CEJOR (2012) 20:319–336
DOI 10.1007/s10100-010-0183-2
ORIGINAL PAPER
A system dynamics model for intentional transmission
of HIV/AIDS using cross impact analysis
Chandra Sekhar Pedamallu · Linet Ozdamar ·
Erik Kropat · Gerhard-Wilhelm Weber
Published online: 1 December 2010
© Springer-Verlag 2010
Abstract The system dynamics approach is a holistic way of solving problems
in real-time scenarios. This is a powerful methodology and computer simulation
C. S. Pedamallu
Department of Medical Oncology, Dana-Farber Cancer Institute,
Boston, MA, USA
e-mail: pcs.murali@gmail.com
C. S. Pedamallu
The Broad Institute of MIT and Harvard, Cambridge, MA, USA
L. Ozdamar
Department of Systems Engineering, Yeditepe University, Kayisdagi, Istanbul 34755, Turkey
e-mail: ozdamar.linet@gmail.com; ozdamar@yeditepe.edu.tr
E. Kropat
Department of Informatics, Institute for Theoretical Computer Science,
Mathematics and Operations Research, Universität der Bundeswehr München,
Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany
e-mail: kropat@am.uni-erlangen.de
G.-W. Weber (B)
Departments of Financial Mathematics, Scientific Computing and Actuarial Sciences,
Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey
e-mail: gweber@metu.edu.tr
G.-W. Weber
Faculty of Economics, Business and Law, University of Siegen, Siegen, Germany
G.-W. Weber
Center for Research on Optimization and Control, University of Aveiro, Aveiro, Portugal
G.-W. Weber
Universiti Teknologi Malaysia, Skudai, Malaysia
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modeling technique for framing, understanding, and discussing complex issues and
problems. System dynamics modeling and simulation is often the background of a
systemic thinking approach and has become a management and organizational development paradigm. This paper proposes a system dynamics approach for modeling the
phenomenon of intentional transmission of HIV/AIDS by non-disclosure. The model
is built using the Cross Impact Analysis (CIA) method of relating entities and attributes relevant to the risky conduct of HIV+ individuals in any given community. The
proposed model uses a hypothetical cross impact matrix that relates pairs of attributes.
The factors that impact non-disclosure are identified by simulating the model through
dynamic difference equations. After the simulation results are reviewed, two policies
are introduced and tested in order to observe the progress in the system state.
Keywords HIV transmission by non-disclosure · Cross impact analysis ·
System dynamics
1 Problem background
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+ persons continues to rise despite national and international HIV prevention
policies and dedicated public healthcare strategies (for HIV/AIDS epidemic statistics
please refer to World Health Organization website: http://www.who.int/hiv/data/en/).
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 becomes 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.
Non-disclosure of infection is deemed a criminal act in some countries even if sex
is reported to be consensual. Intentional transmission of the virus also occurs through
unconsensual sex by intimate partners and/or strangers, through prostitution, through
the sales of infected blood for money, through syringe sharing among drug addicts,
through biting and scratching by HIV+ persons, and through homosexual transmission
among prison inmates (Human Rights Watch 2001). However, not much is being done
in legal terms. For instance, a survey on HIV transmission prosecutions in the U.S.A.
is given by Bray (2003). In rape cases, the female gender is the target (ref. to Outwater
et al. 2005 for violence, sexual abuse and intentional transmission of AIDS to women in
South Africa) whereas in consensual sex and prostitution both genders become victims.
Cases of deliberate syringe injection may target both passers-by who are subjected
to robbery and security people who wish to enforce the law. Transmission through
syringe injection and biting might usually target security people and others while settling disputes physically. Bray (2003) analyzes 316 cases from the U.S.A. and arrives
at the conclusion that prosecutions are few and they are not likely to serve the public
health purpose of reducing HIV transmission from those who know they are infected.
An on-line survey conducted by UKC (UK Coalition of People living with HIV and
AIDS) suggests that prosecution or criminalization of HIV transmission would not
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induce HIV disclosure. This view is also shared by some others (Kenney 1992;
Hermann 1990). However, all parties agree that more stringent legal actions should be
adopted against rape crimes (whether the crime is committed by an intimate partner,
husband or stranger) in order to reduce HIV transmission. The latter is important
in developed countries (Maman et al. 2000) as well as developing ones (Jewkes
and Abrahams 2002) where sexual abuse is common and socio-economic and sexual empowerment of women is weak.
Here, we propose to build a cross impact analysis (CIA) model to describe the intentional transmission of HIV/AIDS in correlation to socio-economic factors related to
the individual and the individual’s social network. The goal in developing this model
is to predict the effects of some factors on the intentional 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 our knowledge, the intentional transmission of HIV by nondisclosure has not been analyzed by a formal Operations Research method. The CIA
that is proposed here is expected to fill this gap. However, since detailed non-disclosure
data were not available to us, the cross impact matrix utilized in our simulations is
hypothetical. Consequently, our results should be justified with non-disclosure survey
data. If such data were available, then, better intervention policies can be developed.
Several versions of the CIA have been developed by researchers and applications
exist in various social areas (Gordon and Hayward 1968; Mclean 1976; Sarin 1978;
Novak and Lorant 1978; Wissema and Benes 1980; Helmer 1981; Gordon 1994;
Pedamallu et al. 2010). In the CIA model, the cross impact among impacting factors
(attributes) are measured by pairwise correlation analysis. Correlation parameters that
might be extracted from survey data are normalized into a step function for each attribute pair and a cross impact matrix is built. The elements of the cross impact matrix
are fed into difference equations that change the level of the model variables during the
simulation. The significance of the model variables are then identified and analyzed.
The CIA model is then augmented with policy variables that are designed to impact
the viral transmission rate, and the effects of these policies are observed.
The proposed cross impact model relates the intentional transmission of HIV to factors such as the donor’s income level, injection drug usage, educational background,
strength of family ties, promiscuity, criminal records and health care infrastructure.
Some of these factors are analyzed in Atun et al. (2007), Goncalves et al. (2004),
Odimegwu (2003), and in the annual HIV monitoring reports of the U.S. Center for
Disease Control (2008).
In the subsequent sections, we provide a literature survey on the Operations
Research methods designed to model the transmission of the virus. Then we explain
the CIA method and simulate the model using a hypothetical cross impact matrix
where two policies that might alleviate the system are introduced. We summarize the
study with concluding remarks.
2 Literature survey
Many studies exist on HIV transmission, however, mostly, these studies do not differentiate between intentional and non-intentional transmission. Early surveys of mathematical and statistical methods developed for HIV/AIDS transmission are found in Isham
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(1988), Anderson and May (1991), Schwager et al. (1989) and Fusaro et al. (1989).
An early approach used for the prediction of AIDS cases is extrapolation (Morgan
and Curran 1986; Karon et al. 1989) where separate curves and extrapolations are
carried out for various risk groups. Advantages of extrapolation are its simplicity and
ease of use, but it does not consider factors such as behavioral changes or saturation
in the high risk groups. Another approach is back calculation (Brookmeyer and Gail
1988) which is a deconvolution process that uses a given AIDS incidence up to time
t and an estimated distribution for the AIDS incubation period to estimate the HIV
incidence up to time t. Similar to extrapolation, back calculation does not yield any
information on the HIV transmission dynamics. May and Anderson (1988) develop
HIV transmission dynamics models that represent the progression from HIV+ status to
AIDS where the population is divided into categories of progressive infectious stages.
These models translate the movements between these stages into difference equations
in the deterministic case and into state transition probabilities in the stochastic case.
Comparisons between deterministic and stochastic models are made using expected
values and simulation (Mode et al. 1989) and reviews on both types of models exist in
the literature (e.g., Anderson and May 1991). Tan (1991) proposes general stochastic
models for the simulation of various transmittable diseases including AIDS. Hyman
and Stanley (1989) consider both continuous and discrete HIV/AIDS models with
heterogeneity and different mixing structures that analyze the spread from high to low
risk groups, the effects of variable infectivity and the instability of the back calculation procedure. Dietz and Hadeler (1988) consider the dynamics of pairs of individuals
and the duration of their partnerships explicitly in their dynamic pair formation and
dissolution models. This approach is distinctly different from the contact rate mixing
matrix models used by many researchers. A more detailed review on these early models developed for HIV transmission in the United States is available in http://biotech.
law.lsu.edu/cphl/Models/aids/. A web site that summarizes such analytical tools and
models with an extensive bibliography is maintained by International Aids Economics
Network (IAEN: http://www.iaen.org/models/).
HIV researchers have long appreciated the need to understand the social and behavioral determinants of HIV related risk behavior, and the methods for this type of analysis are well established. For instance, Rauner and Brandeau (2001) present an overview
of the key issues related to developing useful AIDS policy models and highlight geographical area, setting, target groups, affordability and effectiveness of interventions,
and economic analysis as important issues in policy development.
A model that computes HIV / AIDS transmission rates is given by Cassels et al.
(2008). Ajay et al. (2009) develop a mathematical model with explicit behavioral
foundations to explore an array of policy interventions related to HIV transmission among injection drug users. Since the epidemic began, injection drug use has
directly and indirectly accounted for more than 36% of AIDS cases in the United
States (http://www.cdc.gov/hiv/resources/Factsheets/idu.htm). The distributing trend
appears to continuing. Kimbir and Oduwole (2008) propose a mathematical model
of HIV/ AIDS transmission dynamics considering counseling and antiretroviral therapy as major means of control of infection. Analytical and numerical results obtained
indicate that Antiretroviral therapy (ART) and counseling could be effective methods in the control and eradication of HIV/ AIDS. Abdulkarim and Ndakwo (2007)
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323
study the susceptible-exposed-infectious-aids, epidemic model for vertical transmission of HIV / AIDS in a homogeneous mixing population. Rauner (2005) present a
discrete-event simulation model that evaluates the relative benefits of two potentially
affordable interventions aimed at preventing mother-to-child transmission of HIV,
namely anti-retroviral treatment at childbirth and/or bottlefeeding strategies. Huiyu
et al. (2009) develop an extended CIA simulation model to study the dynamical behaviors of HIV/AIDS transmission. The model incorporates heterogeneity into agents’
behaviors. Agents have various attributes such as infectivity and susceptibility, varying
degrees of influence on their neighbors and different mobilities.
Although mathematical models on viral transmission exist, they are rather vague
about accounting for the psyche and behavioral pattern of the HIV+ person. Under
what conditions would an infected person disclose his/her condition to a potential
partner or to work colleagues without damaging his social and financial status? The
current approach about the disclosure of HIV status in many developing countries is to
keep it under cover because of the possible physical isolation of the disclosing person
and discrimination to an extreme extent, job loss, and so on. This does not change
even when public awareness of transmission channels and protection against the virus
is high (Odimegwu 2003). Stigmatization and the anger it creates on the part of the
infected person, and the absence of learned social responsibility and selfishness all
lead to the cover up of the infected person in the majority of homosexual/heterosexual
casual/non-casual sexual encounters. It is possible to identify many non-disclosure
cases in marital sex that lasts until the symptoms of the illness become too apparent
to hide (Niccolai et al. 1999). In these cases, the patient’s family may get infected
with the disease and also become financially depleted and outcast from their community.
Previous studies have shown concerns on continuing risky sexual behavior among
HIV+ individuals. Stein et al. (1998) interview 203 consecutive patients presenting
for HIV at Boston city hospital and Rhode Island Hospital and found that 40% of
a sample of HIV+ patients recorded at urban hospitals have not disclosed their HIV
status to all sexual partners, and that among participants who do not disclose their
status to their sexual partners, 57% use condoms less than all the time. Similarly,
Singh et al. (1993) find that more than half of HIV+ men and nearly half of HIV+
women studied have not disclosed their HIV status to a sex partner in the previous six
months. Among those who report practicing unprotected anal or vaginal intercourse
during their most recent sexual encounter, 57% of men and 71% of women have not
disclosed their HIV status to that partner. 59% of men and 60% of women report that
their most recent partner with whom they had unprotected intercourse is not known
to be HIV+. Wenger et al. (1994) report that 22% of HIV+ study participants state
that their last partner is known to be HIV negative or that they do not know their partner’s status, and that they had unprotected vaginal, anal, or oral sex with that partner.
Among those whose partners who are HIV negative, 24% of the partners are unaware
that the subject is HIV+, while 41% of partners of unknown status are aware that
the subject was HIV+. Another study shows that 23% of HIV+ subjects reported not
using a condom with a person to whom their status is not disclosed (Niccolai et al.
1999).
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3 The cross impact method and system definition for intentional transmission
and spread of HIV/AIDS
The CIA is one of the most popular systems thinking approaches developed for identifying the relationship among the variables defining systems. We first describe the
steps to be followed for understanding and building a systems model (Julius 2002;
Monto et al. 2005).
3.1 Definition of the system
Systems are made of entities, which interact with each other and produce some outputs, which are either designed or natural. A system receives inputs and converts them
through a process and produces outputs. All the outputs of a system do not need to be
desirable. In the present context, the system represents the intentional transmission of
HIV/AIDS.
3.1.1 Environment
Every system functions in an Environment, which provides inputs to the system and
receives outputs from the system. In our context, the Environment is the society.
3.1.2 Structure
All systems have a Structure. The ‘body’ of a system’s structure is represented by
the entities of the system and their interrelationships or linkages or connections. The
entities in our system are defined as follows.
1.
2.
3.
4.
HIV+: Informed and disclosing HIV+ patient,
FM: Family Members of HIV+ patient,
WC: Work Colleagues of HIV+ patient,
HI: Healthcare Infrastructure, i.e., hospital, clinic, lab facilities used by HIV+
patients in the sample,
5. LC: Local Community in which HIV+ patient lives.
3.1.3 Linkages
The linkages among entities may be physical (e.g., sexual contacts, medical equipment, syringes and so on), electro-magnetic (e.g., electrical, electronic systems, and so
on), and information-based (e.g., mass communications systems, legal systems, and so
on). It is important to try and understand, what linkages make the system’s structure,
which entities are linked with each other, and the implications of these linkages on
the behavior of the entities in particular. The entity relationship diagram of the system
is transformed into a matrix for better illustration of relationships among the entities
identified above. The entity relationship diagram of our system is illustrated in Fig. 1.
Exchange of matter, information and/or spirit between two entities causes a change in
the state of both entities. This is reflected as system behavior.
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Fig. 1 Entity relationship diagram for the system
3.2 System entities and relationships equations
The dynamic change of the system state is referred to as system behavior. The state
of a system is an instantaneous snapshot of levels of the relevant attributes possessed
by the entities that constitute the system. In all systems, every entity possesses many
attributes, but only a few attributes are ‘relevant’ with reference to the problem at
hand. Some attributes are of immediate or short-term relevance while others may be
of relevance in the long run. The choice of relevant attributes has to be made carefully,
keeping in mind both the short-term and long-term consequences of solutions (decisions). All attributes can be associated with given levels that may indicate quantitative
or qualitative possession.
The set of attributes identified for the model are given below. These attributes are
selected based on previous studies that analyze the prevalence of HIV/AIDS in different categories of the population, such as the poor and the uneducated, and those who
lack of access to proper health care (UNAIDS 2008; U.S. Center for Disease Control 2008, WHO website: http://www.who.int/hiv/data/global_data/en/index.html;
http://www.who.int/hiv/data/Table4.1_2010.png). The prevalence of HIV in those
who inject drugs is discussed in Singh et al. (1993), Atun et al. (2007), and in Ajay
et al. (2009) whereas sexual behavior patterns and HIV spread are found in Hyman
and Stanley (1989) and Niccolai et al. (1999). Prostitution is another impacting factor (Talbott 2007) as well as level of promiscuity (Goncalves et al. 2004). Finally,
many individuals contract HIV by rape (Maman et al. 2000; Outwater et al. 2005).
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We assume that similar factors and the social stigmatization of the donor defined by
family, colleagues, sexual partners, etc. affect non-disclosure of HIV status.
Entity 1. HIV+ Patient:
1.1. Level of income (HIV-LIC).
1.2. Level of awareness related to viral transmission channels (HIV-LAVTC).
1.3. Level of anti-social attitude and intentional harming potential (records of vandalism, psychological disturbances and aggressive behavior exhibited in public,
and so on) and criminal record (non-existence of, or, if existent, then the number
of crimes and their types, imprisonment) (HIV-LATI).
1.4. Level of education (HIV-LE).
1.5. Level/frequency of sexual activity pattern, duration of relationships
(HIV-LSAP).
1.6. Type of relationship: homosexual/men (HIV-LTHOM).
1.7. Type of relationship: homosexual/women (HIV-LTHOW).
1.8. Type of relationship: heterosexual (HIV-TRH).
1.9. For females: Level of empowerment in sexual relationships and ability to avoid
unconsensual and unsafe sex with intimate partner (HIV-ESR).
1.10. Usage of common drug syringes (HIV-UCY).
1.11. Level of good family relationships (HIV-LGFR).
1.12. Frequency of unsafe sexual intercourse and non-disclosure patterns
(HIV-FUSID).
Entity 2. Family Members (FM):
2.1. Level of awareness related to viral transmission channels (FM-LAVTC).
2.2. Level of psychological and financial support for family members with HIV+
(FM-LPFFH).
2.3. Level of education (FM-LE).
2.4. Usage of common drug syringes (FM-UCY).
Entity 3. Work Colleagues (WC):
3.1.
3.2.
3.3.
3.4.
3.5.
Level of awareness related to viral transmission channels (WC-LAVTC).
Level of competition and professional aggression (WC-LCPA).
Level of social conscience (e.g., involvement in social aid activities) (WC-LSC).
Usage of common drug syringes (WC-UCY).
Level of exposure to previous infection cases (WC-LEPI).
Entity 4. Healthcare Infrastructure (HI):
4.1. Level of sterility and facility maintenance (HI-LSFM).
4.2. Level of healthcare worker awareness and negligence and sloppy behavior at
work (HI-LHABS).
4.3. Level of support and acceptance for HIV+ patients with known status
(HI-LSAH).
4.4. Frequency that the facility accepts HIV+ patients (HI-FFH).
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327
Entity 5. Local Community (LC):
5.1. Level of awareness of HIV infection and AIDS (LC-LAHA).
5.2. Level of community solidarity and compassion (LC-LCSC).
5.3. Level of taboo in the community – conservatism caused by religion and other
social factors (LC-LTC).
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 behavior. 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. Therefore, interrelationships among variables should be analyzed carefully before introducing new
policies.
The following conjectures are valid in the systems approach (the following subsection is motivated from Julius 2002).
a. Modeling and forecasting the behavior of complex systems are necessary if we
are to exert some degree of control over them.
b. Properties of variables and interactions in large scale system variables are bounded
such that:
1. System variables are bounded. It is now widely recognized that the level of
any variable cannot increase indefinitely. There must be distinct limits. In an
appropriate set of units these can always be set to one and zero.
2. A variable level increases or decreases according to whether the net impact
of the other variables is positive or negative.
3. A variable’s response to a given impact decreases to zero as that variable
approaches its upper or lower bound. It is generally found that bounded growth
and decay processes exhibit this sigmoidal character.
4. All other things being equal, a variable (attribute) will produce greater impact
on the system as it grows larger (ceteris paribus).
5. Complex interactions are described by a looped network of binary interactions
(this is the basis of cross impact analysis).
With these conditions in mind consider the following mathematical structure. Since
state variables (xi (t)) are bounded from above and below, they can be rescaled to the
range between zero and one. Thus for each variable we have
0 < xi (t) < 1,
for i = 1, 2, . . ., N and ∀t > 0
(1)
where, xi (t) is the level of variable i in period t.
To preserve boundedness, xi (t + t) is calculated by the transformation
xi (t + t) = xi (t) Pi (t) ∀i, t,
(2)
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where the exponent Pi (t) is given by
Pi (t) =
1+
1+
t
2
t
2
N
j=1 (|αi j | −
αi j )x j (t)
j=1 (|αi j | +
αi j )x j (t)
N
∀ i, t,
(3)
where αi j are cross impact matrix elements indicating the impact of variable x j on xi
and t is the time period of one iteration of the system’s simulation.
Equation (3) guarantees that Pi (t) > 0 for all i = 1, 2, . . . , N and all t > 0. Thus
the transformation (2) maps the open interval (0, 1) onto itself, preserving boundedness of the state variables (condition 1 above). Equation (3) can be made somewhat
clearer if we write it in the following form:
Pi (t) =
1 + t|sum of negative impacts on xi |
.
1 + t|sum of positive impacts on xi |
(4)
The rate of change in the level of an attribute is defined as:
d xi
= −
αi j x j (t)xi (t)ln(x j (t)).
dt
N
(5)
j=1
4 Simulating the system using cross impact analysis
4.1 Steps in the simulation
There are four steps to follow while implementing the CIA in our case. We first conduct the simulation without any policy intervention and observe where the attribute
states such as HIV-FUSID (HIV+ patients who do not disclose their status) converge.
Then, we simulate the system again after augmenting the cross impact matrix with
selected policy variables. The four steps of the simulation are explained below.
Step 1. Set the initial attribute levels.
Step 2. Build a cross impact matrix with the selected attributes. The parameters αi j
can be determined by using a pairwise correlation matrix extracted from
survey data, and adjusting the correlation values by subjective assessment.
Table 1 illustrates how qualitative impacts can be quantified subjectively as
a step function.
Step 3. Simulate the system for ‘m’ iterations using Eqs. (2) and (3), and tabulate the
state level of each attribute in every iteration. Plot the results on a worksheet.
Step 4. Identify a policy variable to achieve the desired system state and augment
the cross impact matrix with this policy variable with qualitative assessment
of pairwise attribute interactions. Observe the system for ‘m’ iterations, and
check if the desired state is achieved by introducing the policy variable. Compare the results.
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Table 1 Impact rates of
attributes
Representation of impact
Value
329
Description
++++
0.8
Very strong positive effect
+++
0.6
Strong positive effect
++
0.4
Moderate positive effect
+
0.2
Mild positive effect
0
0
Neutral
_
−0.2
Mild negative effect
__
−0.4
Moderate negative effect
___
−0.6
Strong negative effect
____
−0.8
Very strong negative effect
4.2 An illustrative example
Here, we present a hypothetical example to illustrate the CIA method.
Implementation of Step 1:
The initial attribute levels that are needed to start up the simulation by Eqs. (2) and
(3) can be hypothesized by talking to experts in the field. For the current hypothetical
simulation, an initial value of 0.25 is assigned to each attribute.
Implementation of Step 2:
In Table 2, we provide the hypothetical cross impact matrix between the variables.
This matrix is built by grading the influence of one variable over another. For example, having good family relationships, HIV-LGFR, has a strong negative effect on
anti-social behavior, HIV-LATI.
Implementation of Step 3:
In our implementation, we execute a simulation of m = 20 iterations.
Implementation of Step 4:
Two intervention policies are suggested here for illustrative purpose.
P1. A symbolic payment to the patient that is conditional on compulsory monthly
visits to a clinic where a urine test is conducted for drug abuse among other tests.
These visits would also provide a means to provide psychological support to the HIV+
individual. During a brief confidential session with the psychologist, the patient might
be asked questions on promiscuity and unsafe sex. This policy would be expensive for
Medicare if the person is uninsured. This policy might improve the HIV+ individual’s
responsibility perception towards society.
P2. More aggressive community awareness programs for the society to remove the
AIDS stigma. Adjustment in school curricula for including aids in relevant books,
regular adult seminars in non-urban areas, free voluntary blood test campaigns might
increase the awareness in viral transmission while mobilizing social empathy towards
AIDS patients. This policy has a wider target and might improve social responsibility
towards AIDS patients.
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0
0
0
0
0
WC-LEPI
HI-LSFM
HI-LHABS
0
FM-UCY
0
0
FM-LE
WC-UCY
0.2
FM-LPFFH
0
0
FM-LAVTC
WC-LSC
0
HIV-FUSID
0.2
0.2
HIV-LGFR
0.2
−0.2
HIV-UCY
WC-LCPA
0
0
HIV-ESR
WC-LAVTC
0
0
HIV-TRH
0
0
0
−0.2
−0.4
−0.4
0
−0.6
−0.2
0.2
0
0
0
0
0
−0.4
0.4
0.2
0
−0.2
0.2
0
0.2
0.2
0.2
0
0
−0.6
0
−0.2
−0.4
0
0
0
0
0
0
0
0.2
0
0
0
0
−0.4
−0.2
0.4
−0.2
0
0
0
0.2
0
0.2
0
0
0
0.2
0
0.2
0
0
HIV-LTHOW −0.2
0
0
HIV-LSAP
HIV-LTHOM −0.2
0.4
0.4
−0.2
HIV-LE
HIV-LATI
0
0
HIV-LAVTC
0.2
0
HIV-LIC
0
0
0
0
0
0
−0.2
0.2
0
0
0
0.6
−0.2
0.4
−0.4
0.2
0.2
0.6
0.2
0
0.4
−0.4
0
0
0
0
0
0
0
0
0
0
0
0
0.4
0
0
−0.4
0
0
0
0
0
0.6
−0.2
0
0
0
0
0
0
0
0
0
0
0
0
0.2
0
0
−0.4
0
0
0
0.2
0
0.2
−0.2
0
0
0
0
0
0
0
0
0
0
0
0
0.2
0
0
0
0
−0.2
−0.4
0
0
0.4
0.4
0
0
0
0
0
0
0.4
0
0
0.4
0
0.2
0
−0.2
0
−0.2
0.2
0.4
0
0
0
0
0
0
0
−0.2
−0.2
0.2
−0.2
0.2
−0.2
0
−0.4
0.2
0.4
−0.4
−0.4
−0.2
0.2
−0.4
0.2
0
0.2
−0.2
0
−0.2
0
0
0
0
0
0.2
0.2
−0.4
0.2
−0.4
0
0.2
−0.2
0.4
0.4
−0.2
−0.2
−0.2
−0.2
−0.2
0.4
−0.4
0.4
0.4
0.4
0
0
0
0
0.2
−0.2
0
0
−0.2
−0.2
0
0
0.4
−0.4
0
0.2
0.4
0.4
−0.2
0.6
−0.4
0
0
0
0
0
0
0
0
0
0.4
0
0
0
0
0
0
0
0
0
0
0
0
0.2
0.2
−0.4
0
0
0
0
0
0
−0.4
0.2
0
0
0
0
0
0
0.2
−0.2
−0.2
0
0
−0.4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.2
HIV-LIC HIV-LAVTC HIV-LATI HIV-LE HIV-LSAP HIV-LTHOM HIV-LTHOW HIV-TRH HIV-ESR HIV-UCY HIV-LGFR HIV-FUSID FM-LAVTC FM-LPFFH FM-LE
Table 2 Hypothetical cross impact matrix for pairs of attributes (without policies)
330
C. S. Pedamallu et al.
0
0
LC-LAHA
LC-LCSC
LC-LTC
0
0
0
0.2
0
0
0
0
0
0.2
HIV-LTHOM 0.2
HIV-LTHOW 0
0
0
0
0
0
HIV-LSAP
HIV-TRH
HIV-ESR
HIV-UCY
HIV-LGFR
HIV-FUSID
0
HIV-LE
0
0
0
−0.2
FM-LPFFH
FM-LE
0
FM-LAVTC −0.2
0
0
0
0
0
0
0
HIV-LATI
0
0
0.2
HIV-LAVTC −0.2
HIV-LIC
0
HI-FFH
0
−0.6
0
0
−0.4
0.6
0
0
−0.4
−0.4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
FM-UCY WC-LAVTC WC-LCPA WC-LSC WC-USY WC-LEPI
0
0
HI-LSAH
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
HI-LSFM
0
0
0
0
0
0
0
0
0
0
0
0
0.2
0
−0.2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
HI-LHABS HI-SAH HI-FFH
0
0
0
0
0
0.6
−0.4
−0.4
−0.2
−0.2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.4
0.2
−0.4
−0.2
0
0
0.2
0.2
0
0
−0.2
0
0
0
0
−0.2
0.2
−0.2
0
0.2
0
0
−0.2
0.2
−0.2
0
0
0
0
0
0
0
0
0
Policy
0
0.4
0
0
0
−0.4
0.4
0
0.4
0
0
0.2
−0.2
0.2
0
−0.4
0
−0.2
0
LC-LTC
0
0
0.2
0
0
0.2
0
LC-LAHA LC-LCSC
0
0
0
0
0
HIV-LIC HIV-LAVTC HIV-LATI HIV-LE HIV-LSAP HIV-LTHOM HIV-LTHOW HIV-TRH HIV-ESR HIV-UCY HIV-LGFR HIV-FUSID FM-LAVTC FM-LPFFH FM-LE
Table 2 continued
A system dynamics model for intentional transmission of HIV/AIDS
331
123
123
0
0
0
0.2
0
0
0
0.2
−0.2
0
−0.2
0
0
WC-UCY
WC-LEPI
HI-LSFM
HI-LHABS
HI-LSAH
HI-FFH
LC-LAHA
LC-LCSC
LC-LTC
0
0
0
0
0.2
0
0.4
0
0
0
0
WC-LSC
0
WC-LAVTC
WC-LCPA
0
FM-UCY
0.2
−0.2
−0.2
0
0
0
0
0
0
0
0
0
0
0
0.2
0
0
0
0
0
−0.2
0
0
0
0
0
0
0
0
−0.4
−0.2
0
−0.2
0.2
0
−0.2
0
0
0.2
0
0
0.4
−0.4
0
0
−0.2
0
0
−0.4
0
0.2
0
0
0
0
0.2
0
0
0.2
0
−0.4
0
0
0
0
0
0
0
0
0
0
−0.2
0
0
0
0
0
0
0
0
−0.6
0
0.2
0
0.4
−0.4
0
0
0
0
0
0
−0.6
0.2
0.4
0
−0.4
0
0
0
0
0
0
0
0
0
0
0
0
−0.2
0
0
0
0.2
0
0
0
−0.6
0
0.4
0
0
0
0
0
−0.2
0
0
0
0
0
0
0
0
−0.4
0
0
−0.4
0
−0.2
0
0
0
0
0
0
0
−0.2
−0.2
0
0
0.2
0
0
0
0
FM-UCY WC-LAVTC WC-LCPA WC-LSC WC-USY WC-LEPI HI-LSFM HI-LHABS HI-SAH HI-FFH LC-LAHA LC-LCSC LC-LTC Policy
Table 2 continued
332
C. S. Pedamallu et al.
A system dynamics model for intentional transmission of HIV/AIDS
333
Table 3 The number of iterations where each variable reaches its desirable and undesirable state under
every policy during the simulation
Attributes
Initial state
Undesirable state
Desirable state
NO_P
P1
P2
HIV-LIC
0.25
0
1
20
17
20
18
HIV-LAVTC
0.25
0
1
6
20
HIV-LATI
0.25
1
0
20
5
6
HIV-LE
0.25
0
1
20
19
19
HIV-LSAP
0.25
1
0
19
7
8
HIV-LTHOM
0.25
1
0
20
20
20
HIV-LTHOW
0.25
1
0
20
20
20
HIV-TRH
0.25
1
0
20
20
20
HIV-ESR
0.25
0
1
5
20
20
HIV-UCY
0.25
1
0
20
5
6
HIV-LGFR
0.25
0
1
9
20
20
HIV-FUSID
0.25
1
0
20
6
8
FM-LAVTC
0.25
0
1
7
20
20
FM-LPFFH
0.25
0
1
9
20
20
FM-LE
0.25
0
1
18
20
20
FM-UCY
0.25
1
0
20
7
6
WC-LAVTC
0.25
0
1
7
20
20
WC-LCPA
0.25
1
0
20
20
20
20
WC-LSC
0.25
0
1
6
20
WC-UCY
0.25
1
0
20
20
9
WC-LEPI
0.25
1
0
20
20
17
20
HI-LSFM
0.25
0
1
20
20
HI-LHABS
0.25
1
0
20
7
4
HI-LSAH
0.25
0
1
7
20
20
20
HI-FFH
0.25
0
1
7
20
LC-LAHA
0.25
0
1
6
20
14
LC-LCSC
0.25
0
1
6
20
20
LC-LTC
0.25
1
0
20
6
4
Bold font indicate the iteration where a variable stabilizes at its undesirable state
Regular font indicate the iteration where a variable stabilizes at its desirable state
Discussion of simulation results
We summarize the simulation results in Table 3 where we define the desirable
and undesirable states for attributes. For instance, the desirable state of HIV-LATI
(anti-social behavior) is zero, whereas the desirable state for HIV-LATVTC (awareness of viral transmission channels) is one. The desirable state for HIV-FUSID (nondisclosure) is zero. We describe the simulation results in terms of the number of
iterations that an attribute reaches its desirable or undesirable state. In Table 3, the
numbers of iterations in which attributes reach their undesirable states are indicated
123
334
C. S. Pedamallu et al.
in bold against gray background. We report three sets of simulation results: No Policy
(NO_P), Policy 1 (P1) and Policy 2 (P2).
We observe that under column NO_P, all attributes reach their undesirable states
sooner or later during the simulation. Policy P1 has no effect on work colleague
related attributes, the sterility of health care facilities and the sexual preferences of
HIV+ individuals. This is an expected result, because sexual preference attributes are
inserted into the system to differentiate promiscuity levels of males and females. Sterility of a facility is a principle and not a result of AIDS patients that frequent the
facility. Unlike family members, work colleagues are not affected by the hospital visits, because family members might accompany the HIV+ individual during hospital
visits. More aware family members might increase empathy in their local community
by telling about their experiences to neighbors. Therefore, policy P1 has a positive
effect on these attributes. Policy P2 is more effective because it covers the all segments
of the society. Work colleagues would be positively affected by policy P2, except their
competitiveness. Ambition for more monetary reward is an attribute that most human
beings cannot trim. The two policies can further be compared upon the criterion of
the speed of reaching to a desirable state. For instance, policy P1 drives HIV-FUSID
(the attribute for non-disclosure) to its desirable state faster than policy P2, because
policy P1 is directly targeted at the HIV+ individual.
All these remarks should be read with care, being mindful of the fact that the cross
impact matrix is hypothetical. If survey data were available, then, it would be possible
to discuss the costs and benefits of policies more precisely.
5 Summary and conclusion
A cross impact model is developed here to study the intentional transmission of HIV by
non-disclosure of status in various risky situations. The cross impact matrix illustrates
the influence of one attribute over the others and it also has a provision to identify
the impact variables (i.e., policy variables) to control this epidemic. 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.
For illustrative purpose, we build a hypothetical cross impact matrix for the selected
attributes and simulate the system. We observe that if intervention policies are not
implemented, the system deteriorates to an undesirable state. Therefore, we introduce
two policies, the first involving monthly clinic visits to support the HIV+ individual
and the second involving a more aggressive public awareness program in order to
improve the AIDS stigma. Obviously, the first policy is more expensive than the second policy, but it affects the HIV+ individual’s disclosure status faster. Despite that the
second policy has a wider impact that affects all segments of the HIV+ individual’s
social network.
We remind the reader that these remarks are not conclusive and need to be supported
by survey data to build a more realistic cross impact matrix.
Through the illustrative example, we can see that it is not difficult to identify intervention policies that might have strong positive impact on disclosure status. It would
123
A system dynamics model for intentional transmission of HIV/AIDS
335
then be possible to conduct cost and benefit analysis by observing how fast they impact
the HIV+ individual’s sexual behavior. Furthermore, cross impact matrix may be further improved by using some recent approaches that are developed to solve real-time
problems with and without uncertainties (Weber et al. 2009, 2010).
Acknowledgments The authors wish to thank the anonymous referees for their constructive comments
and Mrs. Anupama Pedamallu for helping us with data entry.
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