Simple Dynamics In A Vector-borne Disease Model.
Volume 5 Number 0401
ISSN 1979-3898
Journal of
Theoretical and Computational
Studies
Simple Dynamics in a Vector-Borne Disease Model
A.K. Supriatna and E. Soewono
J. Theor. Comput. Stud. 5 (2008) 0401
Received: July 7th , 2008; Accepted for publication: September 8th , 2008
Published by
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c 2008 GFTI & MKI
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ISSN 1979-3898
J. Theor. Comput. Stud. Volume 5 (2008) 0401
Simple Dynamics in a Vector-Borne Disease Model
Asep K. Supriatna
Department of Mathematics, Universitas Padjadjaran, Jl. Raya Bandung-Sumedang Km 21, Jatinangor 45363,
Indonesia
Edy Soewono
Industrial and Financial Mathematics Group, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132,
Indonesia
Abstract : In this paper we review a simple model of an infectious disease transmission. In general the rate of
incidences can be model by mass action principle, so that its functional is bilinear. In some circumstances, disease
transmission might be more complicated involving different species, for example in the case of the transmission of the
disease required a vector (vector-borne disease), such as in malaria and dengue infection cases, the rate of incidences
takes a nonlinear functional form. In this paper we show the conditions needed for the endemic equilibrium in the model
to exist and to be stable. The analysis reveals that there is a simple transcritical bifurcation in the dynamics of the
model, despite the complex interaction of the disease transmission.
Keywords : Epidemic model, host-vector transmission, basic reproduction number, population dynamics
E-mail : aksupriatna@bdg.centrin.net.id
Received: July 7th , 2008; Accepted for publication: September 8th , 2008
1
INTRODUCTION
Despite recognized as an abstract science, mathematics has proved to be useful in helping to solve many
problems arising in daily life and problems from other
disciplines, such as industrial, environmental, and biological sciences (see [17, 16] and the reference therein
for examples). The inter-relations between mathematics and other disciplines not merely have given benefits to the disciplines served by mathematics, but in
many cases, there also fruitfulness to mathematics itself. There are some mathematical concepts and theories inspired from these inter-relations. Sometimes the
intimate connection between mathematics and other
discipline gives rise to a new discipline, such as mathematical bio-economics [5], mathematical conservation
theory [4] and mathematical epidemiology [1]. In this
paper we will review an application of mathematics in
controlling the transmission of an infectious disease.
The first documented work on the application of
mathematics in controlling an infectious disease dates
back to the 18th century when Daniel Bernoulli used
a mathematical method to evaluate the effectiveness
of the techniques of variolation against smallpox [2].
Among the aims of his work was to influence the pubc 2008 GFTI & MKI
⃝
lic health authority in deciding the effectiveness of
the infectious disease control at the time. He showed
that the techniques of inoculation practiced by the
society could increase the number of survivors per
year or increase the average life expectancy in an epidemic episode, if it is implemented universally to the
whole population (known as a universal inoculation
method). Current review on his work can be found in
[9].
Early scientists postulated that the course of an epidemic depends on the rate of contact between susceptible and infected individuals. Generally, they model
this phenomenon through the mass action principle.
Among them is Ross [15], a medical doctor who served
as a colonel in the British Army. He used this principle, identified the main factors in malaria transmission, calculated the number of new infection, and concluded that no need to eradicate all of the mosquitoes
to eradicate malaria, because there exists a critical
density of mosquito, below which the disease will vanish. This result is usually known as the mosquito theorem or the theory of critical level density. His work
is then generalized extensively by Kermack and McKendrick [11]. Current review on their work can be
found in [8, 10, 3].
0401-1
2
Simple Dynamics in a Vector-Borne...
A concept similar to the theory of critical level density is introduced by McDonald [14] who coined it as
the basic reproduction rate. In a modern notation
this concept is symbolized by R0 and is defined as
the expected number of secondary cases produced, in
a completely susceptible population, by a typical infected individual during its entire period of infectiousness [7, 6]. In the following sections we will discuss a
simple epidemic model and show that in some circumstances nonlinearity may often occur as a result of a
complex epidemiological phenomenon.
2
2.1
SIMPLE EPIDEMIC MODELS
A model without demographic parameter
A fairly simple epidemic model is an SIR model in
which we divide the population into three compartments, namely the susceptible (S), infected (I) and
recover (R) sub-populations. Here we assume that the
total population (N ) is constant with N = S + I + R.
If it is further assumed that the force of infection is β
and removal rate is γ then the dynamics of the disease
transmission is given by
dS
= −βIS,
dt
(1)
dI
= βIS − γI,
(2)
dt
dR
= γI,
(3)
dt
with initial conditions S(0) = S0 > 0, I(0) = I0 > 0,
and R(0) = 0.
Note from (2) that in the beginning of the epidemic,
there will be a build up of infection rises to an outbreak only if S0 > ρ = γ/β, otherwise the epidemic
will die out. Hence, S0 = ρ is a threshold density
of susceptible. Here γ/β is the relative removal rate.
This threshold can be reformulated as follows. The
condition S0 > ρ = γ/β is equivalent to βγ S0 > 1. In
this regard, R0 = βγ S0 is called the basic reproduction
number of the system. If the value of this basic reproduction number is more than one, then the number of
infected population increases, otherwise the number
of infected population decreases.
The general solution of the system can be found
dI
=
in the SI-plane by looking at the equation dS
βIS−γI
= −1 + ρ/S. Next, by considering N0 =
−βIS
S0 + I0 , the solution through (S0 , I0 ) is given by
I = N − S + ρ ln SS0 , having its maximum infection
Imax = N − ρ + ρ ln(ρ/S0 ) at S = ρ. Furthermore, it can be shown that I(∞) = 0 and S(∞) >
0 found as the smallest positive root of equation
N − S(∞) + ρ log [S (∞)/S0 ] = 0. This means that
eventually the disease will die out leaving a portion
of uninfected population, regardless the value of the
initial conditions (see Figure 1 for illustration).
The general solution of the system can also be found
dS
=
in the SR-plane by looking at the equation dR
−βIS
S
(−R/ρ)
=
−
which
gives
S
=
S
e
as
the
solution
0
γI
ρ
through (S0 , I0 ). If the initial infection is relatively
low, I0 ≈ 0 then considering the steady state solution
of (1)-(3), we have R ≈ 2ρ(1 − ρ/S0³). Furthermore,
if
´
ε
≈ 2ε. ConS0 = ρ + ε with ε → 0 then R ≈ 2ρ ρ+ε
sidering the intensity of the epidemic is measured by
R(∞) the expression R ≈ 2ε means that the epidemic
has succeeded in reducing the density of susceptible
from the initial condition ρ + ε to the final condition
ρ − ε. This is known as the Kermack-McKendrick
threshold theorem.
2.2
A model with demographic parameters
To increase realism, demographic parameters, such as
birth rate and death rate, are incorporated into the
model in the previous section. Suppose that the birth
rate is a constant B, and the death rate is proportional to the population density, with the constant of
proportionality µ. Hence, the system becomes
dS
= B − βIS − µS,
dt
(4)
dI
= βIS − γI − µI,
(5)
dt
dR
= γI − µR.
(6)
dt
Now, from (5), there will be a build up of infection rises to an outbreak only if S0 > ρµ = (γ +
µ)/β, otherwise the epidemic will die out. In either case, there are two equilibrium solutions possible
to
endemic-free equilibrium state (S0∗ , I0∗ ) =
´
³ occur,
B
and endemic equilibrium state (S1∗ , I1∗ ) =
´´
³
γ+µ 1
B
. However, the endemic equilibβ , β S1∗ − µ
´
³
rium state only appears when SB∗ − µ > 0 or equiv1
alently its basic reproduction number is more than
Bβ
one, that is R0 = µ(γ+µ)
> 1. This basic reproduction number is also a threshold for stability, in the
sense that the endemic equilibrium state exists and is
stable if R0 > 1 while the endemic-free equilibrium
is unstable, otherwise the endemic equilibrium state
does not exist while the endemic-free equilibrium state
is stable.
Note that the condition of R0 > 1 is a sufficient
condition for the endemic equilibrium to occur, while
the inclusion demographic parameter B is a necessary
condition. Meanwhile, the inclusion of demographic
³µ
0401-2
,0
3
Simple Dynamics in a Vector-Borne...
1
In(t)
1
0,8
In(t)
0,6
0,6
0,4
0,4
0,2
0,2
0
0
0
0,2
0,4
0,6
0,8
1
0
S(t)
parameter µ changes the value of the threshold density of susceptible. An illustration of the solutions for
various initial conditions can be seen in Figure 2.
Many diseases required a vector to spread. For example mosquitoes are responsible in dengue and malaria
transmission. In regards to the transmission of a
vector-borne disease, the previous models ignore the
presence of vectors in spreading the disease. In this
section we generalize the previous model with demographic parameters to include a vector in the model.
Let us assume SH (t) be the density of susceptible
human population, IH (t) be the density of infected
human population, RH (t) be the density of removed
and immune human population, SV (t) be the density of susceptible vector population, and IV (t) be the
density of infected vector population. The governing
equations for the transmission of the disease in the
presence of demographic parameters are
dIH
= βH IV SH − IH (γH + µH ) ,
dt
dRH
= γH IH − µH RH ,
dt
0,4
0,6
0,8
1
Figure 2: The trajectory of the SIR system in the SIplane with the inclusion of demographic parameters and
R0 > 1. It shows that the disease will be endemic eventually. The parameter values in the figure are the same
as in figure 1 with additions B = 0.08 and µ = 0.1. The
resulting basic reproduction number is R0 = 2 indicating
the endemicity of the disease.
A MODEL FOR VECTOR-BORNE DISEASE TRANSMISSION
dSH
= B − βH IV SH − µH SH ,
dt
0,2
S(t)
Figure 1: The trajectory of the SIR system without demographic parameter in the SI-plane. It shows that eventually the disease will die out regardless the value of the
initial conditions. The parameter values in the figure are
γ = 0.3 and β = 1 with various initial values of S and I.
3
0,8
dSV
= BV − βV IH SV − µV SV ,
dt
(10)
dIV
= βV IH SV − µV IV ,
dt
(11)
where, as before, we also assume SH + IH + RH = NH
and SV + IV = NV .
Following [12], we further assume that the vector
dynamics runs on a much faster time scale than the
human dynamics. Hence the vector population can be
considered to be at its equilibrium with the respect to
the changes in human population. Equilibrium solutions for the vector is given by SV∗ = βV IBHV+µV and
BV βV IH
. Substituting these values to the
IV∗ = µV (β
V IH +µV )
human dynamics yields approximation equations to
the original vector-borne disease transmission
(7)
βH BV βV IH SH
dSH
=B−
− µH SH ,
dt
µV (βV IH + µV )
(12)
dIH
βH BV βV IH SH
=
− IH (γH + µH ) ,
dt
µV (βV IH + µV )
(13)
(8)
dRH
= γH IH − µH RH .
dt
(9)
Compared to the direct transmission model, in which
the incidence rate is a bilinear function, here the inci-
0401-3
(14)
4
Simple Dynamics in a Vector-Borne...
1
In(t)
1
0,8
In(t)
0,8
0,6
0,6
0,4
0,4
0,2
0,2
0
0
0
0,2
0,4
0,6
0,8
1
0
0,2
0,4
0,6
S(t)
0,8
1
S(t)
Figure 3: The trajectory of the SIR system for the hostvector model in the SI-plane with various initial values of
S and I. The parameter values in the figure are γH = 0.3,
βH = 1, BH = 0.08, µH = 0.1, βV = 1, BV = 0.1, and
µV = 0.5. The basic reproduction number is R0 = 0.8
indicating the vanishing of the disease.
Figure 4: The trajectory of the SIR system for the hostvector model in the SI-plane with various initial values of
S and I, with the same parameters as in figure 3 except
a lower vector mortality rate, i.e. µV = 0.1. The resulting basic reproduction number is R0 = 20 indicating the
endemicity of the disease.
dence rate
Next, following [13], we derive lemmas relating the
basic reproduction number R0 with the properties of
the nonlinear incidence rate f (SH , IH ) at the equlibria
of (12)-(14).
Lemma 3.1: At the endemic-free equilibrium state
∗
∗
, IH0
), we have
p∗0 = (SH0
f (IH , SH ) =
βH BV βV IH SH
µV (βV IH + µV )
(15)
∗
∗
is nonlinear
in IH . It is clear that P0∗ = (SH0
, IH0
)=
´
³
B
µH , 0 is the endemic-free equilibrium solution. The
∗
∗
endemic equilibrium solution is given by p∗e (SHe
, IHe
)
in which
¡
¢
BβV µV + µ2H (γH + µH )
∗
SHe =
,
(16)
(BV βV βH + µH µV βV )
∗
IHe
= (R0 − 1) X ∗ ,
(17)
X∗ =
R0
µH µ2V (γH + µH )
(γH + µH ) (BV βV βH + µH µV βV )
¶
,
∗
∗
∂f (SH0
, IH0
)
≤ γH + µH ,
∂IH
R0 > 1 ⇒
∗
∗
∂f (SH0
, IH0
)
> γH + µH .
∂IH
Proof: In relation to the incidence rate in (15), the
basic reproduction number in (19) can be written as
with
µ
R0 ≤ 1 ⇒
(18)
=
=
BBV βV βH
µH µ2V (γH + µH )
¯
1
∂f (IH , SH ) ¯¯
.
¯
(γH + µH )
∂IH
(IH ,SH )=(I ∗ ,S ∗ )
0
BBV βV βH
R0 =
.
µH µ2V (γH + µH )
(19)
It is easy to see that the nonlinear incidence rate
BV βV IH SH
has the following properf (SH , IH ) = βµH
V (βV IH +µV )
ties:
P1: f (0, IH ) = f (SH , 0) = 0,
H ,IH )
H ,IH )
P2: ∂f (S
> 0 and ∂f (S
> 0,
∂SH
∂IH
P3:
∂ 2 f (SH ,IH )
2
∂IH
≤ 0.
0
Hence, the proof of the lemma is clear. ¤
Lemma 3.2: At the endemic equilibrium state p∗e =
∗
∗
(SHe
, IHe
), we have
R0 > 1 ⇒
Furthermore,
∗
∗
∂ 2 f (SHe
,IHe
)
2
∂IH
0401-4
∗
∗
∂f (SHe
, IHe
)
≤ γH + µH .
∂IH
∗
∗
∂f (SHe
,IHe
)
∂IH
= 0.
=
γH + µH only if
5
Simple Dynamics in a Vector-Borne...
Proof: We observe, from (13), that the en∗
∗
demic equilibrium state satisfies f (SHe
, IHe
) =
∗
∗
˜
(γH + µH ) IHe . Let us define f (IH ) := f (SHe , IH ).
∗
∗
∂f (SHe
,IHe
)
=
Suppose that, in the contrary, we have
∂IH
∗
df˜(IHe
)
dIH
> γH + µH . By considering P1, the mean
value theorem guarantees that there is a point IH1 ∈
∗
(0, IHe
) such that
df˜(IH1 )
dIH
=
=
∗
f˜(IHe
) − f˜(0)
∗
IHe − 0
∗
(γH + µH ) IHe
−0
= γH + µH .
∗
IHe
Stability of the endemic equilibrium state is clear
from the following theorem.
Theorem 2: If R0 > 1 then the endemic equilib∗
∗
rium Pe∗ = (SHe
, IHe
) of the system (12)-(14) exists
and is locally asymptotically stable.
Proof: It is obvious by (17) the endemic equilibrium
exists only if R0 > 1. The stability of the endemic
equilibrium state is investigated by observing the Jacobian matrix M of the system (12)-(14). At this
equilibrium state, we have the characteristic equation
λ2 + a1 λ + a2 = 0, with
a1
df˜(I ∗ )
Moreover, since we have assumed dIHHe > γH + µH ,
then again by the mean value theorem there exist a
∗
point IH0 ∈ (IH1 , IHe
) such that
∂
2
∗
f (SHe
, IH0 )
2
∂IH
a2
d f˜(IH0 )
2
dIH
=
2
=
=
=
=
∗
˜(IH1 )
df˜(IHe
)
− dfdI
dIH
H
∗ −I
IHe
H1
∗
df˜(IHe
)
− (γH + µH )
dIH
∗ −I
IHe
H1
This contradicts P3, and hence
∗
∗
∂f (SHe
,IHe
)
Furthermore,
=
∂IH
∗
df˜(IHe )
df˜(IH1 )
= dIH for all IH ∈
dIH
∗
∗
∂f (SHe
,IHe
)
∂IH
det(M )
µ
¶
∂f
(γH + µH )
∂SH
µ
¶
∂f
+µH (γH + µH ) −
.
∂IH
From Lemma 3.2 and P2 we conclude a1 > 0 and
a2 > 0 which confirms that the endemic equilibrium
state is asymptotically stable. ¤
> 0.
≤ γH +µH .
γH + µH holds only if
∗
∗
(0, IHe
) and SH = SHe
.
¤
In the following section we will investigate the stability of these equilibria in relation to the basic reproduction number R0 .
3.1
= −trace(M )
¶
¶ µ
µ
∂f
∂f
,
+ µH + (γH + µH ) −
=
∂SH
∂IH
Stability of the equilibrium states
ACKNOWLEDGMENTS
Financial support was provided by The Royal Netherlands Academy of Arts and Sciences (KNAW). Earlier
version of the paper was presented in the Workshop of
Nonlinear Phenomena, Bandung 11 December 2007.
JTCS
REFERENCES
The stability results are summarized in the following
theorems.
Theorem 1:If R0 < 1 then the endemic-free equi∗
∗
librium p∗0 = (SH0
, IH0
) of the system (12)-(14) is
locally asymptotically stable, otherwise it is unstable.
Proof: The stability of the endemic-free equilibrium
is easily identified through the investigation of the Jacobian matrix M of the system (12)-(14). At the
endemic-free equilibrium, we have the characteristic
equation λ2 + a1 λ + a2 = 0, with
a1 = −trace(M ) = µH + (γH + µH ) (1 − R0 ) ,
a2 = det(M ) = µH (γH + µH ) (1 − R0 ) .
This means that the endemic-free equilibrium is stable
if R0 < 1 and is unstable if R0 > 1. The endemicfree equilibrium undergoes a transcritical bifurcation
at R0 = 1. ¤
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Simple Dynamics in a Vector-Borne...
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Presented at Workshop on Nonlinear Phenomena 2K7,
Sumedang, Indonesia, December 11th , 2007.
0401-6
ISSN 1979-3898
Journal of
Theoretical and Computational
Studies
Simple Dynamics in a Vector-Borne Disease Model
A.K. Supriatna and E. Soewono
J. Theor. Comput. Stud. 5 (2008) 0401
Received: July 7th , 2008; Accepted for publication: September 8th , 2008
Published by
Indonesian Theoretical Physicist Group
http://www.opi.lipi.go.id/situs/gfti/
Indonesian Computational Society
http://www.opi.lipi.go.id/situs/mki/
Journal of Theoretical and Computational Studies
Journal devoted to theoretical study, computational science and its cross-disciplinary studies
URL : http://www.jurnal.lipi.go.id/situs/jtcs/
Editors
A. Purwanto (ITS)
A. S. Nugroho (BPPT)
A. Sopaheluwakan (LabMath)
A. Sulaksono (UI)
B. E. Gunara (ITB)
B. Tambunan (BPPT)
F.P. Zen (ITB)
H. Alatas (IPB)
I.A. Dharmawan (UNPAD)
I. Fachrudin (UI)
Honorary Editors
B.S. Brotosiswojo (ITB)
M. Barmawi (ITB)
M.S. Ardisasmita (BATAN)
Guest Editors
H. Zainuddin (UPM)
T. Morozumi (Hiroshima)
J.M. Tuwankotta (ITB)
L.T. Handoko (LIPI)
M. Nurhuda (UNIBRAW)
M. Satriawan (UGM)
P. Nurwantoro (UGM)
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Indonesian Theoretical Physicist Group
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c 2008 GFTI & MKI
⃝
ISSN 1979-3898
J. Theor. Comput. Stud. Volume 5 (2008) 0401
Simple Dynamics in a Vector-Borne Disease Model
Asep K. Supriatna
Department of Mathematics, Universitas Padjadjaran, Jl. Raya Bandung-Sumedang Km 21, Jatinangor 45363,
Indonesia
Edy Soewono
Industrial and Financial Mathematics Group, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132,
Indonesia
Abstract : In this paper we review a simple model of an infectious disease transmission. In general the rate of
incidences can be model by mass action principle, so that its functional is bilinear. In some circumstances, disease
transmission might be more complicated involving different species, for example in the case of the transmission of the
disease required a vector (vector-borne disease), such as in malaria and dengue infection cases, the rate of incidences
takes a nonlinear functional form. In this paper we show the conditions needed for the endemic equilibrium in the model
to exist and to be stable. The analysis reveals that there is a simple transcritical bifurcation in the dynamics of the
model, despite the complex interaction of the disease transmission.
Keywords : Epidemic model, host-vector transmission, basic reproduction number, population dynamics
E-mail : aksupriatna@bdg.centrin.net.id
Received: July 7th , 2008; Accepted for publication: September 8th , 2008
1
INTRODUCTION
Despite recognized as an abstract science, mathematics has proved to be useful in helping to solve many
problems arising in daily life and problems from other
disciplines, such as industrial, environmental, and biological sciences (see [17, 16] and the reference therein
for examples). The inter-relations between mathematics and other disciplines not merely have given benefits to the disciplines served by mathematics, but in
many cases, there also fruitfulness to mathematics itself. There are some mathematical concepts and theories inspired from these inter-relations. Sometimes the
intimate connection between mathematics and other
discipline gives rise to a new discipline, such as mathematical bio-economics [5], mathematical conservation
theory [4] and mathematical epidemiology [1]. In this
paper we will review an application of mathematics in
controlling the transmission of an infectious disease.
The first documented work on the application of
mathematics in controlling an infectious disease dates
back to the 18th century when Daniel Bernoulli used
a mathematical method to evaluate the effectiveness
of the techniques of variolation against smallpox [2].
Among the aims of his work was to influence the pubc 2008 GFTI & MKI
⃝
lic health authority in deciding the effectiveness of
the infectious disease control at the time. He showed
that the techniques of inoculation practiced by the
society could increase the number of survivors per
year or increase the average life expectancy in an epidemic episode, if it is implemented universally to the
whole population (known as a universal inoculation
method). Current review on his work can be found in
[9].
Early scientists postulated that the course of an epidemic depends on the rate of contact between susceptible and infected individuals. Generally, they model
this phenomenon through the mass action principle.
Among them is Ross [15], a medical doctor who served
as a colonel in the British Army. He used this principle, identified the main factors in malaria transmission, calculated the number of new infection, and concluded that no need to eradicate all of the mosquitoes
to eradicate malaria, because there exists a critical
density of mosquito, below which the disease will vanish. This result is usually known as the mosquito theorem or the theory of critical level density. His work
is then generalized extensively by Kermack and McKendrick [11]. Current review on their work can be
found in [8, 10, 3].
0401-1
2
Simple Dynamics in a Vector-Borne...
A concept similar to the theory of critical level density is introduced by McDonald [14] who coined it as
the basic reproduction rate. In a modern notation
this concept is symbolized by R0 and is defined as
the expected number of secondary cases produced, in
a completely susceptible population, by a typical infected individual during its entire period of infectiousness [7, 6]. In the following sections we will discuss a
simple epidemic model and show that in some circumstances nonlinearity may often occur as a result of a
complex epidemiological phenomenon.
2
2.1
SIMPLE EPIDEMIC MODELS
A model without demographic parameter
A fairly simple epidemic model is an SIR model in
which we divide the population into three compartments, namely the susceptible (S), infected (I) and
recover (R) sub-populations. Here we assume that the
total population (N ) is constant with N = S + I + R.
If it is further assumed that the force of infection is β
and removal rate is γ then the dynamics of the disease
transmission is given by
dS
= −βIS,
dt
(1)
dI
= βIS − γI,
(2)
dt
dR
= γI,
(3)
dt
with initial conditions S(0) = S0 > 0, I(0) = I0 > 0,
and R(0) = 0.
Note from (2) that in the beginning of the epidemic,
there will be a build up of infection rises to an outbreak only if S0 > ρ = γ/β, otherwise the epidemic
will die out. Hence, S0 = ρ is a threshold density
of susceptible. Here γ/β is the relative removal rate.
This threshold can be reformulated as follows. The
condition S0 > ρ = γ/β is equivalent to βγ S0 > 1. In
this regard, R0 = βγ S0 is called the basic reproduction
number of the system. If the value of this basic reproduction number is more than one, then the number of
infected population increases, otherwise the number
of infected population decreases.
The general solution of the system can be found
dI
=
in the SI-plane by looking at the equation dS
βIS−γI
= −1 + ρ/S. Next, by considering N0 =
−βIS
S0 + I0 , the solution through (S0 , I0 ) is given by
I = N − S + ρ ln SS0 , having its maximum infection
Imax = N − ρ + ρ ln(ρ/S0 ) at S = ρ. Furthermore, it can be shown that I(∞) = 0 and S(∞) >
0 found as the smallest positive root of equation
N − S(∞) + ρ log [S (∞)/S0 ] = 0. This means that
eventually the disease will die out leaving a portion
of uninfected population, regardless the value of the
initial conditions (see Figure 1 for illustration).
The general solution of the system can also be found
dS
=
in the SR-plane by looking at the equation dR
−βIS
S
(−R/ρ)
=
−
which
gives
S
=
S
e
as
the
solution
0
γI
ρ
through (S0 , I0 ). If the initial infection is relatively
low, I0 ≈ 0 then considering the steady state solution
of (1)-(3), we have R ≈ 2ρ(1 − ρ/S0³). Furthermore,
if
´
ε
≈ 2ε. ConS0 = ρ + ε with ε → 0 then R ≈ 2ρ ρ+ε
sidering the intensity of the epidemic is measured by
R(∞) the expression R ≈ 2ε means that the epidemic
has succeeded in reducing the density of susceptible
from the initial condition ρ + ε to the final condition
ρ − ε. This is known as the Kermack-McKendrick
threshold theorem.
2.2
A model with demographic parameters
To increase realism, demographic parameters, such as
birth rate and death rate, are incorporated into the
model in the previous section. Suppose that the birth
rate is a constant B, and the death rate is proportional to the population density, with the constant of
proportionality µ. Hence, the system becomes
dS
= B − βIS − µS,
dt
(4)
dI
= βIS − γI − µI,
(5)
dt
dR
= γI − µR.
(6)
dt
Now, from (5), there will be a build up of infection rises to an outbreak only if S0 > ρµ = (γ +
µ)/β, otherwise the epidemic will die out. In either case, there are two equilibrium solutions possible
to
endemic-free equilibrium state (S0∗ , I0∗ ) =
´
³ occur,
B
and endemic equilibrium state (S1∗ , I1∗ ) =
´´
³
γ+µ 1
B
. However, the endemic equilibβ , β S1∗ − µ
´
³
rium state only appears when SB∗ − µ > 0 or equiv1
alently its basic reproduction number is more than
Bβ
one, that is R0 = µ(γ+µ)
> 1. This basic reproduction number is also a threshold for stability, in the
sense that the endemic equilibrium state exists and is
stable if R0 > 1 while the endemic-free equilibrium
is unstable, otherwise the endemic equilibrium state
does not exist while the endemic-free equilibrium state
is stable.
Note that the condition of R0 > 1 is a sufficient
condition for the endemic equilibrium to occur, while
the inclusion demographic parameter B is a necessary
condition. Meanwhile, the inclusion of demographic
³µ
0401-2
,0
3
Simple Dynamics in a Vector-Borne...
1
In(t)
1
0,8
In(t)
0,6
0,6
0,4
0,4
0,2
0,2
0
0
0
0,2
0,4
0,6
0,8
1
0
S(t)
parameter µ changes the value of the threshold density of susceptible. An illustration of the solutions for
various initial conditions can be seen in Figure 2.
Many diseases required a vector to spread. For example mosquitoes are responsible in dengue and malaria
transmission. In regards to the transmission of a
vector-borne disease, the previous models ignore the
presence of vectors in spreading the disease. In this
section we generalize the previous model with demographic parameters to include a vector in the model.
Let us assume SH (t) be the density of susceptible
human population, IH (t) be the density of infected
human population, RH (t) be the density of removed
and immune human population, SV (t) be the density of susceptible vector population, and IV (t) be the
density of infected vector population. The governing
equations for the transmission of the disease in the
presence of demographic parameters are
dIH
= βH IV SH − IH (γH + µH ) ,
dt
dRH
= γH IH − µH RH ,
dt
0,4
0,6
0,8
1
Figure 2: The trajectory of the SIR system in the SIplane with the inclusion of demographic parameters and
R0 > 1. It shows that the disease will be endemic eventually. The parameter values in the figure are the same
as in figure 1 with additions B = 0.08 and µ = 0.1. The
resulting basic reproduction number is R0 = 2 indicating
the endemicity of the disease.
A MODEL FOR VECTOR-BORNE DISEASE TRANSMISSION
dSH
= B − βH IV SH − µH SH ,
dt
0,2
S(t)
Figure 1: The trajectory of the SIR system without demographic parameter in the SI-plane. It shows that eventually the disease will die out regardless the value of the
initial conditions. The parameter values in the figure are
γ = 0.3 and β = 1 with various initial values of S and I.
3
0,8
dSV
= BV − βV IH SV − µV SV ,
dt
(10)
dIV
= βV IH SV − µV IV ,
dt
(11)
where, as before, we also assume SH + IH + RH = NH
and SV + IV = NV .
Following [12], we further assume that the vector
dynamics runs on a much faster time scale than the
human dynamics. Hence the vector population can be
considered to be at its equilibrium with the respect to
the changes in human population. Equilibrium solutions for the vector is given by SV∗ = βV IBHV+µV and
BV βV IH
. Substituting these values to the
IV∗ = µV (β
V IH +µV )
human dynamics yields approximation equations to
the original vector-borne disease transmission
(7)
βH BV βV IH SH
dSH
=B−
− µH SH ,
dt
µV (βV IH + µV )
(12)
dIH
βH BV βV IH SH
=
− IH (γH + µH ) ,
dt
µV (βV IH + µV )
(13)
(8)
dRH
= γH IH − µH RH .
dt
(9)
Compared to the direct transmission model, in which
the incidence rate is a bilinear function, here the inci-
0401-3
(14)
4
Simple Dynamics in a Vector-Borne...
1
In(t)
1
0,8
In(t)
0,8
0,6
0,6
0,4
0,4
0,2
0,2
0
0
0
0,2
0,4
0,6
0,8
1
0
0,2
0,4
0,6
S(t)
0,8
1
S(t)
Figure 3: The trajectory of the SIR system for the hostvector model in the SI-plane with various initial values of
S and I. The parameter values in the figure are γH = 0.3,
βH = 1, BH = 0.08, µH = 0.1, βV = 1, BV = 0.1, and
µV = 0.5. The basic reproduction number is R0 = 0.8
indicating the vanishing of the disease.
Figure 4: The trajectory of the SIR system for the hostvector model in the SI-plane with various initial values of
S and I, with the same parameters as in figure 3 except
a lower vector mortality rate, i.e. µV = 0.1. The resulting basic reproduction number is R0 = 20 indicating the
endemicity of the disease.
dence rate
Next, following [13], we derive lemmas relating the
basic reproduction number R0 with the properties of
the nonlinear incidence rate f (SH , IH ) at the equlibria
of (12)-(14).
Lemma 3.1: At the endemic-free equilibrium state
∗
∗
, IH0
), we have
p∗0 = (SH0
f (IH , SH ) =
βH BV βV IH SH
µV (βV IH + µV )
(15)
∗
∗
is nonlinear
in IH . It is clear that P0∗ = (SH0
, IH0
)=
´
³
B
µH , 0 is the endemic-free equilibrium solution. The
∗
∗
endemic equilibrium solution is given by p∗e (SHe
, IHe
)
in which
¡
¢
BβV µV + µ2H (γH + µH )
∗
SHe =
,
(16)
(BV βV βH + µH µV βV )
∗
IHe
= (R0 − 1) X ∗ ,
(17)
X∗ =
R0
µH µ2V (γH + µH )
(γH + µH ) (BV βV βH + µH µV βV )
¶
,
∗
∗
∂f (SH0
, IH0
)
≤ γH + µH ,
∂IH
R0 > 1 ⇒
∗
∗
∂f (SH0
, IH0
)
> γH + µH .
∂IH
Proof: In relation to the incidence rate in (15), the
basic reproduction number in (19) can be written as
with
µ
R0 ≤ 1 ⇒
(18)
=
=
BBV βV βH
µH µ2V (γH + µH )
¯
1
∂f (IH , SH ) ¯¯
.
¯
(γH + µH )
∂IH
(IH ,SH )=(I ∗ ,S ∗ )
0
BBV βV βH
R0 =
.
µH µ2V (γH + µH )
(19)
It is easy to see that the nonlinear incidence rate
BV βV IH SH
has the following properf (SH , IH ) = βµH
V (βV IH +µV )
ties:
P1: f (0, IH ) = f (SH , 0) = 0,
H ,IH )
H ,IH )
P2: ∂f (S
> 0 and ∂f (S
> 0,
∂SH
∂IH
P3:
∂ 2 f (SH ,IH )
2
∂IH
≤ 0.
0
Hence, the proof of the lemma is clear. ¤
Lemma 3.2: At the endemic equilibrium state p∗e =
∗
∗
(SHe
, IHe
), we have
R0 > 1 ⇒
Furthermore,
∗
∗
∂ 2 f (SHe
,IHe
)
2
∂IH
0401-4
∗
∗
∂f (SHe
, IHe
)
≤ γH + µH .
∂IH
∗
∗
∂f (SHe
,IHe
)
∂IH
= 0.
=
γH + µH only if
5
Simple Dynamics in a Vector-Borne...
Proof: We observe, from (13), that the en∗
∗
demic equilibrium state satisfies f (SHe
, IHe
) =
∗
∗
˜
(γH + µH ) IHe . Let us define f (IH ) := f (SHe , IH ).
∗
∗
∂f (SHe
,IHe
)
=
Suppose that, in the contrary, we have
∂IH
∗
df˜(IHe
)
dIH
> γH + µH . By considering P1, the mean
value theorem guarantees that there is a point IH1 ∈
∗
(0, IHe
) such that
df˜(IH1 )
dIH
=
=
∗
f˜(IHe
) − f˜(0)
∗
IHe − 0
∗
(γH + µH ) IHe
−0
= γH + µH .
∗
IHe
Stability of the endemic equilibrium state is clear
from the following theorem.
Theorem 2: If R0 > 1 then the endemic equilib∗
∗
rium Pe∗ = (SHe
, IHe
) of the system (12)-(14) exists
and is locally asymptotically stable.
Proof: It is obvious by (17) the endemic equilibrium
exists only if R0 > 1. The stability of the endemic
equilibrium state is investigated by observing the Jacobian matrix M of the system (12)-(14). At this
equilibrium state, we have the characteristic equation
λ2 + a1 λ + a2 = 0, with
a1
df˜(I ∗ )
Moreover, since we have assumed dIHHe > γH + µH ,
then again by the mean value theorem there exist a
∗
point IH0 ∈ (IH1 , IHe
) such that
∂
2
∗
f (SHe
, IH0 )
2
∂IH
a2
d f˜(IH0 )
2
dIH
=
2
=
=
=
=
∗
˜(IH1 )
df˜(IHe
)
− dfdI
dIH
H
∗ −I
IHe
H1
∗
df˜(IHe
)
− (γH + µH )
dIH
∗ −I
IHe
H1
This contradicts P3, and hence
∗
∗
∂f (SHe
,IHe
)
Furthermore,
=
∂IH
∗
df˜(IHe )
df˜(IH1 )
= dIH for all IH ∈
dIH
∗
∗
∂f (SHe
,IHe
)
∂IH
det(M )
µ
¶
∂f
(γH + µH )
∂SH
µ
¶
∂f
+µH (γH + µH ) −
.
∂IH
From Lemma 3.2 and P2 we conclude a1 > 0 and
a2 > 0 which confirms that the endemic equilibrium
state is asymptotically stable. ¤
> 0.
≤ γH +µH .
γH + µH holds only if
∗
∗
(0, IHe
) and SH = SHe
.
¤
In the following section we will investigate the stability of these equilibria in relation to the basic reproduction number R0 .
3.1
= −trace(M )
¶
¶ µ
µ
∂f
∂f
,
+ µH + (γH + µH ) −
=
∂SH
∂IH
Stability of the equilibrium states
ACKNOWLEDGMENTS
Financial support was provided by The Royal Netherlands Academy of Arts and Sciences (KNAW). Earlier
version of the paper was presented in the Workshop of
Nonlinear Phenomena, Bandung 11 December 2007.
JTCS
REFERENCES
The stability results are summarized in the following
theorems.
Theorem 1:If R0 < 1 then the endemic-free equi∗
∗
librium p∗0 = (SH0
, IH0
) of the system (12)-(14) is
locally asymptotically stable, otherwise it is unstable.
Proof: The stability of the endemic-free equilibrium
is easily identified through the investigation of the Jacobian matrix M of the system (12)-(14). At the
endemic-free equilibrium, we have the characteristic
equation λ2 + a1 λ + a2 = 0, with
a1 = −trace(M ) = µH + (γH + µH ) (1 − R0 ) ,
a2 = det(M ) = µH (γH + µH ) (1 − R0 ) .
This means that the endemic-free equilibrium is stable
if R0 < 1 and is unstable if R0 > 1. The endemicfree equilibrium undergoes a transcritical bifurcation
at R0 = 1. ¤
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Simple Dynamics in a Vector-Borne...
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Presented at Workshop on Nonlinear Phenomena 2K7,
Sumedang, Indonesia, December 11th , 2007.
0401-6