a mathematical model for corruption

8/16/2019 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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....Nothing more than just clouds.


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....Laten we naar buiten gaan!


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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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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


Corruption !


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Problem


Corruption !What is the question?

.............

.............


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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


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


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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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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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?



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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.



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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.

a mathematical model for corruption

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