A Mathematical Model for Corruption
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Transcript of A Mathematical Model for Corruption
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A mathematical model for corruption
(some explanations from Islamic point of view)
Agus Yodi Gunawan
Maret 2012 M
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Intro
Aims of the talk:
to give some feelings in constructing mathematical model
to derive and explain the existing corruption model from the Islamic point
of view
to play some parameter scenarios and make interpretations
The subject is taken from the book of Grass et al. (see References).
I myself did not derive the governing equations (except for some parts in the
extended model). However, I try to explain the derived model as simple as
possible for common public purposes.
Also, I try to relate the model into what Qur’an/Hadits quoted.
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Mathematical Modelling
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What do you see? And then?
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What do you see? And then?
....Nothing more than just clouds.
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What do you see? And then?
....Nothing more than just clouds.
....Laten we naar buiten gaan!
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What do you see? And then?
....Nothing more than just clouds.
....Laten we naar buiten gaan!
How I love to watch the clouds ;Peacefully, peacefully drifting by Silently
upon the breeze; They ease across the clear blue sky...etc. (by Craig
Nicholson,http:// poetry.wholesomebalance.com/Loving_Clouds.html)
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What do you see? And then?
....Nothing more than just clouds.
....Laten we naar buiten gaan!
How I love to watch the clouds ;Peacefully, peacefully drifting by Silently
upon the breeze; They ease across the clear blue sky...etc. (by Craig
Nicholson,http:// poetry.wholesomebalance.com/Loving_Clouds.html)
AYG: It could be some mathematical problems in there. (Kelvin-Helmholtz
instability)
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What is a mathematical modelling?
• When mathematics is applied to real-life problems, a translation is needed
to put the subject into mathematically tractable form.
• This process is usually referred to as mathematical modelling (the descrip-tion of an experimentally verifiable phenomenon by means of the mathe-
matical language).
• The phenomenon to be described will be called the system, and the math-ematics used, together with its interpretation in the context of the system,
will be called the mathematical model.
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Two classes of quantities
In general mathematical models contain two classes of quantities:
1. variables :
• dependent variables,• independent variables.
2. parameters:• constant, e.g. gravity acceleration,• adjusted parameter, e.g. temperature of chemical reactions.
For example,
N (t) = N (0)e
µt
.Here, N(t) is the population of spider at time t, and µ is the growth rate.
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Steps of modelling
1. Identify the problem (What exactly are you going to answer or solve?)
2. Make assumptions: classify the variables, hypothesizing relationships
among the variables.
3. Solve or interpret the model: analytical approach, or numerical analysis.
4. Verify the model (qualitatively or quantitatively).
5. Implement the model.
6. Maintain, generalize or refine the model.
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Simplifying or refining the model
Model simplification Model refinement
Restrict problem identification Expand the problemNeglect variables Consider additional variables
Conglomerate effects of several variables Consider each variable in detail
Set some varibles to be constant Allow variation in the variables
Assume simple (linear) relationships Consider nonlinear relationships
Incorporate more assumptions Reduce the number of assumptions
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Nature of models
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To conclude, A model is a simplified representation of reality, not a perfect representa-
tion.
...don’t be surprised! an intricate problem can lead to a simple model, or
the other way around.
Some models are constructed in order to understand a certain phenomenon
(this is what we are talking now)
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A Corruption Model
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Selingan
QS 2: 183, Hai orang-orang yang beriman, diwajibkan atas kamu berpuasasebagaimana diwajibkan atas orang-orang sebelum kamu agar kamu bertakwa.
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Selingan lagi
QS 2: 178, Hai orang-orang yang beriman, diwajibkan atas kamu qishaashberkenaan dengan orang-orang yang dibunuh; orang merdeka dengan orang
merdeka, hamba dengan hamba, dan wanita dengan wanita.......dst
Qishaash ialah mengambil pembalasan yang sama. qishaash itu tidak dilakukan,
bila yang membunuh mendapat kema’afan dari ahli waris yang terbunuh Yaitudengan membayar diat (ganti rugi) yang wajar.
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Problem
What is the problem?
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Problem
What is the problem?
Corruption !
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Problem
What is the problem?
Corruption !What is the question?
.............
.............
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Problem
What is the problem?
Corruption !What is the question?
.............
.............
How to control it !
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Background Knowledge
What
physical Laws, or
Mathematical postulates, or
principles of Legal Community, or
...........etc
are involved?
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Fact 1: berpasangan.
QS: 51:49,"dan segala sesuatu Kami ciptakan berpasang-pasangan supaya kamu
mengingat kebesaran Allah". Tafsir Ibnu Katsir:
So, There exists honest (good) and corrupt(bad) people.
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Fact 2: cinta harta.
QS: 3:14,". dijadikan indah pada (pandangan) manusia kecintaan kepada apa-
apa yang diingini, Yaitu: wanita-wanita, anak-anak, harta yang banyak dari
jenis emas, perak, kuda pilihan, binatang-binatang ternak dan sawah ladang.....".Umdatul Qaariy Bab ar-riqaq (syarah shahih Bukhariy):
So, financial could stimulate someone to corrupt.
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Fact 3: Pertemanan.
"Sesungguhnya perumpamaan teman yang baik (shalihah) dan teman yang jahat
adalah seperti pembawa minyak wangi dan peniup api pandai besi. Pembawaminyak wangi mungkin akan mencipratkan minyak wanginya itu atau engkau
membeli darinya atau engkau hanya akan mencium aroma harmznya itu. Sedan-
gkan peniup api tukang besi mungkin akan membakar bajumu atau engkau akan
mencium darinya bau yang tidak sedap" (Riwayat Bukhari).
Almushohabatu tasyriqu at-thobi’ah (Pertemanan itu mencuri tabiat).
So, more corrupt people tends to increase corruption.
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Assumptions
• Community is divided into two groups: Honest and Corrupt people("berpasangan").Let x(t) be the proportion of people who are corrupt at time t; 1 − x(t) isthe proportion of honest people.
• Temptation to become corrupt is usually financial ("Cinta harta"). So theincomes of corrupt people are assumed to be higher than those who are
honest, by some constant amount per unit of time: wc > wh.Let us write w = wc − wh > 0.
• More corrupt people tends to increase corrupt population ("Pertemanan").• It must be a control ("Legal community, religious understanding"), i.e. a
formal corruption control program or anti-corruption efforts. Let u be aparameter describing the sanction risk, could be dependent or independent
of time t.
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Equation
Model:
dxdt
= k1wx(t)− k2(u0 + u(t)),x(0) = x0.
Here, k1 and k2 are (dimensional) positive constants, u0 is the standard controlwhere the active control u(t) is not applied yet, and x0 > 0 is the initial corruptpopulation.
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Equation
Model:
dxdt
= k1wx(t)− k2(u0 + u(t)),x(0) = x0.
Here, k1 and k2 are (dimensional) positive constants, u0 is the standard controlwhere the active control u(t) is not applied yet, and x0 > 0 is the initial corruptpopulation.
Equilibrium population. For t →∞, x(t) → x,
x =
k2(u0 + u(t))
k1w
.
Note that since k1, k2, w > 0 then x ≥ 0. So, x = 0, when..............
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Local analysis
What happens if corrupt population x(t) is close to
x?
Intuitively?..........when corrupt population is a bit larger (lower) than x.....
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Local analysis
What happens if corrupt population x(t) is close to
x?
Intuitively?..........when corrupt population is a bit larger (lower) than x.....Write as x(t) = x + y(t) where y(t) 1.dy
dt = k1w(x(t)− x).
When x(0) = x0 0, it means
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Local analysis
What happens if corrupt population x(t) is close to
x?
Intuitively?..........when corrupt population is a bit larger (lower) than x.....Write as x(t) = x + y(t) where y(t) 1.dy
dt = k1w(x(t)− x).
When x(0) = x0 x, then dy/dt > 0, it means the number of corrupt peopleincreases.
Equilibrium x is so called unstable.
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Simulations: x(t) vs t
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Top-left:u0 + u = 0, Top-right:0 < u0 + u < w, Bottom:u0 + u = w.
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Interpretation
• u0 + u = 0: No sanctions: u0 + u = 0. Only a society that is totally non-corrupt will remain honest, and even that is an unstable situation. When-ever corruption appears, x(t) > 0, corruption will increase exponentiallyuntil everyone is corrupt.
• 0 < u0 + u < w: Medium sanctions. Depending on the actual proportionof corrupt people, corruption will increase for x(t) >
x or decrease for
x(t)
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Interpretation
• u0 + u = 0: No sanctions: u0 + u = 0. Only a society that is totally non-corrupt will remain honest, and even that is an unstable situation. When-ever corruption appears, x(t) > 0, corruption will increase exponentiallyuntil everyone is corrupt.
• 0 < u0 + u < w: Medium sanctions. Depending on the actual proportionof corrupt people, corruption will increase for x(t) >
x or decrease for
x(t)
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Extended model
"Taushiyah": an interaction between honest and corrupt people.
Model:
dx
dt = k1wx(t)(1− x(t))− k2(u0 + u(t)),
x(0) = x0.
Equilibrium population. For t →∞, x(t) → x, x− = 1−
√ D
2 and x+ = 1 +
√ D
2 .
where D = 1− 4k2(u0 + u(t))k1w
..
Si i ( )
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Simulations: x(t) vs t
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Mild/Medium Taushiyah:
• Left: it is still effective when a small group of corrupt people is just at theonset to grow (when x(0)
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Can we do something more?
What is the idea?
C d thi ?
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Can we do something more?
What is the idea? when x− = x+?Remember x± = 1±
√ D
2 , where D = 1− 4k2(u0 + u(t))
k1w .
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
k1
D > 0
w
If x− = x+, then D = 0. Say, we know w (salary). If we start with the redbox, then we must go back so that we arrive at the black box. What does it
mean?
Can we do something more?
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Can we do something more?
What is the idea? when x− = x+?Remember x± = 1±
√ D
2 , where D = 1− 4k2(u0 + u(t))
k1w .
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
k1
D > 0
w
If x− = x+, then D = 0. Say, we know w (salary). If we start with the redbox, then we must go back so that we arrive at the black box. What does it
mean? Reducing k1 −→ Intensive taushiyah.
Simulations: x(t) vs t
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Simulations: x(t) vs t
Comparison:
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
1.2
D < 0
Most intensive
with high sanction(alMaaidah 38)
D > 0
medium
D = 0
Intensive
Summary
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Summary
What I called as a toy model is the model that may give some insights to
understand our phenomena qualitatively. There may be other models than this that you can derive and translate it to
your own words.
Religious (Islam) understanding must be enhanced (as self-control).
Taushiyah is our (moslem) obligation.
to close this topic, QS 2:179:
"Dan dalam qishash itu ada (jaminan kelangsungan) hidup bag-
imu, hai orang-orang yang berakal (Ulul alBaab), supaya kamu
bertakwa".
References
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F.R. Giordano, M.D. Weir, W.P. Fox, A first course in Mathematical mod-
elling, Thomson-Brooks/Cole, 2003.
R. M. M. Mattheij, J. Molenaar, Ordinary differential equations in theory
and practice (Chapt. XII), SIAM, 2002.
R. M. M. Mattheij, S.W. Rienstra, J. H. M. ten Thije Boonkkamp, Partial
Differentia Equations: Modeling, Analysis, Computation, SIAM, 2005.
D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler, D. A. Behrens, OptimalControl of Nonlinear Processes With Applications in Drugs, Corruption
and Terror , Springer, 2008.
Free software: Quran, Shahih Bukhari.