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Local Stability Analysis of Onchocerciasis Transmission Dynamics With Nonlinear Incidence

Functions in Two Interacting Populations

K. M. Adeyemo
Department of Mathematics, Hallmark University Ijebu-Itele, Ogun State, Nigeria

∗Correspondence: mikyade2019@gmail.com

Abstract. A deterministic compartmental model for the transmission dynamics of onchocerciasis withnonlinear incidence functions in two interacting populations is studied. The model is qualitatively an-alyzed to investigate its local asymptotic behavior with respect to disease-free and endemic equilibria.It is shown, using Routh-Hurwitz criteria, that the disease-free equilibrium is locally asymptoticallystable when the associated basic reproduction number is less than the unity. When the basic repro-duction number is greater than the unity, we prove the existence of a locally asymptotically stableendemic equilibrium.

1. Introduction
Onchocerciasis is one of the neglected tropical diseases caused by the parasite OnchocercaVolvulus, a filarial nematode [2]. The disease is transmitted from one person to another by repeatedbites of black flies. The disease is endemic in Sub-saharan Africa. Many researchers have workedon many ways to reduce the spread of the disease. For instance, Remme et al. [10] used skin snipsurvey in West Africa to investigate the impact of controlling black flies by larviciding. Plaisieret al. [9] used micro simulation model to determine the period required for combining annualivermectin treatment and vector control in the onchocerciasis Control Programme in West Africa.Alley et al. [1] used a computer simulation model to study prevention of onchocerciasis by usingmacrofilaricide which kills the adult worms. Asha Hassan & Nyimvua Shaban [3] investigated theeffects of four control strategies on the spread of the disease.In this paper, we consider onchocerciasis transmission dynamics with nonlinear incidence functions.The human population is sub-divided into four compartments and the vector population is sub-divided into three compartments. We show local asymptotic behaviour in disease-free and endemicequilibria. The rest of the paper is organized as follows: the description of the model and theorems
Received: 27 Apr 2023.
Key words and phrases. basic reproduction number; diseases free equilibrium; onchocerciasis epidemic model; non-linear incidence function. 1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.22 2
on positivity of solutions are given in section 2 while section 3 is devoted to the proof local stabilitytheorems.

2. Model Description
Two interacting populations are considered; the humans and the black-flies populations. Thehuman population is partitioned into four compartments: the susceptible human compartment;

Sh„ the exposed compartment; Eh, the infectious human compartment; Ih and the recoveredhuman compartment; Rh. The black-fly population is partitioned into three compartments:susceptible vector; Sv , the exposed vector compartment; Ev and the infective vector compart-ment. The total human and vector populations at any given time, t, are respectively given by;
N = Sh(t) + Eh(t) + Ih(t) + Rh(t) and V = Sv (t) + Ev (t) + Iv (t). We assume that thetransmission of onchocerciaisis in susceptible hosts is only through contact with infectious vector.We also assume that susceptible vector becomes infectious as a result of contact with infectioushosts during blood meal. The population under study is assumed to be large enough to bemodelled deterministically. The following system of non-linear ordinary differential equations,with non-negative initial conditions, describes the dynamics of onchocerciaisis epidemics.

dSh(t,xi )
dt

= Ψh(xi ) −
∑L
i=0

δλh(xi )Sh(t,xi )Iv (t)
1+νh(xi )Iv (t)

−µh(xi )Sh + w(xi )Rh(t,xi ))
dEh(t,xi )

dt
=
∑L
i=0

δλh(xi )Sh(t,xi )Iv (t)
1+νh(xi )Iv (t)

− (αh(xi ) + µh(xi ))Eh(t,xi )
dIh(t,xi )
dt

=
∑L
i=0 αh(xi )Eh − (r(xi ) + γh(xi ) + µh(xi ))Ih(t,xi )

dRh(t,xi )
dt

=
∑L
i=0 r(xi )Ih − (µh(xi ) + w(xi ))Rh(t,xi )

dSv
dt

= Ψv −
δλv (xi )Sv (t)Ih(xi,t)

1+νvIh(xi,t)
−µvSv (t)

dEv
dt

=
δλv (xi )Sv (t)Ih(xi,t)

1+νvIh(xi,t)
− (αv + µv )Ev (t)

dIv
dt

= αvEv (t) − (µv + γv )Iv (t)


(2.1)

subject to the following initial conditions:
Sh(0,xi ) = S0h(xi ),Eh(0,xi ) = E0h(xi ),

Ih(0,xi ) = I0h(xi ),Rh(0,xi ) = R0h(xi )

Sm(0) = S0m,Em(0) = E0m, Im(0) = I0m (2.2)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.22 3
Symbols Definitionss
Sh(t,xi ) Number of susceptible humans at time t and discrete age xi
Eh(t,xi ) Number of exposed humans at time t and discrete age xi
Ih(t,xi ) Number of infectious humans at time t and discrete age xi
Rh(t,ai) Number of recovered humans at time t and discrete age xi
Sv (t) Number of susceptible black-flies at time t
Ev (t) Number of exposed black-flies at time t
Iv (t) Number of infectious black-flies at time t

Ψh(xi ) Recruitment term of the susceptible humans at discrete age xi
Ψv Recruitment term of the susceptible vectors
δ Biting rate of the vector

λh(xi ) Probability that a bite by an infectious vector results in transmission of disease to human at discrete age xi
λv Probability that a bite results in transmission of parasite to a susceptible vector

µh(xi ) Per capita death rate of humans at discrete age xi
µv Per capita death rate of vector

γh(xi ) Disease-induced death rate of humans at discrete age xi
γv Disease-induced death rate of vectors

αh(xi ) Per capita rate of progression of humans from the exposed state to the infectious state at discrete age xi
αv Per capita rate of progression of vectors from the exposed state to the infectious state
r(xi ) Per capita recovery rate for humans from the infectious state to the recovered state due to treatment at discrete age xi
ω(xi ) Per capita transition rate of recovered humans to the susceptible state at discrete age xi
νh(xi ) Humans disease-inhibiting factor at discrete age xi
νv Vectors disease-inhibiting factor

Model assumptionsThe formulation of the compartmental model is based on the following assumptions:
1. That all humans are born susceptible. That is, humans are liable to contract the disease.2. That the susceptible humans, when infected, becomes exposed humans who are not yetinfectious.3. That the exposed humans progress to become infectious only.4. That the infectious humans may either die naturally or as a result of the disease, and ifnot, they become recovered humans due to treatment.5. That the recovered humans become susceptible again.6. All black-flies are born susceptible.7. That the susceptible black-flies, when infected, becomes exposed black-flies who are notyet infectious.8. That the exposed black-flies progress to become infectious only.9. That the infectious black-flies remain infectious for life. That is, there is no recovered classfor black-fly population.

2.1. Existence and Positivity of Solutions. In this section, we analyse the general properties ofthe system (2.1) with positive initial conditions. It describes the population dynamics both in humanand black-fly populations. The system is biologically relevant in the set given by
Ω = (Sh(t,xi ),Eh(t,xi ), Ih(t,xi ),Rh(t,xi )) ∈R4+ : Nh ≤

L∑
i=0

Ψh(xi )

µh(xi )
, (Sv (t),Ev (t), Iv (t)) ∈R3+ : Nv ≤

Ψv
µv

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Eur. J. Math. Anal. 10.28924/ada/ma.3.22 4
Here, the following results are provided which guarantee that the model governed by system (2.1)is mathematically well-posed in a feasible region Ω defined by:

Ω = Ωh × Ωv ⊂R4 ×R3

Theorem 1:There exists a domain Ω in which the solution set Sh(t,xi ),Eh(t,xi ), Ih(t,xi ),Rh(t,xi ),Sv (t),Ev (t), Iv (t)is contained and bounded.
ProofIf the total human population size is given by Nh = Sh(t,xi ) + Eh(t,xi ) + Ih(t,xi ) + Rh(t,xi ), andthe total size of black-fly population is Nv = Sv (t) + Ev (t) + Iv (t). From model (2.1), we havethat

dNh(t,xi )
dt

≤ Ψh(xi ) −
L∑
i=0

µh(xi )Nh(t,xi ) (2.3)
and

dNv
dt
≤ Ψv −µvNv (2.4)It follows from (2.3) and (2.4) that

Nh(t,xi ) ≤
Ψh(xi )

µh(xi )
[1 −e1−µh(xi )t]+Nh(0,xi )e

−µh(xi )t]

and
Nv ≤

Ψv
µv

[1 −e−µvt] + Nv (0)e−µvt

Taking the lim sup as t → ∞ gives Nh ≤ Ψh(xi )µh(xi ) and Nv ≤ Ψvµv . This shows that all solu-tions of the humans population only are confined in the solution set Ωh and all solutions of theblack-fly population are confined in Ωv . It also suffices to say that Ω is positively invariant as
Nh(t,xi ) ≤

∑L
i=0

Ψh(xi )
µh(xi )

whenever Nh(0,xi ) ≤ Ψh(xi )µh(xi ) and Nv (t) ≤ Ψvµv if Nv (0) ≤ Ψvµv , Therefore thesolution set for the model (2.1) exists and is given by Ω = Ωh × Ωv ⊂R4+ ×R3+ 2It remains to show that the solutions of system (2.1) are nonnegative in Ω for any time t > 0 sincethe variables represent human and black-fly populations.
Theorem 2:The solutions, Sh(t,xi ), Eh(t,xi ), Ih(t,xi ), Rh(t,xi ), Sv (t), Ev (t), Iv (t), of model (2.1) with non-negative initial conditions in Ω, remain nonnegative in Ω for all t > 0.
Proof: Given that the initial conditions, S0h(xi ), E0h(xi ), I0h(xi ), R0h(xi ), S0v,E0v,I0v , are non-negative and from (2.1),

dSh(t,xi )
dt

+

L∑
i=0

[
bλh(xi )Iv (t)

1 + νh(xi )Iv (t)
+ µh(xi )

]
Sh(t,xi ) ≥ 0

so that
d

dt

[
L∑
i=0

Sh(t,xi )exp

(∫ t
0

bλh(xi )Iv (η)

1 + νh(xi )Iv (η)
dη + µh(xi )t

)]
≥ 0, (2.5)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.22 5
Integrating (2.5), we have

L∑
i=0

Sh(t,xi ) ≥
L∑
i=0

S0h(xi )exp

[
−
(∫ t

0

bλh(xi )Iv (η)

1 + νh(xi )Iv (η)
dη + µh(xi )t

)]
≥ 0,

which implies that for all t > 0 and for all a ∈R+, we have
Sh(t,xi ) ≥

L∑
i=0

S0h(xi )exp

[
−
(∫ t

0

bλh(xi )Iv (η)

1 + νh(xi )Iv (η)
dη + µh(xi )t

)]
≥ 0.

Hence, Sh(t,xi ) > 0 for any arbitrary xi . Also, we have
dEh(t,xi )

dt
+

L∑
i=0

((αh(xi ) + µh(xi )))Eh(t,xi ) ≥ 0

so that
d

dt

[
L∑
i=0

Eh(t, (xi ))exp(αh(xi ) + µh(xi )t)

]
≥ 0 (2.6)

Integrating (2.6), we have for all t > 0 and for all a ∈ mathbbR+, that
Eh(t,a) ≥

L∑
i=0

E0h(xi )exp [−(αh(xi ) + µh(xi ))t]

Hence, Eh(t,xi ) > 0 for any arbitrary xi Also we have
dIh(t,xi )
dt

≥−
L∑
i=0

(r(xi ) + γh(xi ) + µh(xi ))Ih(t)

so that
d

dt
[Ih(t)exp(r(xi ) + γh(xi ) + µh(xi ))t] ≥ 0 (2.7)

Similarly, (2.7) becomes
Ih(t,a) ≥

L∑
i=0

I0hexp [−(r(xi ) + γh(xi ) + µh(xi ))t] > 0for all t > 0 for all a ∈R+
Hence, Ih(t,xi ) > 0 for any arbitrary xi . Also from (2.1), we have

dRh(t,xi )
dt

+

L∑
i=0

(µh(xi ) + w(xi ))Rh(t,xi ) ≥ 0

and we have
d

dt

[
L∑
i=0

Rh(t,xi )exp((µh(xi ) + w(xi ))t

]
≥ 0 (2.8)

Integrating (2.8), we have, for all t > 0 and a ∈R, that
Rh(t,a) ≥

L∑
i=0

R0h(xi )exp(−(µh(xi ) + w(xi ))t) > 0

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Eur. J. Math. Anal. 10.28924/ada/ma.3.22 6
Hence, Rh(t,xi ) > 0 for any arbitrary xi . In a similar manner, we have

dSv
dt

+

[
L∑
i=0

bλvIh(t))

1 + νvIh(t)
+ µv

]
Sv (t) ≥ 0

so that
d

dt

[
Sv (t)exp

(∫ t
0

bλvIh(η))

1 + νvIh(η)
d(η) + µvt

)]
≥ 0 (2.9)

Integrating (2.9), we have
Sv (t) ≥ S0vexp

[
−
(∫ t

0

bλvIh(η))

1 + νvIh(η)
d(η) + µvt

)]
> 0 ∀ t > 0

Also we have
dEv
dt
≥−(αv + µv )Ev (t)

which on integration gives
Ev (t) ≥ Ev (0)exp [−(αv + µv )t] > 0 ∀ t > 0 (2.10)

And finally, we have
dIv
dt

+ (µv + γv )Iv (t)

so that
d

dt
[Iv (t)exp(µv + γv )t] ≥ 0 (2.11)

And we have
Iv (t) ≥ Iv (0)exp [−(µv + γv )t] > 0, ∀ t > 0

This completes the proof 2
3. Existence and stability of the equilibrium points

3.1. Disease-free equilibrium. The disease-free equilibrium (DFE) points are steady state solu-tions that depict the absence of infection in both the human host and black-fly vector populations,i.e, onchocerciasis does not exist in the population. Thus, the disease-free equilibrium point, E0, forthe model (2.1) implies that S∗(xi )h 6= 0, E∗h(xi ) = I∗h = 0(xi ) = R∗h(xi ) = 0, S∗v 6= 0, Ev = Iv = 0and putting these into (2.1), we have S∗(xi )h = Ψh(xi )µh(xi ) and S∗v = Ψvµv . Consequently we obtain E0as
E0 =

(
Ψh(xi )

µh(xi )
, 0, 0, 0,

Ψv
µv
, 0, 0

) (3.1)
A key notion in the analysis of infectious disease models is the basic reproduction number R0 , anepidemiological threshold that determines whether disease dies out or persists in the population.Thebasic reproduction number R0 of the system (2.1) is computed using the next generation matrixmethod and is given by

R0 =
√
RhRv

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Eur. J. Math. Anal. 10.28924/ada/ma.3.22 7
where Rh = ∑Li=0 δαhλh(xi )Ψh(xi )µh(xi )(αh(xi )+µh(xi ))(r(xi )+γh(xi )+µh(xi )) and Rv = δαvλv Ψvµv (αv +µv )(γv +µv ) . The basicreproduction number R0, determines whether onchocerciasis dies out or persists in the population.Therefore, Rh describes the number of humans that one infectious black-fly infects over its expectedinfectious period in a completely susceptible humans population, while Rv is the number of blac-flies infected by one infectious human during the period of infectiousness in a completely susceptibleblack-fly population.
3.2. Local Stability of the Disease-free Equilibrium Point E0. Using the basic reproduction num-ber obtained for the model (2.1), we analyse the stability of the equilibrium point in the followingresult.
Theorem 3:The disease-free equilibrium point, E0, is locally asymptotically stable if R0 < 1, and unstable if
R0 > 1.
Proof: The Jacobian matrix of the system (2.1) evaluated at the disease-free equilibrium point E0,is obtained as

M(E0) =



M11 0 0 M14 0 0 M17

0 M22 0 0 0 0 M27

0 M32 M33 0 0 0 0

0 0 M43 M44 0 0 0

0 0 M53 0 M55 0 0

0 0 M63 0 0 M66 0

0 0 0 0 0 M76 M77


where M11 = −µh(xi ), M14 = w(a1), M17 = −∑Li=0 δλh(xi )Ψh(xi )µh(xi ) , M22 = −(αh(xi ) + µh(xi )),
M27 =

∑L
i=0

δλh(xi )Ψh(xi )
µh(xi )

, M32 = αh(xi ), M33 = −(r(xi ) + γh(xi ) + µh(xi )), M43 = r(xi ), M44 =
−(µh(xi ) + w(xi )), M53 = −δλv Ψvµv , M55 = −µv , M63 = δλv Ψvµv , M66 = −(αv + µv ), M76 = αv ,
M77 = −(µv + γv ) We need to show that all the eigenvalues of M(E0) are negative. As the firstand fifth columns form the two negative eigenvalues, h(xi ) and −v , the other five eigenvalues canbe obtained from the sub-matrix, M1(E0), formed by excluding the first and fifth rows and columnsof M(E0). Hence

M1(E0) =


M′11 0 0 0 M

′
15

αh(xi ) M
′
22 0 0 0

0 r(xi ) M
′
33 0 0 0

0 δλv Ψv
µv

0 −(αv + µv ) 0
0 0 0 αv −(µv + λv )


In the same way, the third column of M1(E0) contains only the diagonal term which forms anegative eigenvalue, (µh(xi ) + w(xi )). The remaining four eigenvalues are obtained from the

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Eur. J. Math. Anal. 10.28924/ada/ma.3.22 8
sub-matrix M2(E0) given by

M2(E0) =


M′′11 0 0 M

′′
14

αh(xi ) M
′
22 0 0

0 δλv Ψv
µv

−(αv + µv ) 0
0 0 αv −(µv + λv )


Thus, the eigenvalues of the matrix M2(E0) are the roots of the characteristic equation of the form

(ξ + αh(xi ))(ξ + r(xi ) + γh(xi ) + µh(xi ))(ξ + µv + γ)−
L∑
i=0

δ2αh(xi )λh(xi )Ψh(xi )vλv Ψv
µh(xi )µv

= 0 (3.2)
If we let Y1 = αh(xi ) + µh(xi ), Y2 = r(xi ) + γh(xi ) + µh(xi ), Y3 = αv + µv , and Y4 = µv + γv , then(3.2) becomes

X4ξ
4 + X3ξ

3 + X2ξ
2 + X1ξ + X0 = 0, (3.3)

where
X4 = 1

X3 = Y1 + Y2 + Y3 + Y4

X2 = (Y1 + Y2)(Y2 + Y4) + Y1Y2 + Y3Y4

X1 = (Y1 + Y2)Y3Y4 + (Y3 + Y4)Y1Y2

X0 = Y1Y2Y3Y4 −
∑L
i=0

δ2αh(xi )λh(xi )Ψh(xi )vλv Ψv
µh(xi )µv


(3.4)

Expressing X0 in terms of reproduction number R0, we have
X0 = Y1Y2Y3Y4(1 −R20) (3.5)

We can see from (3.4) that X1 > 0, X2 > 0, X3 > 0, X4 > 0, since all Yis are positive. Moreover,if R0 < 1, it follows from (3.5) that X0 > 0. Thus, using the Routh-Hurwitz criterion, we have
H1 = X3 > 0

H2 =
∣∣∣∣∣ X3 X4X1 X2

∣∣∣∣∣ = Y1(Y2 + Y3 + Y4)(Y1 + Y2 + Y3 + Y4) + (Y2 + Y3)(Y2 + Y4)(Y3 + Y4) > 0Similarly
we have H3 > 0 and H4 > 0 where H3 =

∣∣∣∣∣∣∣∣
X3 X4 0

X1 X2 X3

0 X0 X1

∣∣∣∣∣∣∣∣and H4 =
∣∣∣∣∣∣∣∣∣∣∣
X3 X4 0 0

X1 X2 X3 X4

0 X0 X1 X2

0 0 0 X0

∣∣∣∣∣∣∣∣∣∣∣
Theref ore,alltheeigenvaluesof theJacobianmatrixM(E0) have negative real parts when R0 <
1 and the disease-free equilibrium point is locally asymptotically stable. However, when R0 > 1,we see that X0 < 0 and there is one eigenvalue with positive real part and therefore the disease-free equilibrium point is unstable 2

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Eur. J. Math. Anal. 10.28924/ada/ma.3.22 9
3.3. Endemic Equilibrium Point Ee. We shall show that the formulated model (2.1) has an endemicequilibrium point, Ee. The endemic equilibrium point is a positive steady state solution where thedisease persists in the population.
Theorem 4: The model (2.1) has a unique endemic equilibrium Ee whenever R0 > 1.
Proof: Let Ee = (S′′h(xi ),E′′h (xi ), I′′h (xi ),R′′h(xi ),S′′v ,E′′v , I′′v ) be a nontrivial equilibrium of the model(2.1). That is, all components of Ee are positive. Then the onchocerciasis model (2.1) at steady-statebecomes

Ψh(xi ) −
L∑
i=0

(
δλh(xi )S

′′
h(xi )Iv

1 + νh(xi )I
′′
v

−µh(xi )S′h(xi ) + ω(xi )R
′′
h(xi )

)
= 0 (3.6)

L∑
i=0

(
δλh(xi )S

′′
h(xi )Iv

1 + νh(xi )I
′′
v

− (αh(xi ) + µh(xi ))E′′h (xi )
)

= 0 (3.7)
L∑
i=0

(αh(xi )E
′′
h (xi ) − (r(xi ) + µh(xi ) + γh(xi ))I

′′
h (xi )) = 0 (3.8)

L∑
i=0

r(xi )I
′′
h (xi ) − (µh(xi ) + ω(xi ))R

′′
h(xi ) = 0 (3.9)

Ψv −
δλvS

′′
vIh(xi )

1 + νv (xi )I
′′
h

(xi )
−µvS′′v = 0 (3.10)

δλvS
′′
vIh(xi )

1 + νv (xi )I
′′
h

(xi )
− (αv + µv )E′′v = 0 (3.11)

αvE
′′
v − (µv + γv )I

′′
v = 0 (3.12)From the last three equations, we have

I′′v =
αvE

′′
v

µv + γv
(3.13)

E′′v =
δλvS

′′
vIh(xi )

1 + νv (xi )I
′′
h

(xi )(αv + µv )
(3.14)

and
S′′v =

Ψv
δλvS′′vIh(xi )

1+νv (xi )I
′′
h

(xi )
+ µv

(3.15)
Substituting (3.14) and (3.15) into (3.13) yields

I′′v =
RvµvI′′h (xi )

µv + (δλv + µvνv )I
′′
h

(xi )
(3.16)

From (3.8) and (3.9), we have
E′′h (xi ) =

L∑
i=0

(r(xi ) + µh(xi ) + γh(xi ))Ih(xi )

αh(xi )
(3.17)

and
R′′h(xi ) =

L∑
i=0

r(xi )I
′′(xi )

µh(xi ) + ω(xi )
(3.18)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.22 10
If we put (3.16) and (3,17) in (3.7) in terms of R0, we have

S′′h(xi ) =

∑L
i=0 Ψh(xi )[µv + (δλv + µvνv + νh(xi )µvRv )I

′′
h (xi )]

µh(xi )µvR20
(3.19)

Finally, using (3.16), (3.18) and (3.19) in (3.7), we have
I′′h (xi ) =

L∑
i=0

µh(xi )µv Ψh(xi )(µh(xi ) + ω(xi ))
(R20 − 1)

ρ
(3.20)

where
ρ =

L∑
i=0

(µh(xi )+ω(xi ))[δλh(xi )µvRv +Ψh(xiµh(xi )(δλv +µvνv +νh(xi )µv )Rm)]−
L∑
i=0

µh(xi )µvω(xi )r(xi )R20.

(3.21)If in (3.20), ω(xi ) = 0 then ρ > 0. From this, one sees that model (2.1) has no positive solutionwhen R0 < 1. However, with ω(xi ) = 0, a unique endemic equilibrium exists when R0 > 1. Thiscompletes the proof. 2
Remark 1: It is important to have a remark that positive solution exists for the model (2.1) in acase where ρ < 0 and R0 < 1. This implies that the disease-free equilibrium co-exists with theendemic equilibrium state when R0 is slightly less than unity resulting into a phenomenon ofsubcritical (backward) bifurcation.

References
[1] W.S. Alley, B.A.B. Boatin, N.J.D.N. Nagelkerke, Macrofilaricides and onchocerciasis control, mathematical modellingof the prospects for elimination. BMC Public Health. 1 (2001) 12.[2] U. Amazigo, M. Noma, J. Bump, B. Bentin, B. Liese, L. Yameogo, H. Zouré, and A. Seketeli, Onchocerciasis Diseaseand Mortality in Sub Saharan Africa, Chapter 15, World Bank, Washington, DC, 2006.[3] A. Hassan, N. Shaban, Onchocerciasis dynamics: modelling the effects of treatment, education and vector control,J. Biol. Dyn. 14 (2020) 245-268.[4] E.M. Poolman, A.P. Galvani, Modeling targeted ivermectin treatment for controlling river blindness, Amer. J. Trop.Med. Hygiene, 75 (2006) 921–927.[5] J.P. Mopecha, H.R. Thieme, Competitive dynamics in a model for onchocerciasis with cross-immunity, Can. Appl.Math. Quart. 11 (2003) 339–376.[6] M.G. Basanez, M. Boussinesq, Population biology of human onchocerciasis, Phil. Trans. R. Soc. Lond. B: Biol. Sci.354 (1999) 809–826.[7] M.G. Basanez, J. Ricardez-Esquinca, Models for the population biology and control of human onchocerciasis, TrendsParasitol. 17 (2001) 430–438.[8] J.D. Murray, Mathematical biology I., an introduction. 3rd ed. Heidelberg: Springer-Verlag Berlin, 2002.[9] A.P. Plaisier, E.S. Alley, G.J. van Oortmarssen, B.A. Boatin, J.D.F Habbema, Required duration of combined annualivermectin treatment and vector control program in West Africa, Bull. World Health Organ. 75 (1997) 237-245.[10] J. Remme, G. De Sole, G.J. van Oortmarssen, The predicted and observed decline in onchocerciasis infection during14 years of successful control of black flies in West Africa, Bull. World Health Organ. 68 (1990) 331–339.

https://doi.org/10.28924/ada/ma.3.22


Eur. J. Math. Anal. 10.28924/ada/ma.3.22 11
[11] S.I. Omade, A.T. Omotunde, A.S. Gbenga, Mathematical modeling of river blindness disease with demography usingeuler method, Math. Theory Model. 5 (2015), 75–85.[12] World Health Organization, African programme for onchocerciasis control: Meeting of national onchocerciasis taskforces, September 2012, Weekly Epidemiol. Record 87(49–50) (2012), pp. 494–502.

https://doi.org/10.28924/ada/ma.3.22

	1. Introduction
	2.  Model Description
	2.1. Existence and Positivity of Solutions

	3. Existence and stability of the equilibrium points
	3.1.  Disease-free equilibrium
	3.2. Local Stability of the Disease-free Equilibrium Point E0
	3.3.  Endemic Equilibrium Point Ee

	References

