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. SoewonoJ. Theor. Comput. Stud. 5 (2008) 0401
Received: July 7th, 2008; Accepted for publication: September 8th, 2008
Published by
Indonesian Theoretical Physicist Group Indonesian Computational Society
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) J.M. Tuwankotta (ITB)
A. S. Nugroho (BPPT) L.T. Handoko (LIPI)
A. Sopaheluwakan (LabMath) M. Nurhuda (UNIBRAW)
A. Sulaksono (UI) M. Satriawan (UGM)
B. E. Gunara (ITB) P. Nurwantoro (UGM)
B. Tambunan (BPPT) P. W. Premadi (ITB)
F.P. Zen (ITB) R.K. Lestari (ITB)
H. Alatas (IPB) T. Mart (UI)
I.A. Dharmawan (UNPAD) Y. Susilowati (LIPI)
I. Fachrudin (UI) Z. Su’ud (ITB)
Honorary Editors
B.S. Brotosiswojo (ITB) M.O. Tjia (ITB)
M. Barmawi (ITB) P. Anggraita (BATAN)
M.S. Ardisasmita (BATAN) T.H. Liong (ITB)
Guest Editors
H. Zainuddin (UPM) K. Yamamoto (Hiroshima)
T. Morozumi (Hiroshima)
Coverage area
1. Theoretical study: employing mathematical models and abstractions of a particular field in an attempt to explain known or predicted phenomenon. E.g. : theoretical physics, mathematical physics, biomatter modeling, etc.
2. Computational science: constructing mathematical models, numerical solution techniques and using computers to analyze and solve natural science, social science and engineering problems. E.g. : numerical simulations, model fitting and data analysis, optimization, etc.
3. Cross-disciplinary studies: inter-disciplinary studies between theoretical study and computational science, including the development of computing tools and apparatus. E.g. : macro development of Matlab, paralel processing, grid infrastructure, etc.
Types of paper
1. Regular: an article contains an original work.
2. Comment: an article responding to another one which has been published before.
3. Review: an article which is a compilation of recent knowledges in a particular topic. This type of article is only by invitation. 4. Proceedings: an article which has been presented in a scientific meeting.
Paper Submission
The submitted paper should be written in English using the LATEX template provided in the web. All communication thereafter should be done only through the online submission page of each paper.
Referees
All submitted papers are subject to a refereeing process by an appropriate referee. The editor has an absolute right to make the final decision on the paper.
Reprints
Electronic reprints including covers for each article and content pages of the volume are available from the journal site for free.
Indonesian Theoretical Physicist Group Indonesian Computational Society
Secretariat Office : c/o Group for Theoretical and Computational Physics, Research Center for Physics -LIPI, Kompleks Puspiptek Serpong, Tangerang 15310, Indonesia
http://www.opi.lipi.go.id/situs/gfti/ http://www.opi.lipi.go.id/situs/mki/
c
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, mathemat-ics 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 bi-ological sciences (see [17, 16] and the reference therein for examples). The inter-relations between mathemat-ics and other disciplines not merely have given bene-fits to the disciplines served by mathematics, but in many cases, there also fruitfulness to mathematics it-self. There are some mathematical concepts and theo-ries inspired from these inter-relations. Sometimes the intimate connection between mathematics and other discipline gives rise to a new discipline, such as mathe-matical bio-economics [5], mathemathe-matical 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
pub-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 epi-demic 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 epi-demic depends on the rate of contact between suscep-tible 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 prin-ciple, identified the main factors in malaria transmis-sion, calculated the number of new infection, and con-cluded 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 van-ish. This result is usually known as the mosquito the-orem or the theory of critical level density. His work is then generalized extensively by Kermack and McK-endrick [11]. Current review on their work can be found in [8, 10, 3].
c
2 Simple Dynamics in a Vector-Borne...
A concept similar to the theory of critical level den-sity 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 in-fected individual during its entire period of infectious-ness [7, 6]. In the following sections we will discuss a simple epidemic model and show that in some circum-stances nonlinearity may often occur as a result of a complex epidemiological phenomenon.
2 SIMPLE EPIDEMIC MODELS
2.1 A model without demographic parameter
A fairly simple epidemic model is an SIR model in which we divide the population into three compart-ments, namely the susceptible (S), infected (I) and recover (R) sub-populations. Here we assume that the total population (N) is constant withN =S+I+R. If it is further assumed that the force of infection isβ
and removal rate isγthen the dynamics of the disease transmission is given by
dS
Note from (2) that in the beginning of the epidemic, there will be a build up of infection rises to an out-break only ifS0 > ρ = γ/β, otherwise the epidemic
will die out. Hence, S0 = ρ is a threshold density
of susceptible. Hereγ/β is the relative removal rate. This threshold can be reformulated as follows. The conditionS0> ρ=γ/β is equivalent to βγS0 >1. In
this regard,R0= βγS0is called the basic reproduction
number of the system. If the value of this basic repro-duction 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 in the SI-plane by looking at the equation dI
dS = βIS−γI
−βIS = −1 + ρ/S. Next, by considering N0 =
S0 +I0 , the solution through (S0, I0) is given by
I = N −S+ρln S
S0, having its maximum infection
Imax = N −ρ+ρln(ρ/S0) at S = ρ.
Further-more, it can be shown that I(∞) = 0 and S(∞) >
0 found as the smallest positive root of equation
N −S(∞) +ρlog [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 in the SR-plane by looking at the equation dS
dR =
through (S0, I0). If the initial infection is relatively
low,I0≈0 then considering the steady state solution
of (1)-(3), we haveR≈2ρ(1−ρ/S0). Furthermore, if sidering the intensity of the epidemic is measured by
R(∞) the expressionR≈2εmeans that the epidemic has succeeded in reducing the density of susceptible from the initial conditionρ+εto 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 propor-tional to the population density, with the constant of proportionalityµ. Hence, the system becomes
dS
Now, from (5), there will be a build up of infec-tion rises to an outbreak only if S0 > ρµ = (γ +
µ)/β, otherwise the epidemic will die out. In ei-ther case, ei-there are two equilibrium solutions possible to occur, endemic-free equilibrium state (S∗
0, I0∗) =
³
B µ,0
´
and endemic equilibrium state (S∗
1, I1∗) =
. However, the endemic
equilib-rium state only appears when³SB∗
1 −µ
´
>0 or equiv-alently its basic reproduction number is more than one, that is R0 = µ(Bβγ+µ) > 1. This basic
reproduc-tion 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 parameterBis a necessary condition. Meanwhile, the inclusion of demographic
3 Simple Dynamics in a Vector-Borne...
0,2 0
In(t)
S(t) 1
1 0,8
0,6
0,8 0,4
0,2
0,6 0
0,4
Figure1: The trajectory of theSIRsystem without de-mographic parameter in theSI-plane. It shows that even-tually the disease will die out regardless the value of the initial conditions. The parameter values in the figure are
γ= 0.3 andβ= 1 with various initial values ofS andI.
parameter µ changes the value of the threshold den-sity of susceptible. An illustration of the solutions for various initial conditions can be seen in Figure 2.
3 A MODEL FOR VECTOR-BORNE DIS-EASE TRANSMISSION
Many diseases required a vector to spread. For exam-ple 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 demo-graphic 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
den-sity of susceptible vector population, andIV(t) be the
density of infected vector population. The governing equations for the transmission of the disease in the presence of demographic parameters are
dSH
dt =B−βHIVSH−µHSH, (7) dIH
dt =βHIVSH−IH(γH+µH), (8) dRH
dt =γHIH−µHRH, (9)
0,2 0
In(t)
S(t) 1
1 0,8
0,6
0,8 0,4
0,2
0,6 0
0,4
Figure 2: The trajectory of the SIRsystem in the SI -plane with the inclusion of demographic parameters and
R0 >1. It shows that the disease will be endemic
even-tually. The parameter values in the figure are the same as in figure 1 with additionsB = 0.08 and µ= 0.1. The resulting basic reproduction number isR0= 2 indicating
the endemicity of the disease.
dSV
dt =BV −βVIHSV −µVSV, (10) dIV
dt =βVIHSV −µVIV, (11)
where, as before, we also assumeSH+IH+RH=NH
andSV +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 solu-tions for the vector is given by S∗
V = B
V
βVIH+µV and
I∗
V =
BVβVIH
µV(βVIH+µV). Substituting these values to the human dynamics yields approximation equations to the original vector-borne disease transmission
dSH
dt =B−
βHBVβVIHSH
µV (βVIH+µV)
−µHSH, (12)
dIH
dt =
βHBVβVIHSH
µV (βVIH+µV)
−IH(γH+µH), (13)
dRH
dt =γHIH−µHRH. (14)
Compared to the direct transmission model, in which the incidence rate is a bilinear function, here the
4 Simple Dynamics in a Vector-Borne...
Figure3: The trajectory of theSIRsystem for the host-vector model in theSI-plane with various initial values of
SandI. The parameter values in the figure areγH= 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.
dence rate
is the endemic-free equilibrium solution. The endemic equilibrium solution is given byp∗
e(SHe∗ , IHe∗ ) It is easy to see that the nonlinear incidence rate
f(SH, IH) = µβHVB(βVVβIVHI+HµSVH) has the following
proper-Figure4: The trajectory of theSIRsystem for the host-vector model in theSI-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
result-ing basic reproduction number is R0 = 20 indicating the
endemicity of the disease.
Next, following [13], we derive lemmas relating the basic reproduction number R0 with the properties of
the nonlinear incidence ratef(SH, IH) at the equlibria
of (12)-(14).
Lemma 3.1: At the endemic-free equilibrium state
p∗
Proof: In relation to the incidence rate in (15), the basic reproduction number in (19) can be written as
R0 = BBVβVβH
Hence, the proof of the lemma is clear. ¤
Lemma 3.2: At the endemic equilibrium statep∗
5 Simple Dynamics in a Vector-Borne...
Proof: We observe, from (13), that the en-demic equilibrium state satisfies f(S∗
He, IHe∗ ) = value theorem guarantees that there is a pointIH1∈
(0, I∗ then again by the mean value theorem there exist a pointIH0∈(IH1, IHe∗ ) such that
In the following section we will investigate the sta-bility of these equilibria in relation to the basic repro-duction numberR0.
3.1 Stability of the equilibrium states
The stability results are summarized in the following theorems.
Theorem 1:If R0 <1 then the endemic-free equi-librium p∗
0 = (SH∗0, IH∗0) 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 Ja-cobian matrix M of the system (12)-(14). At the endemic-free equilibrium, we have the characteristic equationλ2+a
1λ+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
endemic-free equilibrium undergoes a transcritical bifurcation atR0= 1. ¤
Stability of the endemic equilibrium state is clear from the following theorem.
Theorem 2: If R0 >1 then the endemic equilib-rium P∗
e = (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 Ja-cobian matrix M of the system (12)-(14). At this equilibrium state, we have the characteristic equation
λ2+a
a2 >0 which confirms that the endemic equilibrium
state is asymptotically stable. ¤
ACKNOWLEDGMENTS
Financial support was provided by The Royal Nether-lands 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
[1] R.M. Anderson and R.M. May, Infectious Dis-eases of Human: Dynamics and Control, Oxford University Press (1991).
[2] S. Blower,Rev. Med. Virol., 14(2004) 275. [3] F. Brauer,Math. Biosc.198(2005) 119.
[4] M.A. Burgman, S. Ferson and H.R. Akcakaya,
Risk Assessment in Conservation Biology, Chap-man & Hall, London (1994).
[5] C.W. Clark, Mathematical Bioeconomics: The optimal Management of Renewable Resources, Wiley Interscience, New Jersey (1990).
[6] O. Diekmann and J.A.P Heesterbeek, Mathemat-ical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wi-ley & Son, Chichester (2000).
[7] O. Diekmann, J.A.P. Heesterbeek and J.J. Metz,
J. Math. Biol.28(1990) 365.
6 Simple Dynamics in a Vector-Borne...
[8] O. Diekmann, J.A.P. Heesterbeek and J.J. Metz,
The legacy of Kermack and McKendrick (1995) 95-115.
[9] K. Dietz and J.A.P. Heesterbeek, Math. Biosc. 180(2002) 1.
[10] H. Inaba, Kermack and McKendrick revisited: The variable susceptibility model for infectious diseases, Technical Paper, University of Tokyo (2000) 11.
[11] W.O. Kermack and A.G. McKendrick,Proc. Roy. Soc. Med.A115(1927) 700.
[12] J.C. Koella and R. Antia,Malaria J.2(2003) 1. [13] A. Korobeinikov and P.K. Maini, Math. Med.
Biol. 22(2005) 113.
[14] G. McDonald,Trop. Dis. Bull.49(1952) 813. [15] R. Ross,The Prevention of Malaria, John
Mur-ray, London (1911).
[16] E. Soewono and A.K. Supriatna,Paradox of vac-cination predicted by a simple dengue disease model, Industrial Mathematics, M.C. Joshi, A.K. Pani and S.V. Sabnis Eds., Narosa Publ. House, New Delhi, (2006) 459.
[17] A.K. Supriatna and H.P. Possingham, Bull. Math. Biol.60(1998) 49.
Presented at Workshop on Nonlinear Phenomena 2K7, Sumedang, Indonesia, December 11th, 2007.
