Research Journal of Mathematics and Statistics 3(4): 136-140, 2011
ISSN: 2040-7505
© Maxwell Scientific Organization, 2011
Submitted: October 02, 2011
Accepted: November 04, 2011
Published: December 15, 2011
Mathematical Modelling on the CDTI Prospects for Elimination of
Onchocerciasis: A Deterministic Model Approach
1
L. Jibril and 2M.O. Ibrahim
Department of Mathematics, Collage of Agriculture and Animal Sciences,
Bakura, Zamfara-Nigeria
2
Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria
1
Abstract: The study incorporated host heterogeneous exposure to Onchocerciasis into four simple
compartmental of both human and vector host regulatory processes, which is of initial value problems of a
systems of ordinary differential equations, with newborns being susceptible, in which both births and deaths
occurs at equal rate. The control of Onchocerciasis currently focuses on Community Directed Treatment with
Ivermectin, CDTI, which effectively kills the Onchocerca volvulus microfilarae in the human hosts. In this
study, the asymptotically stability analysis is determined for the disease free equilibrium and endemic
equilibrium states. Furthermore, on using principles of linearised stability, the study established that the state
of total eradication of Onchocerciasis would be stable and becomes reality if the CDTI control strategies are
maintained throughout the targeted period, hence the population is sustainable.
Key words: Heterogeneous, ivermectin, linearised, Microfilaria, Oncocerca-volvulus, prospects
up to 50% of adults in the most endemic communities.
The fear of blindness resulted in depopulation of the
fertile river valleys, and this made onchocerciasis a major
obstacle to socio-economic development in West African
savannah regions. The socio-economic importance of the
disease was the main reason for creation of the
Onchocerciasis Control Programme in West Africa (OCP)
in 1975 in West Africa, the Onchocerciasis Elimination
Program in the Americas (OEPA), and the African
Program for Onchocerciasis Control (APOC). All three
programmes have come to rely on the regular (OEPA
semi-annually, OCP both annually and semi-annually, and
APOC annually) distribution of ivermectin (Mectizan) to
lower the microfilarial load in affected individuals and
thereby reduce transmission and mitigate the clinical
manifestations of the infection. In addition, since 1975,
OCP has made intensive use of vector control by means
of aerial larviciding. This has led to the virtual elimination
of the parasite from many formerly endemic areas
Njepoume et al. (2009).
In most endemic parts of the world, Onchocerciasis
(river blindness) control relies, or will soon rely,
exclusively on mass treatment with the Microfilaricide
ivermectin. Macrofilaricidal drugs are currently being
developed for human use. Unfortunately, the number of
safe effective alternative treatments is limited.
Diethylcarbamazime, also a Microfilaricide, causes severe
side effects in Onchocerciasis patients. Suramin, the only
currently available highly effective Macrofilaricide has
INTRODUCTION
Onchocerciasis is an insidious nonfatal filarial
disease that causes blindness, lifelong human suffering,
and grave socioeconomic problems. It is a cause of
clinical and epidemiological burden of skin disease in
Africa. An estimated 40 million people are afflicted
worldwide with about 2 million blind. About 85.5 million
people in 35 countries live in endemic areas. It is endemic
in 28 countries in Africa, 6 countries in the Americas, and
in Yemen. Some 18 million people are estimated to be
infected (over 99% of them living in Africa) Basanez and
Boussinesq (1999). In 1875, O'Neill first reported the
presence of filaria in craw-craw as onchocerciasis is
called in West Africa. In 1919, Robles described in the
French literature an anterior uveitis and keratitis
associated with acute and chronic skin changes. Budden
reported Onchocerciasis as an important cause of
blindness in many parts of Northern Nigeria.
Onchocerciasis, or river blindness, is caused by
infection with the filarial parasite Onchocerca volvulus.
The parasite is transmitted by Simulium species
(blackflies) that breed in fast flowing streams. Until
recently the blindness and skin pathology caused by
heavy infections, constituted a major public health
problem in many parts of tropical Africa, Yemen, and
Latin America. In the West African savannah, the risk of
Onchocercal blindness used to be very high along the
rivers, where the vector breeds, and blindness could affect
Corresponding Author: L. Jibril, Department of Mathematics, Collage of Agriculture and Animal Sciences, Bakura, ZamfaraNigeria
136
Res. J. Math. Stat., 3(4): 136-140, 2011
then released to other parts of the body. Their movement
under the skin causes severe itching.
The population of Human infective HI, grows at the
rate .HsVI i.e., directly proportional to the product of
human susceptible, HS(t) and vector infective,VI (called
the incidence rate of infections); where . is the emigration
rates from human susceptible, HS into the human
infective, HI population at time t. Similarly, human
susceptible population decrease at the same rate, at time
t.
The average life-span of adults worms is
approximately 10 years. Repeated treatments of
Ivermectin/Mectizan seem to have some permanent effect
on the fertility of adults worms, this effect manifests itself
only slowly after years of treatment. And therefore it was
assumed that infected human, HI can recover and becomes
human susceptible, HS again at a rate "$HI; where $ is
the movement rate from human infective, HI class into
human susceptible, HS class and " was assumed to be
proportion of Ivermectin/Mectizan treatments successfully
cure the patient.
Similarly, the disease is transmitted at *VHI rate;
where *V is the transmission rate fromVS human infective,
HI into vector susceptible VS. When a susceptible vector
VS (black fly) takes blood meals (bites) from infective
human, HI (suffering from river blindness). Here,
susceptible vector, VS when sucking the human infective,
HI blood, tiny filarial worms are also taken in at the rate
(VS ; where ( is the movement rate of vector susceptible,
VS into the vector infective, VI class at time t, and viceversa. By natural death the individuals in human
susceptible, HS class and human infective, HI class are
removed permanently at a rate JH and JI, respectively.
And F is the mortality rate of human infective causes as
a result of chronic river blindness disease. Also, natural
death results in vector population compartment at a rate DS
and DI, respectively.
ψ
H
ξ
τ
Hs
aβ
H
δ
τ
σ
δV
V
Vs
V
γ
ρs
ρ
Fig. 1: Showing host heteregonous compartments
even more serious side effects. Large scale Nodulectomy,
which has been attempted in Latin America, is impractical
and will never succeed in eliminating all adult worms
Barend et al. (2005).
Ivermectin is an Antimicrofiliarial agent that acts as
secondary prevention in infected individuals, reducing
Microfiliarial survival and hence the burden of disease.
Ivermectin also provides primary prevention for the
community by decreasing the number of Microfilariae
picked up by Simulium during blood feedings, thus
reducing transmission. Since the establishment of the
Mectizan Donation Program (MDP) by Merck and Co.,
Inc., more than 525 million tablets of ivermectin have
been distributed, mostly through programs that use
community directed Treatment with Ivermectin (CDTI),
an untargeted allocation strategy. Ivermectin has no
significant Macrofilaricidal activity and only moderate
long-term effects on Macrofilarial fertility, necessitating
repeated treatment. Ivermectin alone is unlikely to
achieve onchocerciasis eradication in many parts of
Africa, due to operational difficulties and epidemiologic
conditions in hyperendemic communities Eric and Alison
(2006). In view from the above, the study developed a
new deterministic compartmental model for analyzing and
eradicating the spread of Onchocerciasis in communities
settlement, where in the worst affected areas the impact of
the disease can be so extreme that the fertile valley/areas
are depopulated with serious socio-security and
environmental preservation.
ONCHOCERCIASIS TRANSMISSION
MODEL DIAGRAM
Mathematical formulation of the models: The model
consists of four ordinary differential equation describing
the rate of change with respect to time of the compartment
of human susceptible HS at time t , HS(t); the compartment
of human infective HI at time t , HI (t) ; The compartment
of vector susceptible at time t , VS(t) ; and lastly the
compartment of vector infective VI at time t,VI(t).
From the diagram in Fig. 1 the initial value problems
for model with vital dynamics are:
STATEMENT OF THE MODELS
FORMULATION
The river blindness disease is transmitted at the rate
*HVI, when infective black flyVI. Bites a susceptible
person (human) HS ; where *H is the transmission rate
from vector infective, VI into human susceptible, HS on
biting, infective vector (black fly) VI transfer the tiny
worms which grow into adult worms. These adult worms
go into lumps, called Nodules (Oncocermata) their they
reproduce other tiny worms, and these tiny worms are
dHs (t )
= ψ + δ HVI + αβH I − ξHSVI − τ H HS
dt
137
(1)
Res. J. Math. Stat., 3(4): 136-140, 2011
dH I (t )
= ξ HSVI − αβ H I − τ I H I − σ H I − δV H I
dt
Substituting for x4 in Eq. (13):
(2)
⇒ x3 = 0
dVS (t )
= δV H I − ρ SVI − γVS
dt
(17)
Also, substituting (15) for x4 in Eq. (14):
(3)
⇒ x2 = 0
dVI (t )
= δVS − ρ IVI − δ HVI
dt
Substituting (18) for x2 and x4 in (9):
(4)
⇒ x1 =
NHS (0) = NHS ( 0) > 0, NH I (0) = NI 0 > 0, NR(0)
(5)
= NR(0) > 0
(18)
ψ
τH
(19)
ˆ the disease free equilibrium state is:
VS (0) = VSo > 0, VI (0) = VIO = 0
(6)
dHS (t ) dH I (t ) dVS (t ) dVI (t )
=
=
=
=0
dt
dt
dt
dt
(7)
⎛ψ
⎞
( x1 , x2 , x3 , x4 ) = ⎜
, 0, 0, 0⎟
⎝τH
⎠
(20)
Hence, the endemic equilibrium state is given by:
Hs (t ) = x1 , H I (t ) = x2 , Vs (t ) = x3 , VI (t ) = x4
(8)
ψ + αβx2 + δ H x4 − (ξx4 + τ H ) x1 = 0
x1 =
(9)
ξ x1 x4 − (αβ + τ I + σ + δV ) x2 = 0
(10)
δV x2 − ( ρs + γ ) x3 = 0
(11)
γ x3 − ( ρ I + δ H ) x4 = 0
(12)
x2 =
x3 =
Solving systems of the model equation in Eq. (9), (10),
(11) and (12) for the diseases free equilibrium state.
Therefore, Eq. (12) becomes:
⇒ x3 =
( ρ I + δ H ) x4
( ρS + γ )( ρ I + δ H ) x4
(14)
γδV
[
]
( ρ I + δ H )[ψξγδV − τ H ( ρS + γ )( ρ I + δ H )(αβ + τ I + σ + δV )]
[
γ ξ ( ρ I + δ H )[ ( ρS + γ )(τ I + σ ) + ρS δV ] + ξγρ I δV
]
] (22)
(23)
ψξγδV − τ H ( ρS + γ )( ρ I + δ H )(αβ + τ I + σ + δV )
(24)
ξ ( ρ I + δ H )((τ I + σ )( ρS + γ ) + ρS δV ) + ξγρ I δV
The model characteristics equation: From the models
in: Eq. (9), (10), (11) and (12). We obtain the Jacobean
matrix of the system of equations as presented by
Egbetade (2007):
Substituting Eq. (14) in (10):
0 δ H − ξ x1
⎛ − (ξ x4 + τ H ) αβ
⎜
− (αβ + τ I + σ + δV ) 0
ξx
ξ x1
J = ⎜⎜ 4
− ( ρS + γ ) 0
0
δV
⎜⎜
− (ρI + δ H )
0
γ
⎝0
⎡
(αβ + τ I + σ + δV )( ρS + γ )( ρ I + δ H ) ⎤
x4 ⎢ξx1 −
⎥=0
γδV
⎣
⎦
⇒ x4 = 0
[
γδV ξ ( ρ I + δ H )[( ρS + γ )(τ I + σ ) + ρ SδV ] + ξγρ I δV
Stability analysis of the disease free equilibrium states:
On obtaining the equilibrium states, next we investigate
the stability of the Disease Free Equilibrium by examining
the behaviour of the model population near the
equilibrium state as presented by Enagi (2011).
Substituting Eq. (13) in (11):
∴x2 =
(21)
ξγδ v
( ρ I + δ H )( ρS + γ ) ψξγδV − τ H ( ρ S + γ )( ρ I + δ H )(αβ + τ I + σ + δV )
x4 =
(13)
γ
( ρS + γ 0( ρ I + δ H )(αβ + τ I + σ + δV )
(15)
⎞
⎟
⎟ (25)
⎟
⎟⎟
⎠
or
ξ x1 −
(αβ + τ I + σ + δV )( ρ S + γ )( ρ I + δ H )
γδV
=0
Substituting, value for the disease free equilibrium as
given in Eq. (20). We obtained:
(16)
138
Res. J. Math. Stat., 3(4): 136-140, 2011
⎡
⎢− τ H
⎢
⎢
⎢ 0
⎢
⎢ 0
⎢
⎢⎣ 0
αβ
0
− (αβ + τ I + σ + δV )
0
δV
0
− (ρS + γ )
γ
⎛ ψ ⎞⎤
δ H − ξ⎜ ⎟ ⎥
⎝ τ H ⎠⎥
⎛ψ ⎞ ⎥
ξ⎜ ⎟ ⎥
⎝τH ⎠ ⎥
⎥
0
⎥
− ( ρ I + δ H ) ⎥⎦
(26)
The characteristics equation is obtained from the Jacobean determinant with the Eigen values:
8: det (J - 8I) = 0
(27)
Hence, we have:
⎡
⎢− τ H − λ
⎢
⎢
det ⎢ 0
⎢
⎢ 0
⎢
⎢⎣ 0
⎛ψ ⎞ ⎤
δ H − ξ⎜ ⎟ ⎥
⎝τH ⎠ ⎥
⎥
⎛ψ ⎞
⎥=0
ξ⎜ ⎟
⎥
⎝τH ⎠
⎥
0
⎥
− ( ρ I + δ H ) − λ ⎥⎦
(28)
ψ
⎤
) ⎥
τH ⎥
ψ
⎥=0
0
ξ( )
⎥
τH
⎥
+γ + λ)
0
⎥
γ
− ( ρ I + δ H + λ ) ⎥⎦
(29)
αβ
0
− (αβ + τ I + σ + δV ) − λ
0
δV
0
− (ρS + γ ) − λ
γ
⎡
αβ
⎢− (τ H + λ )
⎢
0
− (αβ + τ I + σ + δV + λ )
det ⎢⎢
⎢
δV
0
− (ρS
⎢
⎢⎣
0
0
δH − ξ(
0
This gives:
− (αβ + τ I + σ + δ V + λ )
− (τ H + λ )
δV
0
ψ
)
τH
− (ρ S + γ + λ )
=0
0
γ
− (ρ I + δ H + λ )
ξ(
0
Expanding the Jacobean matrix determinant; We get:
[
]
− (τ H + λ ) − (αβ + τ I + σ + δV + λ )[ ( ρ S + γ + λ )( ρ I + δ H + λ ) − 0 − 0[δV ( ρ I + δ H + λ ) − 0] +
(
(τ H + λ ) (αβ + τ I + σ + δ V + λ )( ρ S + γ + λ )( ρ I + δ H + λ )) +
ψξ
(γδV − 0)] = 0
τH
ψξγδ V
=0
τH
(30)
This gives:
λ1 = −τ H
(31)
139
Res. J. Math. Stat., 3(4): 136-140, 2011
⎛ ψξγδV
⎞
+ αβ + τ I + σ + δV ⎟
⎠
(32)
⎛ ψξγδV
⎞
+ ρS + γ ⎟
⎠
(33)
⎛ ψξγδV
⎞
+ ρI + δ H ⎟
⎠
(34)
λ2 = − ⎜
⎝ τH
λ3 = − ⎜
⎝ τH
λ4 = − ⎜
⎝ τH
A Deterministic Model Approach. The stability analysis
carried out shows that the disease free equilibrium which
is locally and asymptotically stable, would lead to
eradication of onchocerca volvulus infection in a finite
time. And therefore, the treatment of onchocerciasis
disease with Ivermectin drugs is an effective treatment
strategy that could lead to the total eradication of the
disease. Hence, the population is sustainable.
REFERENCES
We obtained the value for 81, 82, 83 and 84 and are all
negatives, therefore the disease free equilibrium state is
stable.
Basanez, M.G. and M. Boussinesq, 1999. Population
biology of human Onchocerciasis. Philos Trans. R.
Soc. Lond. B. Biol. Sci., 354: 809-826.
Barend, M.B., R. Alfons, T. Virginia, N.T. Vincent,
E. David and J.T. Alexander, 2005. Repeated high
doses of avermectins cause Prolonged Sterilization,
but do not kill, Onchocerca ochengi adult worms in
African cattle. Filaria J., 4:8
Egbetade, S.A., 2007. Stability analysis of equilibrium
state of a mathematical model of HIV infection in tcells. J. Mathe. Assoc. Nigeria, 44: 103-105.
Enagi, A., 2011. Mathematical model of tuberclosis. Phd
Thesis, UD USokoto.
Eric, M.P. and P.G. Alison, 2006. Modelling targeted
Ivermectin treatment for controlling river blindness:
The American. J. Trop. Med. Hyg., 75(5): 921-927.
Njepoume, N., P. Ogbu-Pearce, C. Okoronkwo and
M.A. Igbe, 2009. Controlling Onchocerciasis: The
Nigerian Experience. The Internet J. Parasitic Dis.,
4:1
DISCUSSION
It was observed that, the stability analysis carried out
in the models above, shows that, the four roots of the
characteristics equations have non -positive real parts (all
have negative real parts). According to the principles of
linearised stability, that stated as: The systems are
asymptotically stable if and only if all eigen-values of a
system have negative real parts; and hence, the disease
free equilibrium states are locally asymptotically stable.
This study agrees that, the disease steady state for the
eradication of river blindness is globally asymptotically
stable.
CONCLUSION
In this study, we presented a Mathematical Modelling
on the CDTI Prospects for Elimination of Onchocerciasis:
140