in Nigeria. The present estimates had suggested that 7–10 million Nigerians are infected withOnchocerca volvulus, and approximately 40 million are at risk of the disease [1]. More than 120,000 cases of Onchocerciasis related blindness have been reported [2–4], 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.Figure1 explains the life cycle of the parasiteOnchocerca volvulus.Figure 1
Life cycle ofOnchocerca volvulus [8].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 microfilaria ofOnchocerca 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 forOnchocerca 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.


## 2. Model Formulation

In this section, we define the parameters used in the model below as follows:β1 is the contact rate between HS and HE; β2 is the contact rate between HS and HI; β3 is the contact rate between HE and HI; β4 is the contact rate between HI and VS; β5 is the contact rate between VS and VI; γ is the transmission rate from VI to HS; ρ and δ are the proportions of Ivermectin/Mectizan treatment that cured the patient; ζ is the rate of movement of humans from the infective HI compartment into the human susceptible HS compartment; λ is the rate of movement of human from the human infective HI compartment into the latent (incubation) period HE compartment.


## 3. Model Diagram

Figure2 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 HS; human latent (incubation) period compartment HE; human infective compartment HI; vector susceptible compartment VS; and the vector infective compartment VI.Figure 2
Model transfer diagram for onchocerciasis.


## 4. Mathematical Formulation of the Model

From the model transfer diagram (Figure2), we have the following set of equations:
(1)dHSdt=ρζ+γVI-(β1+β2)HS,(2)dHEdt=δλ+β1HS-β3HE,(3)dHIdt=β2HS+β3HE-ρζ-δλ-β4HI,(4)dVSdt=β4HI-β5VS,(5)dVIdt=β5VS-γVI.
To find the steady state, we assume that the compartments VS, VI, HE, HS, and HI do not depend on time. Therefore,
(6)dHS-dt=dHE-dt=dHI-dt=dVS-dt=dVI-dt=0.
We now solve the differential equations dHS-/dt=dHE-/dt=dHI-/dt=dVS-/dt=dVI-/dt=0 to find the value for HS-, HE-, HI-, VS-, and VI-.From (4),
(7)β4HI-β5VS=0,HI-=β5β4VS-,HI-=AVS-.
From (5),
(8)β5VS-γVI=0,VI-=β5VS-γ=BVS-.
From (1),
(9)ρζ+γVI-(β1+β2)HS=0,(β1+β2)HS=ρζ+γVI.
But
(10)VI-=β5VS-γ,(β1+β2)HS-=ρζ+γβ5VS-γ,HS-=ρζ+β5VS-β1+β2=CVS-.
From (2),
(11)δλ+β1HS-β3HE=0,β3HE-=δλ+β1HS-,but,HS-=ρζ+β5VS-β1+β2,β3HE-=δλ+β1(ρζ+β5VS-β1+β2),HE-=δλ(β1+β2)+β1ρζ+β1β5VS-β3=DVS-.
Lastly, from (3),
(12)β2HS+β3HE-ρζ-δλ-β4HI=0,β2HS-+β3HE--ρζ-δλ-β4HI-=0,β2CVS-+β3DVS--ρζ-δλ-β4AVS-=0.
We have that
(13)VS-=ρζ+δλβ2C+β3D-β4A,
provided that β2C+β3D>β4A.Therefore, the disease free equilibrium (DFE) is given as(14)E0=(ρζβ1+β2,0,0,0,0)
and the endemic equilibrium is given as
(15)HS-=ρζ+β5β1+β2((ρζ+δλ)-β4(β5β4))-1×((δλ(β1+β2)+β1ρζ+β1β5β3)β2(ρζ+β5β1+β2)+β3×(δλ(β1+β2)+β1ρζ+β1β5β3)-β4(β5β4)(δλ(β1+β2)+β1ρζ+β1β5β3))-1)=((ρζ+β5)(ρζ+δλ))×(β2(ρζ+β5)+(β1+β2)×[δλ(β1+β2)+β1ρζ+β1β5]-(β1+β2)β5)-1,(16)HE-=δλ(β1+β2)+β1ρζ+β1β5β3×((ρζ+δλ)(ρζ+β5β1+β2)+β1ρζ+β1β5-β5)-1(β2(ρζ+β5β1+β2)+δλ(β1+β2)(ρζ+β5β1+β2)+β1ρζ+β1β5-β5)-1)=([δλ(β1+β2)+β1ρζ+β1β5](ρζ+δλ))×((β2β3)(ρζ+β5)β1+β2+δλ(β1+β2)β3(β2β3)(ρζ+β5)β1+β2+β1β3ρζ+β1β3β5-β3β5)-1,(17)HI-=β5β4((ρζ+δλ)×(δλ(β1+β2)+β1ρζ+β1β5β3)-β4(β5β4))-1×((δλ(β1+β2)+β1ρζ+β1β5β3)β2(ρζ+β5β1+β2)+β3(δλ(β1+β2)+β1ρζ+β1β5β3)-β4(β5β4)(δλ(β1+β2)+β1ρζ+β1β5β3))-1×(δλ(β1+β2)+β1ρζ+β1β5β3)-β4(β5β4))-1)=(β5(ρζ+δλ))(β2β4(ρζ+β5β1+β2)+β4δλ(β1+β2)(ρζ+β5β1+β2)+β1β4ρζ+β1β4β5-β4β5)-1,(18)VS-=((ρζ+δλ)-β4(β5β4))-1×(β2(ρζ+β5β1+β2)(δλ(β1+β2)+β1ρζ+β1β5β3)+β3(δλ(β1+β2)+β1ρζ+β1β5β3)-β4(β5β4)(δλ(β1+β2)+β1ρζ+β1β5β3))-1),(19)VI-=β5γ((ρζ+δλ)-β4(β5β4))-1×(β2(ρζ+β5β1+β2)(δλ(β1+β2)+β1ρζ+β1β5β3)+β3(δλ(β1+β2)+β1ρζ+β1β5β3)-β4(β5β4)(δλ(β1+β2)+β1ρζ+β1β5β3))-1)=(β5(ρζ+δλ))×(γβ2(ρζ+β5β1+β2)+γδλ(β1+β2)(ρζ+β5β1+β2)+γβ1ρζ+γβ1β5-γβ5)-1.


## 5. 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:
(20)J=(-(β1+β2)000γβ1-β3000β2β3-β40000β4-β50000β5-γ).
Applying the eigenvalue
(21)λ:det⁡(J-λI)=0,
we have(22)det⁡[-(β1+β2)-λ000γβ1-β3-λ000β2β3-β4-λ0000β4-β5-λ0000β5-γ-λ]=0,det⁡[-(β1+β2+λ)000γβ1-(β3+λ)000β2β3-(β4+λ)0000β4-(β5+λ)0000β5-(γ+λ)]=0,-(β1+β2+λ)[-(β3+λ)000β3-(β4+λ)000β4-(β5+λ)000β5-(γ+λ)]+γ[β1-(β3+λ)00β2β3-(β4+λ)000β4-(β5+λ)000β5]=0,-(β1+β2+λ)[-(β3+λ)|-(β4+λ)00β4-(β5+λ)00β5-(γ+λ)|]+γ[β1[β3-(β4+λ)00β4-(β5+λ)00β5]+(β3+λ)[β2-(β4+λ)00β4-(β5+λ)00β5]]=0,-(β1+β2+λ)[-(β3+λ)[-(β4+λ)|-(β5+λ)0β5-(γ+λ)|]]+γβ1β3β4β5+γ(β3+λ)β2β4β5=0,-[(β1+β2+λ)(β3+λ)(β4+λ)(β5+λ)(γ+λ)]+γβ1β3β4β5+γβ2β3β4β5+γλβ2β4β5=0.Hence, we obtain from-(β1+β2+λ)=0(23)λ1=-(β1+β2).
From -(β3+λ)=0,
(24)λ2=-β3.
From -(β4+λ)=0,
(25)λ3=-β4.
From (β5+λ)=0,
(26)λ4=-β5.
From (γ+λ)=0,
(27)λ5=-γ.
Given that the results of λ1, λ2, λ3, λ4, and λ5 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.


## 6. Application of the Homotopy Decomposition Method (HDM)

In this section, we apply the HDM [15–17] to solve these mathematical sets of equations representing the onchocerciasis disease model as follows:
(28)HS(t)-HS(0)=∫0t(ρζ+γVI(τ)-(β1+β2)HS(τ))dτ,HE(t)-HE(0)=∫0t(δλ+β1HS(τ)-β3HE(τ))dτ,HI(t)-HI(0)=∫0t(β2HS(τ)+β3HE(τ)-ρζ-δλ-β4HI(τ))dτ,VS(t)-VS(0)=∫0t(β4HI(τ)-β5VS(τ))dτ,VI(t)-VI(0)=∫0t(β5VS(τ)-γVI(τ))dτ.
We then assume that the solutions of the above integral equations can be put in the following form, for P∈(0,1),
(29)HS(t)=∑n=0∞PnHSn(t),HE(t)=∑n=0∞PnHEn(t),HI(t)=∑n=0∞PnHIn(t),VS(t)=∑n=0∞PnVSn(t),VI(t)=∑n=0∞PnVIn(t).
Replacing the above expressions in the integral equations, we obtain the following expressions:
(30)∑n=0∞PnHSn(t)-HS(0)=P∫0t[(ρζ+γ∑n=0∞PnVIn(τ)-(β1+β2)∑n=0∞PnHSn(τ))dτ],∑n=0∞PnHEn(t)-HE(0)=P∫0t[(δλ+β1∑n=0∞PnHSn(τ)-β3∑n=0∞PnHEn(τ))dτ],∑n=0∞PnHIn(t)-HI(0)=P∫0t[(β2∑n=0∞PnHSn(τ)+β3∑n=0∞PnHEn(τ)-ρζ-δλ-β4∑n=0∞PnHIn(τ))dτ],∑n=0∞PnVSn(t)-VS(0)=P∫0t[(β4∑n=0∞PnHIn(τ)-β5∑n=0∞PnVSn(τ))dτ],∑n=0∞PnVIn(t)-VI(0)=P∫0t[(β5∑n=0∞PnVSn(τ)-γ∑n=0∞PnVIn(τ))dτ].
We compared the terms of the same power of P and obtained the following:
(31)P0:HS0(t)=HS(0),P0:HE0(t)=HE(0),P0:HI0(t)=HI(0),P0:VS0(t)=VS(0),P0:VI0(t)=VI(0),P1:HS1(t)=∫0t(ρζ+γVI0(τ)-(β1+β2)HS0(τ))dτ,P1:HE1(t)=∫0t(δλ+β1HS0(τ)-β3HE0(τ))dτ,P1:HI1(t)=∫0t(β2HS0(τ)+β3HE0(τ)-ρζ-δλ-β4HI0(τ))dτ,P1:VS1(t)=∫0t(β4HI0(τ)-β5VS0(τ))dτ,P1:VI1(t)=∫0t(β5VS0(τ)-γVI0(τ))dτ.
Continuing for n≥2, we have
(32)HSn(t)=∫0t(γVI(n-1)(τ)-(β1+β2)HS(n-1)(τ))dτ,HEn(t)=∫0t(β1HS(n-1)(τ)-β3HE(n-1)(τ))dτ,HIn(t)=∫0t(β2HS(n-1)(τ)+β3HE(n-1)(τ)HI(n-1)(τ)-β4HI(n-1)(τ))dτ,VSn(t)=∫0t(β4HI(n-1)(τ)-β5VS(n-1)(τ))dτ,VIn(t)=∫0t(β5VS(n-1)(τ)-γVI(n-1)(τ))dτ.
We now integrate the above equations to obtain the following:
(33)HI0(t)=0,HE0(t)=0,HS0[t]=N,VS0[t]=M,VI0[t]=0,HI1(t)=t(-δλ-ζρ+Nβ2),HS1[t]=t(ζρ-N(β1+β2)),HE1(t)=t(δλ+Nβ1),VI1[t]=Mtβ5,VS1[t]=-Mtβ5,HI2(t)=12t2ζρβ2-12Nt2β1β2-12Nt2β22+12t2δλβ3+12Nt2β1β3+12t2δλβ4+12t2ζρβ4-12Nt2β2β4,HE2(t)=12t2ζρβ1-12Nt2β12-12Nt2β1β2-12t2δλβ3-12Nt2β1β3,HS2[t]=-12t2ζρβ1+12Nt2β12-12t2ζρβ2+Nt2β1β2+12Nt2β22+12Mt2γβ5,VS2[t]=-12t2δλβ4-12t2ζρβ4+12Nt2β2β4+12Mt2β52,VI2[t]=-12Mt2γβ5-12Mt2β52,HI3(t)=-16t3ζρβ1β2+16Nt3β12β2-16t3ζρβ22+13Nt3β1β22+16Nt3β23+16t3ζρβ1β3-16Nt3β12β3-16Nt3β1β2β3-16t3δλβ32-16Nt3β1β32-16t3ζρβ2β4+16Nt3β1β2β4+16Nt3β22β4-16t3δλβ3β4-16Nt3β1β3β4-16t3δλβ42-16t3ζρβ42+16Nt3β2β42+16Mt3γβ2β5,HE3(t)=-16t3ζρβ12+16Nt3β13-16t3ζρβ1β2+13Nt3β12β2+16Nt3β1β22-16t3ζρβ1β3+16Nt3β12β3+16Nt3β1β2β3+16t3δλβ32+16Nt3β1β32+16Mt3γβ1β5,HS3[t]=16t3ζρβ12-16Nt3β13+13t3ζρβ1β2-12Nt3β12β2+16t3ζρβ22-12Nt3β1β22-16Nt3β23-16Mt3γ2β5-16Mt3γβ1β5-16Mt3γβ2β5-16Mt3γβ52,VS3[t]=16t3ζρβ2β4-16Nt3β1β2β4-16Nt3β22β4+16t3δλβ3β4+16Nt3β1β3β4+16t3δλβ42+16t3ζρβ42-16Nt3β2β42+16t3δλβ4β5+16t3ζρβ4β5-16Nt3β2β4β5-16Mt3β53,VI3[t]=16Mt3γ2β5-16t3δλβ4β5-16t3ζρβ4β5+16Nt3β2β4β5+16Mt3γβ52+16Mt3β53,HS4[t]=-124t4ζρβ13+124Nt4β14-18t4ζρβ12β2+16Nt4β13β2-18t4ζρβ1β22+14Nt4β12β22-124t4ζρβ23+16Nt4β1β23+124Nt4β24+124Mt4γ3β5+124Mt4γ2β1β5+124Mt4γβ12β5+124Mt4γ2β2β5+112Mt4γβ1β2β5+124Mt4γβ22β5-124t4γδλβ4β5-124t4γζρβ4β5+124Nt4γβ2β4β5+124Mt4γ2β52+124Mt4γβ1β52+124Mt4γβ2β52+124Mt4γβ53,HE4(t)=124t4ζρβ13-124Nt4β14+112t4ζρβ12β2-18Nt4β13β2+124t4ζρβ1β22-18Nt4β12β22-124Nt4β1β23+124t4ζρβ12β3-124Nt4β13β3+124t4ζρβ1β2β3-112Nt4β12β2β3-124Nt4β1β22β3+124t4ζρβ1β32-124Nt4β12β32-124Nt4β1β2β32-124t4δλβ33-124Nt4β1β33-124Mt4γ2β1β5-124Mt4γβ12β5-124Mt4γβ1β2β5-124Mt4γβ1β3β5-124Mt4γβ1β52,HI4(t)=124t4ζρβ12β2-124Nt4β13β2+112t4ζρβ1β22-18Nt4β12β22+124t4ζρβ23-18Nt4β1β23-124Nt4β24-124t4ζρβ12β3+124Nt4β13β3-124t4ζρβ1β2β3+112Nt4β12β2β3+124Nt4β1β22β3-124t4ζρβ1β32+124Nt4β12β32+124Nt4β1β2β32+124t4δλβ33+124Nt4β1β33+124t4ζρβ1β2β4-124Nt4β12β2β4+124t4ζρβ22β4-112Nt4β1β22β4-124Nt4β23β4-124t4ζρβ1β3β4+124Nt4β12β3β4+124Nt4β1β2β3β4+124t4δλβ32β4+124Nt4β1β32β4+124t4ζρβ2β42-124Nt4β1β2β42-124Nt4β22β42+124t4δλβ3β42+124Nt4β1β3β42+124t4δλβ43+124t4ζρβ43-124Nt4β2β43-124Mt4γ2β2β5-124Mt4γβ1β2β5-124Mt4γβ22β5+124Mt4γβ1β3β5-124Mt4γβ2β4β5-124Mt4γβ2β52VS4[t]=-124t4ζρβ1β2β4+124Nt4β12β2β4-124t4ζρβ22β4+112Nt4β1β22β4+124Nt4β23β4+124t4ζρβ1β3β4-124Nt4β12β3β4-124Nt4β1β2β3β4-124t4δλβ32β4-124Nt4β1β32β4-124t4ζρβ2β42+124Nt4β1β2β42+124Nt4β22β42-124t4δλβ3β42-124Nt4β1β3β42-124t4δλβ43-124t4ζρβ43+124Nt4β2β43+124Mt4γβ2β4β5-124t4ζρβ2β4β5+124Nt4β1β2β4β5+124Nt4β22β4β5-124t4δλβ3β4β5-124Nt4β1β3β4β5-124t4δλβ42β5-124t4ζρβ42β5+124Nt4β2β42β5-124t4δλβ4β52-124t4ζρβ4β52+124Nt4β2β4β52+124Mt4β54,VI4[t]=-124Mt4γ3β5+124t4γδλβ4β5+124t4γζρβ4β5-124Nt4γβ2β4β5+124t4ζρβ2β4β5-124Nt4β1β2β4β5-124Nt4β22β4β5+124t4δλβ3β4β5+124Nt4β1β3β4β5+124t4δλβ42β5+124t4ζρβ42β5-124Nt4β2β42β5-124Mt4γ2β52+124t4δλβ4β52+124t4ζρβ4β52-124Nt4β2β4β52-124Mt4γβ53-124Mt4β54,HS5[t]=1120t5ζρβ14-1120Nt5β15+130t5ζρβ13β2-124Nt5β14β2+120t5ζρβ12β22-112Nt5β13β22+130t5ζρβ1β23-112Nt5β12β23+1120t5ζρβ24-124Nt5β1β24-1120Nt5β25-1120Mt5γ4β5-1120Mt5γ3β1β5-1120Mt5γ2β12β5-1120Mt5γβ13β5-1120Mt5γ3β2β5-160Mt5γ2β1β2β5-140Mt5γβ12β2β5-1120Mt5γ2β22β5-140Mt5γβ1β22β5-1120Mt5γβ23β5+1120t5γ2δλβ4β5+1120t5γ2ζρβ4β5+1120t5γδλβ1β4β5+1120t5γζρβ1β4β5-1120Nt5γ2β2β4β5+1120t5γδλβ2β4β5+160t5γζρβ2β4β5-160Nt5γβ1β2β4β5-160Nt5γβ22β4β5+1120t5γδλβ3β4β5+1120Nt5γβ1β3β4β5+1120t5γδλβ42β5+1120t5γζρβ42β5-1120Nt5γβ2β42β5-1120Mt5γ3β52-1120Mt5γ2β1β52-1120Mt5γβ12β52-1120Mt5γ2β2β52-160Mt5γβ1β2β52-1120Mt5γβ22β52+1120t5γδλβ4β52+1120t5γζρβ4β52-1120Nt5γβ2β4β52-1120Mt5γ2β53-1120Mt5γβ1β53-1120Mt5γβ2β53-1120Mt5γβ54,HE5[t]=-1120t5ζρβ14+1120Nt5β15-140t5ζρβ13β2+130Nt5β14β2-140t5ζρβ12β22+120Nt5β13β22-1120t5ζρβ1β23+130Nt5β12β23+1120Nt5β1β24-1120t5ζρβ13β3+1120Nt5β14β3-160t5ζρβ12β2β3+140Nt5β13β2β3-1120t5ζρβ1β22β3+140Nt5β12β22β3+1120Nt5β1β23β3-1120t5ζρβ12β32+1120Nt5β13β32-1120t5ζρβ1β2β32+160Nt5β12β2β32+1120Nt5β1β22β32-1120t5ζρβ1β33+1120Nt5β12β33+1120Nt5β1β2β33+1120t5δλβ34+1120Nt5β1β34+1120Mt5γ3β1β5+1120Mt5γ2β12β5+1120Mt5γβ13β5+1120Mt5γ2β1β2β5+160Mt5γβ12β2β5+1120Mt5γβ1β22β5+1120Mt5γ2β1β3β5+1120Mt5γβ12β3β5+1120Mt5γβ1β2β3β5+1120Mt5γβ1β32β5-1120t5γδλβ1β4β5-1120t5γζρβ1β4β5+1120Nt5γβ1β2β4β5+1120Mt5γ2β1β52+1120Mt5γβ12β52+1120Mt5γβ1β2β52+1120Mt5γβ1β3β52+1120Mt5γβ1β53,HI5[t]=-1120t5ζρβ13β2+1120Nt5β14β2-140t5ζρβ12β22+130Nt5β13β22-140t5ζρβ1β23+120Nt5β12β23-1120t5ζρβ24+130Nt5β1β24+1120Nt5β25+1120t5ζρβ13β3-1120Nt5β14β3+160t5ζρβ12β2β3-140Nt5β13β2β3+1120t5ζρβ1β22β3-140Nt5β12β22β3-1120Nt5β1β23β3+1120t5ζρβ12β32-1120Nt5β13β32+1120t5ζρβ1β2β32-160Nt5β12β2β32-1120Nt5β1β22β32+1120t5ζρβ1β33-1120Nt5β12β33-1120Nt5β1β2β33-1120t5δλβ34-1120Nt5β1β34-1120t5ζρβ12β2β4+1120Nt5β13β2β4-160t5ζρβ1β22β4+140Nt5β12β22β4-1120t5ζρβ23β4+140Nt5β1β23β4+1120Nt5β24β4+1120t5ζρβ12β3β4-1120Nt5β13β3β4+1120t5ζρβ1β2β3β4-160Nt5β12β2β3β4-1120Nt5β1β22β3β4+1120t5ζρβ1β32β4-1120Nt5β12β32β4-1120Nt5β1β2β32β4-1120t5δλβ33β4-1120Nt5β1β33β4-1120t5ζρβ1β2β42+1120Nt5β12β2β42-1120t5ζρβ22β42+160Nt5β1β22β42+1120Nt5β23β42+1120t5ζρβ1β3β42-1120Nt5β12β3β42-1120Nt5β1β2β3β42-1120t5δλβ32β42-1120Nt5β1β32β42-1120t5ζρβ2β43+1120Nt5β1β2β43+1120Nt5β22β43-1120t5δλβ3β43-1120Nt5β1β3β43-1120t5δλβ44-1120t5ζρβ44+1120Nt5β2β44+1120Mt5γ3β2β5+1120Mt5γ2β1β2β5+1120Mt5γβ12β2β5+1120Mt5γ2β22β5+160Mt5γβ1β22β5+1120Mt5γβ23β5-1120Mt5γ2β1β3β5-1120Mt5γβ12β3β5-1120Mt5γβ1β2β3β5-1120Mt5γβ1β32β5+1120Mt5γ2β2β4β5-1120t5γδλβ2β4β5-1120t5γζρβ2β4β5+1120Mt5γβ1β2β4β5+1120Mt5γβ22β4β5+1120Nt5γβ22β4β5-1120Mt5γβ1β3β4β5+1120Mt5γβ2β42β5+1120Mt5γ2β2β52+1120Mt5γβ1β2β52+1120Mt5γβ22β52-1120Mt5γβ1β3β52+1120Mt5γβ2β4β52+1120Mt5γβ2β53,VS5[t]=1120t5ζρβ12β2β4-1120Nt5β13β2β4+160t5ζρβ1β22β4-140Nt5β12β22β4+1120t5ζρβ23β4-140Nt5β1β23β4-1120Nt5β24β4-1120t5ζρβ12β3β4+1120Nt5β13β3β4-1120t5ζρβ1β2β3β4+160Nt5β12β2β3β4+1120Nt5β1β22β3β4-1120t5ζρβ1β32β4+1120Nt5β12β32β4+1120Nt5β1β2β32β4+1120t5δλβ33β4+1120Nt5β1β33β4+1120t5ζρβ1β2β42-1120Nt5β12β2β42+1120t5ζρβ22β42-160Nt5β1β22β42-1120Nt5β23β42-1120t5ζρβ1β3β42+1120Nt5β12β3β42+1120Nt5β1β2β3β42+1120t5δλβ32β42+1120Nt5β1β32β42+1120t5ζρβ2β43-1120Nt5β1β2β43-1120Nt5β22β43+1120t5δλβ3β43+1120Nt5β1β3β43+1120t5δλβ44+1120t5ζρβ44-1120Nt5β2β44-1120Mt5γ2β2β4β5-1120Mt5γβ1β2β4β5+1120t5ζρβ1β2β4β5-1120Nt5β12β2β4β5-1120Mt5γβ22β4β5+1120t5ζρβ22β4β5-160Nt5β1β22β4β5-1120Nt5β23β4β5+1120Mt5γβ1β3β4β5-1120t5ζρβ1β3β4β5+1120Nt5β12β3β4β5+1120Nt5β1β2β3β4β5+1120t5δλβ32β4β5+1120Nt5β1β32β4β5-1120Mt5γβ2β42β5+1120t5ζρβ2β42β5-1120Nt5β1β2β42β5-1120Nt5β22β42β5+1120t5δλβ3β42β5+1120Nt5β1β3β42β5+1120t5δλβ43β5+1120t5ζρβ43β5-1120Nt5β2β43β5-160Mt5γβ2β4β52+1120t5ζρβ2β4β52-1120Nt5β1β2β4β52-1120Nt5β22β4β52+1120t5δλβ3β4β52+1120Nt5β1β3β4β52+1120t5δλβ42β52+1120t5ζρβ42β52-1120Nt5β2β42β52+1120t5δλβ4β53+1120t5ζρβ4β53-1120Nt5β2β4β53-1120Mt5β55,VI5[t]=1120Mt5γ4β5-1120t5γ2δλβ4β5-1120t5γ2ζρβ4β5+1120Nt5γ2β2β4β5-1120t5γζρβ2β4β5+1120Nt5γβ1β2β4β5-1120t5ζρβ1β2β4β5+1120Nt5β12β2β4β5+1120Nt5γβ22β4β5-1120t5ζρβ22β4β5+160Nt5β1β22β4β5+1120Nt5β23β4β5-1120t5γδλβ3β4β5-1120Nt5γβ1β3β4β5+1120t5ζρβ1β3β4β5-1120Nt5β12β3β4β5-1120Nt5β1β2β3β4β5-1120t5δλβ32β4β5-1120Nt5β1β32β4β5-1120t5γδλβ42β5-1120t5γζρβ42β5+1120Nt5γβ2β42β5-1120t5ζρβ2β42β5+1120Nt5β1β2β42β5+1120Nt5β22β42β5-1120t5δλβ3β42β5-1120Nt5β1β3β42β5-1120t5δλβ43β5-1120t5ζρβ43β5+1120Nt5β2β43β5+1120Mt5γ3β52-1120t5γδλβ4β52-1120t5γζρβ4β52+1120Mt5γβ2β4β52+1120Nt5γβ2β4β52-1120t5ζρβ2β4β52+1120Nt5β1β2β4β52+1120Nt5β22β4β52-1120t5δλβ3β4β52-1120Nt5β1β3β4β52-1120t5δλβ42β52-1120t5ζρβ42β52+1120Nt5β2β42β52+1120Mt5γ2β53-1120t5δλβ4β53-1120t5ζρβ4β53+1120Nt5β2β4β53+1120Mt5γβ54+1120Mt5β55.
In general, by the repeated application of the above procedure, we have that
(34)HSn=tnn!a1n,HEn=tnn!a2n,HIn=tnn!a3n,VSn=tnn!a4n,VIn=tnn!a5n,
where a1n,a2n,a3n,a4n,a5n depend on fixed set of empirical parameters. Hence, the approximate solution of the model above is given as
(35)HS(t)=∑n=0∞tnn!a1n,HE(t)=∑n=0∞tnn!a2n,HI(t)=∑n=0∞tnn!a3n,VS(t)=∑n=0∞tnn!a4n,VI(t)=∑n=0∞tnn!a5n.
We therefore use the following theoretical parameters β1=0.5, β2=0.5, β3=0.5, β4=0.5, β5=0.5, γ=1, ζ=1, N=10, M=10, ρ=1, δ=1, and λ=1 to examine the behavior of the five different compartmental numerical solutions in the system. Figure 3 shows the behavior of the numerical solution.Figure 3
Numerical solution of the model.


## 7. 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 parasiteOnchocerca 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.



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*Source: 631658-2014-10-22.xml*