Mathematical analysis of the dynamics of visceral leishmaniasis in
Sudan
Ibrahim M.ELmojtaba
Faculty of Mathematical Sciences, University of Khartoum
P.O.Box 321, Khartoum, Sudan
J.Y.T. Mugisha∗
Department of Mathematics, Makerere University
P.O.Box 7062, Kampala, Uganda
Mohsin H.A.Hashim
Faculty of Mathematical Sciences, University of Khartoum
P.O.Box 321, Khartoum, Sudan
Abstract
In this paper a mathematical model was developed to study the dynamics of visceral leishmaniasis
(VL) the in Sudan. To develop this model, the dynamics of the disease between three different populations, human, reservoir and vector populations was considered. The model analysis is done and
the equilibrium points are analyzed to establish their stability. The threshold of the disease, the basic
reproduction number, is established for which the disease can be eliminated. Results show that the
disease can be eliminated under certain conditions. It is also concluded that human treatment helps
in disease control, but it is not sufficient to eliminate the disease and it should be followed by vector
control in order to eradicate the disease from the human population.
Keywords: Visceral leishmaniasis, PKDL, animal reservoir, sandfly, basic reproduction number.
∗
Corresponding Author: [email protected]
1
1
Introduction
Leishmaniasis is caused by a protozoal parasite of the genus leishmania which multiplies in certain
vertebrates that act as reservoirs of the disease. The parasite is transmitted to humans through the
bite of sandflies that have previously fed on an infected reservoir or an infected human. So far, at least
20 types of leishmania are identified as being pathogenic to humans [36], but there are two basic forms
of the disease, namely cutaneous leishmaniasis and visceral leishmaniasis depending on the species of
leishmania responsible and the immune response to infection.
The cutaneous form tends to heal spontaneously leaving scars which, depending on the species
of leishmania responsible, may evolve into diffuse cutaneous leishmaniasis, recidivate leishmaniasis,
or mucocutaneous leishmaniasis, with disastrous aesthetic consequences for the patient. Visceral
leishmaniasis has two major forms L.donovani, and L.infantum, and it is the most severe form and it
is fatal in almost all cases if left untreated. It may cause epidemic outbreaks with a high mortality
rate. A varying proportion of visceral cases may evolve into a cutaneous form known as post kalaazar dermal leishmaniasis (PKDL), which requires lengthy and costly treatment [36], but fortunately
after the infection with leishmaniasis, the human host and the reservoir host will gain a permanent
immunity[13].
Leishmaniasis is endemic in 88 countries in the world and 350 million people are considered at
risk. An estimated 14 million people are infected, and each year about two million new cases occur.
The disease contributes significantly to the propagation of poverty, because treatment is expensive and
hence either unaffordable or it imposes a substantial economic burden, including loss of wages[12].
Each year, there are some 500 000 cases of visceral leishmaniasis (90% in Bangladesh, Brazil,India,
Nepal and Sudan), with an estimated more than 50 000 deaths, and 1 500 000 cases of cutaneous
leishmaniasis (90% in Afghanistan, Algeria, Brazil, Islamic Republic of Iran, Peru, Saudi Arabia and
Sudan). The global mortality from visceral leishmaniasis can only be estimated, because in many
countries the disease is not notifiable or is frequently undiagnosed, especially where there is no access
to medication. In some cases, for cultural reasons and lack of access to treatment, the case-fatality
rate is three times higher in women than in men. The disease burden is calculated at 2 090 000
disability adjusted life years (1 249 000 in men and 840 000 in women), a significantly high rank
among communicable diseases [38]. The number of cases is increasing, mostly because of gradually
more transmission in cities, displacement of populations, exposure of people who are not immune,
deterioration of social and economic conditions in outlying urban areas, malnutrition (with consequent
weakening of the immune system), and co-infection with HIV, and Malaria. In 34 of the 88 countries in
which the disease is endemic, cases of co-infection have been reported [37], also there are some animals
(dog, the Nile grass rat, spiny mouse, and some other mammals) helping in the disease transmission
by playing the role of reservoir [20, 35].
First-line treatment, especially for visceral leishmaniasis, is expensive and needs to be administered, by injection, in hospital. As a rule, patients have to overcome major logistic problems in order
2
to access treatment: long distances to the treatment center, lack of transport, treatment is unaffordable, or its costs pose a serious financial burden. For these reasons, patients may not comply with
treatment (if they began) and drug resistance may emerge [32]. There is a shortage of information
on the actual cost of leishmaniasis, although it is known that in some parts of Asia a family in which
there is a case of leishmaniasis is three times more likely than an unaffected family to have sold its
cow or rice field, plunging it into a vicious circle of disease-poverty-malnutrition-disease.
Improved control reduces both mortality and morbidity, it also reduces the role of humans as a
reservoir in the disease cycles and makes it possible to avert progression of the disease to complicated
cutaneous forms. The combination of active case detection and treatment is the key to control, because
the vaccine against visceral leishmaniasis is not so effective yet [24], and the control of sandflies is very
difficult (it is more difficult than the control of Anopheles mosquitoes), but some studies showed that
using treated bed nets reduce the sandfly’s biting rates by 64% - 100% [4]. Nevertheless, even that
seemingly simple approach faces major obstacles. Although during their initial phases, leishmaniases
respond well to treatment, many patients are unaware of the initial symptoms. Furthermore, health
systems are frequently either poorly staffed and lack equipment or are non-existent in remote rural
areas where contact with sandflies is most common.
Visceral leishmaniasis is one of the most important endemic diseases in the Sudan, and Sudan
is considered one of the foci of the disease. Occasionally, sever epidemics have claimed the live of
thousands of people, and in recent years the disease spreads out in a wide areas of the country, like
Atbara river in the north-east and along the Sudanese-Ethiopian borders, south of Sobat river, Nasir,
Malakal, and also extended west across the White Nile up to Nuba mountains and Darfur region [29].
There were some models developed to study the dynamics of cutaneous and visceral leishmaniasis
[3, 6, 7, 9, 13, 21], but only Burattini et.al [6] considers humans as a source of infection for sandflies.
In this model, humans were considered as a source of infection for sandflies and the role of PKDL in
the transmission process was also considered. The model is simulated with data from previous studies
[16, 24, 29] carried out on leishmaniasis in the Sudan.
2
Model formulation and equations
To formulate this model, the transmission of the disease between three different populations, human
host population, NH (t), reservoir host population, NR (t), and vector population, NV (t) was considered. Let the human host population be divided into four categories, susceptible individuals SH (t),
those who are infected with visceral leishmaniasis, IH (t), ignoration of the incubation period followed
Dye [13] and because the incubation period can be as short as 10 days [18], those who develop PKDL
after the treatment of visceral leishmaniasis, PH (t), and those who are recovered and have permanent
immunity, RH (t). This implies that
3
NH (t) = SH (t) + IH (t) + PH (t) + RH (t).
Similarly, let the reservoir host population be divided into two categories, susceptible reservoir, SR (t),
and infected reservoir, IR (t), such that
NR (t) = SR (t) + IR (t)
and the vector population have two categories, susceptible sandflies, SV (t), and infected sandflies,
IV (t), such that
NV (t) = SV (t) + IV (t)
It is assumed that susceptible individuals are recruited into the population at a constant rate ΛH
and acquire infection with leishmaniasis following contacts with infected sandflies at a per capita
rate ab NIVH , where a is the per capita biting rate of sandflies on humans (or reservoirs), and b is the
transmission probability per bite per human (as the case for malaria, [27, 30]). The per capita biting
rate of sandflies a is equal to the number of bites received per human from sandfly due to conservation
of bites mechanism [5, 28]. Infected humans die due to leishmaniasis at an average rate δ, or get
treatment at an average rate α1 , and a fraction σ of those get treated recover and acquire permanent
immunity, and the fraction (1 − σ) develop PKDL. Humans with PKDL get treated at an average rate
α2 , or recover naturally at an average rate β, and acquire permanent immunity in both cases. There
is a per capita natural mortality rate µh in all human sub-population.
Susceptible reservoirs are recruited into the population at a constant rate ΛR , and acquire
infection with leishmaniasis following contacts with infected sandflies at a rate ab NIVH where a and b
as described above. It is assumed that the transmission probability per bite is the same for human
and reservoir because sandflies do not distinguish between humans and reservoirs. It is also assumed
that reservoirs do not die due to the disease, but a per capita natural mortality rate µr occurs in the
reservoir population.
Susceptible sandflies are recruited at a constant rate ΛV , and acquire leishmaniasis infection
following contacts with human infected with leishmaniasis or human having PKDL or reservoir infected
PH
with leishmaniasis at an average rate equal to ac NIHH + ac N
+ ac NIRR , where a is the per capita biting
H
rate, and c is the transmission probability for sandfly infection. Sandflies suffer natural mortality at a
per capita rate µv regardless of their infection status. The dynamics of the disease in human, reservoir
and sandfly populations is as described in Figure 1.
From the description of the terms and Figure 1, the following system of differential equations was
derived:
4
ΛH
-
λh
SH
-
?
µh
PH
?
?
(α2 + β)
σα1 - RH (µh + δ)
Human population
-
(1 − σ)α1
IH
µh
?
µh
Vector population
IV
λv
SV
-
ΛV
?
µ?
v
ΛR
µv
-
λr
SR
IR
Reservoir population
?
µ?
r
µr
Figure 1: Compartmental model
SH
− µ h SH
NH
0
SH
= ΛH − abIV
0
IH
= abIV
PH0
= (1 − σ)α1 IH − (α2 + β + µh )PH
0
RH
= σα1 IH + (α2 + β)PH − µh RH
SH
− (α1 + δ + µh )IH
NH
0
SR
= ΛR − abIV
0
IR
= abIV
SR
+ µ r SR
NR
SR
− µ r IR
NR
SV0
= ΛV − acSV
IV0
= acSV
IH
PH
IR
− acSV
− acSV
SV − µ v SV
NH
NH
NR
IH
PH
IR
+ acSV
+ acSV
S V − µ v IV
NH
NH
NR
with
0
NH
= ΛH − µh NH − δIH
5
(1)
NR0 = ΛR − µr NR
NV0
= Λ V − µ v NV
Invariant region
All parameters of the model are assumed to be nonnegative, furthermore since model (1) monitors
living populations, it is assumed that all the state variables are nonnegative at time t = 0. It is noted
that in the absence of the disease (δ = 0), the total human population size, NH → ΛH /µh as t → ∞,
also NR → ΛR /µr and NV → ΛV /µv as as t → ∞. This shows that the biologically-feasible region:
ΛH
8 : S , I , P , R , S , I , S , I ≥ 0, N
Ω = {(SH , IH , PH , RH , SR , IR , SV , IV ) ∈ R+
H H
H
H
R R
V
V
H ≤ µ h , NR ≤
ΛR
ΛV
µr , NV ≤ µv } is positively-invariant domain, and thus, the model is epidemiologically and mathematically well posed, and it is sufficient to consider the dynamics of the flow generated by (1) in this
positively-invariant domain Ω.
3
Analysis of the model
In this Section, system (1) was analyzed to obtain the equilibrium points of the system and their
stability. The equations for the proportions was considered by first scaling the sub-populations for
NH , NR and NV using the following set of new variables
sh =
SH
IH
PH
RH
SR
IR
SV
IV
, ih =
, ph =
, rh =
, sr =
, ir =
, sv =
, and iv =
;
NH
NH
NH
NH
NR
NR
NV
NV
NV
and let m = N
be the female vector-human ratio defined as the number of female sandflies per
H
human host (see similar definition in malaria models, in [2, 15]). Note that the ratio m is taken as a
constant because it is well known (see [34] pages 218-220) that a vector takes a fixed number of blood
V
meals per unit time independent of the population density of the host. Similarly, we let n = N
NR be
the female vector-reservoir ratio defined as the number of female sandflies per reservoir host.
Differentiating with respect to time t and simplifying gives the following reduced system of differential equations:
s0h
=
ΛH
ΛH
− abmiv +
− δih sh
NH
NH
i0h = abmiv sh − α1 + δ +
ΛH
− δih ih
NH
p0h = (1 − σ)α1 ih − α2 + β +
rh0 = σα1 ih + (α2 + β2 )ph −
6
ΛH
− δih ph
NH
ΛH
− δih rh
NH
(2)
s0r =
ΛR
ΛH
− abniv sr −
sr
NR
NH
i0r = abniv sr −
s0v =
ΛH
ir
NH
ΛV
ΛV
− acih + acph + acir +
sv
NV
NV
i0v = acih sv + acph sv + acir sv −
ΛV
iv
NV
with the feasible region (i.e where the model makes biological sense)
Γ = {(sh , ih , ph , rh , sr , ir , sv , iv ) ∈ R+ 8 : 0 ≤ sh , ih , ph , rh , sh + ih + ph + rh ≤ 1, 0 ≤ sr , ir , sr + ir ≤
1, 0 ≤ sv , iv , sv + iv ≤ 1}.
It can be shown that the above region is positively invariant with respect to the system(2),
where R+ 8 denotes the non-negative cone of R8 including its lower dimensional faces. We denote the
◦
boundary and the interior of Γ by ∂Γ and Γ, respectively.
3.1
Stability analysis of the disease-free equilibrium E0
The disease-free equilibrium (DFE) of the system(2) is given by E0 = (s0h , i0h , p0h , rh0 , s0r , i0r , s0v , i0v ) =
(1, 0, 0, 0, 1, 0, 1, 0).
To study the stability of the disease-free equilibrium, first we have to find the basic reproduction
number R0 . It is defined as the number of secondary infections that occur when an infected individual
is introduced into a completely susceptible population [11, 22]. To calculate the basic reproduction
number we will use the next generation approach [11, 33]. We first re-arrange system(2) beginning
with the infected classes as follows
ΛH
0
ih = abmiv sh − α1 + δ +
− δih ih
NH
p0h
ΛH
= (1 − σ)α1 ih − α2 + β +
− δih ph
NH
i0r = abniv sr −
ΛH
ir
NH
i0v = acih sv + acph sv + acir sv −
s0h =
ΛV
iv
NV
ΛH
ΛH
− abmiv +
− δih sh
NH
NH
rh0 = σα1 ih + (α2 + β2 )ph −
7
ΛH
− δih rh
NH
s0r =
ΛR
ΛH
− abniv sr −
sr
NR
NH
s0v =
ΛV
ΛV
− acih + acph +
sv
NV
NV
and then distinguish the new infections from all other class transitions in the population. The infected
class are ih , ph in the human population, ir in the reservoir population, and iv in the sandfly population. Using the next generation operator method, we define F as the column-vector of rates of the
appearance of new infections in each compartment; V = V + + V − , where V + is the column-vector of
rates of transfer of individuals into the particular compartment; and V − is the column-vector of rates
of transfer of individuals out of the particular compartment. Hence, from our model, and because we
have only 4 infected compartments, we have




F =


and




V =


abmiv sh
0
aciv sr
ac(ih + ph + ir )sv







(α1 + δ + µh )ih
(α2 + β + µh )ph − (1 − σ)α1 ih
µ r ir
µ v iv




.


then the matrices F and V from the partial derivatives of F and V with respect to the infected classes
computed at the DFE are given by




F=



0 0 0 abm

0 0 0
0 

,
0 0 0 abn 

ac ac ac
0

and



V=


(α1 + δ + µh )
0
0 0
−(1 − σ)α1 (α2 + β+ µh ) 0 0
0
0
µr 0
0
0
0 µv







Then the basic reproduction number, R0 defined as the spectral radius of matrix FV−1 ;
R0 = ρ(FV
−1
)=
s
ac[µr abm(α2 + β + µh + (1 − σ)α1 ) + abn(α1 + δ + µh )(α2 + β + µh )]
µr µv (α1 + δ + µh )(α2 + β + µh )
The basic reproduction number calculated above is biologically meaningful, because the term
ac
represents the number of secondary infections caused by one infected sandfly in the population of
µv
humans or in the population of reservoirs (since sandfly dose not distinguish between the two hosts).
8
The second term has two terms; the first one
abm(α2 + β + µh + (1 − σ)α1 )
represents the number
(α1 + δ + µh )(α2 + β + µh )
of secondary infections in the population of sandflies caused by one infected human, and the second
term abn
µr represents the number of secondary infections in the population of sandflies caused by one
infected reservoir, and hence the term
µr abm(α2 + β + µh + (1 − σ)α1 ) + abn(α1 + δ + µh )(α2 + β + µh )
µr (α1 + δ + µh )(α2 + β + µh )
represents the number of secondary infections in the population of sandflies caused by one infected
human or one infected reservoir.
Since system(2) satisfies axioms (A1)– (A5) of Theorem 2 in van den Driessche and Watmough
[33], and since R0 which is calculated above is biologically meaningful, we have the following result:
Lemma 3.1 The disease-free equilibrium is locally asymptotically stable if R0 < 1 and unstable if
R0 > 1.
The following theorem shows that the DFE is globally asymptotically stable if R0 < 1.
Theorem 3.1 The disease-free equilibrium point is globally asymptotically stable if R0 < 1
Proof:
Consider the following Liapunov function
L = µr ac(α2 + β + µh + (1 − σ)α1 )ih + µr ac(α1 + δ + µh )ph + ac(α1 + δ + µh )(α2 + β + µh )ir
+µr (α1 + δ + µh )(α2 + β + µh )iv
with derivative
L0 = µr ac(α2 + β + µh + (1 − σ)α1 )[abmiv (1 − ih − ph − rh ) − (α1 + δ + µh )ih ]
+µr ac(α1 + δ + µh )[(1 − σ)α1 ih − (α2 + β + µh )ph ]
+ac(α1 + δ + µh )(α2 + β + µh )[abniv (1 − ir ) − µr iR ]
+µr (α1 + δ + µh )(α2 + β + µh )[ac(ih + ph + ir )(1 − iv ) − µv iv ]
= µr a2 bcm(α2 + β + µh + (1 − σ)α1 )iv − µr a2 bcm(α2 + β + µh + (1 − σ)α1 )ih iv
−µr a2 bcm(α2 + β + µh + (1 − σ)α1 )ph iv − µr a2 bcm(α2 + β + µh + (1 − σ)α1 )rh iv
−µr ac(α1 + δ + µh )(α2 + β + µh + (1 − σ)α1 )ih + µr ac(α1 + δ + µh )(1 − σ)α1 ih
−µr ac(α1 + δ + µh )(α2 + β + µh )ph + a2 bcn(α1 + δ + µh )(α2 + β + µh )iv
−a2 bcn(α1 + δ + µh )(α2 + β + µh )ir iv − µr ac(α1 + δ + µh )(α2 + β + µh )ir
+µr ac(α1 + δ + µh )(α2 + β + µh )ih + µr ac(α1 + δ + µh )(α2 + β + µh )ph
9
+µr ac(α1 + δ + µh )(α2 + β + µh )ir − µr ac(α1 + δ + µh )(α2 + β + µh )ih iv
−µr ac(α1 + δ + µh )(α2 + β + µh )ph iv − µr ac(α1 + δ + µh )(α2 + β + µh )ir iv
−µr µv (α1 + δ + µh )(α2 + β + µh )iv
≤
µr a2 bcm(α2 + β + µh + (1 − σ)α1 ) + a2 bcn(α1 + δ + µh )(α2 + β + µh )
−µr µv (α1 + δ + µh )(α2 + β + µh ) iv − µr a2 bcm(α2 + β + µh + (1 − σ)α1 )ih iv
−µr a2 bcm(α2 + β + µh + (1 − σ)α1 )ph iv − µr ac(α1 + δ + µh )(α2 + β + µh )ih
−µr ac(α1 + δ + µh )(α2 + β + µh )ph − a2 bcn(α1 + δ + µh )(α2 + β + µh )ir iv
−µr ac(α1 + δ + µh )(α2 + β + µh )ir + µr ac(α1 + δ + µh )(α2 + β + µh )ih
+µr ac(α1 + δ + µh )(α2 + β + µh )ph + −µr ac(α1 + δ + µh )(α2 + β + µh )ir
−µr ac(α1 + δ + µh )(α2 + β + µh )ih iv − µr ac(α1 + δ + µh )(α2 + β + µh )ph iv
−µr ac(α1 + δ + µh )(α2 + β + µh )ir iv
= µr µv (α1 + δ + µh )(α2 + β + µh )(R20 − 1)iv
−µr a2 bcm(α2 + β + µh + (1 − σ)α1 )ih iv − µr a2 bcm(α2 + β + µh + (1 − σ)α1 )ph iv
−a2 bcn(α1 + δ + µh )(α2 + β + µh )ir iv − µr ac(α1 + δ + µh )(α2 + β + µh )ih iv
−µr ac(α1 + δ + µh )(α2 + β + µh )ph iv − µr ac(α1 + δ + µh )(α2 + β + µh )ir iv
≤ µr µv (α1 + δ + µh )(α2 + β + µh )(R20 − 1)iv
≤ 0
if R0 < 1. We observe that our system has the maximum invariant set for L0 = 0 if and only if R0 ≤ 1
holds and ih = ph = rh = iv = ir = 0. By Liapunov-Lasallés Theorem [19], all the trajectories starting
in the feasible region where the solutions have biological meaning approach the positively invariant
subset of the set where L0 = 0 , which is the set where ih = ph = rh = iv = ir = 0. In this set,
s0h =
ΛH
NH
ΛH
− abmiv + N
−δih sh , s0r =
H
ΛR
NR
ΛR
−abniv sr − N
sr , and s0v =
R
ΛV
NV
ΛV
− acih +acph +acir + N
sv ,
V
so that as t → ∞, sh (t) → 1,sr (t) → 1, and sv (t) → 1 . This shows that all solutions in the set where
ih = ph = rh = iv = ir = 0, go to the disease-free equilibrium E0 . Thus, R0 < 1 is the necessary and
sufficient condition for the disease to be eliminated from the community.
3.2
Existence and stability analysis of the endemic equilibrium E1
In order to prove the existence of E1 we equate the right hand sites of system (2) to zeros, and
substitute sh = 1 − ih − ph − rh , sr = 1 − ir and sv = 1 − iv , to obtain
abmiv (1 − ih − ph − rh ) = (α1 + δ + µh )ih
10
(3)
(1 − σ)α1 ih = (α2 + β + µh )ph
(4)
σα1 ih + (α2 + β)ph = µh rh
(5)
abniv (1 − ir ) = µr ir
(6)
(acih + acph + acir )(1 − iv ) = µv iv
(7)
From equations (4) and (5) we have
p∗h =
(1 − σ)α1 ∗
i
(α2 + β + µh ) h
(8)
rh∗ =
α1 (α2 + β + σµh ) ∗
i
µh (α2 + β + µh ) h
(9)
Substituting equations (6), (7), (8), and (9) into (3) we get either i∗h = 0 (DFE), or i∗h satisfying the
following equation:
A(i∗h )2 + Bi∗h + C = 0
(10)
where
A = ac(1 + R)[nG2 + abmnG(1 + F + R) − M µr G(1 + F + R) − abm2 µr (1 + F + R)2 ]
B = nG2 (ac + µv ) + a2 bcmnGF + acmµr + 4a2 bcm2 µr (1 + R)(1 + F + R) − mµr µv G(1 + F + R)
C = mµr µv G − a2 bcm2 µr (1 + R) − a2 bcmnG,
and R =
(1 − σ)α1
α (α + β + σµh )
,F = 1 2
, G = (α1 + δ + µh ).
(α2 + β + µh )
µh (α2 + β + µh )
For a positive endemic equilibrium E1 > 0 we seek A > 0 and C < 0. We note that
A = ac(1 + R)[nG(G + abm(1 + F + R)) − mµr (1 + F + R)(G + abm(1 + F + R))]
= ac(1 + R)(G + abm(1 + F + R))(nG − mµr (1 + F + R))
= ac(1 + R)(G + abm(1 + F + R))(n(α1 + δ + µh ) − mµr
= ac(1 + R)(G + abm(1 + F + R))
(α1 + µh )
)
µh
(α1 + µh )
(nµh − mµr ) + nδ
µh
and
11
(α1 + µh )
(nµh − mµr ) + nδ =
µh
m(α1 + µh ) NR (α1 + δ + µh )
µ h − µr
µh
NH
(α1 + µh )
which is clearly greater than 0, and hence A > 0.
Also
C = mµr µv (α1 + δ + µh ) − a2 bcmn(α1 + δ + µh ) −
=
a2 bcm2 µr (α2 + β + µh + (1 − σ)α1 )
α2 + β + µ h
−m
a2 bcn(α1 + δ + µh )(α2 + β + µh ) + a2 bcmµr (α2 + β + µh + (1 − σ)α1 )
α2 + β + µ h
−µr µv (α1 + δ + µh )(α2 + β + µh )
= −mµr µv (α1 + δ + µh )(R20 − 1)
and hence C < 0 when R0 > 1. From the theory of quadratic equations if ih1 and ih2 are the solutions
C
for the quadratic equation(10) then ih1 ih2 = C
A ; and if A < 0 then we have one positive solution of
equation(10). From the computation above, A > 0 and C < 0 when R0 > 1, hence we have one and
only one positive solution for equation(10) when R0 > 1, and then we have the following lemma:
Lemma 3.2 The system(2) has precisely one positive endemic equilibrium E1 given by
E1 = (s∗h , i∗h , p∗h , rh∗ , s∗r , i∗r , s∗v , i∗v ) and i∗h satisfying equation (10), when R0 > 1.
The following theorem states the local stability of the endemic equilibrium.
Theorem 3.2 The Endemic Equilibrium (E1 ) is locally asymptotically stable whenever it exits.
Proof:
Consider the jacobian of the system (2) at the endemic equilibrium E1









J (E1 ) = 








−J1
δs∗h
0
0
0
0
0
−abms∗h

abmi∗v
−J2
0
0
0
0
0
abms∗h 


0
(1 − σ)α1 + δp∗h
−J3
0
0
0
0
0



∗
0
δrh
(α2 + β) −J4
0
0
0
0


0
0
0
0
−J5
0
0
−abns∗r 


0
0
0
0
abni∗v −J6
0
abns∗r 


0
−acs∗v
−acs∗v
0
0
−acs∗v
−J7
0

0
acs∗v
acs∗v
0
0
acs∗v ac(i∗h + p∗h + i∗r )
−µv
12
where
J1 = abmi∗v + µh
J2 = α1 + µh + δ(1 − δi∗h )
J3 = α2 + β + µh
J4 = µ h
J5 = abni∗v + µr
J6 = µ r
J7 = ac(i∗h + p∗h + i∗r ) + µv
Since the diagonal elements of the matrix J (E1 ) are negative and the eigenvalues of any square
matrix A are the same as those of AT , an easy argument using Gers̆gorin discs [26] shows that
J (E1 ) is stable if it is diagonally dominant in columns.
Setting µ = max {g1 , g2 , g3 , g4 , g5 , g6 , g7 , g8 } where
g1 = −J1 + abmi∗v = −µh
g2 = δs∗h − J2 + (1 − σ)α1 + δp∗h + δrh∗ − acs∗v + acs∗v = −(µh + σα1 )
g3 = −J3 + α2 + β = −µh
g4 = −J4 = −µh
g5 = −J5 + abni∗v = −µr
g6 = −J6 = −µr
g7 = −J7 + ac(i∗h + p∗h + i∗r ) = −µv
g8 = −µv
gives µ = max
−µh , −(µh + σα1 ), −µr , −µv
< 0; which implies diagonal dominance as claimed,
and concludes the proof.
Due to computational complexity needed, it is not easy to analytically determine the global
behavior of the endemic equilibrium of the model. Numerical simulation to study the model
behavior within a selected set of parameters was used.
4
Numerical Simulation
In this section we give numerical simulation for our model. Most of the parameters were obtained
from literature, and some of them were assumed or made varying to have more realistic simulation
results.
The parameters values used are in Table 1, with NH = 200, 000, NR = 1, 000 and NV = 50, 000.
13
parameter
parameter description
value
ΛH
ΛR
ΛV
µh
µr
µv
a
b
c
α1
1−σ
σ
δ
α2
β
Human recruitment rate
Reservoir recruitment rate
Vector recruitment rate
Natural mortality rate of humans
Natural mortality rate of reservoirs
Natural mortality rate of vectors
Biting rate of sandflies
Progression rate of VL in sandfly
Progression rate of VL in human and reservoir
Treatment rate of VL
Developing PKDL rate after treatment
Recovery rate from VL infection after treatment
Death rate due to VL
PKDL recovery rate without treatment
PKDL recovery rate after treatment
0.0015875×NH day −1
0.073×NR day −1
0.299×NV day −1
0.00004 day −1
0.000274 day −1
0.189 day −1
0.2856 day −1
0.22 day −1
0.0714 day −1
Variable
0.36 day −1
0.64 day −1
0.011 day −1
0.00556 day −1
0.033 day −1
Source
[17]
Assumed
[23]
[8]
Assumed
[23]
[14]
[14]
[25]
Variable
[16]
[32]
[32]
[16]
[16]
Table 1: parameter values (and their sources)
We discuss the simulation results of the model to establish the best treatment scenario, and control
measures.
(a) When the treatment rate is high: There are two major drugs used against VL in the
Sudan, pentamidine and sodium stibogluconate. Despite of its toxicity and expensiveness,
sodium stibogluconate is the preferable one because it is very effective (its cure rate is about
65% [29]). By high treatment rate we mean that most of VL infected humans go to hospital
and get treatment. We notice that when the treatment rate is high, say α1 = 0.9, there is a
huge reduction in the fraction of human infected with visceral leishmaniasis(Figure 2(a)), also
there will be some reduction in the PKDL infected human as noted from (Figure 2(b)), but it
is not that high because no matter how high the treatment, there is still a fraction 1 − σ
developing PKDL. On the other hand there will be endemicity in the human population,
reservoir population and vector population(Figures 2(a), 2(c) and 2(d)) because this strategy
dose not affect the reservoir population nor the vector population.
(b) When the treatment rate is low: In contrary to high treatment rate, by low treatment rate
we mean that most of VL infected humans either do not get treatment at all because they are
far a way from hospitals, or they get a wrong treatment because of misdiagnosis. When the
treatment rate is low, say α1 = 0.08, we notice that there will be some reduction over time in
the fraction of human infected with visceral leishmaniasis but not that high compared with the
case when the treatment rate is high(Figure 3(a)). The scenarios in the fraction of human
14
infected with PKDL, the fraction of infected reservoir, and the fraction of infected vectors are
the same as in the case of high treatment rate( Figures 3(b), 3(c) and 3(d)).
(c) When the treatment is high and effective: Effective treatment means that most of the VL
infected humans recover and do not develop PKDL after treatment, which is very rare to
happen because developing PKDL after VL treatment depends on immune system response
during drug treatment [16], however if we assume that the treatment is effective and it is in of
high rate, i.e α1 = 0.9, σ = 0.9, then we notice that the fraction of humans infected with
visceral leishmaniasis and the fraction of humans infected with PKDL are reaching very low
level, as in Figures 4(a) and 4(b), respectively.
(d) When we remove the reservoirs from the system: Studies [29] have proposed that the
maximum distance that a sandfly can fly searching for a blood-meal is 1 km, and hence if we
separate reservoirs from humans for a least 1 km the effect will be the same as if we remove the
reservoirs from the system. When we remove the reservoirs from the system , i.e NR = 0, and
treat humans at a high rate, α1 = 0.9, we notice that a big reduction over time in the fraction
of humans infected with visceral leishmaniasis, in the fraction of humans infected with PKDL,
and in the fraction of infected vectors as well as seen from (Figures 5(a), 5(b), and 5(c)),
respectively. This is because high treatment rate reducing the fraction of humans infected with
visceral leishmaniasis, and treating humans while removing reservoirs from the system reduce
the fraction of infected sandflies, which gives a good control strategy against the disease.
(5) When we control the vector: Vector control means reducing the biting rate. Vector control
can be achieved by several ways like using treated bed-nets [4], wearing clothes that cover
almost all of the body and house spraying [10]. When we simulate using small value for the
biting rate and high value for the treatment rate, i.e a = .08, α1 = 0.9, we notice that the
fraction of humans infected with visceral leishmaniasis approaches very small values after a
finite time(Figure 6(a)), also the same scenario happens for the fraction of humans infected
with PKDL(Figure 6(b)), and we notice, for the first time, a reduction in the fraction of
infected reservoirs(Figure 6(c)), and as expected a reduction in the fraction of infected
vectors(Figure 6(d)). This gives the best scenario for the control of VL as suggested by
literature [4, 10, 13, 31].
15
a
0.15
0.3
0.1
0.2
0.05
0
0.1
0
50
150
200
0.6
0.4
0.2
0
50
100
Time(days)
0
0
50
150
150
200
150
200
0.6
0.5
0.4
0.3
0.2
0.1
0
200
100
Time(days)
d
0.7
Fraction of infected vectors
Fraction of infected reservoirs
100
Time(days)
c
0.8
0
b
0.4
Fraction of PKDL infected humnas
Fraction of VL infected humnas
0.2
0
50
100
Time(days)
a
0.5
0.4
0.3
0.2
0.1
0
0
50
100
Time(days)
150
0.3
0.25
0.2
0.15
0.1
0.05
200
Fraction of infected vectors
0.6
0.4
0.2
0
50
100
Time(days)
0
0
50
150
150
200
150
200
0.6
0.4
0.2
0
200
100
Time(days)
d
0.8
0.8
0
b
0.35
c
1
Fraction of infected reservoirs
Fraction of PKDL infected humans
Fraction of VL infected humans
Figure 2: Simulation results for high-treatment rate, α1 = 0.9, R0 = 3.405
0
50
100
Time(days)
Figure 3: Simulation results for low-treatment, α1 = 0.08, R0 = 9.004
16
b
0.35
Fraction of PKDL infected humans
Fraction of VL infected humans
a
0.25
0.2
0.3
0.25
0.15
0.15
0.1
0.05
0
0
50
100
Time(days)
150
200
0
0
50
0.8
0.6
0.4
0.2
0
50
100
Time(days)
150
200
150
200
d
c
Fraction of infected vectors
Fraction of infected reservoirs
0.1
0.05
1
0
0.2
100
Time(days)
150
0.8
0.6
0.4
0.2
200
0
50
100
Time(days)
Figure 4: Simulation results for high and effective treatment, α1 = 0.9, σ = 0.9, R0 = 3.06
b
a
Fraction of PKDL infected humans
Fraction of VL infected humans
0.4
0.3
0.2
0.1
0
0
50
100
150
200
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
50
Time(days)
150
200
c
0.8
Fraction of infected vectors
100
Time(days)
0.6
0.4
0.2
0
0
50
100
Time(days)
150
200
Figure 5: Simulation results for reservoir control with high treatment, (NR = 0), α1 = 0.9, R0 =
2.341
17
b
0.1
0.05
0
0
50
100
Time(days)
150
200
Fraction of PKDL infected humans
Fraction of VL infected humans
a
0.15
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
50
200
150
200
0.7
Fraction of infected vectors
Fraction of infected reservoirs
150
d
c
1
0.8
0.6
0.4
0.2
0
100
Time(days)
0
50
100
150
200
Time(days)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
Time(days)
Figure 6: Simulation for vector control and high treatment rate, a = 0.01, α1 = 0.9, R0 = 1.018
18
5
Discussion
In this paper a mathematical analysis for the model of visceral leishmaniasis dynamics was done. It
is shown that it is possible to eradicate the disease from the community if R0 < 1 despite of the
number of infected humans, reservoirs and vectors, and if R0 > 1, then the disease will be an
endemic disease. Also it is shown numerically that human treatment is the key parameter in the
disease control among human population, but is not sufficient to eliminate the disease, even if it is
high and effective. To eradicate the disease from the community high rate of human treatment
should be followed by vector control strategies by either using treated bed-nets [4] or direct killing to
sandflies, and reservoir control strategies either by killing all infective reservoirs (which is not
applicable in many countries and especially in the Sudan because it needs to test all possible
reservoirs for leishmaniasis), or by separating all reservoir from humans for a safe distance.
Acknowledgments
The research of Ibrahim M. ELmojtaba is supported by a grant from Faculty of Mathematical
Sciences - University of Khartoum. The authors would like to thank Professor A.B. Gumel, of
University of Manitoba for his helpful comments. One of the authors (Ibrahim M. ELmojtaba)
would like to thank the Department of Mathematics at Makerere University for their hospitality, and
especial thanks to Ms. B. Nannyonga for her help.
19
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