a "toy" model for corruption problem (some explanations from islamic point of view)

JJ J N I II 1/26JJ J N I II 1/26

A "Toy" Model for Corruption Problem(some explanations from Islamic point of view)

Agus Yodi Gunawan

=======

Kajian 2 Mingguan

Mesjid Heezerweg, Eindhoven

May,15, 2010


Intro

Aims of the talk:

• to give some feelings in constructing mathematical model

• to derive and explain the existing corruption model from the Islamicpoint 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 theextended model). However, I try to explain the derived model as simple aspossible for common public purposes.

Also, I try to relate the model into what Qur’an/Hadits quoted.


What is a mathematical modelling?

• When mathematics is applied to real-life problems, a translation isneeded to put the subject into mathematically tractable form.

• This process is usually referred to as mathematical modelling (the de-scription of an experimentally verifiable phenomenon by means of the mathe-matical language).

• The phenomenon to be described will be called the system, and themathematics used, together with its interpretation in the context of thesystem, will be called the mathematical model.


Steps of modelling

1. Identify the problem (What exactly are you going to answer or solve?)

2. Make assumptions: classify the variables, hypothesizing relationshipsamong 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.


Nature of models


To conclude,

✍ A model is a simplified representation of reality, not a perfect represen-tation.

✍ ...don’t be surprised! an intricate problem can lead to a simple model, orthe other way around.

✍ Some models are constructed in order to understand a certain phe-nomenon (this is what we are talking now)


A Corruption Model


Problem

What is the problem?


Problem


Corruption !


Problem


Corruption !What is the question?

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

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


Problem


Corruption !What is the question?

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

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

How to control it !


Relations among variables

What

☞ physical Laws, or

☞ Mathematical postulates, or

☞ principles of Legal Community, or

☞ ...........etc

are involved?


Fact 1: "Pairs" law/ Hukum berpasangan.

QS: 51:49,"dan segala sesuatu Kami ciptakan berpasang-pasangan supayakamu mengingat kebesaran Allaah". Tafsir Ibnu Katsir:

So, There are honest (good) and corrupt(bad) people.


Fact 2: Love of treasure/Cinta harta.

QS: 3:14,". dijadikan indah pada (pandangan) manusia kecintaan kepadaapa-apa yang diingini, Yaitu: wanita-wanita, anak-anak, harta yang banyakdari jenis emas, perak, kuda pilihan, binatang-binatang ternak dan sawahladang.....". Umdatul Qaariy Bab ar-raqaq (syarah shahih Bukhariy):

So, financial could stimulate someone to corrupt.


Fact 3: Companions/Pertemanan.

"Sesungguhnya perumpamaan teman yang baik (shalihah) dan teman yangjahat adalah seperti pembawa minyak wangi dan peniup api pandai besi.Pembawa minyak wangi mungkin akan mencipratkan minyak wanginya ituatau engkaumembeli darinya atau engkau hanya akanmencium aroma har-mznya itu. Sedangkan peniup api tukang besi mungkin akan membakarbajumu atau engkau akan mencium darinya bau yang tidak sedap" (RiwayatBukhari).

Almushohabatu tasyriqu at-thobi’ah (Pertemanan itu mencuri tabiat).

So, more corrupt people tends to increase corruption.


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)is the 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 arehonest, 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 ("Pertem-anan").

• It must be a control ("Legal community, religious understanding"), i.e. aformal corruption control program or anti-corruption efforts. Let u be aparameter describing the sanction risk, could be dependent or indepen-dent of time t.


Equation

Model: dx

dt= k1wx(t)− k2(u0 + u(t)),

x(0) = x0.

Here, k1 and k2 are (dimensional) positive constants, u0 is the standard con-trol where the active control u(t) is not applied yet, and x0 > 0 is the initialcorrupt population.


Equation

Model: dx

dt= k1wx(t)− k2(u0 + u(t)),

x(0) = x0.

Here, k1 and k2 are (dimensional) positive constants, u0 is the standard con-trol where the active control u(t) is not applied yet, and x0 > 0 is the initialcorrupt population.

Equilibrium population. For t→∞, x(t)→ x̂,

x̂ =k2(u0 + u(t))

k1w.

Note that since k1, k2, w > 0 then x̂ ≥ 0. So, x̂ = 0, when u0 + u(t) = 0(control is not needed).


Local analysis

What happens if corrupt population x(t) is initially close to x̂?Intuitively?..........when corrupt population is a bit larger (lower) than x̂.....


Local analysis


Write as x(t) = x̂ + y(t) where y(t)� 1.

dy

dt= k1w(x(t)− x̂).

When x(0) = x0 < x̂, then dy/dx < 0, it means

When x(0) = x0 > x̂, then dy/dx > 0, it means


Local analysis


Write as x(t) = x̂ + y(t) where y(t)� 1.

dy

dt= k1w(x(t)− x̂).

When x(0) = x0 < x̂, then dy/dx < 0, it means the proportion of corruptpeople decreases.

When x(0) = x0 > x̂, then dy/dx > 0, it means the number of corruptpeople increases.

Equilibrium x̂ is so called unstable.


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 3.00.0

0.2

0.4

0.6

0.8

1.0

1.2

Figure 1: Top-left:u0+u = 0, Top-right:0 < u0+u < w, Bottom:u0+u = w.For convenience, k1 = k2 = 1.


Interpretation

• u0 + u = 0: No sanctions: u0 + u = 0. Only a society that is totallynon-corrupt will remain honest, and even that is an unstable situation.Whenever corruption appears, x(t) > 0, corruption will increase expo-nentially until everyone is corrupt.

• 0 < u0 + u < w: Medium sanctions. Depending on the actual propor-tion of corrupt people, corruption will increase for x(t) > x̂ or decreasefor x(t) < x̂.

• u0 + u = w: High sanctions. The risk of sanctions exceeds the expectedprofit/salary for any given proportion x and corrupt people become hon-est. We end up with a totally honest population regardless of the initialstate. A small fraction of honest people suffices to convert the wholepopulation into one that is entirely free of corruption.


Interpretation

• u0 + u = 0: No sanctions: u0 + u = 0. Only a society that is totallynon-corrupt will remain honest, and even that is an unstable situation.Whenever corruption appears, x(t) > 0, corruption will increase expo-nentially until everyone is corrupt.

• 0 < u0 + u < w: Medium sanctions. Depending on the actual propor-tion of corrupt people, corruption will increase for x(t) > x̂ or decreasefor x(t) < x̂.

• u0 + u = w: High sanctions. The risk of sanctions exceeds the expectedprofit/salary for any given proportion x and corrupt people become hon-est. We end up with a totally honest population regardless of the initialstate. A small fraction of honest people suffices to convert the wholepopulation into one that is entirely free of corruption.

AlMaaidah 38: laki-laki yang mencuri dan perempuan yang mencuri, po-tonglah tangan keduanya (sebagai) pembalasan bagi apa yang mereka ker-jakan dan sebagai siksaan dari Allaah. dan Allaah Maha Perkasa lagi MahaBijaksana.


Extended model

"Taushiyah" (recommend one to the truth): an interaction between honestand corrupt people.

QS 3:110: "kamu adalah umat yang terbaik yang dilahirkan untuk manu-sia, menyuruh kepada yang ma’ruf, dan mencegah dari yang munkar, danberiman kepada Allaah".

Remember: 1− x(t) is the proportion of honest people.


Extended model


Model: dx

dt= k1wx(t)(1− x(t))− k2(u0 + u(t)),

x(0) = x0.

Here, parameter k1 represents how effective taushiyah is.

Bigger k1, increasing x(t) (corrupt people), then taushiyah is less effective.

To control corruption, we need a small value of k1 (means "intensivetaushiyah)".


Extended model


Model: dx

dt= k1wx(t)(1− x(t))− k2(u0 + u(t)),

x(0) = x0.

Equilibrium population. For t→∞, x(t)→ x̂,

x̂− =1−

√D

2and x̂+ =

1 +√

D

2.

where D = 1− 4k2(u0 + u(t))

k1w..



0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 800.0

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 atthe onset to grow (when x(0) < x̂−).

• Middle: it is still useful even when a medium group of corrupt peoplewas already present, at least some people can be saved not being corrupt(when x̂− < x(0) < x̂+).

• Right: When all (or almost) are initially corrupt, medium taushiyah isstill useful to stop some for being corrupt, but not so effective (whenx(0) > x̂+).


Can we do something more?

What is the idea?



What is the idea? when x̂− = x̂+?

Remember x̂± =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 x̂− = x̂+, 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 itmean?



What is the idea? when x̂− = x̂+?

Remember x̂± =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 x̂− = x̂+, 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 itmean? Reducing k1 −→ Intensive taushiyah.



Comparison:

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

1.2

D < 0

Most intensivewith high sanction(alMaaidah 38)

D > 0

medium

D = 0

Intensive


Summary✌ What I called as a toy model is the model that may give some insights tounderstand our phenomena qualitatively.

✌ There may be other models than this that you can derive and translate itto 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 kamubertakwa".


References✐ 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 theoryand practice (Chapt. XII), SIAM, 2002.

✐ R. M. M. Mattheij, S.W. Rienstra, J. H. M. ten Thije Boonkkamp, PartialDifferentia Equations: Modeling, Analysis, Computation, SIAM, 2005.

✐ D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler, D. A. Behrens, Opti-mal Control of Nonlinear Processes With Applications in Drugs, Corruptionand Terror, Springer, 2008.

✐ Free software: Quran, Shahih Bukhari.


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a "toy" model for corruption problem (some explanations from islamic point of view)

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