Dynamics under Various Assumptions on Time and Uncertainty

Gerhard-Wilhelm Weber 1*,

Özlem Defterli 2, Linet Özdamar 3, Zeynep Alparslan-Gök 4, Chandra Sekhar Pedamallu 5,

Büşra Temoçin 1, 6, Azer Kerimov 1, Ceren Eda Can 7, Efsun Kürüm 1,

1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey 2 Department of Mathematics and Computer Science, Cankaya Uniiversity, Ankara, Turkey 3 Department of Systems Engineering, Yeditepe University, Istanbul, Turkey 4 Department of Mathematics, Süleyman Demirel University, Isparta, Turkey 5 Department of Medical Oncology, Dana-Farber Cancer Institute, Boston, MA, USA, The Broad Institute of MIT and Harvard, Cambridge, MA, USA 6 Department of Statistics, Ankara University, Ankara, Turkey 7 Department of Statistics, Hacettepe University, Ankara, Turkey

* Faculty of Economics, Management and Law, University of Siegen, Germany Center for Research on Optimization and Control, University of Aveiro, Portugal Universiti Teknologi Malaysia, Skudai, Malaysia Advisor to EURO Conferences

6th International Summer School

National University of Technology of the Ukraine Kiev, Ukraine, August 8-20, 2011

Let Us Predict Dynamics under Various Assumptions on Time and Uncertainty

http://www.iam.metu.edu.tr/EUROPT/index.html

http://www.ntu-kpi.kiev.ua/

Introduction and Motivation

System Dynamics

Primary Education (PE)

Simulation in PE in Turkey and Its Results

Simulation in PE in India and Its Results

Transmission of HIV in Developing Countries

Time-Continuous and Time-Discrete Models

Gene-Environment Networks

Numerical Example and Results

Regulatory Networks under Uncertainty

Ellipsoidal Model

Stochastic Model

Stochastic Hybrid Systems

Portfolio Optimization subject to SHSs

Lévy Processes, Simulation and Asset Price Dynamics

Conclusion

Outline

Introduction and Motivation

http://www.gene-chips.com/sample1.html

101 million children of primary school age are out of school Number of primary-school-age children not in school, by region (2007)

http://www.childinfo.org/education.html

Education Index (according to 2007/2008, Human Development Report)

System Dynamics Primary Education

http://www.childinfo.org/education.html

http://en.wikipedia.org/wiki/Human_Development_Report

http://en.wikipedia.org/wiki/File:Education_index_UN_HDR_2007_2008.PNG

The first stage of compulsory education is primary or elementary education.

In most countries, it is compulsory for children to receive primary education,

though in many jurisdictions it is permissible for parents to provide it. The transition from elementary school to secondary school or high school is

somewhat arbitrary, but it generally occurs at about eleven or twelve years of age. The major goals of primary education are achieving basic literacy and

numeracy amongst all pupils, as well as establishing foundations

in science, geography, history and other social sciences. The relative priority of various fields, and the methods used to teach them,

are an area of considerable political debate. Some of the expected benefits from primary education are the reduction of

the infant mortality rate, the population growth rate, of the

crude birth and death rate, and so on.


Because of the importance of primary education, there are several models proposed to study the factors influencing the primary school enrollment and progressions. Various models developed to analyze issues in basic education are as follows:

• logistic regression models (Admassu (2008)),

• Poisson regression models (Admassu (2008)), • system dynamics models

(Altamirano and van Daalen (2004), Karadeli et al. (2001), Pedamallu (2001), Terlou et al. (1999)),

• behavioral models (Benson (1995), Hanushek et al. (2008)), in the contexts of different countries.

Several factors have been identified which influence

the school enrollment and drop outs.


To design a good policy, issues to be considered / understood are:

- reasons behind enrollment, drop-outs and repeaters in school,- perceived quality of teaching by students and parents,

- educational level and income level of parents, - expectations from school by parents, - perceived quality of teaching by the District Educational Officer (DEO), - needs of different infrastructure facilities (space, ventilation, sanitation, etc.) in the school, - and much more … .


to analyze importance of infrastructural facilities on the quality of the

primary education system and its correlation to attributes related to the

primary education system and environment including parents, teachers, infrastructure and so on (cross impact matrix), to predict the effects of infrastructural facilities on the quality of primary education system so that a discussion can be started on how to develop policies, to develop intervention policies using societal problem handling methods

such as COMPRAM (DeTombe 1994, 2003), once the outputs of the model become known with available data.


1,2,....,0 ( ) , 0,1 i ix t N tand( ) .ix t i twhere is the level of variable in period

( )( ) ( ) , iP ti ix t t x t

To preserve boundedness :

1 | sum of on |( ) .

1 | sum of on

i

ii

t impact xP t

t positiv im

negat

pac

i

e t x |

ve

where

Julius (2002)

Modeling and forecasting the behaviour of complex systems are necessary

if we are to exert some degree of control over them. Properties of variables and interactions in our large-scale system:

- System variables are bounded:

- A variable increases or decreases according to whether the net impact

of the other variables is positive or negative.


A variable’s response to a given impact decreases to 0 as that variable approaches its upper or lower bound. It is generally found that bounded growth and decay processes exhibit

this sigmoidal character. All other issues being constant, a variable (attribute) produces a greater impact on the system as it grows larger (ceteris paribus). Complex interactions are described by a looped network of

binary interactions (this is the basis of cross-impact analysis).


• When entities interact through their attributes, the levels of the attributes might change, i.e., the system behaves in certain directions.

• Some changes in attribute levels may be desirable while others may not be so. • Each attribute influences several others, thus creating a web of

complex interactions that eventually determine system behaviour. In other terms, attributes are variables that vary from time to time.

• They can vary in an unsupervised way in the system. • However, variables can be controlled directly or indirectly, and partially

by introducing new intervention policies. • However, interrelations among variables should be analyzed carefully

before introducing new policies.

o The choice of relevant attributes has to be made carefully, keeping in mind both

short-term and long-term consequences of solutions (decisions). o All attributes can be associated with given levels that may indicate

quantitative or qualitative possession.

System Dynamics

System Dynamics 4 Steps of Implementation

Step 1. Set the initial values to identified attributes obtained from published sources and surveys conducted.Step 2. Build a cross-impact matrix with the identified relevant attributes.

a. Summing the effects of column attributes on rows indicates the effect of each attribute in the matrix.

b. The parameters -ij can be determined by creating a pairwise correlation matrix after collecting the data, and adjusted by subjective assessment.

c. Qualitative impacts are quantified subjectively as shown in the table. Qualitative impacts can be extracted from a published sources and survey data set.

d. The impact of infrastructural facilities on primary school enrollments and progression become visible by running the simulation model.

e. An exemplary partial cross-impact matrix with the attributes and their hypothetical values above is illustrated as follows:

System Dynamics 4 Steps of Implementation

System Dynamics Problems with Migrant Primary School Students in

Turkey

Problems with Migrant Primary School Students in Turkey

• Lack of Turkish language• Lack of parental interest• Lack of adequate shelter, food• Not able to assimilate in urban life and society

http://en.wikipedia.org/wiki/File:BlankMapTurkeyProvinces.png

Entities in the System Dynamics Model1. student,2. teacher,3. family,4. school environment, 5. Ministry of Education.


Turkey

Entity Relationship diagram

Attributes under each entity

Entity 1: Student:F1.1 Not believing in education F1.2 Dislike of school books F1.3 Lack of good Turkish language skillsF1.4 Dislike of school F1.5 Weak academic self confidence F1.6 Frequent disciplinary problem

Entity 2: Teacher:F2.1 Relating to guidance teacher

Entity 3: Family:F3.1 Economic difficultyF3.2 Child laborF3.3 Large families F3.4 Malnutrition F3.5 Homes with poor infrastructure (e.g., lack of heating, lack of room)F3.6 Different home language F3.7 Parent interest

Entity 4: School environment:F4.1 Violence at schoolF4.2 Alien school environment F4.3 Lecture not meeting student needs


Turkey

The Proposed Policies to remediate the system

P1. Daily distribution of milk P2. Distribution of coal, food, etc. to Turkish green card holders P3. The three-children per family aspiration by government P4. School infrastructure - renovationP5. Providing adult education for poor parents in migrant communities P6. Providing one year of Turkish language class before migrant student attends urban school P7. Combined policy: P2 + P4 + P5 + P6


Turkey

While the mere basics of survival in the city are maintained by P2, policy P5 would enable the parents to find better employment and improve the migrant family’s economic conditions. Policies P2 and P5 support parents and, hence, their impacts on their offspring are indirect. On the other hand, policies P4 and P6 target the academic performance of students directly. The combined policy P7 would naturally produce the best overall impact on the system if the budget of the Ministry of Education can afford it.

Implementation Results

Variables Ini. V. Undesirable value

Desired value NO_P P1 P2 P3 P4 P5 P6 P7

F1.1 0.693 1 0 15 19 21 15 31 39 23 23

F1.2 0.405 1 0 18 18 20 18 50 21 50 20

F1.3 0.509 1 0 14 15 16 14 17 19 31 28

F1.4 0.261 1 0 16 24 21 16 36 22 30 22

F1.5 0.707 1 0 19 20 28 19 50 50 25 14

F1.6 0.5 1 0 25 25 25 25 26 25 25 28

F2.1 0.5 0 1 12 12 12 12 23 12 14 43

F3.1 0.815 1 0 13 16 24 11 15 35 15 36

F3.2 0.5 1 0 20 27 50 16 25 50 20 29

F3.3 0.774 1 0 17 18 21 12 21 50 17 33

F3.4 0.729 1 0 16 41 35 13 21 47 17 30

F3.5 0.799 1 0 14 18 30 12 18 29 15 32

F3.6 0.658 1 0 16 17 19 16 23 27 43 25

F3.7 0.5 0 1 10 16 37 6 44 50 12 23

F4.1 0.5 1 0 16 17 18 16 25 19 20 44

F4.2 0.619 1 0 15 19 17 15 32 17 23 33

F4.3 0.5 1 0 19 20 21 19 50 23 36 12


Turkey

The following simulation is based on data from Gujarat (India).

System Dynamics Problems with Primary Education in India

http://en.wikipedia.org/wiki/File:India_Gujarat_location_map.svg

http://en.wikipedia.org/wiki/File:India_Gujarat_locator_map.svg

Level of enrollment (loe)

Before implementation of policy variables (red line): - sharp increase at the beginning phase of the simulation

(first 12 iterations), and then there is steady decrease

after a certain period of time (first 12 iterations).

After implementation of policy variables (blue line):- steady increase from initial value (0.71) to unity.

Analysis:

- increase in level of enrollment in first 12 iterations happens because of the response lag in population to bad quality school system,

- instant impact on the level enrollment after implementation of policy variables because students and parents are more eager to have the children attend a nice looking healthy school.


Level of repeaters (lr)

Before implementation of policy variables (red line): - steady increase from 0.05 to unity in 50 iterations.

After implementation of policy variables (blue line):- steady increase from initial value 0.05 to 0.12 in first 14

iterations and then declined to zero in 50 iterations.

Analysis:

- level of repeaters has instant impact on level of repeaters

if the infrastructural facilities are bad (assumption: teachers are doing their best in teaching – this variable is static - so variable impacted here is teaching aids),

- increase in level of repeaters in first 14 iterations is because improvement in the infrastructure does not have an instant impact on the level of repeaters.


DNA microarray chip experiments

prediction of gene patterns based on

with

M.U. Akhmet, H. Öktem

S.W. Pickl, E. Quek Ming Poh

T. Ergenç, B. Karasözen

J. Gebert, N. Radde

Ö. Uğur, R. Wünschiers

M. Taştan, A. Tezel, P. Taylan

F.B. Yilmaz, B. Akteke-Öztürk

S. Özöğür, Z. Alparslan-Gök

A. Soyler, B. Soyler, M. Çetin

S. Özöğür-Akyüz, Ö. Defterli

N. Gökgöz, E. Kropat

... Finance

Environment

Health Care

MedicineBio-Systems

Ex.: yeast data

GENE / time 0 9.5 11.5 13.5 15.5 18.5 20.5

'YHR007C' 0.224 0.367 0.312 0.014 -0.003 -1.357 -0.811

'YAL051W' 0.002 0.634 0.31 0.441 0.458 -0.136 0.275

'YAL054C' -1.07 -0.51 -0.22 -0.012 -0.215 1.741 4.239

'YAL056W' 0.09 0.884 0.165 0.199 0.034 0.148 0.935

'PRS316' -0.046 0.635 0.194 0.291 0.271 0.488 0.533

'KAN-MX' 0.162 0.159 0.609 0.481 0.447 1.541 1.449

'E. COLI #10' -0.013 0.88 -0.009 0.144 -0.001 0.14 0.192

'E. COLI #33' -0.405 0.853 -0.259 -0.124 -1.181 0.095 0.027

http://genome-www5.stanford.edu/

DNA experiments

http://genome-www/



Analysis of DNA experiments

Modeling & Prediction

)(: nE

0)0(,)( EEEEME

)(: nnM

prediction, anticipation least squares – max likelihood

statistical learning

expression data

matrix-valued function – metabolic reaction

Tn tetetetEE ))(,...,)(,)(()( 21

Expression

kkk EE M1

1 ( ( )) ,k k k kE I h M E E 2

21 2( )k

k k khE I h M M E

Ex.:

)(Μ jik em M

We analyze the influence of em -parameters on the dynamics (expression-metabolic).

Ex.: Euler, Runge-Kutta

Modeling & Prediction

Genetic Network

, 1

E M E h

)()()()(

)()()()(

)()()()(

)()()()(

34333231

24232221

14131211

04030201

tEtEtEtE

tEtEtEtE

tEtEtEtE

tEtEtEtE

080170255

25570180255

050200255

2550250255

2001

039.02.00

0061.04.0

0000

M

Ė 4Ė 2

Ė 0

Ė 5

Ė 1

Ė 3

0123456789

0 2 4 6 8

Time, t

Ex

pre

ss

ion

lev

el,

Ė

Ex. :

gene2

gene3

gene1

gene4

0.4 x1

0.2 x2 1 x1

Genetic Network

Gene-Environment Networks

1:

0i j

if gene j regulates gene i

otherwise , i i

The Model Class

d-vector of concentration levels of proteins and of certain levels of environmental factors

continuous change in the gene-expression data in time

is the firstly introduced time-autonomous form, where

nonlinearitiesinitial values of the gene-exprssion levels

: experimental data vectors obtained from microarray experiments

and environmental measurements : the gene-expression level (concentration rate) of the i th gene at time t

denotes anyone of the first n coordinates in thed-vector of genetic and environmental states.

is the set of genes.

Weber et al. (2008c), Chen et al. (1999), Gebert et al. (2004a),Gebert et al. (2006), Gebert et al. (2007), Tastan (2005), Yilmaz (2004), Yilmaz et al. (2005),Sakamoto and Iba (2001), Tastan et al. (2005)

(i) is an constant (nxn)-matrix is an (nx1)-vector of gene-expression levels(ii) represents and t the dynamical system of the n genes and their interaction alone. : (nxn)-matrix with entries as functions of polynomials, exponential, trigonometric, splines or wavelets containing some parameters to be optimized.

(iii)

environmental effects

n genes , m environmental effects

are (n+m)-vector and (n+m)x(n+m)-matrix, respectively.

Weber et al. (2008c), Tastan (2005), Tastan et al. (2006),Ugur et al. (2009), Tastan et al. (2005), Yilmaz (2004), Yilmaz et al. (2005),Weber et al. (2008b), Weber et al. (2009b)

(*)

The Model Class

The Model Class

In general, in the d-dimensional extended space,

with

: (dxd)-matrix

: (dx1)-vectors

Ugur and Weber (2007), Weber et al. (2008c),Weber et al. (2008b), Weber et al. (2009b)

The Time-Discretized Model

- Euler’s method, - Runge-Kutta methods, e.g., 2nd-order Heun's method

3rd-order Heun's method is introduced by Defterli et al. (2009)

we rewrite it as

where

Ergenc and Weber (2004), Tastan (2005), Tastan et al. (2006), Tastan et al. 2005)

The Time-Discretized Model

in the extended space denotes the DNA microarray experimental data and the data of environmental items obtained at the time-level

the approximations obtained by the iterative formula above

initial values

k th approximation or prediction is calculated as:

(**)

Matrix Algebra

are (nxn)- and (nxm)-matrices, respectively

(n+m)x(n+m) -matrix

are (n+m)-vectors

Applying the 3rd-order Heun’s method to the eqn. (*) gives the iterative formula (**), where

Final canonical block form of : = .

Matrix Algebra

Optimization Problem

mixed-integer least-squares optimization problem:

subject to

Ugur and Weber (2007),Weber et al.(2008c),Weber et al. (2008b),Weber et al. (2009b), Gebert et al. (2004a), Gebert et al. (2006), Gebert et al. (2007).

Boolean variables

, : th : the numbers of genes regulated by gene (its outdegree), by environmental item , or by the cumulative environment, resp..

The Mixed-Integer Problem

: constant (nxn)-matrix with entries representing the effect which the expression level of gene has on the change of expression of gene

Genetic regulation network

mixed-integer nonlinear optimization problem (MINLP):

subject to

: constant vector representing the lower bounds

for the decrease of the transcript concentration.

Binary variables :

Numerical Example Consider our MINLP for the following data:

Gebert et al. (2004a)

Apply 3rd-order Heun method:

Take

using the modeling language Zimpl 3.0, we solveby SCIP 1.2 as a branch-and-cut framework, together with SOPLEX 1.4.1 as our LP-solver

Numerical Example

Apply 3rd-order Heun’s time discretization :

Results of Euler Method for all genes:

____ gene A........ gene B_ . _ . gene C- - - - gene D

Results of 3rd-order Heun Method for all genes:

____ gene A........ gene B_ . _ . gene C- - - - gene D

Errors uncorrelated Errors correlated Fuzzy values

Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics

θ1

θ2


Errors uncorrelated Errors correlated Stochastics

Interval arithmetics Ellipsoidal calculus Hybrid systems

θ1

θ2



Interval arithmetics Ellipsoidal calculus Hybrid systems

θ1

θ2



Interval arithmetics Ellipsoidal calculus Lévy processes

θ1

θ2


Interaction of Target & Environmental Clusters

Determine the degree of connectivity.


Clusters and Ellipsoids:

Target clusters: C1 , C2 ,…,CR Environmental clusters: D1, D2,…, DS

Target ellipsoids: X1, X2,…, XR Xi = E (μi , Σi) Environmental ellipsoids: E1 , E2,…, ES Ej = E (ρj , Πj)

Center Covariance matrix


Time-Discrete Model

Targetcluster

Environmental cluster

Target - Target Environment - Target

Target - Environment Environment - Environment

R

r = 1A

TTj r X r

(k)ξ j0 +( ) +

S

s = 1A

ETj s E s

(k)( )=X j(k + 1)

R

r = 1A

TEi r X r

(k)ζ i0 +( ) +

S

s = 1A

EEis E s

(k)( )=E i(k + 1)

Determine system matrices and intercepts.




Time-Discrete Model

Emissionclusters

Environ-mental cluster

-

control factor

Prediction of CO2-emissions

+

+

0

u(k)

r = 1A

TTj r X r

(k)ξ j0 + +

s = 1A

ETj s E s

(k)=X j

(k + 1)

R

r = 1A

TEi r X r

(k)ζ i0 + ( )+

S

s = 1A

EEis E s

(k)( )=E i(k + 1)

( () )


The Mixed-Integer Regression Problem:

Maximize

≤j jαTT

deg(C )TE ≤j jαTE

deg(D )ET ≤i iαET

deg(D )EE ≤i iαEE

deg(C )TT

X r(k)^ X r

(k)−∩-- -- E s

(k)^ E s(k)−

∩+ -- --ΣR

r = 1ΣS

s = 1ΣT

k = 1


bounds on outdegrees

s.t.

The Continuous Regression Problem:

Maximize

s.t. P TT ( )TT ≤j r jαTT

≤ jαTE

≤ iαET

≤ iαEE

bounds on outdegrees

ATT

j r, ξ

j0

P TE ( )TE j r

ATE

j r, ξ

j0

P ET ( )ET AET

i s, ζ

i0

P EE ( )EE AEE

i s

ΣR

r = 1

, ζ i0

ΣR

r = 1

ΣR

s = 1

ΣR

s = 1

Continuous Constraints /Probabilities

is

is

X r(k)^ X r

(k)−∩-- -- E s

(k)^ E s(k)−

∩+ -- --ΣR

r = 1ΣS

s = 1ΣT

k = 1


Stock Markets

http://en.wikipedia.org/wiki/File:Photos_NewYork1_032.jpg

http://en.wikipedia.org/wiki/File:Paternoster_Square.jpg

http://en.wikipedia.org/wiki/File:Shanghai_Stock_Exchange_Building.jpg

http://en.wikipedia.org/wiki/File:IE_Real_SandP_Prices,_Earnings,_and_Dividends_1871-2006.png

http://en.wikipedia.org/wiki/File:Boerse-ffm021.jpg

( , ) ( , ) t t t tdX a X t dt b X t dW

(0, ) ( [0, ])tW N t t T

Ex.: price, wealth, interest rate, volatility

processes

drift and diffusion term

Financial Dynamics

Milstein Scheme :

and, based on our finitely many data:

21 1 1 1 1

1ˆ ˆ ˆ ˆ ˆ( , )( ) ( , )( ) ( )( , ) ( ) ( )2 j j j j j j j j j j j j j j j jX X a X t t t b X t W W b b X t W W t t

2( )( , ) ( , ) 1 2( )( , ) 1 .

j jj j j j j j j

j j

W WX a X t b X t b b X t

h h

Financial Dynamics

Hybrid Stochastic Control

• standard Brownian motion

• continuous state

Solves an SDE whose jumps are governed by the discrete state.

• discrete state Continuous time Markov chain.

• control

Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard

Applications

• Engineering: Maintain dynamical system in safe domain for maximum time.

• Systems biology: Parameter identification.

• Finance: Optimal portfolio selection.

hybrid

Method: 1st step

1. Derive a PDE satisfied by the objective function in terms of the generator:

• Example 1: If

then

• Example 2:If

then

hybrid

2. Rewrite original problem as deterministic PDE optimization program

3. Solve PDE optimization program using adjoint method

Simple and robust …

hybridMethod: 2nd and 3rd step

References

Thank you very much for your attention!

[email protected]

http://www3.iam.metu.edu.tr/iam/images/7/73/Willi-CV.pdf

mailto:[email protected]

The epidemic of AIDS has been steadily spreading for the past two decades, and now affects every country in the world.

Each year, more people die, and the number of HIV+ people continues to rise despite national and international HIV prevention policies and dedicated public healthcare strategies.

Well-known modes of transmission of HIV are sexual contact, direct contact with HIV-infected blood or fluids and perinatal transmission from mother to child.

Transmission can be classified into intentional and unintentional transmission

- intentional if the infected person knows that he/she is HIV+ and he/she does not disclose it when there is a risk of transmission,

- otherwise, transmission is not intentional.

Estimated prevalence of HIV among young adults (15-49) per country at the end of 2005 (source: http://en.wikipedia.org/wiki/AIDS)

System Dynamics Epidemics of HIV Appendix

http://en.wikipedia.org/wiki/AIDS

To design good policy, issues to be considered / understood are:

- psyche and behavioral pattern of HIV+,

- interactions between the society and HIV+,

- socio and economic situation in which HIV+ are living,

- awareness level of society about HIV and its transmission channels,

- circumstances that make HIV people to do intentional transmission,

- and much more.

Different issues to consider:


Goals: to analyze the phenomenon of intentional transmission of HIV/AIDS and its correlation to

attributes related to the virus carrier and his/her community including family members, work colleagues and healthcare infrastructure (cross impact matrix),

to predict the effects of intentional transmission on the spread of HIV so that a discussion can be started on how to develop policies that induce openness about the HIV+ status of individuals,

to develop intervention policies using societal problem handling methods such as COMPRAM (DeTombe 1994, 2003) once the outputs of the model become known with available data.

Possible Solutions:Some popular approaches are:

system thinking approach (cross impact analysis),

system dynamics,

mathematical modeling,

etc..


basic entities and their relationships with the behavior of HIV+ persons are described,

and a list of attributes are provided, the data requirements for the model are summarized along with a questionnaire

to collect the data, the cross impact among attributes can be measured by pairwise correlation analysis, partial cross impact matrix that conveys information on the influence of one variable

over the other is illustrated using qualitative judgement, parameters are then fed into mathematical equations that change the level of variables

throughout simulation iterations, significance of intentional infection transmission rates due to various sources

can then be identified and analyzed.

Proposed approach:

A cross impact analysis method has been proposed here to study the behaviour of HIV+ persons with respect to the disclosure of their status:


System Dynamics Epidemics of HIV

A cross-impact model is developed here to study the intentional transmission of HIV by non-disclosure of status in various risky situations.

Here, we identify certain entities and attributes that might affect an HIV+ patient’s attitude towards disclosure.

We describe the simulation method and the data to be collected if such a model is to be executed. Two policy variables may be proposed as intervention to non-disclosure:

- The first could be investing funds in improving hygiene and preventative measures in healthcare institutions.

- However, such a policy should be accompanied by supporting the HIV+ individuals with economic aid if they are unemployed, free access to special AIDS clinics and access to housing units where they will not be subjected to any harassment.

The proposed cross impact model enables the identification of important factors that result in non-disclosure and could invoke new intervention policies and regulations to prevent the intentional transmission of HIV/AIDS in different countries and societal environments.

Conclusions:

Appendix

Stochastic Control of Hybrid Systems

We consider a financial market consisting of:

• one risk-free asset with price satisfying:

• risky assets with prices following:

0 0 t TS t

0 0 0 0( ) ( , ( )) ( ) , 0 , dS t r t X t S t dt S s

0n t T

S t

1

( ) ( , ( )) ( ) ( ) ( , ( )) ( ), (0) 0.

M

n n n n nm m n nm

dS t t X t S t dt S t t X t dB t S s

1N

Appendix

: a filtered probability space with filtration . F.. . . . .

1( ) ( ), , ( ) : T

MB t B t B t Ma standard - dimensional Brownian motion.

( , ) : r t x the riskless interest rate.

0

1

( ) ( ( ))

, , .

t T

n

X t t X t

S s s

is the state at time of a Markov process

on a finite state space

1 ,1( , ) ( ( , )) : nm n N m Mt x t x the matrix of risky assets' volatilities.

We assume:

Appendix Stochastic Control of Hybrid Systems

We assume:

v chain

.

( , ), ( ,

on

S

r t x t

t

this makes the Markov process an extension to continuous The Markov process transitions

time of a Maroccur only on integer values of time ;

o k

) ( , ) 0, ,

.

( , ) > 0 0, .

( , ) ( , ) 0,

T

x t x T

s S

r t x t T s S

t x t x t T

and are deterministic continuous functions on

for every

for all and every

is nonsingular for Lebesgue almost all

and

2

01 1

( , ) .

N M T

n m

t x dt

satisfies the integrability condition

Wealth Process Appendix Stochastic Control of Hybrid Systems

Be

0

0

0

( ) :

( ) a.s. :

( )

t T

t

n

c t

c t dt

t

a progressively measurable nonnegative process such

proces

that

.sconsumption

0 1

0

: at time .

( ) ( ( ), ( ), , ( )) ,

( ) 1 ( 0, ) :

t T n

TN

N

nn

S t

t t t t

t t T

the fraction of the investor's wealth allocated to the asset

where

proc .essportfolio

AppendixWealth Process


0 00

( ) ( 0, ):

( ) ( ) ( ) ( ) ( ( ) ( )) ,

( )

:

( ) :

N t t

nn

n n

W t t T

s W sW t w dS s i s c s ds

S s

w

i t

the defined by

the investor's initial w

w

ealth process

, the

ealth in

0 ( ) .

T

i t dt

, a deterministic function with

The integral equation above can be rewritten as differthe enti

co

al

me

equ atihybrid

01

1 1

( ) ( ) ( ) , ( ) ( ) ( , ( )) ( )

( ) ( ) ( , ( )) ( ) ( 0, ) .

N

n nn

N M

n nm mn m

dW t i t c t t r t X t t t X t W t dt

t W t t X t dB t t T

o : n

Be



[0, ]t T

The investor is faced with the problem of finding strategies that maximize the utility of

(i) his consumption for all , and

(ii) his terminal wealth at time T.



0( ) 0, 0, 1 20

( ) sup , ,

( )

TS Px S wV x E E U c s s ds U W T

x

The investor's goal is to

where is the set of all decision strategies.

his

admissible

A

A

maximize expected utility

Wealth Process Stochastic Control of Hybrid Systems Appendix

Stock Prices and Indices

d-fine, 2002DAX Empirical and Simulation

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

Emp.

Sim.

Ö. Önalan, 2006

Appendix

Root Monte Carlo Simulation B. Dupire, 2007 Appendix

Definition: A cadlag adapted processes

defined on a probability space (Ώ, F, P) is said to be a

Lévy processes, if it possesses the following properties:

( | 0) tL L t

Lévy Processes Appendix

ί )

ίί ) For is independent of , i.e., has independent increments.

ίίί) For any is equal in distribution to (the distribution of does not depend on t ); has stationary increments.

ίv) For every , , i.e., is stochastically continuous.

In the presence of ( ί ), ( ίί ), ( ίίί ), this is equivalent to the condition

0 0 1. P L

st LLts ,0sF

L

0 , t ss t L L stL

sst LL

L

, 0 0 s t and 0lim

st

tsLLP

L

0

lim 0.

tt

P L


There is strong interplay between Lévy processes and infinitely divisible distributions.

Proposition: If is a Lévy processes, then is infinitely divisible for each .

Proof : For any and any :

Together with the stationarity and independence of increments we conclude that the random variable is infinitely divisible.

L tL

0tI n

0t

2 ( 1)... . t t n t n t n t n t nL L L L L L

tL


Moreover, for all and all we define

hence, for rational : .

Ru 0t

( ) log . ti u L

tΨ u E e

0t

1( ) ( ) ttΨ u Ψ u


For every Lévy process, the following property holds:

1 ( ) ti u L tΨ uE e e

2

1exp 1 1 ( ) ,2

R

iuxx

u ct ibu e iux v dx

where is the characteristic exponent of . )()( 1 uu XL 1


The triplet is called the Lévy or characteristic triplet;

is called the Lévy or characteristic exponent.

Here, : drift term,

: diffusion coefficient, and

: Lévy measure.

vcb ,,

Rb

Rc

v

Lévy Processes

dxvxuiecu

buiu xxui 1

2

112

)(

Appendix

The Lévy measure is a measure on which satisfies

.

0R

2 1 ( ) x v dxR

This means that a Lévy measure has no mass at the origin, but infinitely many jumps can occur around the origin.

The Lévy measure describes the expected number of jumps of a certain height in a time in interval length 1.


The sum of all jumps smaller than some does not converge. However, the sum of the jumps compensated by their mean does converge. This pecularity leads to the necessity of the compensator term .

If the Lévy measure is of the form , then f (x) is called the Lévy density.

In the same way as the instant volatility describes the local uncertainty of a diffusion, the Lévy density describes the local uncertainity of a pure jump process.

The Lévy-Khintchine Formula allows us to study the distributional properties of a Lévy process. Another key concept, the Lévy-Ito Decomposition Theorem, allows one to describe the structure of a Lévy process sample path.

0

11 x

iux

dxxfdxv


Every Lévy process is a combination of a Brownian motion with drift and a possibly infinite sum of independent compound Poisson processes. This also means that every Lévy process can be approximated with arbitrary precision by a jump-diffusion process. In particular, the Lévy measure v describes arrival rates for jumps of every size for each component of Lt .

Jumps of sizes in the set A occur according to a Poisson process

with intensity parameter , A being a any interval bounded away from 0.

The Lévy measure of the process L may also be defined as

A pure jump Lévy process can have a finite activity (aggregate jump arrival rate: finite)

or infinite activity (infinitely many jumps possibly occuring in any finite time interval).

A

dxv

A0 1

1 , .

-s s s s

s

v A E ΔL ΔL L L


A sample path of an NIG Lévy process is drawn with jumps being of irregular size

which implies the very jagged shape of the picture.

Source: Barndorff Nielsen and N.Shephard (2002) , pp. 22.


Normal Inverse Gaussian Distribution

The normal inverse Gaussian (NIG) distributions were introduced by Barndorff-Nielsen (1995) as a subclass of generalized hyperbolic laws with , so that .

The normal inverse Gaussian (NIG) distribution is defined by the following

probability density function:

,

2

1

),,,,21;(),,,;( xfxf GHNIG

2)(2

2)(2(1)(22exp),,,;(

x

xKxxNIGf

Appendix

where , 0 ≤│β│≤ α

and K1 is the modified Bessel function of third kind with index 1:

The tail behavior of the NIG density is characterized by the following asymptotic relation:

, , 0 R Rx

22

10

( ) exp ( ).4 4

R .y y

K y t t dt yt

3 2 ( ); , , , ( ). - mα + β xNIGf x α β μ δ x e x

Normal Inverse Gaussian Distribution Appendix

NIG-density for different values of ; here, and . 0 1


NIG-density for different values of ; here, and . 5 0


NIG Lévy Asset Price Model

Under the probability measure P, we consider the modeling of the asset price process as the exponential of an Lévy process

where L is an NIG Lévy process.

The NIG distribution is infinitely divisible and, hence, it generates a Lévy process.

The log-returns of the model have independent and stationary increments.

0 e , tLtS S

log log , , , . t s t sX S S L L NIG t s t s

Appendix

Any infinitely divisible distribution X generates a Lévy process . According to the construction, increments of length 1 have distribution X , i.e., .

But, in general, none of the increments of length different from 1 has a distribution of the same class.

The price process is the solution of the stochastic differential equation

where is an NIG Lévy process and

is the jump of L at time t. The solution of above SDE is given by

0ttL

XLd

1

e 1 , tL

t t ttd S S d L L

0ttL

ttt LLL

0 exp .t tS S L

NIG Lévy Asset Price Model Appendix

is the unique solution of the following equation: . Under the corresponding probability , the process is again a Lévy process

which is called the Esscher transform. This Esscher equivalent martingale measure is given by .

0

2 2 2 2( ) ( 1) r

Q

Q


Simulating NIG Distributed Random Variables

We consider the simulation algorithm for sampling from an NIG (α, β, μ, δ) distributed random variable L.

o Sample Z from .

o Sample Y from N (0,1) .

o Return

An IG variate Z can be sampled as follows,

firstly drawing a random variable ,

which is distributed, defining a random variable:

222 , IG

. L Z Z Y

V

)1(2

T.H. Rydberg (1997)

Appendix

and then letting

being uniform distributed and . We see that to sample, a standard normal Y, a distributed and a uniform appear.

22222

2

422

VVV

W

11

21 1 ;

UU

WW

Z WW

1U 22/

2 V 1U

Simulating NIG Distributed Random Variables Appendix

The probability density function of the inverse Gaussian distribution:

The parameters a and b are functions of α and β : , .

2

3 2 ( ); , exp ( 0).

22

a a b xIG x a b x x

bxb

2 2 a 22 b

Simulating NIG Distributed Random Variables Appendix

DAX Empirical and Simulation

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

Emp.

Sim.


Asset Price Dynamics

Dynamics of Jump Models:

• Jump-Diffusion Models,

• Infinite Activity Lévy Models.

Appendix

Jump Diffusion Models

• Special cases of exponential-Lévy models.• Frequency of jumps is finite.• Stock price follows a geometric Brownian motion between jumps. S t

Appendix

• Merton’s Model

• Tries to capture skewness and excess kurtosis of the log return density.

• Stock price has the following SDE:

with : a Brownian motion, and

: a compound Poisson process with i.i.d. lognormally distributed jump sizes.

, dS t S t dt S t dW t S t dJ t

1

N t

ii

J t V

W t

Jump Diffusion Models Appendix

• Kou’s Model

• Employed to produce analytical solutions for path-dependent options, (barrier, lookback) because of the memoryless property of exponential density.

• Stock price has the following SDE:

with : a Brownian motion, and : a Poisson process with asymmetric double exponentially distributed logarithmic jump sizes.

1

1 ,N t

ii

dS t S t dt S t dW t S t d V

W t

N t


• Bates’s Model

• Combines compound Poisson jumps and stochastic volatility.• Easy to simulate and efficient MC methods can be employed for

pricing path-dependent options.• Dynamics of the stock price is given by the following SDEs:

with , : Brownian motions with non-zero correlation, and

: a compound Poisson process.• Can also be considered as a SV extension of the Merton’s model. • Jumps of the log-price process do not have to be Gaussian.

1 ,

tt t t

t

dSdt V dW dQ

S

2 ,t t t tdV V dt V dW

1W t 2W t

Q t


Infinite Activity Lévy Models

• Infinitely many jumps in any finite time interval.

• Empirical performance of these models generally is not improved by adding a diffusion component.

• Easier to calibrate than JD models since• the global error declines rapidly,• all the parameters vary simultaneously from the initial ones

during the calibration.• High activity is accounted for by a large/infinite number of small jumps.

Appendix

http://upload.wikimedia.org/wikipedia/commons/d/d7/Variance-Gamma-process.png

Variance Gamma Process

• The characteristic function is given by

• Infinitely divisible distribution. • Can also be defined by considering as time-changed Brownian motion

with drift

1

2 21; , , 1 .

2

VG u iu u

. VGt ttX W

Appendix

Normal Inverse Gaussian Process


• Can also be defined by Inverse Gaussian time-changed Brownian motion.• Negative and positive values of result in negative and positive skewness,

respectively.

22 2 2; , , , exp .

NIG u iu iu

Appendix

Meixner Process


• Special case of the Generalized z (GZ) distributions.• Moments of all orders exist.• Has semi-heavy tails.

2

cos2

; , , , exp .cosh

2

Meixner u iuu i

Appendix

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Tastan , M., Pickl, S.W., and Weber, G.-W., Mathematical modeling and stability analysis of gene-expression patterns in an extended space and with Runge-Kutta discretization, Proceedings of Operations Research, Bremen, 2006, 443-450.Wunderling, R., Paralleler und objektorientierter Simplex Algorithmus, PhD Thesis. Technical Report ZIB-TR 96-09. Technische Universitat Berlin, Berlin, 1996.Weber, G.-W., Alparslan -Gök, S.Z., and Dikmen, N., Environmental and life sciences: Gene-environment networks-optimization, games and control - a survey on recent achievements, deTombe, D. (guest ed.), special issue of Journal of Organizational Transformation and Social Change 5, 3 (2008) 197-233.Weber, G.-W., Taylan, P., Alparslan-Gök, S.Z., Özögur, S., and Akteke-Öztürk, B., Optimization of gene-environment networks in the presence of errors and uncertainty with Chebychev approximation, TOP 16, 2 (2008) 284-318.Weber, G.-W., Alparslan-Gök, S.Z ., and Söyler, B., A new mathematical approach in environmental and life sciences: gene-environment networks and their dynamics,Environmental Modeling & Assessment 14, 2 (2009) 267-288.Weber, G.-W., and Ugur, O., Optimizing gene-environment networks: generalized semi-infinite programming approach with intervals, Proceedings of International Symposium on Health Informatics and Bioinformatics Turkey '07, HIBIT, Antalya, Turkey, April 30 - May 2 (2007).Yılmaz, F.B., A Mathematical Modeling and Approximation of Gene Expression Patterns by Linear and Quadratic Regulatory Relations and Analysis of Gene Networks, MSc Thesis, Institute of Applied Mathematics, METU, Turkey, 2004.

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Dynamics under Various Assumptions on Time and Uncertainty

Technology

primary school enrollment

school number of primary

theprimary education

basic education

elementary education

children of primary

system variables

elementary school