




































©2024 Ada Academica https://adac.eeEur. J. Math. Anal. 4 (2024) 9doi: 10.28924/ada/ma.4.9
Global Stability Analysis of Onchocerciasis Transmission Dynamics with Vigilant Compartment

in Two Interacting Populations

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

mikyade2019@gmail.com

Abstract. A deterministic compartmental model for the transmission dynamics of onchocerciasis withvigilant compartment in two interacting populations is studied. The model is qualitatively analyzedto investigate its global asymptotic behavior with respect to disease-free and endemic equilibria. Itis shown, using a linear Lyapunov function, that the disease-free equilibrium is globally asymptoti-cally stable when the associated basic reproduction number, R0 < 1. When the basic reproductionnumber R0 > 1, under some certain conditions on the model parameters, we prove that the endemicequilibrium is globally asymptotically stable with the aid of a suitable nonlinear Lyapunov function.

1. Introduction
Onchocerciasis is one of the neglected tropical diseases caused by the parasite OnchocercaVolvulus, a filarial nematode [3]. The disease is transmitted from one person to another by re-peated bites of black flies. The disease is endemic in Sub-saharan Africa. Many researchers haveworked on many ways to reduce the spread of the disease. For instance, Remme et al. [14] usedskin snip survey in West Africa to investigate the impact of controlling black flies by larviciding.Plaisier et al. [13] used micro simulation model to determine the period required for combiningannual ivermectin treatment and vector control in the onchocerciasis Control Programme in WestAfrica. Alley et al. [3] used a computer simulation model to study prevention of onchocerciasis byusing macrofilaricide which kills the adult worms. Asha Hassan & Nyimvua Shaban [5] investigatedthe effects of four control strategies on the spread of the disease.In this paper, we consider global stability analysis of onchocerciasis transmission dynamics withvigilant compartment. The human population is sub-divided into four compartments and the vec-tor population is sub-divided into three compartments. We show global asymptotic behaviour indisease-free and endemic equilibria. This is an extension of the work done in [1] where the authorworked on the local stability of the model without the vigilant compartment.
Received: 18 Jan 2024.
Key words and phrases. Onchocerciasis epidemic model; Vigilant compartment; Global dynamics; Lyapunov function.1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.9


Eur. J. Math. Anal. 10.28924/ada/ma.4.9 2The case of onchocerciasis model presented in this paper incorporates a new class of humancompartment called vigilant individuals denoted by Vh(t, xi). The individuals in the compartmentare assumed to be tired of onchocerciasis and guide against it by strictly adhering to the vectorcontrol measures such as: regular indoor residual spraying (IRS), insecticide-treated bed-nets(ITNs), clearing of stagnant water bodies and drainages and the use of head-nets in the outdoor.The rest of the paper is organized as follows: the description of the model and theorems on positivityof solutions and reproduction number are given in section 2 while section 3, we explored the globalasymptotic stability of the disease-free equilibrium and endemic equilibrium with a concludingremark.
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 vigilant compartment;
Vh. The black-fly population is partitioned into three compartments: susceptible vector; Sv , theexposed vector compartment; Ev and the infective vector compartment. The total human and vectorpopulations at any given time, t, are respectively given by; N = Sh(t) +Eh(t) + Ih(t) +Vh(t) and
V e = Sv (t)+Ev (t)+Iv (t). We assume that the transmission of onchocerciaisis in susceptible hostsis only through contact with infectious vector. We also assume that susceptible vector becomesinfectious as a result of contact with infectious hosts during blood meal. The population understudy is assumed to be large enough to be modelled deterministically. The following systemof non-linear ordinary differential equations, with non-negative initial conditions, describes thedynamics of onchocerciaisis epidemics.

dSh(t,xi )
dt =

∑L
i=0(1− τ)Ψh(xi)− δλh(xi )σh(t,xi )Iv (t)

Nh(t,xi )
− µh(xi)Sh(t, xi)

dEh(t,xi )
dt =

∑L
i=0

δλh(xi )σSh(t,xi )Iv (t)
Nh(t,xi )

− (αh(xi) + µh(xi))Eh(t, xi)
d Ih(t,xi )
dt =

∑L
i=0(1− θ)αh(xi)Eh − (γ(xi) + µh(xi))Ih(t, xi)

dVh(t,xi )
dt = τΨh(xi)N(t, xi) + θαh(xi)Eh(t, xi) + γ(xi)Ih(t, xi)− µh(xi)Vh(t, xi)

dSv
dt = Ψv − δλv (xi )Sv (t)Ih(t,xi )

Nh(t,xi )
− µvSv (t)

dEv
dt = δλv (xi )Sv (t)Ih(t,xi )

N(t,xi )
− (αv + µv )Ev (t)

d Iv
dt = αvEv (t)− µv 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), Vh(0, xi) = V0h(xi) (2.2)
Sv (0) = S0v , Ev (0) = E0v , Iv (0) = I0v

https://doi.org/10.28924/ada/ma.4.9


Eur. J. Math. Anal. 10.28924/ada/ma.4.9 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
Vh(t, ai) Number of vigilant host 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 transmissionof disease to humanat discrete age xi
λv Probability that a bite results in transmission of parasiteto 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 theinfectious stateat discrete age xi
αv Per capita rate of progression of vectors from the exposed state to theinfectious state
νh(xi) Humans disease-inhibiting factor at discrete age xi
νv Vectors disease-inhibiting factor
τ(xi) Proportion of human population that is born vigilant at discrete age xi
θ(xi) Proportion of exposed humans that becomes vigilant at discrete age xi
γ(xi) Per capita recovery rate of infectious humans to the vigilant state at discrete age xi

Model assumptionsThe formulation of the compartmental model is based on the following assumptions:
1. That only humans are vigilant.2. That humans are born either susceptible or vigilant.3. That exposed humans progress to either become infectious or vigilant. The assumption thatexposed humans can become vigilant is motivated by the possibility of treating Plasmodiumvivax infection which is at the dormant liver stage4. That all infectious humans become vigilant upon recovery due to treatment5. That strict adherence to vector control measures by the vigilant humans does not result intore-infection.

https://doi.org/10.28924/ada/ma.4.9


Eur. J. Math. Anal. 10.28924/ada/ma.4.9 46. 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.10. That a proportion of susceptible humans is infected by infectious mosquitoes and thatsusceptible mosquitoes become infected when in contact with a proportion of infectioushumansTo carry out the analysis of the formulated model (2.1), it is convenient to rescale the variablesby dividing the number of the individuals in the subpopulations by their respective total numberof populations Nh(t, xi) and Nv (t). This process is achieved by making the following change ofvariables:
S̄h(t, xi) = Sh(t,xi )

Nh(t,xi )
, Ēh(t, xi) = Eh(t,xi )

Nh(t,xi )
, Īh(t, xi) = Ih(t,xi )

Nh(t,xi )
, V̄h(t, xi) = Vh(t,xi )

Nh(t,xi )
,

S̄v (t, xi) = Sv (t,xi )
Nv (t,xi )

, Ēv (t, xi) = Ev (t,xi )
Nv (t,xi )

, Īv (t, xi) = Iv (t,xi )
Nv (t,xi )so that̄

Sh(t, xi) + Ēh(t, xi) + Īh(t, xi) + V̄h(t, xi) = 1 and S̄v (t, xi) + Ēv (t, xi) + Īv (t, xi) = 1

The consequence of this, we have Ψh(xi) = µh(xi), Ψv (xi) = µv and σ = Nv (t)
Nh(t,xi )

. After droppingof bars (̄), model (2.1) gives rise to the following system of equations:
dSh(t,xi )

dt = (1− τ)Ψh(xi)−
∑L
i=0 δλh(xi)σh(t, xi)Iv (t)− µh(xi)Sh(t, xi)

dEh(t,xi )
dt =

∑L
i=0 δλh(xi)σSh(t, xi)Iv (t)− (αh(xi) + µh(xi))Eh(t, xi)

d Ih(t,xi )
dt =

∑L
i=0(1− θ)αh(xi)Eh − (γ(xi) + µh(xi))Ih(t, xi)

dVh(t,xi )
dt = τΨh(xi) + θαh(xi)Eh(t, xi) + γ(xi)Ih(t, xi)− µh(xi)Vh(t, xi)

dSv
dt = Ψv − δλv (xi)Sv (t)Ih(t, xi)− µvSv (t)
dEv
dt = δλv (xi)Sv (t)Ih(t, xi)− (αv + µv )Ev (t)
d Iv
dt = αvEv (t)− µv Iv (t)


(2.3)

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

Ih(0, xi) = I0h(xi), Vh(0, xi) = V0h(xi) (2.4)
Sv (0) = S0v , Ev (0) = E0v , Iv (0) = I0v

3. Global Stability Analysis
Here, we explore the global asymptotic stability of the DFE and EE for the special case with noloss of immunity acquired by the recovered individuals. We use the concept of Lyapunov functionsto analyze the global stability

https://doi.org/10.28924/ada/ma.4.9


Eur. J. Math. Anal. 10.28924/ada/ma.4.9 53.1. Global Stability of Disease-free Equilibrium. The following result establishes the globalasymptotic behavior of system (2.1) around E0 which is determined by the basic reproductionnumber R0.
Theorem 3:The disease-free equilibrium (2.12) of model (2.1) is globally asymptotically stable in Ω whenever
R0 ≤ 1

Proof:Consider the linear Lyapunov function of the form
M = d1Eh(t, xi) + d2Ih(t, xi) + d3Ev (t) + d4Iv (t) (3.1)

where
d1 =

αh(xi)(1− θ)

(αh(xi) + µh(xi))(γ(xi) + µh(xi))

d2 =
1

(γ(xi) + µh(xi))

d3 =
1

δλv

d4 =
αv + µv
δλvαvIn what follows, the time derivative of M given by (3.1) along the solutions of the model (2.3)yields

Ṁ =
αh(xi)(1− θ)[δλh(xi)σh(t, xi)Iv − (αh(xi)Iv + µh(xi))Eh(t, xi)]

(αh(xi) + µh(xi))(γ(xi) + µh(xi))

+

L∑
i=0

(γ(xi) + µh(xi))[(1− θ)αh(xi)Eh(t, xi)− (γ(xi) + µh(xi))Ih(t, xi)]

+
1

δλv
[δλvSv Ih(t, xi)− (αv + µv )Ev ] +

αv + µv
δλvαv

[αvEv − µv Iv ]

=

L∑
i=0

δλh(xi)σαh(xi)(1− θ)Sh(t, xi)Iv
(αh(xi))(γ(xi) + µh(xi))

−
αh(xi)(1− θ)Eh(t, xi)

γ(xi) + µh(xi)

+

L∑
i=0

(1− θ)Eh(t, xi)

(γ(xi) + µh(xi))
− Ih(t, xi) + Sv Ih(t, xi)−

(αv + µv )µv Iv
δλvαv

≤
L∑
i=0

δλh(xi)σαh(xi)(1− θ)(1− τ)Iv
(αh(xi) + µh(xi))(γ(xi) + µh(xi))

−
(αv + µv )µv Iv

δλvαv

=

[
L∑
i=0

δλh(xi)σαh(xi)(1− θ)(1− τ)

(αh(xi) + µh(xi))(γ(xi) + µh(xi))
−

(αv + µv )µv
δλvαv

]
Iv

=
(αv + µv )µv
δλvαv

[R20 − 1]Iv

https://doi.org/10.28924/ada/ma.4.9


Eur. J. Math. Anal. 10.28924/ada/ma.4.9 6We have that Ṁ ≤ 0 whenever R0 ≤ 1 with Ṁ = 0 if and only if Iv = 0. We also see that
(Sh(t, xi), Eh(t, xi), Ih(t, xi), Vh(t, xi), Sv (t), Ev (t)) tends to ((1− τ), 0, 0, 0, 1, 0) as t →∞ since
Iv (t) → 0 as t → ∞. By LaSalle’s principle [7], one concludes that every solution of the model(2.3) in Ω approaches the disease-free equilibrium, E0, as t →∞.We have that Ṁ ≤ 0 whenever R0 ≤ 1 with Ṁ = 0 if and only if Iv = 0. We also see that
(Sh(t, xi)), Eh(t, xi), Ih(t, xi), Vh(t, xi), Sv (t), Ev ()) Hence E0 is globally asymptotically stable in
Ω if R0 ≤ 1 2The global asymptotic stability analysis of the endemic equilibrium is considered next forthe special case with τ = θ = 0. The disease-present (endemic) equilibrium of the model(2.3) is referred to the steady-state solution where at least one of the infected compartmentsis nonzero. Let the arbitrary endemic equilibrium of the model (2.1) be represented by Ee =

(S∗∗h (xi), E
∗∗
h (xi), I

∗∗
h (xi), V

∗∗
h (xi), S

∗∗
m , E

∗∗
m , I

∗∗
m ) In order to do this, nonlinear Lyapunov function isused of Goh-Volterra type [6, 15].

Theorem 4: The unique endemic equilibrium,Ee , of the model (2.3) is globally asymptoticallystable if R0 > 1.
Proof: Let R0 > 1 so that there exists a unique endemic equilibrium and consider the nonlinearLyapunov function defined by
M =

(
Sh(t, xi)− S∗∗h (xi)− S∗∗h (xi) ln

Sh(t, xi)

S∗∗h (xi)

)
+

(
Eh(t, xi)− E∗∗h (xi)− E∗∗h (xi) ln

Eh(t, xi)

E∗∗h (xi)

)
+

L∑
i=0

αh(xi) + µh(xi)

αh(xi)

[
Ih(t, xi)− I∗∗h (xi)− I∗∗h (xi) ln

Ih(t, xi)

I∗∗h (xi)

]
+

(
Sv − S∗∗v − S∗∗v ln

Sv
S∗∗v

)
+

(
Ev − E∗∗v − E∗∗v ln

Ev
E∗∗v

)
+
αv + µv
αv

[
Iv − I∗∗v − I∗∗v ln

Iv
I∗∗v

]
With Lyapunov time-derivative given as
Ṁ = Ṡh(t, xi)−

S∗∗h (xi)

Sh(xi)
Ṡh(t, xi) + Ėh(t, xi)−

E∗∗h (xi)

Eh(xi)
Ėh(t, xi)

+

L∑
i=0

αh(xi) + µh(xi)

αh(xi)

(
İh(t, xi)−

I∗∗h (xi)

Ih(xi)
İh(t, xi)

)
+ Ṡv −

S∗∗v
Sv

Ṡv + Ėv −
E∗∗v
Ev

Ėv +
αv + µv
αm

(
İv −

I∗∗v
Iv
İv

) (3.2)

https://doi.org/10.28924/ada/ma.4.9


Eur. J. Math. Anal. 10.28924/ada/ma.4.9 7Using equations of the model (2.3) in (3.3) we obtain
Ṁ = (1− τ)Ψh(xi)−

L∑
i=0

δλh(xi)σSh(t, xi)Iv − µh(xi)Sh(t, xi) (3.3)
−

L∑
i=0

S∗∗h (xi)

Sh(t, xi)
(Ψh(xi)− δλh(xi)σSh(t, xi)Iv − µh(xi)Sh(t, xi))

+

L∑
i=0

δλh(xi)Sh(t, xi)Iv + [αh + µh]Eh(t, xi)−
L∑
i=0

E∗∗h (xi)

Eh(t, xi)
(δλh(xi)σSh(t, xi)Iv + [αh + µh]Eh(t, xi))

+

L∑
i=0

αh(xi) + µh(xi)

αh(xi)
× ((1− θ)αh(xi)Eh(t, xi)− [r(xi) + µh(xi) + γh(xi)])Ih(t, xi)

−
L∑
i=0

I∗∗h (xi)(αh(xi) + µh(xi))

Ih(t, xi)αh(xi)
× ((1− θ)αh(xi)Eh(t, xi)− [r(xi) + µh(xi) + γh(xi)])Ih(t, xi)

+ Ψv − δλvSv Ih(t, xi)− µvSv −
S∗∗v
Sv

(Ψv − δλvSv Ih(t, xi)− µvSv ) + δλvSv Ih(t, xi) + [αv + µv ]Ev

− E∗∗v
Eh(t, xi)

(δλh(xi)Sh(t, xi)Iv + [αh + µh]Ev ) +
αv + µv
αv

[
αvEv − [µv + γv ]Iv −

I∗∗v
Iv

(αvEv − [µv + αv ]Iv )

]

Simplifying Ṁ gives
Ṁ =

L∑
i=0

Ψh(xi)

(
1− S∗∗h (xi)

Sh(t, xi)

)
−

L∑
i=0

µh(xi)Sh(t, xi)

(
1− S∗∗h (xi)

Sh(t, xi)

)
+

L∑
i=0

δλh(xi)S
∗∗
h (xi)Iv (3.4)

−
L∑
i=0

E∗∗h (xi)δλh(xi)Sh(t, xi)Iv
Eh(t, xi)

+

L∑
i=0

(αh(xi) + µh(xi))E∗∗h (xi) (3.5)
−

L∑
i=0

(αh(xi) + µh(xi))

αh(xi)
(r(xi) + µh(xi)γh(xi))Ih(t, xi) (3.6)

−
L∑
i=0

(αh(xi) + µh(xi))I∗∗h (xi)Eh(t, xi)

Ih(t, xi)
+

L∑
i=0

αh(xi) + µh(xi)

αh(xi)
(r(xi) + µh(xi) + γh(xi))I∗∗h (xi) (3.7)

+ µv

(
1− S∗∗v

Sv

)
− µvSv

(
1− S∗∗v

Sv

)
+ δλvS

∗∗
v (xi)Ih −

E∗∗v δλvSv Ih
Ev

+ (αv + µv )E∗∗v (3.8)
− (αv + µv )(µv + γv )Iv

αv
− (αv + µv )I∗∗v Ev

Iv
+

(αv + µv )(µv + γv )Iv
αv

(3.9)
At the endemic equilibrium Ee , we get from model (2.4) that

Ψh(xi) =
∑L
i=0 δλh(xi)σ

∗
h(xi)I

∗
v +

∑L
i=0 µh(xi)S

∗
h(xi)

αh(xi) + µh(xi) =
∑L
i=0

δλh(xi )σ
∗
hI
∗
v

E∗h(xi )

µh(xi) + γh(xi) =
∑L
i=0

αh(xi )E
∗
h(xi )

I∗h(xi )

Ψv = δλvS
∗
v I
∗
h + µvS

∗
v

αv + µv =
δλvS∗v I

∗
h

E∗v

µv + γv =
αvE∗v
I∗v


(3.10)

https://doi.org/10.28924/ada/ma.4.9


Eur. J. Math. Anal. 10.28924/ada/ma.4.9 8Using (3.6) in (3.5), we have
Ṁ =

L∑
i=0

µh(xi)σ
∗
h

(
2− S∗h(xi)

Sh(t, xi)
− Sh(t, xi)

S∗h(xi)

)
+

L∑
i=0

δλh(xi)σS
∗
h(xi)I

∗
v (3.11)

−
L∑
i=0

δλh(xi)(S∗h)2I∗v
Sh(xi)

+ δλh(xi)σS
∗
h(xi)Iv −

L∑
i=0

E∗h(xi)δλh(xi)σSh(t, xi)Iv
Eh(t, xi)

+ δλh(xi)σS
∗
h(xi)Iv

−
L∑
i=0

δλh(xi)σS
∗
h(xi)Ih(t, xi)I

∗
v

I∗h(xi)
−

L∑
i=0

δλh(xi)I
∗
h(xi)Eh(t, xi)I

∗
v

E∗h(xi)Ih(t, xi)
+

L∑
i=0

δλh(xi)σS
∗
hI
∗
v + µvS

∗
v

(
2− S∗∗v

Sv
− Sv
S∗v

)
− δλvS∗v I∗h −

δλv (S∗v )2I∗h
Sv

+ δλvS
∗
v Ih −

E∗v δλvSv Ih
Ev

+ δλvS
∗
v Ih

− δλvS
∗
v Iv I

∗
h

I∗v
− δλv I

∗
vEv I

∗
h

E∗v Iv
+ δλvS

∗
v I
∗
h

Simplifying further, we have
Ṁ =

L∑
i=0

µh(xi)S
∗
h

(
2−

S∗h(xi)

Sh(t, xi)
−
Sh(t, xi)

S∗h(xi)

)
+

L∑
i=0

δλh(xi)σS
∗∗
h I
∗
v (3.12)

×
[

4−
S∗h(xi)

Sh(t, xi)
−
E∗h(xi)σSh(t, xi)Iv
Eh(t, xi)σS

∗
hI
∗
v

−
I∗h(xi)Eh(t, xi)

Ih(t, xi)E
∗
h(xi)

−
Ih(t, xi)I

∗
v

I∗h(xi)Iv

]
+

L∑
i=0

δλh(xi)σS
∗
hI
∗
v −

δλ(xi)σS
∗
h(xi)Ih(t, xi)I

∗
v

I∗h(xi
+

L∑
i=0

δλh(xi)σS
∗
h(xi)Ih(t, xi)(I∗v
I∗h(xi)Iv

− δλh(xi)σS
∗
hI
∗∗
v

+ µvS
∗∗
v

(
2−

S∗∗v
Sv
−
Sv
S∗∗v

)
+ δλvS

∗
v I
∗
h ×

[
4−

S∗v
Sv
−
E∗vSh(t, xi)g(Ih)

EvS∗∗v g(I∗∗h )
−
I∗∗v Ev
IvE∗∗v

−
Ivg(I∗∗h )

I∗∗v g(Ih)

]
+ δλvS

∗
v I
∗
h)−

δλvS
∗
v Iv I

∗
h)

I∗v
+
δλvS

∗
v Iv (I∗h(xi)

)

2

I∗v Ih − δλvS∗v I∗h

Further simplification yields
Ṁ = −Ṁ1 − Ṁ2 −

L∑
i=0

δλh(xi)S
∗
h(xi)I

∗
v

[
1−

Iv
I∗v

+
Ih(t, xi)

I∗h(xi)
+
Ih(t, xi)I

∗
v

I∗h(xi)Iv

]
− Ṁ3 − Ṁ4

−
L∑
i=0

δλvS
∗∗
v I
∗
h

[
1−

Ih
I∗∗h

+
Iv
I∗v

+
Iv I
∗
h

I∗v Ih

] (3.13)
Where

M1 =

L∑
i=0

µh(xi)S
∗
h(xi)

(
S∗h(xi)

Sh(t, xi)
+
Sh(t, xi)

S∗h(xi)
− 2

)
,

M2 =

L∑
i=0

δλh(xi)S
∗
h(xi)Iv ×

[
S∗h(xi)

Sh(t, xi)
+
E∗h(xi)Sh(t, xi)Iv

Eh(t, xi)S
∗
hI
∗
v

+
I∗h(xi)Eh(t, xi)

Ih(t, xi)E
∗
h(xi)

+
Ih(t, xi)I

∗
v

I∗h(xi)Iv
− 4

]
,

M3 = µvS
∗
v

(
S∗v
Sv

+
Sv
S∗v
− 2

)
M4 = δλvS

∗
[
S∗v
Sv

+
E∗vSv (t, xi)g(Ih)

EvS∗v I
∗
h

+
I∗vEv
IvE∗v

+
Iv I
∗
h

I∗v Ih
− 4

]

https://doi.org/10.28924/ada/ma.4.9


Eur. J. Math. Anal. 10.28924/ada/ma.4.9 9

Conclusion: In this article, a onchocerciasis transmission dynamics with vigilant compartmentgoverned by system of differential equations has been theoretically analyzed. The analysis iscentered on the global asymptotic behavior of solutions of the system (2.3) around the disease-free and endemic equilibria using Lyapunov functions. The system has a globally asymptoticallystable disease-free equilibrium whenever the basic reproduction R0 < 1. Moreover, the endemicequilibrium of the system, when it exists, is shown to be globally asymptotically stable wheneverthe associated basic reproduction number R0 > 1.
References

[1] K.M. Adeyemo, Local stability of onchocerciasis transmission dynamics with nonlinear incidence functions in twointeracting populations, Eur. J. Math. Anal. 3 (2023) 22.[2] 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.[3] 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.[4] A. Hassan, N. Shaban, Onchocerciasis dynamics: modelling the effects of treatment, education and vector control,J. Biol. Dyn. 14 (2020) 245-268.[5] C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng. 1 (2004)361-404.[6] P. Georgescu, H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse,Appl. Math. Comp. 219 (2013) 8496-8507.[7] J.P. LaSalle, The stability of dynamical systems, SIAM, Philadelphia, 1976.[8] E.M. Poolman, A.P. Galvani, Modeling targeted ivermectin treatment for controlling river blindness, Amer. J. Trop.Med. Hyg. 75 (2006) 921–927.[9] J.P. Mopecha, H.R. Thieme, Competitive dynamics in a model for onchocerciasis with cross-immunity, Canad. Appl.Math. Q. 11 (2003) 339–376.[10] M.G. Basáñez, M. Boussinesq, Population biology of human onchocerciasis, Phil. Trans. R. Soc. Lond. B: Biol. Sci.354 (1999) 809–826.[11] M.G. Basáñez, J. Ricárdez-Esquinca, Models for the population biology and control of human onchocerciasis, TrendsParasitol. 17 (2001) 430–438.[12] J.D. Murray, Mathematical biology I., an introduction. 3rd ed. Springer-Verlag, Berlin, 2002.[13] 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.[14] 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.[15] M.A. Safi, S.M. Garba, Global stability analysis of SEIR model with holling type II incidence function, Comp. Math.Meth. Med. 2012 (2012) 826052.[16] 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.[17] World Health Organization, African programme for onchocerciasis control: Meeting of national onchocerciasis taskforces, September 2012, Weekly Epidemiol. Record 87 (2012) 494–502.

https://doi.org/10.28924/ada/ma.4.9

	1. Introduction
	2.  Model Description
	3. Global Stability Analysis
	3.1. Global Stability of Disease-free Equilibrium

	References

